Prm Handbook III

The Professional Risk Managers Handbook A Comprehensive Guide to Current Theory and Best Practices ___________________________________________________

Edited by Carol Alexander and Elizabeth Sheedy Introduced by David R. Koenig

Volume III: Risk Management Practices

The Official Handbook for the PRM Certification

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The PRM Handbook Volume III

Copyrighted Materials Published by PRMIA Publications, Wilmington, DE No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Publisher. Requests for permission should be addressed to PRMIA Publications, PMB #5527, 2711 Centerville Road, Suite 120, Wilmington, DE, 19808 or via email to [email protected]. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability of fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential or other damages. This book is also available in a Sealed digital format and may be purchased as such by members of the Professional Risk Managers International Association at www.PRMIA.org.

ISBN 0-9766097-0-3 (3 Volume Set) ISBN 0-9766097-3-8 (Volume III)

Copyright ? 2004 The Authors and the Professional Risk Managers International Association


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Contents Introduction .................................................................................................................................................7 Preface to Volume III: Risk Management Practices.......................................................................9 III.0 Capital Allocation and RAPM..........................................................................................................13 III.0.1

Introduction ..........................................................................................................................13

III.0.2

Economic Capital .................................................................................................................17

III.0.3

Regulatory Capital ................................................................................................................24

III.0.4

Capital Allocation and Risk Contributions.......................................................................31

III.0.5

RAROC and Risk-Adjusted Performance........................................................................35

III.0.6

Summary and Conclusions..................................................................................................39

References ................................................................................................................................................40 III.A.1 Market Risk Management .............................................................................................................43 III.A.1.1

Introduction......................................................................................................................43

III.A.1.2

Market Risk .......................................................................................................................43

III.A.1.3

Market Risk Management Tasks....................................................................................45

III.A.1.4

The Organisation of Market Risk Management..........................................................47

III.A.1.5

Market Risk Management in Fund Management........................................................49

III.A.1.6

Market Risk Management in Banking...........................................................................57

III.A.1.7

Market Risk Management in Non-financial Firms .....................................................66

III.A.1.8

Summary............................................................................................................................72

References ................................................................................................................................................73 III.A.2 Introduction to Value at Risk Models.........................................................................................75 III.A.2.1

Introduction......................................................................................................................75

III.A.2.2

Definition of VaR ............................................................................................................76

III.A.2.3

Internal Models for Market Risk Capital......................................................................78

III.A.2.4

Analytical VaR Models....................................................................................................79

III.A.2.5

Monte Carlo Simulation VaR.........................................................................................81

III.A.2.6

Historical Simulation VaR ..............................................................................................86

III.A.2.7

Mapping Positions to Risk Factors ...............................................................................95

III.A.2.8

Backtesting VaR Models .............................................................................................. 109

III.A.2.9

Why Financial Markets Are Not Normal................................................................ 111

III.A.2.10

Summary......................................................................................................................... 112

References ............................................................................................................................................. 113 III.A.3: Advanced Value at Risk Models ............................................................................................. 115 III.A.3.1

Introduction................................................................................................................... 115

III.A.3.2

Standard Distributional Assumptions........................................................................ 117

III.A.3.3

Models of Volatility Clustering ................................................................................... 121



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III.A.3.4

Volatility Clustering and VaR...................................................................................... 126

III.A.3.5

Alternative Solutions to Non-normality.................................................................... 134

III.A.3.6

Decomposition of VaR................................................................................................ 142

III.A.3.7

Principal Component Analysis.................................................................................... 149

III.A.3.8

Summary......................................................................................................................... 154

References ............................................................................................................................................. 155 III.A.4 Stress Testing ............................................................................................................................... 157 III.A.4.1

Introduction................................................................................................................... 157

III.A.4.2

Historical Context......................................................................................................... 158

III.A.4.3

Conceptual Context...................................................................................................... 163

III.A.4.4

Stress Testing in Practice ............................................................................................. 164

III.A.4.5

Approaches to Stress Testing: An Overview............................................................ 166

III.A.4.6

Historical Scenarios ...................................................................................................... 168

III.A.4.7

Hypothetical Scenarios................................................................................................. 174

III.A.4.8

Algorithmic Approaches to Stress Testing ............................................................... 182

III.A.4.9

Extreme-Value Theory as a Stress-Testing Method................................................ 186

III.A.4.10

Summary and Conclusions .......................................................................................... 187

Further Reading.................................................................................................................................... 187 References ............................................................................................................................................. 188 III.B.1 Credit Risk Management ............................................................................................................ 193 III.B.1.1

Introduction................................................................................................................... 193

III.B.1.2

A Credit To-Do List..................................................................................................... 194

III.B.1.3

Other Tasks.................................................................................................................... 207

III.B.1.4

Conclusions.................................................................................................................... 208

References ............................................................................................................................................. 209 III.B.2 Foundations of Credit Risk Modelling..................................................................................... 211 III.B.2.1

Introduction................................................................................................................... 211

III.B.2.2

What is Default Risk? ................................................................................................... 211

III.B.2.3

Exposure, Default and Recovery Processes ............................................................. 212

III.B.2.4

The Credit Loss Distribution...................................................................................... 213

III.B.2.5

Expected and Unexpected Loss ................................................................................. 215

III.B.2.6

Recovery Rates .............................................................................................................. 218

III.B.2.7

Conclusion ..................................................................................................................... 223

References ............................................................................................................................................. 223 III.B.3 Credit Exposure........................................................................................................................... 225 III.B.3.1

Introduction................................................................................................................... 225

III.B.3.2

Pre-settlement versus Settlement Risk....................................................................... 227



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III.B.3.3

Exposure Profiles.......................................................................................................... 228

III.B.3.4

Mitigation of Exposures .............................................................................................. 236

References ............................................................................................................................................. 241 III.B.4 Default and Credit Migration .................................................................................................... 243 III.B.4.1

Default Probabilities and Term Structures of Default Rates ................................. 243

III.B.4.2

Credit Ratings ................................................................................................................ 247

III.B.4.3

Agency Ratings .............................................................................................................. 252

III.B.4.4

Credit Scoring and Internal Rating Models .............................................................. 258

III.B.4.5

Market-Implied Default Probabilities ........................................................................ 262

III.B.4.6

Credit Rating and Credit Spreads ............................................................................... 268

III.B.4.7

Summary......................................................................................................................... 270

References ............................................................................................................................................. 271 III.B.5 Portfolio Models of Credit Loss ............................................................................................... 273 III.B.5.1

Introduction................................................................................................................... 273

III.B.5.2

What Actually Drives Credit Risk at the Portfolio Level?...................................... 276

III.B.5.3

Credit Migration Framework ...................................................................................... 279

III.B.5.4

Conditional Transition Probabilities CreditPortfolioView .................................. 292

III.B.5.5

The Contingent Claim Approach to Measuring Credit Risk ................................. 294

III.B.5.6

The KMV Approach .................................................................................................... 300

III.B.5.7

The Actuarial Approach............................................................................................... 307

III.B.5.8

Summary and Conclusion............................................................................................ 312

References ............................................................................................................................................. 312 III.B.6 Credit Risk Capital Calculation.................................................................................................. 315 III.B.6.1

Introduction.................................................................................................................. 315

III.B.6.2

Economic Credit Capital Calculation ........................................................................ 316

III.B.6.3

Regulatory Credit Capital: Basel I ............................................................................. 320

III.B.6.4

Regulatory Credit Capital: Basel II............................................................................. 324

III.B.6.5

Basel II: Credit Model Estimation and Validation .................................................. 334

III.B.6.6

Basel II: Securitisation.................................................................................................. 336

III.B.6.7

Advanced Topics on Economic Credit Capital ....................................................... 338

III.B.6.8

Summary and Conclusions .......................................................................................... 340

References ............................................................................................................................................. 341 III.C.1 The Operational Risk Management Framework.................................................................... 343 III.C.1.1

Introduction................................................................................................................... 343

III.C.1.2

Evidence of Operational Failures............................................................................... 345

III.C.1.3

Defining Operational Risk........................................................................................... 347

III.C.1.4

Types of Operational Risk.......................................................................................... 348



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III.C.1.5

Aims and Scope of Operational Risk Management ................................................ 351

III.C.1.6

Key Components of Operational Risk ...................................................................... 354

III.C.1.7

Supervisory Guidance on Operational Risk ............................................................. 357

III.C.1.8

Identifying Operational Risk the Risk Catalogue ................................................. 358

III.C.1.9

The Operational Risk Assessment Process............................................................... 359

III.C.1.10

The Operational Risk Control Process...................................................................... 364

III.C.1.11

Some Final Thoughts ................................................................................................... 365

References ............................................................................................................................................. 366 III.C.2 Operational Risk Process Models ............................................................................................. 367 III.C.2.1

Introduction................................................................................................................... 367

III.C.2.2

The Overall Process ..................................................................................................... 369

III.C.2.3

Specific Tools ................................................................................................................ 372

III.C.2.4

Advanced Models ......................................................................................................... 374

III.C.2.5

Key Attributes of the ORM Framework................................................................... 378

III.C.2.6

Integrated Economic Capital Model.......................................................................... 381

III.C.2.7

Management Actions.................................................................................................... 384

III.C.2.8

Risk Transfer.................................................................................................................. 386

III.C.2.9

IT Outsourcing.............................................................................................................. 388

References ............................................................................................................................................. 393 III.C.3 Operational Value-at-Risk.......................................................................................................... 395 III.C.3.1

The Loss Model Approach........................................................................................ 395

III.C.3.2

The Frequency Distribution........................................................................................ 401

III.C.3.3

The Severity Distribution ............................................................................................ 404

III.C.3.4

The Internal Measurement Approach ....................................................................... 407

III.C.3.5

The Loss Distribution Approach ............................................................................... 411

III.C.3.6

Aggregating ORC.......................................................................................................... 413

III.C.3.7

Concluding Remarks .................................................................................................... 415

References ............................................................................................................................................. 416



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Introduction If you're reading this, you are seeking to attain a higher standard. Congratulations! Those who have been a part of financial risk management for the past twenty years, have seen it change from an on-the-fly profession, with improvisation as a rule, to one with substantially higher standards, many of which are now documented and expected to be followed. Its no longer enough to say you know. Now, you and your team need to prove it. As its title implies, this book is the Handbook for the Professional Risk Manager. It is for those professionals who seek to demonstrate their skills through certification as a Professional Risk Manager (PRM) in the field of financial risk management. And it is for those looking simply to develop their skills through an excellent reference source. With contributions from nearly 40 leading authors, the Handbook is designed to provide you with the materials needed to gain the knowledge and understanding of the building blocks of professional financial risk management. Financial risk management is not about avoiding risk. Rather, it is about understanding and communicating risk, so that risk can be taken more confidently and in a better way. Whether your specialism is in insurance, banking, energy, asset management, weather, or one of myriad other industries, this Handbook is your guide. In this volume, the current and best practices of Market, Credit and Operational risk management are described. This is where we take the foundations of Volumes I and II and apply them to our profession in very specific ways. Here the strategic application of risk management to capital allocation and risk-adjusted performance measurement takes hold. After studying all of the materials in the PRM Handbook, you will have read the materials necessary for passage of Exam III of the PRM Certification program. Those preparing for the PRM certification will also be preparing for Exam I on Finance Theory, Financial Instruments and Markets, covered in Volume I of the PRM Handbook, Exam II on the Mathematical Foundations of Risk Measurement, covered in Volume II of the PRM Handbook and Exam IV - Case Studies, Standards of Best Practice Conduct and Ethics and PRMIA Governance. Exam IV is where we study some failed practices, standards for the performance of the duties of a Professional Risk Manager, and the governance structure of our association, the Professional Risk Managers International Association. The materials for this exam are freely available on our web site (see http://www.prmia.org/pdf/Web_based_Resources.htm) and are thus outside of the Handbook.



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At the end of your progression through these materials, you will find that you have broadened your knowledge and skills in ways that you might not have imagined. You will have challenged yourself as well. And, you will be a better risk manager. It is for this reason that we have created the Professional Risk Managers Handbook. Our deepest appreciation is extended to Prof. Carol Alexander and Prof. Elizabeth Sheedy, both of PRMIAs Academic Advisory Council, for their editorial work on this document. The commitment they have shown to ensuring the highest level of quality and relevance is beyond description. Our thanks also go to Laura Bianco, President of PRMIA Publications, who has tirelessly kept the work process moving forward and who has dedicated herself to demanding the finest quality output. We also thank Richard Leigh, our London-based copyeditor, for his skilful and timely work. Finally, we express our thanks to the authors who have shared their insights with us. The demands for sharing of their expertise are frequent. Yet, they have each taken special time for this project and have dedicated themselves to making the Handbook and you a success. We are very proud to bring you such a fine assembly. Much like PRMIA, the Handbook is a place where the best ideas of the risk profession meet. We hope that you will take these ideas, put them into practice and certify your knowledge by attaining the PRM designation. Among our membership are several hundred Chief Risk Officers / Heads of Risk and tens of thousands of other risk professionals who will note your achievements. They too know the importance of setting high standards and the trust that capital providers and stakeholders have put in them. Now they put their trust in you and you can prove your commitment and distinction to them. We wish you much success during your studies and for your performance in the PRM exams! David R. Koenig, Executive Director, Chair, Board of Directors, PRMIA



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Preface to Volume III: Risk Management Practices Section III is the ultimate part of The PRM Handbook in both senses of the word. Not only is it the final section, but it represents the final aims and objectives of the Handbook. Sections I (Finance Theory, Financial Instruments and Markets) and II (Mathematical Foundations of Risk Measurement) laid the necessary foundations for this discussion of risk management practices the primary concern of most readers. Here some of the foremost practitioners and academics in the field provide an up-to-date, rigorous and lucid statement of modern risk management. The practice of risk management is evolving at a rapid pace, especially with the impending arrival of Basel II.

Aside from these regulatory pressures, shareholders and other stakeholders

increasingly demand higher standards of risk management and disclosure of risk. In fact, it would not be an overstatement to say that risk consciousness is one of the defining features of modern business. Nowhere is this truer than in the financial services industry. Interest in risk management is at an unprecedented level as institutions gather data, upgrade their models and systems, train their staff, review their remuneration systems, adapt their business practices and scrutinise controls for this new era. Section III is itself split into three parts which address market risk, credit risk and operational risk in turn. These three are the main components of risk borne by any organisation, although the relative importance of the mix varies. For a traditional commercial bank, credit risk has always been the most significant. It is defined as the risk of default on debt, swap, or other counterparty instruments. Credit risk may also result from a change in the value of a security, contract or asset resulting from a change in the counterpartys creditworthiness. In contrast, market risk refers to changes in the values of securities, contracts or assets resulting from movements in exchange rates, interest rates, commodity prices, stock prices, etc. Operational risk, the risk of loss resulting from inadequate or failed internal processes, people and systems or from external events, is not, strictly speaking, a financial risk. Operational risks are, however, an inevitable consequence of any business undertaking. For financial institutions and fund managers, credit and market risks are taken intentionally with the objective of earning returns, while operational risks are a by-product to be controlled. While the importance of operational risk management is increasingly accepted, it will probably never have the same status in the finance industry as credit and market risk which are the chosen areas of competence. For non-financial firms, the priorities are reversed. The focus should be on the risks associated with the particular business; the production and marketing of the service or product in which



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expertise is held. Market and credit risks are usually of secondary importance as they are a byproduct of the main business agenda. The last line of defence against risk is capital, as it ensures that a firm can continue as a going concern even if substantial and unexpected losses are incurred. Accordingly, one of the major themes of Section III is how to determine the appropriate size of this capital buffer. How much capital is enough to withstand unusual losses in each of the three areas of risk? The measurement of risk has further important implications for risk management as it is increasingly incorporated into the performance evaluation process. Since resources are allocated and bonuses paid on the basis of performance measures, it is essential that they be appropriately adjusted for risk. Only then will appropriate incentives be created for behaviour that is beneficial for shareholders and other stakeholders. Chapter III.0 explores this fundamental idea at a general level, since it is relevant for each of the three risk areas that follow. Market Risk Chapter III.A.1 introduces the topic of market risk as it is practised by bankers, fund managers and corporate treasurers. It explains the four major tasks of risk management (identification, assessment, monitoring and control/mitigation), thus setting the scene for the quantitative chapters that follow. These days one of the major tasks of risk managers is to measure risk using value-at-risk (VaR) models. VaR models for market risk come in many varieties. The more basic VaR models are the topic of Chapter III.A.2, while the advanced versions are covered in III.A.3 along with some other advanced topics such as risk decomposition. The main challenge for risk managers is to model the empirical characteristics observed in the market, especially volatility clustering. The advanced models are generally more successful in this regard, although the basic versions are easier to implement. Realistically, there will never be a perfect VaR model, which is one of the reasons why stress tests are a popular tool. They can be considered an ad hoc solution to the problem of model risk. Chapter III.A.4 explains the need for stress tests and how they might usefully be constructed. Credit Risk Chapter III.B.1 introduces the sphere of credit risk management. Some fundamental tools for managing credit risk are explained here, including the use of collateral, credit limits and credit derivatives. Subsequent chapters on credit risk focus primarily on its modelling. Foundations for modelling are laid in Chapter III.B.2, which explains the three basic components of a credit loss: the exposure, the default probability and the recovery rate. The product of these three, which can be defined as random processes, is the credit loss distribution. Chapter III.B.3 takes a more detailed look at the exposure amount. While relatively simple to define for standard loans,



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assessment of the exposure amount can present challenges for other credit sensitive instruments such as derivatives, whose values are a function of market movements. Chapter III.B.4 examines in detail the default probability and how it can evolve over time. It also discusses the relationship between credit ratings and credit spreads, and credit scoring models. Chapter III.B.5 tackles one of the most crucial issues for credit risk modelling: how to model credit risk in a portfolio context and thereby estimate credit VaR. Since diversification is one of the most important tools for the management of credit risk, risk measures on a portfolio basis are fundamental. A number of tools are examined, including the credit migration approach, the contingent claim or structural approach, and the actuarial approach. Finally, Chapter III.B.6 extends the discussion of credit VaR models to examine credit risk capital. It compares both economic capital and regulatory capital for credit risk as defined under the new Basel Accord. Operational Risk The framework for managing operational risk is first established in Chapter III.C.1. After defining operational risk, it explains how it may be identified, assessed and controlled. Chapter III.C.2 builds on this with a discussion of operational risk process models.

By better

understanding business processes we can find the sources of risk and often take steps to reengineer these processes for greater efficiency and lower risk. One of the most perplexing issues for risk managers is to determine appropriate capital buffers for operational risks. Operational VaR is the subject of Chapter III.C.3, including discussion of loss models, standard functional forms, both analytical and simulation methods, and the aggregation of operational risk over all business lines and event types. Elizabeth Sheedy, Member of PRMIAs Academic Advisory Council and co-editor of The PRM Handbook.



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III.0 Capital Allocation and RAPM Andrew Aziz and Dan Rosen 1

III.0.1

Introduction

We introduce in this chapter the definitions and key concepts regarding capital, focusing on the important role that capital plays in financial institutions. Readers should already be familiar with Chapter I.A.5, which has presented the basic principles behind the capital structure of the firm. There it was argued that the actual capital the physical capital that a firm holds should be distinguished from the optimal level of capital. The optimal capital depends on many things, including capital targets that are associated with a desired level of capital adequacy to cover the potential for losses made by the firm. Capital adequacy is assessed using internal models (for economic capital) and for banks a certain level of capital is imposed by external standards (regulatory capital). Capital adequacy is a measure of a firms ability to remain a going concern. Thus, targeted levels of capital are direct functions of the riskiness of the business activities or, from a balance sheet perspective, the riskiness of the assets. In this chapter you will learn: the role of capital in financial institutions and the different types of capital; the key concepts and objectives behind regulatory capital, as well as the main calculations principles in the Basel I Accord and the current Basel II Accord; the definition and mechanics of economic capital as well as the methods to calculate it; the use of economic capital as a management tool for risk aggregation, risk-adjusted performance measurement and optimal decision making through capital allocation. This introductory section presents the definition of capital and its role in financial institutions. We make the distinction between the various types of capital: book capital, economic capital and regulatory capital. We discuss briefly the use of economic capital as a management tool for risk aggregation, decision making and performance measurement. III.0.1.1

Role of Capital in Financial Institution

Banks generate revenue by taking on exposure to their customers and by earning appropriate returns to compensate for the risk of this exposure. In general, if a bank takes on more risk, it can expect to earn a greater return. The trade-off, however, is that the same bank will, in general, increase the possibility of facing losses to the extent that it defaults on its debt obligations and is forced out of business. Banks that are managed well will attempt to maximise their returns only 1 Algorithmics Inc.



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through risk taking that is prudent and well informed. The primary role of the risk management function in a bank is to ensure that the total risk taken across the enterprise is no greater than the banks ability to absorb worst-case losses within some specified confidence interval. In its pure form, capital represents the difference between the market value of a banks assets and the market value of its liabilities. Because capital can be viewed as a buffer against insolvency, capital adequacy is a measure of a banks ability to remain a going concern under adverse conditions. In contrast to a typical corporation, the key role of capital in a financial institution such as a bank is not primarily one of providing a source of funding for the organisation. Banks usually have ready access to funding through their deposit-taking activities, which can be increased fairly fluidly. Instead, the primary role of capital in a bank, apart from the transfer of ownership, is to act as a buffer to: absorb large unexpected losses; protect depositors and other claim holders; provide enough confidence to external investors and rating agencies on the financial health and viability of the firm. A firms credit rating can be seen as a measure of its capital adequacy and is generally linked to a specific probability that the firm will enter into default over some period of time. If we make the assumption that liabilities are riskless, then the credit rating of a firm becomes a function of the overall riskiness of its assets and the amount of capital that the bank holds. Firms which hold more capital are able to take on riskier assets than firms of similar credit rating which hold less capital. Typically, the sources of risk within the assets of a firm are classified as follows: credit risk losses associated with the default (or credit downgrade) of an obligor (a counterparty, borrower or debt issuer); market risk losses associated with changes in market values; operational risk losses associated with operating failures. Capital represents an ideal metric for aggregating risks across both different asset classes and across different risk types.



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III.0.1.2

Types of Capital

We can broadly classify capital into three types: 2 Economic capital (EC) an estimate of the level of capital that a firm requires to operate its business with a desired target solvency level. Sometimes this is also referred to as risk capital. 3 Regulatory capital (RC) the capital that a bank is required to hold by regulators in order to operate; this is an accounting measure defined by the regulatory authorities to act as a proxy for economic capital. Book capital (BC) the actual physical capital held. While in its strictest definition this should be simply equity capital, more generally this might also include other assets like liquid debt or hybrid instruments In practice, many firms hold book capital in excess of the required economic and even regulatory capital. This reflects both historical and practical business reasons (for example, given their size, they might be too slow to invest it effectively), as well as their more conservative view on the applicability of the models. The combined forces of deregulation and the increased market volatility in the late 1970s motivated many banks to aggressively grow market share and to acquire increasingly riskier assets on their balance sheets. This emphasis on growth precipitated a decline of capital levels throughout the 1980s that led to fears of increasing instability in the international banking system. These concerns motivated the push for the creation of international capital adequacy standards such as those ultimately established by the Basel Committee on Banking Supervision (BCBS). The imposition of the Basel I Accord in 1988 proved to be successful in its objective of increasing worldwide capital levels to desired levels by 1993 and, ultimately, to reduce the overall riskiness of the international banking system. In general, EC is meant to reflect the true fair market value differential between assets and liabilities, and thus it is limited by the ability to mark to market a balance sheet in a manner that is indisputable for all key constituencies the financial institution, the regulators and the investors themselves. As such, the determination of EC has traditionally been highly institution-specific. The foremost objective of regulations, however, is to define an unarguable standard for capital comparison that creates a level playing field across all financial institutions. Thus, regulatory capital has traditionally been defined with respect to accounting book value measures rather than

2 This is a general classification, and there are various alternative definitions of capital and terminology used to describe

them. 3 Some authors have used alternative definitions: for example, Matten (2000, pp. 222223) defines economic capital as risk capital plus goodwill; Perold (2001) defines risk capital in terms of insurance (explained in Section III.0.2.6).



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to market value measures (notwithstanding the fact that, in some cases, accounting practice allows balance sheet items to be reported on a market value basis). To capture the discrepancy between fair values and market values, regulatory capital measures incorporate ad hoc approaches to normalise asset book values to reflect differences in risk. As such, it is recognised that regulatory capital calculations tend to contain a number of inconsistencies, which have led regulators to set prescribed levels on a conservative basis. In some cases, these inconsistencies have led to the notion of regulatory arbitrage, whereby investment is determined not on the basis of riskreward optimisation but on the basis of regulatory capital reward optimisation. III.0.1.3

Capital as a Management Tool

Capital can be used as a powerful business management tool, since it provides a consistent metric to determine: risk aggregation; performance measurement; asset and business allocation. The objective of risk-adjusted performance measurement (RAPM) is to define a consistent metric that spans all asset and risk classes, thereby providing an apples to apples benchmark for evaluating the performance of alternative business opportunities. RAPM thus becomes an ideal tool for capital allocation purposes. By allocating the appropriate amount of EC to each asset, net expected payoffs can then be expressed as returns on capital. Each asset can, therefore, be assessed on a consistent basis, with returns adjusted appropriately in the context of the amount of risk taken on. (This is further discussed in Sections III.0.4 and III.0.5). Risk aggregation generally refers to the development of quantitative risk measures that incorporate multiple sources of risk. The most common approach is to estimate the EC that is necessary to absorb potential losses associated with each of the risks. EC can be seen as a common measure that can be used to summarise and compare the different risks incurred by a firm, across different businesses and activities; different types of risk market risk, credit risk and operational risk. According to a recent study (BCBS, 2003b), the application of risk aggregation and EC methods is still in the early stages of its evolution. While some firms remain sceptical of the value of reducing all risks to a single number, many now believe that there is a need for a common metric



16


that allows riskreturn comparisons to be made systematically across business activities whose mix of risks may be quite different (e.g., insurance versus trading). However, there remains a wide variation in the manner in which aggregated risk measures such as EC are used for risk management decision making in practice today.

III.0.2

Economic Capital

Economic capital acts as a buffer that provides protection against all the credit, market, operational and business risks faced by an institution. EC is set at a confidence level that is less than 100% (e.g., 99.9%), since it would be too costly to operate at the 100% level. The confidence interval is chosen as a trade-off between providing high returns on capital for shareholders and providing protection to the debt holders (and achieving a desired rating) as well as confidence to other claim holders, such as depositors. In so far as EC reflects the amount of capital required to maintain a firms target capital rating, the confidence interval can be defined at a very high quantile of the loss distribution. For example, to achieve a target S&P credit rating of BB, the probability of default over the next year for the firm cannot be greater than 3.0%, so the quantile should be set at least at 97%. In contrast, for the same firm to achieve a target S&P credit rating of BBB, it must lower its probability of default to be at most 0.5%, corresponding to the 99.5% quantile of the loss distribution. Given the desire to achieve a BBB rating and to remain solvent 99.5% of the time, a firm must have enough capital to sustain a 0.5% worst-case loss over a one-year time horizon; that is, 99.5% of the time the future value of the non-defaulted assets must be at least equal to the future value of the liabilities. Example III.0.1 below gives a simple outline of how this could be achieved. III.0.2.1

Understanding Economic Capital

Denote by At and Dt the market values (at time t) of the assets and liabilities, respectively. The available capital Ct for the current time, t = 0 ,and at the end of one year, t = 1, can be expressed as C0 = A0 D0,

(III.0.1)

C1 = A1 D1. If the nominal returns on the assets and liabilities are equal to rA and rD, respectively, then a worst-case loss from all sources, l (i.e., when C1 = 0 for a given confidence interval), would result in the value of assets at t = 1 just being sufficient to cover the value of debt in t = 1. Then C1 = 0 = A0 (1 + rA)(1

l)

D0(1 + rD).



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Thus the maximum amount of debt allowable to sustain solvency under the worst-case scenario cannot exceed D0 = A0 (1 + rA)(1

l )/(1 + rD).

(III.0.2)

Since EC0 is the minimum amount of capital required to sustain such a loss, it is given by: EC0 = A0 (1 [(1 + rA)(1

l )/(1 + rD )]).

(III.0.3)

Hence the minimum amount of EC a financial institution must take on in order to avoid insolvency increases as the level of the worse-case loss l increases. The expected return on EC over the period from t = 0 to t = 1 is given by [E(EC1 ) / EC0]

1.

The expected return on EC reflects the impact of leverage on risk and reward. An increase in expected returns (to compensate for increased risk) is reflected in the numerator, while the increase in risk is reflected in the denominator (the current EC). Equation (III.0.3) is often expressed in terms of value-at-risk notation in the following manner: EC0 = A0 VaR/(1+rD),

(III.0.4)

where VaR represents the Ai value associated with the worst-case loss, l, corresponding to the appropriate (x%) confidence interval. For ease of presentation, consider the case where credit risk is the sole source of business risk to which the firm is exposed. Returning to equation (III.0.3), under the simplifying assumption that the spread between the nominal return on the assets and the return on the liabilities is roughly equal to the expected default loss, u, then, EC0 = A0 {1 (1 + rD)(1 + u)(1 l )/(1 + rD)}

(III.0.5)

= A0 {1 (1 + u)(1 l )}. By then ignoring second-order effects, equation (III.0.5) simplifies to the following more familiar expression for economic capital: EC 0

A0 ( l

u) .

(III.0.6)

This relationship is illustrated with respect to a default loss distribution in Figure III.0.1.



18


Credit Loss Distribution Expected Loss Volatility

Probability

Unexpected Loss VaR(

0

%)

Loss

Figure III.0.1: Credit loss distribution: expected and unexpected losses Expressions (III.0.4) and (III.0.6) highlight the link between VaR measures and EC. The simplifying assumption leading to equation (III.0.6) and illustrated in Figure III.0.1 is the approach commonly taken by practitioners and generally leads to conservative estimates (for a detailed discussion, see Kupiec, 2002). Thus, in its most common definition, EC is defined to absorb only unexpected losses (UL) up to a certain confidence level (i.e., A0(l u)). Credit reserves are traditionally set aside to absorb expected losses (EL) over the period (i.e., A0u). More precisely, equation (III.0.4) shows that the VaR measure appropriate for EC should in fact measure losses relative to the assets initial mark-to-market (MtM) value and not relative to the EL in its end-ofperiod distribution. Also, the VaR measure should explicitly account for the interest payments on the funding debt. While the UL approximation has very little effect on market risk, where the horizon is short (and EL is small) it may have a higher impact in credit risk. Example III.0.1 Consider a BBB-rated firm (or a firm that has targeted a BBB rating). Suppose the firm has liabilities consisting of D0 = $92 million in deposits, with a cost of debt of rD = 5%, which have been invested in A0 = $100 million of assets (40% at a nominal return of 6.75% and 60% at a nominal return of 7%.). The weighted average nominal return across the $100 million in total assets is rA = 6.9%, representing a compounded spread of 1.81%. If the nominal values of the assets and liabilities are equal to the market values, then the current capital for this firm is calculated as C0 = $8 million (the difference between the market value of the assets and the market value of the liabilities).



19


Assume that, in a 0.5% worse-case scenario the firm has a potential for a loss of 15% in the value of total assets. Under this scenario, the firm will become insolvent as the value of the assets will be A1 = $100 million ? 1.069 ? 0.85 = $90.9 million, while the value of the liabilities will be D1 = $92 million ? 1.05 = $96.6 million, giving a capital shortfall of $5.7 million. From equation (III.0.3), the minimum amount of capital the firm must hold to avoid insolvency in the worst-case scenario is EC0 = A0 (1 [(1 + rA)(1

l )/(1 + rD)]) = 100(1 [1.069(1

0.15)/1.05]) = 13.46.

Therefore, for the firm to improve its capital adequacy to the desired level, it must increase its capital from $8 million to $13.46 million. The shareholders should be 99.5% sure that such an increase in capital will ensure solvency from t = 0 to t = 1. III.0.2.2

The Top-Down Approach to Calculating Economic Capital

EC can be seen as a common measure that can be used to summarise and compare the different risks incurred by a firm, across different businesses and activities, and across different types of risk: market, credit and operational risk. At the enterprise level, EC can be estimated based on aggregate information of the firms performance. Such top-down approaches generally use one of two types of information: earnings or stock prices. III.0.2.2.1 Top-Down Earnings Volatility Approach A top-down approach based on a firms earnings makes the simplifying assumption that the market value of capital is equal to the value of a perpetual stream of expected earnings. In other words, by assuming that all expected future earnings of the firm are equal to the next periods expected earnings, the value of capital can be expressed as C0 = Expected earnings / k, where k represents the required return associated with the riskiness of the earnings. As the determination of EC is based on the ability to sustain a worst-case loss associated with a given confidence interval, EC0 = EaR / k, where EaR represents the difference between expected earnings and the earnings under the worst-case scenario, for a given confidence interval (see Saita, 2003). Often this approach relies on the additional assumption that earnings are normally distributed, and thus the confidence interval can be determined as a multiple of the standard deviation.



20


Limitations of the earnings volatility approach include the following: It requires historical performance data for reliable estimates of the mean and standard deviation of earnings; few companies have enough data to yield reliable estimates. It does not link EC directly to the sources of risk. In general, it does not naturally allow capital to be separated out into its market, credit and operational risk components, nor across different business lines or activities. III.0.2.2.2 Top-Down Option-Theoretic Approach A top-down approach based on the BlackScholesMerton (BSM) framework (see Chapter III.B.5) assumes that the market value of capital can be modelled as a call option on the value of the firms assets where the strike price is the notional value of the debt. If the value of assets at the end of the period (t = 1) is greater than the value of the debt, then the value of capital is equal to the difference between the value of the assets and the debt; otherwise (in the case of insolvency), it is equal to zero. Using this approach assumes we have the following information available: the current market value and volatility of the companys net assets; the time horizon (e.g., the average duration of the firms assets); the risk-free interest rate (maturity corresponding to the time horizon); the default threshold (the asset level at which the debt holders demand repayment and bankruptcy can occur). The BSM model allows us to estimate the implied probability of insolvency for the firm over the period from t = 0 to t = 1. The EC can then be determined on the basis of reapplying the BSM model for a level of debt that ensures, even under the worst-case scenario (at a given confidence interval), that the firm remains solvent. An advantage of this approach over the one based on EaR is the availability of stock market data. However, several simplifications regarding the capital structure and model assumptions must be made to apply this tool in practice. Similar to the EaR approach, a key limitation is that it does not allow the separation of capital into different risks such as market, credit and operational risks, nor does it suggest how to allocate it across different business lines or activities. III.0.2.3

The Bottom-Up Approach to Calculating Economic Capital

In this approach, EC is estimated by modelling individual transactions and businesses and then aggregating the risks using advanced statistical portfolio models and stress testing. The bottomup approach has now become best practice and, in contrast to the top-down approaches, provides greater transparency with regard to isolating credit risk, market risk and operational risk Copyright ? 2004 The Authors and the Professional Risk Managers International Association


21


capital. Furthermore, it naturally accommodates various methodologies to allocate capital to individual businesses, activities and transactions. In a bottom-up approach, the estimation of enterprise EC requires consolidation of risks at two levels: First, it computes market risk, credit risk and operational risk at the enterprise level. To achieve this, a firm might use an internal VaR model for market risk, a credit VaR methodology for credit risk and a loss-distribution approach for operational risk. Then, at the second level, the firm must consolidate the capital across these risks. To estimate total capital, it is currently common practice to add up the credit risk capital, market risk capital and operational risk capital. This produces a conservative capital measure (basically assuming that the risks are perfectly positively correlated). However, today, many firms are now devoting considerable effort to measuring the correlations between these risks (and hence the levels of diversification), as well as developing frameworks to measure these risks in a more integrated way. III.0.2.4

Stress Testing of Portfolio Losses and Economic Capital

In addition to the statistical approaches inherent in credit portfolio models, practitioners usually use stress testing as an important part of their EC methodology (see Chapter III.A.4). Commonly, the stress-testing methodology involves the development of one or several specific adverse scenarios, which are judged to be extreme (falling beyond the desired confidence level). Current portfolio losses are then assessed against these specific scenarios. Stress scenarios may be based on historical experience or management judgment. The translation of the specific stress scenario losses, and the combination of stress testing and statistical measures, to develop EC measures is today more an art than a science, largely based on managements objectives and judgement. In essence, firms make their own decision on the relative weights of the statistical and stress test results in estimating the amount of EC required to support a portfolio. For example, an institution might assign the EC for market risk as 50% times the 99% VaR plus 50% the loss outcome from some stress scenarios (thus normally being higher than the actual 99% VaR). III.0.2.5

Enterprise Capital Practices Aggregation

A large firm, such as a bank, acquires different types of financial risk through various businesses and activities. Capital is indeed a powerful tool for understanding, comparing and aggregating



22


different types of risk to determine the overall health of the firm and to support better business decisions. Such an institution is likely to have separate methodologies to measure market risk, credit risk and operational risk. In addition, it is likely that the institution has different methodologies to measure the credit risk of its larger commercial loans or its retail credits. More generally, a firm may have any number of methodologies for various risks and segments. If each type of risk is modelled separately, then the amounts of EC estimated for each need to be combined to obtain an enterprise capital amount. In making the combination, the firm needs to incorporate, either implicitly or explicitly, various correlation assumptions. The methods of aggregation are as follows: Sum of stand-alone capital for each business unit and type of risk. This methodology essentially assumes perfect correlation across business lines and risk types and does not allow for diversification from them. Ad hoc or top-down estimates of cross-business and cross-risk correlation. In order to allow for some cross-business and cross-risk diversification, a firm might aggregate the individual stand-alone capital estimates using analytical models and simple cross-business (asset) correlation estimates. The enterprise aggregation of capital is still in its infancy and is a topic of much research today. III.0.2.6

Economic Capital as Insurance for the Value of the Firm

Standard practice is to define EC as a buffer to cover unexpected losses. Thus it is defined in terms of the tail of the loss distribution, using measures such as VaR. Alternatively, some economists have used the term risk capital to define capital in economic terms (Merton and Perold, 1993; Perold, 2001): risk capital is the smallest amount that can be invested to insure the value of the firms net assets 4 against loss in value relative to a risk-free investment. For example, under this definition, the risk capital of a long US Treasury bond position is the value of a put option with strike equal to the forward price of the bond. As pointed out by Perold (2001), in general, the put option accounts for the full distribution of losses, whereas VaR ignores the magnitude of outcomes conditional on being in the extreme tail of the distribution. When returns are normally distributed, the value of such a put option is approximately proportional to the standard deviation of the return on the bond, and thus is approximately proportional to VaR.

4 Net assets refers to gross assets minus customer liabilities (swaps, insurance contracts, etc.), valued as if these

liabilities are default-free.



23


III.0.3

Regulatory Capital

This section considers the key concepts and objectives behind regulatory capital, as well as the main principles used in regulatory capital calculations in the Basel I Accord and the latest proposals of the current Basel II Accord. III.0.3.1

Regulatory Capital Principles

In this subsection, we focus mainly on the capital regulation in the banking industry. As defined in Section III.0.1, regulatory capital refers to the capital that an institution is required to hold by regulators in order to operate. It is largely an accounting measure defined by the regulatory authorities to act as a proxy for economic capital. Capital adequacy is generally the single most important financial measure used by banking supervisors when examining the financial soundness of an institution. As also mentioned in Section III.0.1, from an internal bank perspective, capital is designed as a buffer to absorb large unexpected losses, protect depositors and other claim holders, and provide enough confidence to external investors and rating agencies on the financial health and viability of the firm. In contrast, from the external perspective of the regulator, capital adequacy requirements fulfil two objectives: Reducing systemic risk: to safeguard the security of the banking system and ensure its ongoing viability. In a sense, national governments act as guarantors. They have an interest in ensuring that banks remain capable of meeting their obligations and in minimising potential systemic effects on the economy. Regulatory capital helps to ensure that banks bear their share of the burden, otherwise borne by national governments. Creating a level playing field: to ensure a more even playing field for internationally active banks, by submitting all banks to (roughly) the same rules. As we review the key concepts in regulatory capital, it is important to highlight two points: Regulatory requirements are continuously changing, and it is vital for practitioners to be familiar with both the latest regulations and the specific requirements in each jurisdiction. While the general intention is to make regulatory capital more risk-sensitive and align it more closely to economic capital, it is important to understand the limitations of using it directly for managing risk, measuring performance and pricing credits. The overall objective should be to set up an enterprise risk management framework, which measures economic capital and reconciles it with regulatory capital.



24


III.0.3.2

The Basel Committee of Banking Supervision and the Basel Accord

A key cornerstone of international banking capital regulation is the Basel Committee on Banking Supervision, which first introduced the framework for international capital adequacy standards. This framework has been adopted as the underlying structure of all bank capital adequacy regulations throughout the G10, as well as many other countries around the world. Today, over 100 countries are expected to implement the latest guidelines set by the BCBS, also referred to as the Basel II accord). Summary Chronology of Banking Regulatory Capital 1988 The BCBS introduces the framework for international capital adequacy standards. It is adopted throughout the G10, as well as in over 100 other countries (BCBS, 1988). Commonly referred to as the Basel I Accord or the BIS I Accord, it was the first step in establishing a level playing field across member countries for internationally active banks. The 1988 accord focused mainly on credit risk. 1995 An amendment to the initial accord further allows banks to reduce creditequivalent exposures when netting agreements are in place (BCBS, 1995). 1996 The 1996 amendment 5 extends the capital requirements to include risk-based capital for the market risk in the trading book (BCBS, 1996). 1999 The BCBS issues a proposal for a new capital adequacy framework to replace the 1988 Basel I Accord. This is commonly referred to as the Basel II Accord or BIS II Accord. The new accord attempts to improve the capital adequacy framework by substantially increasing the risk sensitivity of the minimum capital requirements, and also encompassing a supervisory review and market discipline principles.

Under the

proposal, banks are required specifically to allocate capital against operational risks for the first time. 20002003 The BCBS releases various consultation documents and conducts major data collection exercises called quantitative impact studies (QIS), intended to gather information to assess whether it has met its goals. April 2003 The BCBS releases the third consultative paper (CP3) on the new Basel Accord (BCBS, 2003a).

5 Sometimes referred to as BIS 98, after its date of implementation.



25


June 2004 The final version of the Basel II Accord is published (BCBS, 2004). 20062007 Currently scheduled implementation of Basel II. All the papers from the BCBS can be downloaded from www.bis.org. III.0.3.3

Basel I Regulation

The 1988 accord focused mainly on credit risk, establishing minimum capital standards that linked capital requirements to the credit exposures of banks. Prior to its implementation in 1992, bank capital was regulated through simple, ad hoc capital standards. While generally prescriptive, Basel I left various choices to be made by local regulators, thus resulting in several variations of the implementation across jurisdictions. The 1996 amendment further extended the capital requirements to include risk-based capital for the market risk in the trading book. Basel I does not cover capital charges for operational risk. III.0.3.3.1 Minimum Capital Requirements under Basel I Capital requirements under Basel I are the sum of: credit risk capital charge, which applies to all positions in the trading and banking books (including OTC derivatives and balance sheet commitments); market risk capital charge for the trading book portfolio and off-balance sheet items. For market risk capital, the accord allows, in addition to a standardised method, the use of internal VaR models covering both general market risk (or systemic risk) and specific risk. Specific VaR applies to both equities and bonds. For bonds it covers the risk of defaults, migration and changes in spreads. The reader is referred to Chapter III.A.2 for the basics of market risk VaR. The regulatory charge for banks using internal market risk models is given by Market Risk Capital

[ M MR VaR

M SR SpecificVaR ]

Trigger , 8

(III.0.7)

where VaR and Specific VaR denote, respectively, the 99% market VaR and specific VaR over a 10-day horizon, and MMR and MSR are multipliers designed to adjust the capital to cover for modelling errors and reward the quality of the models. The first one ranges between 3 and 4, and the second one between 4 and 5. Finally, the Trigger is related to quality of controls in the bank. Currently it is set to 8 in North America and between 8 and 25 in the UK.



26


The methodology for credit capital is simple. Minimum capital requirements are obtained by multiplying the sum of all the risk-weighted assets by the capital adequacy ratio of 8% (also referred to as the Cook ratio): Capital

RWAk

8% .

(III.0.8)

k

Thus the calculation of credit regulatory requirements has three steps: converting exposures to credit equivalent assets; computing loan equivalents for off-balance sheet and OTC portfolios; and applying the capital adequacy ratio. This is described in greater detail in Chapter III.B.6. A great strength of Basel I is the simplicity of the framework. This has allowed it to be implemented in countries with different banking and accounting practices. Thus, it has been quite successful in achieving its two general objectives (to safeguard the stability of the banking system and to ensure an level playing field internationally). Its simplicity also has been its major weakness, as the accord does not effectively align regulatory capital requirements closely with an institutions risk. For example, some criticisms on the credit risk capital include the lack of proper differentiation for credit quality and maturity, insufficient incentives for credit mitigations techniques, and lack of recognition of portfolio effects (these are discussed briefly in Chapter III.B.6). III.0.3.3.2 Regulatory Arbitrage under Basel I The lack of differentiation in the accord, together with the financial engineering advances in credit risk over the last decade, have lead to the development of a regulatory capital arbitrage industry. This refers to the process by which regulatory capital is reduced through instruments such as credit derivatives or securitisation, without an equivalent reduction of the actual risk being taken. Through regulatory arbitrage instruments, for example, banks typically transfer lowrisk exposures from their banking book to their trading book, or simply place them outside the regulated banking system. III.0.3.3.3 Meeting Capital Adequacy Requirements Available regulatory credit capital is divided into two categories: Tier 1 capital: essentially shareholder funds equity and retained earnings. Tier 2 capital: long-term subordinated debt, other qualifying hybrid instruments and reserves (such as loan loss reserves).



27


From a regulatory perspective, Tier 1 capital must cover at least 50% of the total capital; that is, Tier 2 cannot exceed Tier 1 capital. In addition, the subordinated debt included in Tier 2 cannot exceed 50% of the Tier 1 capital.6 Capital adequacy is generally expressed as a ratio. For example, an 8% capital ratio means that the total Tier 1 and Tier 2 capital is 8% of the risk-weighted assets (RWA). A 6% Tier 1 capital ratio refers to Tier 1 capital being 6% of the RWA. III.0.3.4

Basel II Accord Latest Proposals

The final version of the Basel II Accord was published in June 2004 (BCBS, 2004). 7 Its implementation will take effect between the end of 2006 and the end of 2007. In this subsection we present a brief summary of the basic principles of Basel II; the key formulae for minimum capital requirements for credit risk are given in Chapter III.B.6. For greater detail, the reader is referred to the BCBS papers. Basel II attempts to improve capital adequacy framework along two important dimensions: First, the development of a capital regulation that encompasses not only minimum capital requirements, but also supervisory review and market discipline. Second, a substantial increase in the risk sensitivity of the minimum capital requirements. The new accord intends to foster a strong emphasis on risk management and to encourage ongoing improvements in banks risk assessment capabilities. This is to be accomplished by closely aligning banks capital requirements with prevailing modern risk management practices, and by ensuring that this emphasis on risk makes its way into supervisory practices and into market discipline through enhanced risk- and capital-related disclosures. The Basel II Accord consists of three pillars: minimum capital requirements, supervisory review, and market discipline. We briefly summarise these below and then present the key principles behind the computation of minimum capital requirements. III.0.3.4.1 Pillar 1 - Minimum Capital Requirements Minimum capital requirements consist of three components: 1. definition of capital (no major changes from the 1988 accord);

6 A Tier 3 capital was introduced with the market risk requirements. Short-term subordinated debt can be used to meet

market risk requirements as well, but not credit risk. 7 A small number of open issues are still to be resolved during 2004.



28


2. definition of RWA; 3. minimum ratio of capital/RWA (remains 8%). Basel II proposes to modify the definition of risk-weighted assets in two areas: substantive changes to the treatment of credit risk relative to the Basel I Accord; the introduction of an explicit treatment of operational risk that will result in a measure of operational risk being included in the denominator of a banks capital ratio. Basel II moves away from a one-size-fits-all approach to the measurement of risk, through the introduction of three distinct options for the calculation of credit risk and three others for operational risk. These approaches present increasing complexity and risk-sensitivity. Banks and supervisors can thus select the approaches that are most appropriate to the stage of development of banks operations and of the financial market infrastructure. Chapter III.B.6 briefly reviews the three credit risk approaches. The operational risk approaches can be found in Chapter III.C.3. III.0.3.4.2 Pillar 2 - Supervisory Review The second pillar is based on a series of guiding principles, which point to the need for banks to assess their capital adequacy positions relative to their overall risks, and for supervisors to review and take appropriate actions in response to those assessments. Banks under internal ratings-based credit models will be required to demonstrate that they use the outputs of those models not only for minimum capital requirements but also to manage their business.

The inclusion of

supervisory review provides benefits through its emphasis on strong risk assessment capabilities by banks and supervisors alike. Important new components of Pillar II also include the treatment of stress testing, concentration risk and the residual risks arising from the use of collateral, guarantees and credit derivatives as well as specific securitisation exposures (these are discussed further in Chapter III.B.6). III.0.3.4.3 Pillar 3 - Market Discipline Also referred to as public disclosure, the third pillar aims to encourage safe and sound banking practices through effective market disclosures of capital levels and risk exposures. This will help market participants assess better a banks ability to remain solvent. III.0.3.5

A Simple Derivation of Regulatory Capital

Recognising that the simple difference between the value of assets and the value of liabilities in accounting value terms is not a good indicator of the true difference in market value terms has led regulators to make appropriate adjustments in the calculation of regulatory capital. In this section, we follow a similar approach to Section III.0.2.1 to understand regulatory capital.



29


For a typical bank balance sheet, the difference between assets and liabilities includes general provisions GP, and reserves R0, as well as the book value of equity E0{BV}: A0{B/S} L0{BV} = GP0 + E0{BV} + R0.

(III.0.9)

The amount of available capital for regulatory purposes, however, can be defined loosely as the difference between total assets (balance sheet as well as non-balance sheet) and only that component of total liabilities where non-payment of returns defines insolvency. The amount of available capital can thus be represented as follows: RC0 = A0{B/S} + A0{non-B/S} D0{BV}.

(III.0.10)

Balance sheet assets are the net of the book value of assets and any special provisions to account for defaulted or nearly-defaulted positions, while non-balance sheet assets consist of any revaluations, RV0, to market value as well as any undisclosed profits, UP0. That is, A0{B/S} = A0{BV} SP0

(III.0.11)

A0{non-B/S} = RV0 + UP0. Those liabilities whose non-payment constitutes insolvency represent the difference between total liabilities and the quasi-debt, QD0, that combines both debt and equity features: 8 D0{BV} = L0{BV} QD0{BV}.

(III.0.12)

Combining these relationships provides a very straightforward definition of regulatory capital in book value terms that is designed to be as good a proxy as possible for true market valuation: RC0 = E0{BV} + R0 + QD0{BV} + GP0 .

(III.0.13)

The first two terms of this definition of regulatory capital are referred to as Tier 1 capital, while the second two terms are referred to as Tier 2 capital. Capital adequacy standards are based on minimum requirements for each of the two tiers of capital. Note that the above adjustments to book value measures still do not adequately capture the true market values and, hence, the true riskiness of the assets. Therefore, standards on regulatory capital are typically expressed as minimum percentages of risk-weighted assets rather than total assets. An RWA is expressed as a percentage of the nominal value of a balance sheet asset. For example, under the first Basel Accord loans to private companies are assigned a risk weight of 100%, while loans to banks in OECD countries are assigned a risk weight of 20%, reflecting the perceived differences in risk between the two categories of borrowers.

8 Non-payment of quasi-debt does not imply insolvency, at least for a period of time.



30


III.0.4

Capital Allocation and Risk Contributions

We discuss the importance for allocating economic capital to different business units in a firm or to the constituents of a portfolio and the key methodologies to compute contributions to EC. III.0.4.1

Capital Allocation

In addition to computing the total EC for a firm or portfolio, it is important to develop general methodologies to attribute this capital a posteriori to various sub-portfolios, such as the firms activities, business units and even individual transactions, and allocate it a priori in an optimal fashion, to maximise risk-adjusted returns. EC allocation down to the portfolio is required for: management decision support and business planning; performance measurement and risk-based compensation; pricing, profitability assessment and limits; building optimal riskreturn portfolios and strategies. From a strategic management perspective, the allocation of EC to business units, activities and transactions is an issue that is receiving significant attention in the industry today. There are two views prevalent among firms: Diversification benefits should not be passed down to the business units. Rather, each unit is expected to operate on a stand-alone basis. An optimal level of group risk taking can be achieved only when diversification benefits are allocated to at least the major business units (and perhaps even to the transaction level). Thus, it is preferable for each business unit to be assigned an EC allocation closer to its marginal contribution to the total EC. In the general case, the sum of the stand-alone EC for each asset or business does not equal the total portfolio EC. Indeed, it is higher, since there are diversification benefits. Thus, it is important to devise a general methodology to assign capital to individual business units, activities and assets, which explicitly allocates the diversification benefits of the portfolio. III.0.4.2

Risk Contribution Methodologies for EC Allocation

There is no unique method to allocate EC down a portfolio, and thus it is important to understand how risk contribution tools can be applied to EC allocation decisions. Whether it is



31


EC contributions to business units, arbitrary sub-portfolios or assets, we can classify the allocation methodologies which are currently used in practice into three categories: stand-alone EC contributions; incremental EC contributions; and marginal EC contributions. 9 Every methodology has its advantages and disadvantages, and might be more appropriate for a particular managerial application. III.0.4.2.1 Stand-alone EC Contributions An individual business or sub-portfolio is assigned the amount of capital that it would consume on a stand-alone basis (e.g., if it were an independent firm). As such, it does not reflect the beneficial effect of diversification. The resulting sum of stand-alone capital for the individual business units, activities or sub-portfolios is generally greater than the total EC for the firm. III.0.4.2.2 Incremental EC Contributions This method is also referred sometimes as the discrete marginal EC allocation method. Under this method the EC allocated to a business unit or sub-portfolio attempts to capture an appropriate amount of risk capital that the unit contributes to the entire firms capital requirements. It is calculated by taking the EC computed for the entire firm (including the business unit or subportfolio) and subtracting from it EC for the firm without the business unit or sub-portfolio. This methodology thus captures exactly the amount of capital that would be released if the business unit were sold or added (everything else remaining the same). Incremental EC is a natural measure for evaluating the risk of acquisitions or divestitures. 10 But, while very intuitive, a disadvantage of this methodology is that it is not additive. While it does capture the benefits of diversification, the sum of incremental EC for all the firms business units (activities or sub-portfolios) is smaller than (or equal to) total EC for the firm. III.0.4.2.3 Marginal EC Contributions It would be useful to obtain measures of risk contributions that are additive. Sometimes referred to as diversified EC contributions, such measures are intended to capture the amount of the firms total capital that should be allocated to a particular business or sub-portfolio when viewed as part of a multi-business firm. They are specifically designed to allocate the diversification benefit among the business units and activities, in the form of reduced EC. Thus, by construction, the

9 The reader is cautioned that there is currently no universal terminology for these methodologies in the literature. As

defined here, incremental capital (risk) contributions are sometimes also referred to as marginal capital (risk) contributions or discrete marginal capital (risk) contributions. Marginal risk contributions, as termed here, are sometimes also referred to as diversified capital (risk) contributions or, more precisely, continuous marginal capital (risk) contributions (Smithson, 2003). 10 See, for example, Perold (2001) note that the author refers to this as marginal EC.



32


sum of diversified EC for all the firms business units and activities is equal to total EC for the firm. There are various methodologies that produce additive risk contributions. The most widespread and, perhaps, practical methodology is the one based on marginal risk contributions. While most explanations of this risk decomposition methodology are based on the use of volatility (or standard deviation) as a risk measure, the methodology is quite general and applicable to other risk measures. Volatility-based contributions are common practice today (see Smithson, 2003). However, such allocations can be ineffective for credit and operational risk, given the nonnormality of their loss distributions. Industry best practices are shifting towards allocations based on VaR or expected shortfall (ES) see Chapters III.A.2, III.A.3 and III.B.5. 11 An additive decomposition of EC is of the form: EC i ,

EC

(III.0.14)

i

where EC i denotes the EC contribution of business unit or sub-portfolio i. We can then define the percentage risk contribution of the ith business unit or sub-portfolio as: EC Contribi

EC i 100% . EC

(III.0.15)

Denoting by xi the size of the ith business unit or sub-portfolio, one can show that for EC based on volatility, VaR or ES: 12 EC i

EC ( x ) xi . xi

(III.0.16)

That is, an EC marginal contribution is the product of the size of business unit i and the rate of change of EC with respect to that position. This product essentially represents the rate of change of EC with respect to a small (marginal) percentage change in the size of the unit. Marginal EC contributions require the computation of the first derivative of the risk measure with respect to the size of each unit. When the risk measure used is volatility, they can be computed analytically and are simply given by the covariance of losses of that business unit with the overall portfolio divided by the volatility of losses (see Praschnik et al., 2001; Smithson, 2003):

11 While VaR is defined as a loss which cannot be exceeded x% of the time (a quantile of the loss distribution), ES is

commonly defined as the expected loss, conditional on reaching at least an x% loss (i.e., it is the average of the x% largest losses). Sometimes ES is also referred to as tail conditional expectation or conditional VaR. For some discussions on the use of ES and VaR for capital allocation, see Kalkbrener et al. (2004) and Mausser and Rosen (2004). 12 More formally, if the risk measure is homogeneous of degree 1 and differentiable, this follows from Eulers theorem. This is a requirement of coherent risk measures (Artzner et al., 1999).



33


EC Contribi

Cov( L i , L )/ ( L ) .

The general theory behind the definition and computation of these derivatives in terms of quantile measures (VaR, ES) has been also developed in the last few years (see Gouri閞oux et al., 2000; Tasche, 2000, 2002; see also Chapter III.B.5). It is important to stress that these contributions must be interpreted on a marginal basis. Marginal EC contributions are very general and are best suited to understand the amount of capital to be consumed by an instrument or portfolio (which really is small compared to the whole firm). They also naturally explain how to move EC from one business to another (on a marginal basis). Proponents of marginal EC approaches point out that incremental EC always under-allocates total firm EC and that, even if the incremental EC allocations were scaled up, the signals are potentially misleading. However, marginal EC is likely to be suboptimal for analysing the addition or removal on an entire business, which is not marginal to the firm. Example III.0.2: Capital Allocation Methods Table III.0.1 illustrates the different capital allocation methods for a simple firm consisting of three business lines. Assume, for simplicity that the total losses over one year for each business are normally distributed, and that they are uncorrelated. The stand-alone capital of each line is, respectively, $50 million, $30 million and $20 million. The total stand-alone capital is thus $100 million. The total economic capital of the firm is simply given by EC

EC 12

EC 22

EC 32

61.64 million.

Table III.0.1: Capital allocation methods for a simple firm Stand-alone % Stand-alone Incremental Capital (m) Contributions Capital (m)

% Incremental Capital Contributions

Marginal

Marginal Capital (m)

Capital Contributions

Business 1

50.00

50.0%

25.59

69.7%

40.56

65.8%

Business 2

30.00

30.0%

7.79

21.2%

14.60

23.7%

Business 3

20.00

20.0%

3.33

9.1%

6.49

10.5%

Total

100.00

100.0%

36.72

100.0%

61.64

100.0%

% EC

162.2%

59.6%

100.0%

The last line of this table gives the total capital as a percentage of EC. Notice that the total standalone capital represents a 62% increase of EC. Column 4 (and 5) in the table give the incremental capital for each business (in money terms and percentage contributions). The sum of incremental Copyright ? 2004 The Authors and the Professional Risk Managers International Association


34


capital is only 59.6% of EC. Finally, the last two columns give the marginal contributions (in money and percentage terms). Marginal contributions add up to the total EC. Note in particular that the stand-alone percentage contributions for each business differ meaningfully from the marginal contributions. While the largest business (business 1) contributes one half of the stand alone capital, it represents almost two-thirds of the EC on a marginal basis. This can be understood from the fact that as the biggest unit, increasing its share of the portfolio also marginally increases the overall risk. To diversify risk in an optimal way, one would rather increase the share of the smaller units. III.0.4.2.4 Alternative Methods for Additive Contributions 13 Recently there has been discussion of several alternative methods arising from game theory, which allocate the diversification benefits across the portfolio and yield additive risk contributions (see Denault, 2001; Koyluoglu and Stoker, 2002). Game-theoretic tools are commonly applied to problems involving the attribution of cost among a group and, hence, can potentially offer a useful framework for identifying fair EC attributions. An example of these tools is the Shapley method, which describes how coalitions can be formed so that a group of units benefits more as a group than if each works separately. In this approach, the EC assigned to a unit becomes a cost, and each unit attempts to minimise its cost. A player, of course, leaves the coalition if it is attributed a larger share of EC than its own stand-alone EC. This method is computationally intensive, and may be impractical for problems with even a small number of business units. A variant called the AumannShapley method further allows for fractional units and requires less computation; thus, it is potentially more practical. Under most (but not all) conditions, both these methods may yield similar results to marginal contributions. While these methods are today receiving some academic attention, they are mostly not yet used in practice by financial institutions.

III.0.5

RAROC and Risk-Adjusted Performance

We describe the objectives of risk-adjusted performance measurement, the role of capital allocation, and the basic principles of risk-adjusted return on capital. III.0.5.1

Objectives of RAPM

Banks traditionally measured their performance relative to their balance sheet assets, either simply with respect to overall asset size, or by bringing in the notion of profitability, with respect to

13 This section is added for completeness and is not mandatory.



35


returns on assets (ROA). There are a number of issues that make these approaches far from ideal, of which two are quite fundamental. The first issue is that by focusing solely on assets, the performance impact of financial leverage is ignored as it pertains to managing risk and return for shareholders. In addition to the leverage effect, today many banks have off-balance sheet exposures that are ignored or, at least, not well captured by the assets as represented on a typical balance sheet. The second fundamental issue is that a simple ROA measure does not distinguish between different classes of assets with varying levels of risk. Recall that balance sheet assets are typically book value based and not market value based. Early attempts to address these issues focused on shifting to performance measures that are defined relative to capital rather than to assets. This approach addresses the first fundamental issue as return on equity (ROE) captures the impact of financial leverage as well as, in theory, the impact of non-balance sheet assets. Both EC models and regulatory capital models attempt to address the second issue by focusing on market valuation directly, as in the case of economic models, or by adjusting book-value measures, as in the case of regulatory models. The objective of a risk-adjusted performance measure is to define a consistent metric that spans all asset and risk classes, thereby providing an apples to apples benchmark for evaluating the performance of alternative business opportunities. RAPMs thus become an ideal tool for capital allocation purposes. RAPMs come in many different forms, but they can all be loosely defined as a return on capital whereby the measurement of asset riskiness is a key component of the derivation of the formula. In most of the more sophisticated applications of RAPM, asset riskiness is modelled explicitly with respect to a distribution of default-adjusted returns directly, or with respect to a distribution of default losses that is then netted against nominal returns on assets. From these distributions, expected, worst-case, and x% confidence interval default losses (or default-adjusted returns) can be defined and applied to either the return measures or the underlying capital measure. Two broad classes of RAPM measures include risk-adjusted return on capital (RAROC) and return on risk-adjusted capital (RORAC). The former applies the risk adjustment to the numerator while the latter applies the risk adjustment to the denominator. Often the distinction between these approaches becomes blurred, with risk adjustments occurring in both the numerator and the denominator, prompting the increasing usage of the term risk-adjusted return on risk-adjusted capital



36


(RARORAC). Nonetheless, common to all these approaches is the principle of incorporating the joint default likelihood of a banks obligors explicitly into a banks RAPM. III.0.5.2

Mechanics of RAROC

All RAROC models follow the simplified general formula RAROC = (revenues costs expected losses) / capital.

(III.0.17)

Revenues include all the nominal returns on assets, while costs include all returns to the liabilities holders of the bank. Note that, in practice, all other sources of revenue, including service fees, and all other sources of costs, including general overhead, would normally be incorporated into the RAROC measure. Expected losses would be determined by a risk assessment of the asset base, capturing losses arising from all sources, including credit risk, market risk and operational risk. The example in the introduction considered a bank whose only activities were the taking in of deposits and the extending of credit. In that case the above equation can be rewritten as, RAROC = (A0 rA D0 rD expected losses) / EC0.

(III.0.18)

While a RAPM measure like RAROC is certainly a better indicator of a firms overall performance relative to other firms than a more traditional ROA or ROE approach, the true benefit for an individual firm is that it provides a consistent metric to evaluate the performance of the firms portfolio of assets, regardless of their unique levels of risk. Taken one step further, it also provides a benchmark for making allocation decisions, while appropriating scarce capital amongst possible new investment opportunities. Example III.0.3: A Simple Model for RAROC We return to Example III.0.1, with a firm with liabilities of D0 =$92 million in deposits at a cost of debt of rD = 5%, which have been invested in A0 = $100 million of assets (40% at a nominal return of 6.75% and 60% at a nominal return of 7%). The weighted average nominal return across the $100 million in total assets is rA = 6.9% (a compounded spread of 1.81%). The current capital for this firm was calculated as E0 = $8 million. The firms balance sheet can be conceptually decomposed into two balance sheets, one associated with each asset class. The amount of debt each asset class can support is determined by the amount of EC that must be allocated to that asset class. Assuming the total worst-case loss in value associated with asset class 1 is 18%, then the economic capital, EC1,0, associated with asset class 1, A1,0, can be determined in a similar manner to the EC for the entire firm:



37


EC1,0 = A1,0{1 (1 + r1,A)(1 = 60{1 1.07(1

l )/(1 + rD)}

0.18)/1.05} = 9.86.

Likewise, assuming the total worst-case loss in value associated with asset class 2 is 10.5%, the EC, EC2,0, associated with asset class two, A2,0, can be determined from equation (III.0.3) as EC2,0 = A2,0{1 (1 + r2,A)(1 = 40 {1 1.0675(1

l )/(1 + rD)} 0.105)/1.05} = 3.60.

For illustrative purposes, the current balance sheet can therefore be decomposed on the basis of asset class as shown in Table III.0.2. Table III.0.2: Current balance sheet Asset Class 1 Asset Class 2

Total

Assets

60

40

100

Debt

50.14

36.4

86.54

EC

9.86

3.8

13.46

Table III.0.3 illustrates the expected balance sheets for each asset at year end and, thus, the RAROC or the return on EC for each asset can be measured by the change in the equity position over the year (RAROC in is calculated as EC1/EC0 1). Table III.0.3: Expected balance sheet Asset Class 1 Asset Class 2

Total

Assets

63.2

42.16

105.36

Debt

52.64

38.22

90.86

EC

10.56

3.94

14.5

RAROC

7.01%

9.5%

7.68%

Of course, the change in equity must reconcile with the more familiar relationship, EC1 EC0 = (A1 D1) (A0 D0 ) = A0 rA D0 rD expected losses. In this example, while asset 1 has the higher nominal return, its RAROC is in fact slightly lower than that of asset 2 as proportionately more EC must be allocated to it to compensate for its higher risk. In other words, the excess nominal return of asset 1 over asset 2 is not quite enough



38


to compensate for its increased risk and therefore, on a risk-adjusted basis, asset 1 is a less desirable investment. Note that in this simple example the sum of the EC of each asset class equals the EC of the firm as a whole. This implies that no diversification exists between the two asset classes because the risk of each asset class on a stand-alone basis is equal to each asset classs contribution to the overall risk of the firm. III.0.5.3

RAROC and Capital Allocation Methodologies

When RAROC is used to measure the performance of an asset classes or business, or to allocate capital, it is important to highlight that the denominator measures the capital contribution of the asset or business to the overall portfolio. Hence, it is directly linked to the asset allocation methodology chosen by the institution (see previous section). Thus, for example, if a firm uses stand-alone risk contributions, the measure of performance will not account for the diversification opportunities that a given asset or business brings to the overall portfolio. In general, it is beneficial to use allocation methods that account for diversification, such as the marginal risk contributions. Example III.0.4: In the simple example above, the capital contribution of each asset was measured as the capital each asset consumes in the scenario that produces the extreme 1% loss. Both asset classes incur a 12% loss in this scenario. This is actually consistent with the marginal risk allocation methodology. 14 Furthermore, in this simple example, the marginal contributions coincide with the stand-alone contributions given the discrete nature of the problem and the high correlation of the asset classes implied by the scenarios.

III.0.6

Summary and Conclusions

The primary role of capital in a firm, apart from the transfer of ownership, is to act as a buffer to absorb large unexpected losses, protect depositors and other claim holders, and give external investors and rating agencies enough confidence in the financial health and viability of the firm. We distinguish between three different types of capital: the actual, physical capital that a firm holds (book capital); the economic capital associated with a targeted level of solvency and 14 One can show that for quantile-based measures such as VaR, the derivative in equation (III.0.16), which leads to the

marginal capital allocated to a given asset class (or sub-portfolio), is given by the expected losses of that asset conditional on the total portfolio losses being equal to VaR that is, the expected losses corresponding to all scenarios which lead to the given VaR (see Gouri閞oux et al., 2000).



39


assessed through the use of internal models; and, for banks, the minimum capital imposed by regulatory authorities (regulatory capital). In practice, many firms hold book capital in excess of the required economic and regulatory capital. This reflects some historical and practical business considerations and a more conservative view on the applicability of the models. Economic capital is a powerful business management tool, since it provides a consistent metric for risk aggregation, performance measurement, and asset and business allocation. The objective of a risk-adjusted performance measure is to define a consistent metric that spans all asset and risk classes, thereby providing an apples to apples benchmark for evaluating the performance of alternative business opportunities. Economic capital management tools generally require a bottom-up approach for its estimation, in order to support a practical, risk-sensitive capital allocation methodology. Regulatory capital and economic capital have differed substantially in the past, particularly for credit risk (and also regulatory capital under Basel I did not cover operational risk). The new Basel II Accord for banking regulation has introduced a closer alignment of regulatory capital with economic capital and current best-practice risk management by introducing operational risk capital and allowing the use of internal models for both credit risk and operational risk. This results in minimum capital requirements that are more risk-sensitive. Finally, with its three-pillar foundation, Basel II focuses not only on the computation of regulatory capital, but also on a holistic approach to managing risk at the enterprise level.

References Artzner, P, Delbaen, F, Eber, J-M, and Heath, D (1999) Coherent measures of risk, Mathematical Finance, 9(3), pp. 203228. Basel Committee on Banking Supervision (1988) International convergence of capital measurement and capital standards. Available at http://www.bis.org Basel Committee on Banking Supervision (1995) Basel capital accord: treatment of potential exposure for off-balance-sheet items. Available at http://www.bis.org Basel Committee on Banking Supervision (1996) Overview of the amendment to the capital accord to incorporate market risk. Available at http://www.bis.org Basel Committee on Banking Supervision (2003a) The new Basel capital accord: Consultative document. Available at http://www.bis.org Basel Committee on Banking Supervision (2003b) Trends in risk integration and aggregation. Working paper, available at http://www.bis.org Basel Committee on Banking Supervision (2004) International convergence of capital measurement and capital standards: A revised framework. Available at http://www.bis.org



40


Denault, M (2001) Coherent allocation of risk capital, Journal of Risk, 4(1), pp. 134. Gouri閞oux, C, Laurent, J-P, and Scaillet, O (2000) Sensitivity analysis of values at risk, Journal of Empirical Finance, 7(34), pp. 225245. Kalkbrener, M, Lotter, H, and Overbeck, L (2004) Sensible and efficient capital allocation for credit portfolios, Risk, pp. S19S24. Koyluoglu, H, and Stoker, J (2002) Honour your contribution, Risk, April, pp. 9094. Kupiec, P, (2002) Calibrating your intuition: Capital allocation for market and credit risk. Working paper 99/02, IMF, available at http://www.gloriamundi.org/picsresources/pkcyi.pdf Matten, C (2000) Managing Bank Capital. Chichester: Wiley. Mausser, H, and Rosen, D (2004) Scenario-based risk management tools. In S W Wallace and W T Ziemba (eds), Applications of Stochastic Programming. Philadelphia: SIAM. Merton, R C, and Perold, A F (1993) Theory of risk capital in financial firms, Journal of Applied Corporate Finance, 6(Fall), pp. 1632. Perold, A F (2001) Capital allocation in financial firms. Harvard Business School Working Paper 98-072, available at http://papers.ssrn.com/paper.taf?abstract_id=267282 Praschnik, J, Hayt, G, and Principato, A (2001) Calculating the contribution, Risk, 14(10), pp. S25S27. Saita, F (2003), Measuring risk-adjusted performances for credit risk. Working Paper 89/03, March, available at http://www.sdabocconi.it/it/ricerca/pubblicazioni/dir2003.html Smithson, C (2003) Economic capital - how much do you really need?, Risk, November, pp. 60 63. Tasche, D (2000) Conditional expectation as quantile derivative. Working paper, Technische Universit鋞M黱chen. Tasche, D (2002) Expected shortfall and beyond. Working paper, Technische Universit鋞 M黱chen.



41




42


III.A.1 Market Risk Management Jacques P閦ier 15

III.A.1.1

Introduction

What is market risk and whom does it concern? What do we mean by market risk management, and what does a market risk manager do in a day at the office? These are not theoretical questions with only right or wrong answers, they are practical questions that every financial as well as non-financial firm must grapple with; what appear to be reasonable answers depends very much on the activity, environment, culture, objectives and organisation of each firm. In this chapter students are introduced to the four major tasks of risk management applied to market risks, namely, the identification, assessment, monitoring and control/mitigation of market risks. The difficulties faced in carrying out these tasks vary according to businesses. We therefore examine three typical activities fund management, banking and manufacturing to illustrate a broad spectrum of problems and state-of-the-art approaches. We aim to develop a conceptual and largely qualitative understanding of the topic. More detailed quantitative analyses are given in subsequent chapters.

III.A.1.2

Market Risk

To facilitate the analysis and understanding of risks faced by financial firms it is common practice to classify them into major types according to their main causes. Thus, banking risks are typically classified as being either market, credit or operational in origin. Broadly speaking, market risk refers to changes in the value of financial instruments or contracts held by a firm due to unpredictable fluctuations in prices of traded assets and commodities as well as fluctuations in interest and exchange rates and other market indices. 16 It is not clear to what extent market risks should be or can be considered for less liquid assets such as real estate or banking loans. Accountants usually shy away from attributing fair values to such assets in the absence of reliable and objective market values, and consequently market risks are difficult to assess on illiquid assets. However, when the core activity of a business is to hold portfolios of illiquid assets, it would be dangerous to ignore their potential change in value.

15 Visiting Professor, ISMA Centre, University of Reading, UK. 16 By contrast, credit risk refers to changes in value of assets due to changes in the creditworthiness of an obligor and,

at the limit, losses due an obligor failing to meet its commitments; operational risk is defined as the risk of loss resulting from inadequate or failed internal processes, people and systems or from external events.



43


Banking supervisors have taken a special interest in codifying risks and in setting standards for their assessment.

Their purpose is essentially prudential: to strengthen the soundness and

stability of the international banking system whilst preserving fair competition among banks. To this end they have designed a set of minimum regulatory capital requirements for all types of traded assets, including some illiquid ones, based on detailed definitions and assessments of risk. 17 By and large, banking supervisors have erred on the side of objectivity rather than comprehensiveness when defining market risks. Banks must allocate their assets to either a banking book or a trading book; market risks, bar exceptional circumstances, are recognised only in the trading book.

A trading book consists of positions in financial instruments and

commodities held either with a trading intent or to hedge other elements of the trading book; all other assets (e.g. loans) must, by default, be placed in the banking book. To be able to receive trading book capital treatment for eligible positions, some further basic requirements must be met such as clearly documented trading policies, daily mark to market (or mark to model) of positions and daily monitoring of position limits. To remain prudent, banking regulators have always inflated the capital treatment of credit risks in the banking book to cover for hidden market risks. III.A.1.2.1

Why Is Market Risk Management Important?

Banking supervisors also hope that regulations will promote the adoption of stronger risk management practices, which they view as a worthwhile goal. 18 Working in collaboration with the industry, they have certainly put these issues in the limelight and progress has been made. However, the development and adoption of better risk management practices in banks remains an objective ultimately beyond the reach of banking supervisors. It requires enlarging the purpose of risk management from a purely prudential objective (setting a limit on insolvency risks) to a broader economic objective (balancing risks and returns); it also requires enlarging the scope of market risk assessment to those areas that have been largely ignored by regulators because accrual accounting practices hide the risks and/or the risks are difficult to quantify objectively. Outside financial services there are no prudential regulations offering guidelines for the management of market risk, but market risk nonetheless remains a major determinant in the success or failure of most economic activities and the welfare of people in free market economies. Suffice to observe how variations in the price of energy affect manufacturing and

17 See Basel Committee on Banking Supervision (BCBS, 1996, 2004a). Insurance companies are subject to different

solvency tests; harmonisation of insurance company solvency tests with minimum capital requirements for banks is a long-term aim for regulators. Pension funds and other funds designed to meet strict liabilities are also subject to solvency tests by the relevant regulatory authorities. 18 See BCBS (2004a, paragraphs 4 and 720).



44


transportation, changes in interest rates affect the cost of mortgages and thereby property prices, and the performance of securities markets affects pensions.

The absence of prudential

regulations for non-financial firms gives an opportunity to reconsider the best way to recognise and tackle market risks. And what should become clear is that, satisfying as it may be to categorise risks according to causes, there is no classification system by which every risk would fall into one causal category and one category only, nor any management system that could control one type of risk without affecting others. III.A.1.2.2

Distinguishing Market Risk from Other Risks

Some examples will illustrate how market, credit and operational risks are interrelated. On a macro-economic scale, consider the technology bubble that burst at the turn of the millennium. In 1996 Alan Greenspan, the Fed Chairman, described as irrational exuberance the expectations placed on new technologies; indeed, they did not perform as well or as rapidly as predicted a business or operational risk problem. A wave of corporate failures followed a credit risk and the Fed as well as many other monetary authorities across the world reacted by lowering interest rates a market risk for bond portfolio holders. For an example on a micro-economic scale, consider a firm exposed to exchange-rate fluctuations a market risk. It may seek cover by entering into a forward exchange-rate agreement with a bank, but it thereby takes a credit exposure on the bank if the bank has to pay the firm under the contract. Or consider a bank that makes a floating rate loan to a firm, thus taking primarily a credit risk on the firm; the bank may seek some degree of credit cover by asking for securities or property to be placed as collateral, but the value of the collateral will be subject to market risk. Distinguishing market risks from other risks and managing them separately from and independently of other risks and profit considerations is therefore only valid up to a point. In any organisation, a balance must be struck between the degree of specialisation of risk management functions, so that market risks can be distinguished from other risks and managed separately, and the degree of interaction and coordination between functions so that they can operate coherently.

III.A.1.3

Market Risk Management Tasks

The Basel Committee on Banking Supervision has given a great deal of thought to the role and organisation of a risk management function. It distinguishes four main tasks that it defines as identification, assessment, monitoring and control/mitigation of risks. 19 These tasks are relevant

19 See publications on the New Basel Accord, in particular BCBS (2004a, paragraphs 725745 and 663(a)).



45


for market risks, as they are for most other risk types and for non-financial institutions as well as for banks. Identification is the necessary first step, but it may be less obvious than first thought. Real-world problems do not come neatly defined as in textbooks; they have to be recognised and delineated. Exposures to market risks can easily be overlooked because of either over-familiarity (risks we have always lived with without doing anything about them) or, at the other extreme, ignorance of new risks. The first case is all too common; for example, many corporates do not hedge currency exposures because they are not sure how to assess them or how to hedge them. In the end, it may be easier for a corporate treasurer to remain passive and blame the currency markets for a loss than to be active and have to explain why a loss was made on a hedge, the two circumstances having about equal probabilities. On the other hand, lack of familiarity may lead to over-cautious reactions. For example, investing in foreign equity markets, even when they are denominated in the same currency as the domestic market, is generally considered more risky than investing in the domestic equity market. But not recognising a new risk or combination of risks may be the greatest danger. 20 The managers of the hedge fund LTCM, including two Nobel prize laureates in economics and finance, knew almost everything that could be known about market risks, but they were caught unawares by an unusual combination of events: the repercussions of the Russian bond crises of August 1998, diminished liquidity in major bond markets because of the withdrawal of a large market maker, and difficulty in raising new funds having just returned some capital to shareholders. Assessment is the second step. The word initially chosen by the BCBS was measurement, but it has wisely been replaced by assessment in more recent publications. Indeed, risks are not like objects that can be measured objectively and accurately with a simple measuring tape. Risks are about future unexpected gains or losses. The term assessment reflects the need for a statistical model, that is, a coherent and relevant set of assumptions and parameters, some being supported by past evidence (e.g. former loss events) and others being chosen for the purpose of the exercise (time horizon, trading strategy, etc.). Banking supervisors have set qualitative and quantitative standards for the assessment of market risks to suit their aim, namely, the determination of prudent minimum capital requirements for banks. But internally, banks and other firms should take a wider view to choose standards that suit their own situation and objectives. In the next sections we shall illustrate how different assessment standards may be suitable for different

20 Readers may remember how the US Secretary of Defence, Donald Rumsfeld, was once derided for trying to explain

at a press conference that there are knowns and unknowns and that among the unknowns there known unknowns and unknown unknowns, the latter being the most dangerous. This is in fact old military lore: in the US Navy unknown unknowns are colloquially called unk-unks and said to rhyme with sunk-sunk. Of course, Socrates had put this point across more elegantly when he said: Wisest is the man who knows what he does not know.



46


businesses lines. But we shall not go into detailed quantitative techniques; these are covered in the following chapters. Monitoring refers to the updating and reporting of relevant information. Exposures and results can be monitored. Risks themselves cannot be monitored, but they should be frequently reassessed. Firms, strategies, markets, competition, technology and regulation all evolve, and therefore market risks also change over time. Even in a steady-state situation more information can be collected over time to develop a better understanding of market risks. Monitoring is particularly important when hedging strategies are in place so that one can verify the efficiency of these strategies and update the corresponding risk models. Finally, control has been replaced by control/mitigation in the latest publications of the BCBS. Control gives too much the impression that market risks are intrinsically bad and therefore must be subject to limits, and a key responsibility of the risk manager is to verify that limits are not exceeded or, if they are, to blow the whistle. Mitigation has a wider meaning than control; it indicates that (i) there is a trade-off between risk and return and an optimal balance should be sought, and (ii) there are ways to manage market risks actively that need to be investigated. How each of these tasks should be carried out by market risk managers and what specific problems they may encounter depend upon the business at hand. It would be ineffectual to give general answers to these questions. Rather, we shall explore three business types and show how the three tasks of market risk identification, assessment and control/mitigation vary between them. 21 But first a few comments about how the risk management function should be organised

III.A.1.4

The Organisation of Market Risk Management

Banking regulators have put forward a few general recommendations for the organisation of the risk management function. They reflect a general consensus in the banking industry and are probably valid as well for many other businesses. 22 (i)

The risk management function should be part of a risk management framework and policies agreed by the board of directors. The board and senior managers should be actively involved in its oversight.

(ii)

The risk management function should operate independently of the risk/profitgenerating units; in particular, it should have its own independent sources of

21 We leave aside in this chapter the more routine and easily understood monitoring task. 22 The following four bullet points are not direct quotes but a summary by the author of recommendations that have

appeared in various BCBS publications.



47


information and means of analysis. The risk management function should be given sufficient resources to carry out its tasks with integrity. (iii)

The risk management function should produce regular reports of exposures and risks to line management, senior management and the board of directors. Noncompliance with the risk management policies should be communicated immediately.

(iv)

The risk management process should be well documented and audited at regular time intervals by both internal and external auditors.

It is clear from these recommendations that the risk management function should be separate from and independent of risk-taking line management functions in the front office and support functions in the back office but should be in close communication with them. This is why most banks locate the risk management function in a separate middle office. The middle office must receive information on exposures from the front office in a timely fashion, but it should use its own independent information sources for prices and derived parameters such as volatilities and correlations, and it should use its own models to assess and forecast market risks. The middle office should produce regular (at least daily for banks) market risk reports for the front office and for senior management, containing detailed and aggregate risk estimates and comparisons against limits. These market risk reports are usually combined with credit exposure reports and profit attribution analyses where exceptional gains or losses as well as potential risks are explained. These reports must be immediately verified and approved by designated front office and senior managers. The middle office is also often responsible for producing statutory risk reports for banking supervisors. The risk management function in financial firms is also normally in charge of preparing market risk management policies to be submitted for the approval of the board and of designing, assessing and recommending hedging strategies. It may also be required to calculate provisions and deferred earnings, to design and implement stress test scenarios, to establish controls and procedures for new products and to verify valuation methodologies and models used by traders. In a few instances, market risk managers may be asked to implement global hedging policies that would not sit naturally within any existing department. More recently, market risk managers have been asked to contribute to the analysis of the optimal level and structure of their firms capital (as compared to the evaluation of minimum regulatory capital) and to risk budgeting (also called economic capital allocation) with the aim of improving risk-adjusted return on capital. Proper resources, independence and lines of communication to senior management and ultimately to an executive director on the main board are crucial for the integrity, credibility and



48


efficiency of the risk management function in financial institutions that aim to derive a profit from taking market risks. Supporting and empowering the risk management function are lesser problems in firms that do not seek market risks but would rather avoid them. However, unless there is a clearly defined and adequately supported market risk management function in these firms, market risks may not be properly appreciated and managed.

III.A.1.5

Market Risk Management in Fund Management

III.A.1.5.1

Market Risk in Fund Management

The core activity of fund management is to take market risks 23 with the expectation of generating adequate returns. 24 In most countries traditional funds are subject to strict regulations (about the type of securities in which they can invest, disclosure requirements, etc.) that are aimed at protecting investors. That is particularly so for pension funds that should provide long-term security to their members and, in return, enjoy certain tax advantages. Fund managers also often choose to limit and specialise themselves further according to market sectors or investment strategies, leaving to investors the choice to allocate their savings among funds and manage their own portfolio diversification; for example, specialist funds may describe themselves as UK equity, capital guarantee or index tracker funds. But specialisation and constraints can only limit potential returns so, starting in the 1970s, private investment pools were created to offer to wealthy individuals, who might be more willing to take risks, the promise of greater returns by avoiding traditional fund constraints and regulations for example, they can use short sales, derivative products and leverage. Interestingly, these new, unfettered funds became known as hedge funds because many of them used trading strategies based on spreads between long and short positions rather than pure directional bets. Their popularity grew in the mid-1990s and even more so when the technology bubble burst and traditional funds returns inevitably tumbled. Total assets under hedge fund management may soon pass the trillion dollar mark. 25 Funds take market risks for the potential benefit of their investors. Fund managers themselves are only indirectly affected by market losses. Their income is usually a set percentage of the value of assets under management (plus some participation in profits in the case of hedge funds); it is not directly affected by losses. The market risks are born by the investors. But the reputation of 23 Funds also take credit and other risks but, bar special cases (e.g. a fund investing in a few high-yield corporate bonds

or a certain emerging market), most funds invest in a large number of liquid, good-quality securities and therefore credit risks are less important than market risks. 24 In some cases returns must be sufficient to meet certain liabilities (e.g. pensions); in others, the fund managers objective will be to maximise some risk-adjusted performance measure. 25 By comparison, in mid-2004, total worldwide bond and equity markets were valued at about $70 trillion; more than half these assets were managed by institutions. In the USA, mutual funds alone managed about $7 trillion of assets.



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fund managers and therefore their ability to retain existing investors and to attract new ones depends on their ability to manage their risks (and their clients). It is therefore crucial that (i) fund managers explain to their clients the risks they are taking, (ii) clients agree formally the terms and conditions of their investment, and (iii) fund managers keep to the terms that have been agreed. III.A.1.5.2

Identification

It should be an easy task for fund managers to identify market risks because they normally have chosen deliberately to take those risks. Nonetheless some risks may be overlooked, especially in funds following sophisticated strategies. In general, funds following spread or arbitrage type strategies will have reduced primary directional risks but will have increased exposures to secondary risks. For example, if a fund is allowed to short securities, the uncertainty in the repo cost incurred over the long term may be quite considerable, as well as difficult to estimate. Likewise, the spread between two similar securities, say A and B shares of a company, may be drastically affected by legal, tax or regulatory changes. It would not be much consolation to decide that such events are operational rather than market risks if they have not been foreseen. Liquidity risk is another relevant concern, particularly for hedge funds. 26 Some assets may not be bought or sold at the anticipated price because the transaction is too large compared to the market appetite. Traditional funds are bound by regulations to hold only highly liquid positions. Hedge funds, on the other hand, do not have such constraints and may end up holding relatively large positions in specific securities. If in addition they are highly leveraged, they can easily be thrown into a momentary cash-flow squeeze or even a terminal problem by a liquidity crisis. We have already referred to LTCM as an example. When funds have to meet specific liabilities and many do27 managers seek to maintain a stable surplus of assets over liabilities and should therefore be concerned by possible market risks on the liability as well as on the asset side. Actuarial practices and accounting standards have generally overlooked or hidden these risks in the past but new rules are now coming into effect that bring them to the fore. For example, in the United Kingdom, Financial Reporting Standard 17 (FRS 17) prescribes 28 that asset and liabilities in company pension schemes be immediately

26 Liquidity risks deserve to be analysed separately from market risks, but the two are closely related. The liquidity of a

security can be characterised by its average daily trading volume. Usually, the bidoffer spread increases rapidly with the size of a transaction relative to the average daily trading volume when that fraction is significant. Exceptionally, there are securities that do not trade regularly and yet can be traded in large single blocks without putting undue pressure on their price. This characteristic is commonly referred to as market depth. 27 For example, funds supporting defined benefit pension schemes, insurance policies (life or property and casualty) or backing the issuance of guaranteed investment contracts (GIC). 28 FRS 17 becomes mandatory in the U.K. at the same time as the new International Accounting Standards (IAS) become mandatory for companies listed on European Union stock exchanges, that is for accounting periods ending on



50


recognised on the company balance sheet at their market value for assets or present value based on relevant gilt rates for liabilities. III.A.1.5.3

Assessment

The assessment of market risk is now a very well-developed activity in the fund management industry. It has become part and parcel of performance assessment and, if not always done thoroughly ex ante by fund managers, it is certainly done ex post by a number of analysts in order to compare the so-called risk-adjusted performance of funds. Ex-post analyses are usually pure statistical analyses of time series of returns. They produce estimates of return distributions. These estimates are then fed into risk-adjusted performance measures (RAPMs), the most common RAPM being the Sharpe ratio or ratio of expected excess return relative to the risk-free interest rate divided by the standard deviation of return, both on an annualised basis. There are some simple arguments why investors should prefer funds with the highest Sharpe ratios. However, Sharpe ratios may lead to unwarranted conclusions if applied to the comparison of funds with significantly different return distributions. For example, whereas a well-diversified traditional fund holding long security positions only may be expected to exhibit approximately log-normally distributed returns, a fund selling out-of-the-money options or implementing a dynamic strategy with similar consequences should exhibit a significant downward skewness and excess kurtosis of long-term returns. The Sharpe ratio would be inadequate to compare the performance of these two funds, but a generalised Sharpe ratio or some other RAPM accounting for skewness and kurtosis might do. See Chapter I.A.1 for a full discussion of RAPMs. Because many traditional funds are limited in their choice of securities and/or investment strategies, they prefer to be judged on relative rather than absolute performance. 29 Depending on their strategy they choose or create a benchmark and estimate their risk-adjusted performance relative to the benchmark. The standard deviation of returns relative to the benchmark is called the tracking error. The choice RAPM is the ratio of the average excess return relative to the benchmark over the tracking error; it is called the information ratio or appraisal ratio. Often

or after January 1, 2005. FRS 17 stipulates immediate recognition of gains and losses on a company pension scheme but in a secondary statement of gains and losses called Total Recognised Gains and Losses rather than in the Profit and Loss account. IAS 19, the international standard relative to employee benefits has moved in the direction of FRS 17. Financial Accounting Statement No. 87 (FAS 87), the equivalent statement under U.S. generally accepted accounting principles (U.S. GAAP) issued in 1985 lags behind IAS 19 and FRS 17 both in the application of fair valuation and the rapid recognition of gains and losses. 29 Although, as Warren Buffet said, You cannot eat a relative performance sandwich, comparison to peers or to a chosen benchmark rather than absolute performance is seen as a clear indicator of skills and a powerful source of motivation.



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the analysis of performance is extended to a full performance attribution analysis to explain which strategies and which changes in market factors have contributed to profits and losses. Ex-post assessments are certainly useful for comparison and analysis of returns as well as to check ex-ante assessments, but they do not replace the need for ex-ante assessments. Ex-post assessments lack reliability and relevance because they are based on limited information typically, a few years of monthly returns and they are not forward-looking. Even if return data were available on a much more frequent basis, daily for instance, they could only lead to a more statistically accurate forecast of short-term returns.

Methods such as exponential moving

averages (EWMA) and GARCH (see Section III.A.3.4) have proved useful to estimate daily risks in financial markets exhibiting time-varying volatilities. But estimates of short-term volatilities have little relevance for long-term risks when these are governed by a specific investment strategy such as capital protection, and consequently short-term returns are not mutually independent. It is only on the basis of ex-ante assessments of risks that fund managers can check and justify that they are adhering to their management mandate as described in a mutual fund prospectus or agreed with trustees or shareholders. Of course, ex-ante assessments are more difficult. One must rely on assumptions about future market behaviour and sometimes introduce a degree of subjectivity. One must also assume a trading strategy complete with limits and contingency plans. Too often ex-ante analyses are carried out using standard commercial models without sufficient questioning of the assumptions contained in these models.

A typical error, for

example, is to assume that future departures from the performance of a benchmark will be small if the tracking error has been small in the past. The problem is that, ex post, the tracking error is usually calculated on de-trended return series, but if the composition of the portfolio being evaluated is significantly different from the composition of the benchmark, the two return series may well have different trends. III.A.1.5.4

Control/Mitigation

If star ratings from one to three were to mark the degree of difficulty of a task, the identification of market risks in fund management should be attributed only one star, risk assessment two stars and control/mitigation three stars. Control of limits is not much of a problem, but the design of a risk mitigation strategy is as complex as the design of the investment strategy itself. In fact, the two cannot be separated except in a few special circumstances. III.A.1.5.4.1

Selective Hedging

As a first special case, some undesirable risks may have been acquired as part of a package and need to be reduced. A typical example is that of a fund investing in a particular industry sector



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worldwide but wishing to maintain currency exposures to a minimum. In this case forward currency contracts can be used as hedges, but these hedges will have to be readjusted as a function of changes in foreign-denominated asset values and rolled over regularly.

The

corresponding costs and residual uncertainties will need to be estimated. A more complex case is that of positioning a bond portfolio to take advantage of some interestrate changes whilst protecting the portfolio against other possible interest-rate changes. Active managers of bond portfolios seek to exploit specific views on interest rates, for example, that an interest-rate term structure will flatten or that rates in two currencies will converge. At the same time they are likely to want to reduce exposures to interest-rate movements that should not affect their strategies, for example, for the two strategies above, parallel shifts of all interest rates. A traditional method to achieve this is to calculate the first- and second-order derivatives of bond values with respect to their yield to maturity (see Section I.B.2.6).

These derivatives or

sensitivities are called value duration and value convexity respectively. 30 Assuming the same changes in yields across all bonds in a portfolio, the portfolio value duration and value convexity are simply obtained by adding up the individual bond value durations and value convexities. Adjusting the composition of a bond portfolio so that these two sensitivities become negligible is often interpreted as immunising the value of the portfolio against parallel shifts in interest rates. An example of immunisation against a parallel shift of bond yields, hedging one bond with another, is given in Section I.B.2.7.

But note that reducing the value duration and value

convexity of a portfolio to zero does not eliminate all interest-rate risks. It is an efficient immunisation only against small parallel shifts in bond yields, which is a relatively unlikely scenario; 31 many other variations of interest rates are possible. A sensible approach to selecting a portfolio of bonds to be immune to some movements in interest rates whilst maximising the profit opportunity from a forecast movement is first to calculate each bond price variation relative to each relevant interest-rate movement and then to choose the portfolio weights so as to maximise the portfolio gain for the forecast interest-rate movement whilst leaving the portfolio value unchanged for the other movements. 30 In the same way as we now say value-at-risk rather than dollar-at-risk, it is time to say value duration rather than

$duration. When minus the value duration is divided by the value of the bond, the result is called the modified duration. Finally, when the bond yield is expressed on an annual basis, duration (or Macaulay duration) is defined as modified duration multiplied by (1 + yield). Historically, Macaulay was the first author to introduce the concept of duration; he chose the name because, for a zero-coupon bond, it is equal to the maturity of the bond, and, for a coupon bond, it is equal to the average maturity of the cash flows weighted by their corresponding discount factors calculated at the bond yield. The second-order sensitivity relative of a bond value with respect to its yield is called value convexity because it relates to the curvature of the value versus yield curve. 31 A parallel shift in bond yields corresponds approximately to a parallel shift in the zero-coupon rate curve. Actual movements of the zero-coupon rate curve are best captured by a principal component analysis (see Section III.A.3.7). The first principal component, which usually explains 75% to 80% of interest rates total variance, is frequently described as a parallel shift when in fact it is often anything but parallel. Medium-term rates (18 months to 3 years) are often more volatile than both short-term and long-term interest rates (see, for example, Alexander, 2001, Table 6.2b, p. 149).



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III.A.1.5.4.2

Momentary Hedging

As a second special case, we have the momentarily undesirable risks. At times, fund managers may fear a correction in the markets that would harm their performance or may simply wish to reduce some exposures for peace of mind because they are momentarily absorbed by other tasks. Closing down some positions for a short while may prove difficult or expensive; an overlay hedge may be less costly and still efficient. Adding offsetting derivative positions does it. Even if the derivatives are not a perfect offset for the undesirable exposures, an approximate global hedge can be achieved. It may work particularly well during brief market crises when correlations between related market factors tend to increase. There are many examples throughout the Handbook of such strategies. For example, the use of call options on bond futures to hedge a bond portfolio is given in Example I.C.6.6. The use of equity swaps is explained in Section I.B.4.7.1.1. III.A.1.5.4.3

Managing for a Risk-Adjusted Performance Target

Coming back to the general case, risk mitigation strategies are inseparable from investment strategies designed to achieve some risk-adjusted performance target. There is but one task for the active fund manager: to follow a policy encompassing level of diversification, leverage, selection of securities, etc. compatible with the objective of investors and his own forecasts. Like customers in a supermarket who want choice and want to know what they buy by reading the labels on the cans, investors want a description of the funds offered to them in terms of composition of assets, strategies, objectives and target risk levels. And fund managers must remain true to the description of their products. The type of assets in which a fund can invest is certainly a major determinant of the volatility of returns. For example, equities are generally regarded as more volatile than bonds; within equities some sectors such as dotcoms and emerging markets are clearly more volatile than, say, utilities in G7 countries, and so on. But other factors have also a large influence on risk; chief among them are the level of diversification and the degree of gearing of the risky assets. By increasing the number of relatively independent securities in a portfolio, diversification reduces total risk by averaging out the effect of specific, independent risks. Diversification can be optimised to obtain the best possible value of the chosen risk-adjusted performance target Sharpe ratio, information ratio or other (see Section I.A.3.4 for details). Gearing up, or leveraging, means increasing the allocation of funds to a risky asset class relative to a risk-free asset class, usually cash deposits, or even borrowing in order to invest more in the risky assets than the equity value of the fund. The expected return above the risk-free rate and the volatility of return vary proportionally to the amount invested in the risky assets relative to the equity value of the fund. Therefore gearing up or down does not affect the Sharpe ratio of a fund (see Section I.A.3.5 for details), but it can be used to adjust the risk level to



54


suit a specific group of investors. Of course, many investors are sophisticated enough to manage themselves their own gearing and diversification. All they want is to have a choice among a wide variety of well-defined funds. This leaves fund managers with the choice of dynamic investment strategies as a means of controlling risk and return. Positions can be actively adjusted either with the objective of achieving a specific return distribution we discuss a couple of examples in the following subsection or simply to take advantage of evolving return forecasts. The latter is particularly difficult to optimise as the cumulative costs of rebalancing more frequently compared to the expected opportunity losses of rebalancing less frequently are difficult to perceive intuitively and to analyse quantitatively. This is a subject of academic interest (see Davis and Norman, 1990), but implementation of systematic dynamic strategies is lagging behind theory. Most active fund managers rely on heuristics to rebalance their portfolios. These are rules of thumb based on trial and error combining profit objectives with multiple limits: stop losses, delta limits, limits on turnover, etc. Only a small proportion of active fund managers essentially those using highly quantitative investment strategies (e.g. cointegration or convertible arbitrage) or those promising protected returns rely on systematic rebalancing rules. III.A.1.5.4.4

Capital Protection

Since the early 1980s, there has been a growing number of funds offering some kind of performance protection in order to attract risk-averse investors. They try to offer the best of two worlds: on the upside a participation in the potentially high returns of a risky asset class typically equities or commodities or, as a minimum on the downside, a capital guarantee or the return on a low-risk investment for example, a deposit or a bond. With few exceptions capital guaranteed products have a stated maturity of a few years. Investors staying until maturity are guaranteed to receive a defined performance. For instance, I read the following offer received today: after 5 years you will be guaranteed 105% of the performance of the FTSE 100 index on your initial investment or your money back, whichever is the highest. The guarantee is from the sponsor of the product or a third party, usually a good-quality insurance company or bank. To add to the popularity of these products and attract small savers to long-term equity investments, tax advantages may also be available at maturity. On the other hand, early withdrawals are not guaranteed or are guaranteed at only a fraction of the initial investment. Although these products may appear as manna from heaven to the unsophisticated investor, they are actually simple to manufacture. In our example, the sponsor could use the initial investment



55


to buy the equivalent face value of a five-year zero-coupon bond and a FTSE 100, at-the-money, five-year over-the-counter (OTC) call option on 105% of the initial investment from a specialist bank. 32 The zero-coupon bond might cost 75% and the call option 18%, leaving 7% to the sponsor to cover his expenses and contribute to profit. We leave to Section III.A.1.6.4 the manufacturing of the call option and, more generally, the dynamic hedging of option portfolios. The main drawback of capital guaranteed investments is their bullet form: one fixed size, fixed maturity issue. Investors like the flexibility of open-ended funds whose shares can be issued or redeemed at any time at their net asset value plus or minus a small commission. Can any form of downside protection be offered on an open-ended fund? This question has exercised the minds of many financial engineers and only approximate answers have been found. In the early 1980s, many funds became interested in the concept of portfolio insurance and tried to implement it by themselves or with the help of consultants. Insurance of an equity portfolio consisted of overlaying short positions in the new equity index futures at critical times. A na飗e strategy, for example, would be to short futures whenever the market index fell below a predefined level and to buy them back whenever the market index recovered above that level. This, in effect, is a very inefficient attempt at replicating a put option; it creates significant residual risks and costs, not to mention implementation difficulties (e.g. index jumps, lack of liquidity). Not surprisingly, during the crash of October 1987, this type of portfolio insurance disappointed and various dynamic portfolio strategies came under criticism (Dybvig, 1988). Portfolio insurance strategies were improved (Black and Jones, 1987) and new concepts emerged, notably, that of constant proportional portfolio insurance (CPPI) (Black and Perold, 1992). Under CPPI a fund would maintain an exposure in a risky asset proportional to the net asset value of the fund above a certain minimum. For example, the risky asset could be an equity index future and the exposure would be 200% of the excess value of the fund above the value of 90% of the initial investment placed in a short-term money market. Thus the fund manager would promise (sometimes with a bank guarantee) as a minimum the money market return on 90% of the initial investment but would raise the expectation of an equity index performance with a leverage of up to 200%. In continuous markets and with frequent (weekly or daily) rebalancing, CPPI would be safe. In practice, negative jumps combined with a poor initial performance of the risky asset may bring the fund value rapidly to its minimum guaranteed level, at which point the fund becomes a pure money market fund. This is what has happened in the early 2000s to many CPPI funds that were launched at the end of the 1990s.

32 Always read the small print on the exact definition of the pay-off; it is often not as straightforward as it first appears.



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Special types of portfolio insurance strategies should be developed for funds that must meet specific liabilities. Future liabilities may be uncertain both in amounts and timing (e.g. insurance claims), nonetheless, fund managers must aim for a stable surplus of assets over liabilities. These issues are often obscured by ad hoc actuarial rules and regulations and are in great need of reexamination. III.A.1.5.4.5

Compliance and Accountability

Investors are becoming increasingly sophisticated and capable of scrutinising the market practices and performance of fund managers.

Beyond compliance with regulations and accounting

standards, beyond the avoidance of market malpractice, 33 fund managers must satisfy investors that they are following strategies compatible with stated performance objectives and risk limits. When results have been disappointing, investors have sued fund managers for negligence or noncompliance with agreed policies. The investor does not necessarily have to lose money for this approach to succeed. In a case that has set new standards of accountability for fund managers, the Chief Investment Officer of the Unilever Superannuation Fund (a pension fund) accused the Mercury Asset Management unit of Merrill Lynch Investment Managers of negligence after the fund had underperformed its benchmark by more than 10% over little more than a year (January 1997 to March 1998). Mercury had agreed a target of 1% per year above the benchmark return with a tracking error of no more than 3%. Although the return on the ?

billion fund had still

been positive over the period, Unilever sought damages of ? 30 million. The case revolved about the improper use of risk assessment models and, crucially, about the delegation of day-today operations to a junior investment officer. Merrill admitted no liability but, in June 2001, settled out of court for a substantial amount. Similar cases have followed since.

III.A.1.6

Market Risk Management in Banking

III.A.1.6.1

Market Risk in Banking

Banks, like hedge funds, take geared positions on various asset classes. They differ in that many of their assets (e.g. loans) are illiquid and some of their sources of funds (e.g. deposits on call) are low-cost but with an indeterminate term outside the banks control. In addition, banks are engaged in a number of fee-earning activities, but that is not of primary interest as far as market risks are concerned. Most market risks are taken by banks voluntarily with a view to benefiting from the exposures. At the same time, deposit taking from clients is based on trust and it is crucial for banks to maintain their reputation of financial stability and competence in managing risks. Otherwise, 33 The fund management industry has recently been the subject of a series of enquiries about conflicts of interests due

to close relationships between investment bankers and fund managers and about market malpractices such as market timing which have resulted in hundreds of millions of dollars of fines, withdrawal of billions of dollars of funds, scores of firings, and the reorganisation of several financial conglomerates.



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clients may slip away, causing funding to become rapidly more expensive, and the business may fall into a downward spiral. But banks are generally well equipped to manage market risks.

It is part of their core

competences; they have powerful systems to analyse and monitor risks and good access to the markets for hedging. Moreover, they operate under the close supervision of banking regulators. III.A.1.6.2

Identification

As mentioned earlier (see Section III.A.1.2), a key distinction is made between liquid assets eligible for capital treatment under trading book regulations and less liquid assets or assets held with a long-term intent that are relegated to the banking book. Only assets in the trading book are subject to detailed statutory assessments and corresponding capital charges. By contrast, market risks in the banking book are largely ignored by banking supervisors, and so are market risks affecting liabilities. In fact, accrual accounting standards tend to hide such risks. Only if common sense indicates that unusually large market risks are present in the banking book will banking supervisors request some ad hoc estimates and monitoring/control procedures and be free to impose additional capital charges. 34 Within the trading book, market risks are traditionally categorised by main markets; so, interestrate, equity, currency and commodity risks are often identified separately. Nonetheless many positions entail risks in several markets, for example, a share denominated in a foreign currency, a convertible bond, a commodity linked loan; such positions will have to be identified under each of the corresponding market risk types. To facilitate market risk analyses, it is also traditional to distinguish primary from secondary risks and general market risks from specific risks. The primary risks are the directional risks resulting from taking a net long or short position in a given class of securities or commodities. The secondary risks are the other risks, deemed a priori to be less important, for example, volatility risk, risk on spread between a security and a futures contract on that security, a dividend risk, a risk on repo costs. Secondary risks may well appear less important than primary risks, except for the fact that many trading books are managed actively with the purpose of reducing primary risks to negligible proportions at the expense of an increase in secondary risks. Likewise, general market risks, such as an exposure to movements of an equity index, may seem more important a priori than specific risks due to unequal variations of prices of shares in that

34 Capital surcharges are left to the discretion of banking supervisors under Pillar II of the Basel Accord. See BCBS

(2004b) for general principles on the management of interest-rate risks.



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index. However, the composition of a portfolio of shares may differ markedly and systematically from their weightings in a reference index, for instance because of lack of diversification or because of the implementation of a particular investment strategy (say, value strategy rather than growth strategy). Consequently, specific risks may be large compared to the general (also called systematic) market risk and should not be overlooked. Finally, one should take into account the increasing proportion of option and option-like instruments in trading books. A financial option is an instrument offering the right but not the obligation for the owner to make a claim by a certain date if some underlying market factor (or combination of market factors) is favourable or, otherwise, to forgo any claim and gain nothing. Thus the pay-off of an option is a non-linear function of some market factors. By extension, instruments that yield a non-linear pay-off in some market factor(s) can be called option-like for example, a bond price is a non-linear function of changes in the discount-rate curve. Because of this non-linearity, the fair value of an option or an option-like instrument depends on the full probability distribution of future values of the underlying market factor(s) and not only on their current or expected future values. Long-term volatilities and correlations affecting the value of options offer new trading opportunities and hence new market risks. A systematic identification of market risks in the trading book should proceed through the identification/selection of key market factors and the construction of models relating the value of instruments in the trading book to these factors, a step that will be essential for the assessment of market risks. Within the banking book, market risks are harder to identify because positions are generally not valued at fair prices and therefore fair price variations are of little concern. But it is common sense that most banks are exposed to large market risks in their banking books assets as well as on their liabilities. In fact, interest-rate maturity transformation, the short-term funding of longer-term loans, has been a traditional banking strategy since time immemorial and entails an exposure to interest-rate rises. The impact of interest-rate changes is also enhanced by the clients behaviour: if there are any prepayment or extension possibilities on loans and deposits, clients will take maximum advantage of these options to make loans cheaper or deposits more attractive and therefore less profitable for the bank. Some option-like positions in the banking book are particularly susceptible to interest-rate changes. For example, a line of credit at a predetermined spread above Libor is economically equivalent to an option on the credit spread of the client: the line is much more likely to be drawn upon when the creditworthiness of the client has declined than when it has been maintained.



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III.A.1.6.3

Assessment

We stressed the multiplicity of market risk factors in the preceding section on identification: there are systematic and specific market risks; primary directional risks and secondary risks; volatilities and correlations. The assessment of market risks is based on the selection of a limited number of market risk factors and a choice of models to describe uncertainties in the future values of these factors, their impact on the value of individual instruments and, consequently, on the values of portfolios. Fortunately, a number of models have been put forward and tested over the last 30 years or so. They fall into three main categories: (i) probabilistic/statistical models describing uncertainties about the future values of market factors; (ii) pricing models relating the prices and sensitivities of instruments to underlying market factors; and (iii) risk aggregation models evaluating the corresponding uncertainties on the future values of portfolios of financial instruments. In the first category are the stochastic processes commonly used to describe the evolution of market factors: geometric Brownian motion, stochastic volatility models, GARCH models, etc. A prime example of the second type is the BlackScholes; option pricing model (see Section I.A.8.7) which, for a given choice of dynamics for the underlying asset price, adds some efficient market assumptions and a hedging argument to yield a risk-free option price.

The third type is

exemplified by value-at-risk (VaR) models which, with the help of a few simplifying assumptions, produce a probability distribution (or at least some statistics) on the future value of a static portfolio at a chosen future time. These are explored in Chapters III.A.2 and III.A.3. We should acknowledge immediately that, no matter how sophisticated mathematical models have become, they are idealisations of reality. They are bound to be approximate at best; at worst they can be misleading. Nonetheless, they are indispensable. There is a large degree of subjectivity in the choice of a model. A balance must be struck between realism and tractability. Depending on the business at hand, different models may suit. Conversely, a particular choice of model is always vulnerable to gaming by traders seeking to construct portfolios that will apparently exhibit little risk. That is why banking supervisors want to exercise some control over the use of models for regulatory risk reporting they want to examine the way a model is used, by whom and for what purpose before recognising its use for regulatory reporting and the determination of capital ratios. The greatest degree of freedom in the choice of models lies at the very first stage in the choice of market factors and the description of their dynamics. For example, to describe interest-rate risks



60


across portfolios of bonds, bond derivatives, swaps and other interest-rate derivatives, one starts with a choice of interest-rate term structure model. It can be one of many single- or multiplefactor models from which all interest rates are derived. The temptation on the front desk is to choose the simplest adequate model for each task at hand and therefore to use different models for different instruments.

But a market risk manager should be concerned with

comprehensiveness and coherence across models so that the effects of a variety of possible interest-rate fluctuations are taken into account realistically and consistently. Pricing models are indispensable for all securities that are not readily priced in the market, for example, most OTC derivative and structured products. They are also necessary for most securities whose prices are readily available in the market because, unless these prices are selected as market factors, we need to know how they would be affected by changes in the value of the selected market factors. For example, we need to know how the prices of bonds would be affected by some fluctuations in the risk-free interest-rate curve, such fluctuations being relative to the selected market factors.

Pricing models should follow logically from the choice of

dynamics in the underlying market factors; however, some simplifications/approximations are usually introduced to obtain realistic prices within a limited computation time. Likewise, risk aggregation models should follow logically from the choice of market factors dynamics and pricing models. However, many further simplifications are introduced for two reasons: (i)

Dependencies between market factors can be very complex; they may differ between normal and extreme market conditions, between short- and long-term horizons.

(ii)

Trading book portfolios are by definition very dynamic, but it would be complex to describe the effects of new business and dynamic hedging strategies.

The conventional wisdom in banks is therefore to concentrate on the short term under normal market conditions where dependencies may be approximated by linear correlations and portfolios may be assumed to remain relatively static. This is what regulators ask banks to do: to estimate the maximum level of market losses that would not be exceeded with a probability of more than 1% on a static portfolio over the following 10 trading days. 35 This number is then back-tested against actual conditions and scaled up to produce a minimum capital requirement for market risks. Stress tests (explained in Chapter III.A.4) are then applied to ensure that the minimum capital requirements are sufficiently safe. Market risk capital requirements for portfolios assessed separately are then simply added together and added to capital requirements for other types of risks to yields the total minimum regulatory capital (MRC). Note that this na飗e aggregation 35 This is what is commonly known as the VaR figure in banking.



61


process is likely to produce a larger total MRC than necessary because it does not recognise the effects of diversification among risks. Unfortunately, however, it is not necessarily safe. 36 In some cases, adding the VaRs may understate the gross risk. Many banks have now taken a wider view of market risks than that requested by banking supervisors. Banks can choose parameters to suit their own internal purposes whether to improve resource allocation, to set up an ideal level of capitalisation or to test strategic plans. They can choose their own time horizon for risk assessment and confidence level for extreme losses. They can incorporate less liquid instruments into the banking book, assume some initial pricing uncertainties and take into account the effects of trading and hedging strategies. In the banking book, they may develop models of customer behaviour in response to changes in interest rates and other market factors. III.A.1.6.4

Control/Mitigation

Banks have the means and the competence to manage most market risks very effectively. First, they have some degree of control over the market risk they take. In the medium term they can shape the risks by modifying the design and pricing of the products they offer to their customers. They can also adjust their liabilities to a large extent to match the risk profiles of their assets. In the short term they can use derivative products to hedge most market risks if they wish to do so. The importance of financial derivatives as market risk hedging instruments needs to be stressed. Financial derivatives have become a huge market. In terms of notional size of the underlying assets, they are twice as large as bond and equity markets combined, about $140 trillion against $70 trillion.37 In terms of trading volumes they are even larger. However, the total market value of these instruments counting the positive side only of each transaction is less than $3 trillion and, after netting exposures to single counterparties, less than $2 trillion. The credit risk created by derivative products is therefore very small, about 40 times smaller than the credit risk created by bonds and equities. Derivatives are also relatively cheap to trade. As a fraction of notional size, the sum of bidoffer spread and commissions on derivatives is typically at least ten times cheaper than for the relevant underlying assets. Derivatives also exist on assets that would not be easy to trade, such as equity indices and notional bonds. These features make derivatives the

36 The addition of VaR figures is not sub-additive in general. In particular, super-additivity may occur when the tail

risks are bigger than if they were normally distributed. 37 Of the $140 trillion total, about $110 trillion is OTC and $30 trillion listed; about 60% of financial derivatives are in the form of interest-rate swaps.



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choice instruments for hedging market risks, 38 and indeed banks are usually found on at least one side of most OTC derivative products and as active participants in listed derivatives markets. The crucial element to set up a market risk control/mitigation strategy in a bank is the definition of the objective. As for fund managers and even more so, there are some undesirable market risks accumulated in the course of normal business; from time to time there may also be risks that should clearly be reduced because of changes in market or management circumstances. But the bulk of market risks taken by banks is still taken willingly with the objective of deriving a profit. The risk management objective must therefore be the optimisation of some risk-adjusted performance measure within the constraints on minimum regulatory capital and various concentration limits imposed by banking supervisors or adopted internally.

This general

objective is translated in the short term and at various hierarchical levels (division/desk/trader) into simpler objectives and limits. The simpler objectives usually also take the form of riskadjusted performance measures, but with a cost of risk (or cost of risk capital) adapted to each management unit. 39 Having assessed market risks, recognised the tools that can be used for their control and defined the objective of market risk management, the design and implementation of a control/mitigation strategy should follow naturally. In reality, there are still some complexities due to people and organisations. First, individual incentives should be aligned with the stated objectives. Rewards cannot be based solely on results without considering and agreeing ex ante the risks being taken. When a market risk hedge is put in place, there is roughly one chance in two that the hedge will generate a loss. All too often, if the rationale for the hedge has not been clearly agreed at the start, a loss on the hedge will reflect badly on the hedger, especially if the risks being covered are risks that the bank used to accept in the past.

Second, risk mitigation is achieved more

economically at a macro than a micro level. The following case illustrates these two points. In the late 1970s, the European markets became flooded with petro-dollars.

International

treasury divisions of banks grew rapidly to handle this hot money that could flow in and out rapidly. Many international treasuries implemented a very cautious micro-hedging strategy: each dollar deposit had to be matched with a corresponding lending and vice versa, thus doubling the size of the balance sheet and losing a bid-offer spread to the market. At the same time in the 38 Note that there are also financial derivatives to cover credit risks. The market for credit default swaps and other

credit derivatives has grown at about 50% per year over the last 10 years and now covers about $2.5 trillion of underlying assets. 39 Risks generated by various management units may have a different impact on global risk. Some may have a diversifying or even hedging effect; others may be highly correlated with global risk. The cost of risk attributed to each unit should therefore reflect the marginal contribution of each unit to global risk to ensure that local optimisations of risk-adjusted performance lead to a global optimisation of risk-adjusted performance. Decomposition of risk is explained more fully in Section III.A.3.6.



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same banks, domestic treasuries were continuing to run significant interest-rate gaps (longer interest-rate maturity schedules on the asset side than on the liability side) without worrying about it because they had always done so. Clearly, there was a lack of consistency between the risk management objectives and mitigation strategies between the domestic and international sides. It could be explained for a while by fear of the unknown on the international side. But eventually a more balanced approach had to be implemented in which the interest-rate maturity gap could be tracked on a net basis at regular intervals (e.g. daily) and managed globally. This achieved today in most banks and, indeed, international and domestic treasuries are now often merged in a single treasury division implementing a coherent market risk management policy across desks. We do not have the space here to detail market risk hedging strategies, but we can highlight a couple of points. First, managers are mostly concerned about reducing primary risks, that is, the risks associated with net long or short positions in various asset classes. Exposures to primary risks are characterised by first-order sensitivities, the deltas to the corresponding market factors. 40 Delta is the word commonly used to describe the first-order sensitivity of or change in an option value per unit of underlying asset relative to a very small change in the underlying asset price, as explained in Section I.A.8.8. Primary risks can be covered (delta hedged) relatively cheaply with futures and forward contracts. In many instances, however, the exposures are not linear in the hedging instruments because of the presence of options or option-like instruments. Therefore, hedges with futures and forwards must be rebalanced over time as prices fluctuate. How often should delta hedges be rebalanced? As a rule of thumb, over a given time interval, the transaction costs of rebalancing a hedge (bid offer spreads and commissions) increase as the square root of the rebalancing frequency, whereas the variance of residual risks decreases as the inverse of the rebalancing frequency. An optimum frequency (or more efficient rules based on actual market movements) can be derived from a choice of trade-off between costs and residual risk and the knowledge of some portfolio and market characteristics (volatility of underlying asset price, gamma or second-order sensitivity of the portfolio to changes in the underlying asset price, unit transaction costs); see, for example, Hodges and Neuberger (1989). The results may surprise traders because it is very difficult for anyone to gain an intuitive view about the right balance between expected costs and residual risks

40 When multiplied by the notional size of the underlying asset we obtain a so-called dollar-delta or delta-equivalent

value that is the value of a position on the underlying asset having the same sensitivity as the option. Historically, other names have been used in other markets, for example modified duration in the bond markets as we have seen in Section III.A.1.5.4.



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that accumulate slowly over time but can reach very large figures and can be very different from one portfolio to another. 41 Second, having more or less delta-hedged a portfolio, one is left with secondary risks now playing a primary role. Key among these risks are exposures to volatility changes and larger than expected underlying asset price movements, also known as gamma risks from the name generally given to second-order sensitivities to market factors. 42 The sensitivity of a portfolio to changes in volatility (assuming volatility can be described by a single parameter) is usually referred to as vega. 43 The two risks (explained in Section I.A.8.8) are related but are not the same. For a single plain vanilla option and using a constant volatility model, vega is equal to minus gamma multiplied by time to maturity. Obviously, for portfolios of options and option-like instruments with different maturities, there is no longer a simple relationship between vega and gamma. If the model for the dynamics of a market factor does not assume a constant volatility source of risk, then volatility may change over time and space (market prices) and one can no longer speak about a single vega, at least not without redefining what is meant by vega. Gamma risk is a concern inasmuch as if large and left unchecked it would require a very active and therefore expensive delta-hedging strategy and still leave the trader exposed to large residual risks in case of sudden market movements. Vega risk is a concern inasmuch as, for many market factors, volatilities may fluctuate rapidly (short-term volatilities may suddenly double or treble in a crisis) and are difficult to predict. Both gamma and vega risks can be controlled with the use of options. However, unless options for hedging can be found with maturities similar to the original exposures, vega hedging will be very crude and almost impossible to combine with gamma hedging. Hedging market risks remains therefore something of an art. To summarise the degrees of difficulty in the identification, assessment and control/ mitigation of market risks in banking, I am minded to give the maximum three-star rating to all three tasks. At least this is my excuse for having given a longer description of these three tasks in the banking section compared to the fund management and non-financial firms sections.

41 As a test, consider two portfolios A and B. A has twice the gamma of B (in dollar terms), twice the volatility and

twice the transaction costs per unit volume. How frequently should B be rebalanced relative to A? The answer is, on average, four times more frequently than A. As a rule of thumb, the optimal rebalancing frequency is proportional to (volatility)2 ? (gamma/unit transaction cost)2/3 , which is not immediately obvious. 42 Because gamma is related to the curvature of the value of a portfolio as a function of a market factor, it is also referred to as convexity. That is certainly the case in bond markets when describing the second-order sensitivity of a bond price with respect to its yield. 43 American traders, having quickly run out of Greek letters, opted for a hot blue star and an easy alliteration (i.e. vega and volatility).



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III.A.1.7

Market Risk Management in Non-financial Firms

III.A.1.7.1

Market Risk in Non-financial Firms

Non-financial firms, whether in service, trading or manufacturing industries, take on market risks in the natural course of their business without seeking such risks to derive a profit. Their core competences lie elsewhere and they would rather unload these risks on to market professionals or hedge them directly in the markets. For example, in our global markets most manufacturers are exposed to foreign currency fluctuations; they affect the cost of raw materials, the price at which finished products can be sold in foreign markets as well as the price of competitive foreign imports. Company reports are full of comments about business being affected by the weakness of one currency or the strength of another, by the cost of energy or raw materials, by the crippling effects of an interest-rate increase and the difficulties in raising capital. These are common market risks but they are often regarded by entrepreneurs as externalities about which they can do little. In fact more and more can be done to reduce these risks or at least smooth out their effects in the short to medium term. The real question is to what extent non-financial firms should design and implement hedging programmes to reduce the impact of market risks. Do such programmes add value to shareholders or are they simply contributing to the profits of banks and other financial intermediaries? There are few guidelines on best practice for market risk management in non-financial firms and no regulations comparable to those in banking or fund management. III.A.1.7.2

Identification

The identification of market risks in non-financial firms is arguably the most difficult of the three risk management tasks, followed by assessment and then control/mitigation. Thus, three stars for identification, two for assessment and only one for control/mitigation. Why? Because the management of financial risks is by definition not among the core competences of non-financial firms and therefore it tends to be neglected. It is a natural tendency that we tend to address the problems we know how to solve and ignore the others. So market risks may be not properly recognised, but if they were they might not be so difficult to evaluate and control. In our modern, globalised and deregulated economies, market risks are very pervasive; they affect firms either directly or indirectly through competition. The three main sources of market risks are interest rates, foreign currency exchange rates and commodity prices. Equities tend to be the exception, except for holding companies and other companies relying heavily on investments in securities.



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Finance directors sometimes ask: What is less risky, borrowing at fixed rates or at floating rates? Here lies the paradox: on an accrual or cost-accounting basis, borrowing at fixed rates is safe, the financing costs are fixed, whereas floating rates are risky; but on a fair accounting basis it is the opposite, the present value of the floating rate debt is almost constant, whereas the present value of the fixed rate debt varies with interest rates like the price of the equivalent bond. 44 The choice of accounting standards or, more generally, the choice of a coherent frame of reference for risk evaluation is critical. In particular, mixing the use of several frames of references can only lead to confusion. Difficult, subjective and inaccurate as it may be, I think that a fair valuation of assets and liabilities is the only acceptable basis for recognising and assessing risks, even though for other good reasons companies use accrual accounting extensively in their reports. We may need two sets of accounting principles: one to report results objectively and accurately, the other to serve as a rational basis for risk management. But the answer to the previous question does not depend only on the choice of accounting standards, it also depends on the business, in particular on the composition of assets and liabilities. It is the uncertainty about the future equity value of the firm that is of concern to shareholders. Thus, if the assets of the firm are perceived to generate returns independent of future interest rates, a fixed rate funding may be the safer option, but if future returns are perceived as being highly correlated with interest rates then a floating rate funding is the safer option. In making these judgements, and considering the medium to long term, it may be helpful to consider inflation indices as intermediate factors. Future inflation rates are uncertain, but the operational profit margin of a business before financing costs can often be related to inflation indices and so are the financing costs. Real interest rates relative to inflation tend to be smaller and more stable than nominal interest rates. There is actually a growing market for inflationlinked bonds and loans that secure this relationship and thus are attractive to both investors and borrowers. A similar approach should be used to recognise foreign exchange risk. Companies select a reporting currency, ideally the currency in which most assets and liabilities are denominated, most revenues and costs are incurred, and to which most shareholders are economically tied. When companies were mostly domestic, the choice was obvious. Now that many companies are truly international, there may be some doubt about the choice of a suitable reference currency. The

44 If a loan is evaluated at a fair price like a bond and future cash flows are discounted at, say, the going Libor rates, it

is easy to verify that the fair value of a properly priced floating rate note at Libor should be close to its face value at each interest-rate payment date. There may be small fluctuations between interest-rate payment dates and the present value of any credit spread, and profit margin above Libor will also fluctuate. The fair value of a fixed rate loan, on the other hand, will fluctuate like the fair value of the equivalent coupon bond, going down when interest rates go up and vice versa.



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easiest reference currency is the one with which most shareholders are comfortable, because the goal is to reduce risks for the shareholders. Thus if the majority of shareholders are based in the UK the preferred reporting currency should be the pound sterling, even if most of the business is conducted outside the UK; foreign exchange risks should be assessed on a sterling value and hedges against sterling should be put in place if the risks are deemed excessive. Commodity and indirect market risks due to competition are often called economic risks or input/output risks rather than market risks, although many can be directly traced to market factors such as exchange rates rather than to non-market factors such as innovation, technology or regulation. They are terribly difficult to recognise and appreciate, but they are important. Even purely domestically based companies may find themselves suddenly uncompetitive because of a flood of cheap imports brought about by the weakening of some foreign currency relative to their own domestic currency. It is difficult to appreciate these dangers in advance but critical to develop some awareness of them rather than ignoring them.

An outsiders view may be

informative. Risk managers can take their cues from equity analysts and rating agencies. They are experienced in detecting threats to individual companies and company sectors caused by possible changes in market conditions. III.A.1.7.3

Assessment

To be approximately correct as a whole is more important than to seek accuracy in some areas and to ignore others. A key decision for assessing market risks in non-financial institutions is the choice of time horizon. For example, some companies assess foreign exchange risks purely on current payables and receivables, that is, to a horizon of perhaps three months. They argue that only those transactional foreign exchange risks can be assessed accurately and hedged accordingly. A more fundamental question is whether the company is already exposed to exchange-rate fluctuations beyond this horizon because, for instance, it will not be able to adjust the price of its products and services within this time frame or it has long-term assets and liabilities denominated in foreign currencies. The latter has been called translation or conversion risk. Note that, unlike transaction risk, translation risk has no impact on cash flows so it is sometimes neglected. Similarly, longer-term transaction risks are sometimes ignored because they are less immediate, less certain and less precise. But an approximate evaluation of these further exposures is preferable to total ignorance. It will be a better basis for deciding not only what hedges to implement in the short term but also what offsets could be taken in the long term. Long-term risk assessments extending to several years are indeed indispensable for deciding on major investments and developing long-term strategic plans. Companies face multiple choices



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that are risk-dependent such as where to locate a production facility, whether to outsource some services, whether to invest in new ventures with payback periods of many years, or whether to invest more now to maintain flexibility of choice at a later stage. The assessment of long-term market risks and their potential impact on a firm therefore goes far beyond the calculation of VaR as carried out in banks. It calls for the application of decision analysis methods. A decision analysis cycle proceeds as follows. We construct a simple model of the objective under scrutiny (e.g. maximising the value of the firm) as a function of a few main market and other risk factors and for a base case strategy. Based on initial estimates of the risk factors, we calculate a base case value of the objective. Next we explore the sensitivity of the base case to changes in initial estimates (typically we consider variations in a (subjectively) realistic range such as a 90% range) to identify the most significant sources of uncertainty as far as the objective is concerned. We also design alternative strategies that might do better depending on the evolution of the uncertain factors. At this stage we should understand what are the critical decisions and the most significant risk factors that would influence the choice of strategy. The following stage consists of introducing probability distributions to describe our state of uncertainty about the significant risk factors, and we deduce probability distributions for the objective value under alternative strategies. A choice of optimal strategy can then be attempted, taking into consideration the risk attitude of stakeholders in the firm. But among the possible choices there are often possibilities to acquire more information about some of the sources of uncertainty or to refine the basic model in order to determine the best strategy with greater accuracy and thereby to improve the objective. The decision analysis cycle should then be repeated with the updated information until no more economically valuable information or refinement can be found. Note two essential points in this approach. First, the assessment of risks is carried out with the specific objective of improving decisions. Risk assessment is intimately combined with risk management. Second, market risk factors are combined with other sources of risks in this type of analysis. For decision-making, it does not matter what labels are put on risks; it is the combination of multiple risks and decisions including responses from competitors that is significant. The role of market risk specialists will be to alert management to the existence of certain risks and to contribute to the description of these risks. They cannot work in isolation; one risk is often contingent on another. A company may acquire a foreign exchange exposure if it wins a contract, but may not be sure to win, and even if it wins the foreign exchange exposure may vary as a function of fluctuating demand. These uncertainties and the corresponding decisions pricing the bid, deciding on hedges, etc. must be analysed simultaneously. Small



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firms may lack the expertise to carry out this type of analysis but there is no shortage of consultancy firms and financial intermediaries ready to help. III.A.1.7.4

Control/Mitigation

The decision analysis method outlined in the previous section is particularly useful for making major strategic choices over the medium to long term. For example, a chemical company producing high-density polypropylene should consider whether naphtha or gas oil would be the more economical feed. A decision analysis may reveal that, due to uncertainties in the future costs of these two feeds, it is worthwhile to make the extra investment in a plant that can accept both feeds. That is a form of long-term market risk management.

Likewise, a Japanese car

manufacturer producing cars for the US market may decide to locate production facilities in the USA to reduce foreign exchange risks rather than in a country with currently lower labour costs. Physical long-term solutions limiting exposure to market risks are usually preferable to financial hedges that could be considered as alternatives. For example, the chemical company could consider building a plant suitable for one feed and, in principle, purchase an OTC option on the excess cost of the second feed relative to the first. Likewise, the Japanese manufacturer could opt for the country offering the lowest production and delivery cost into the US market and hedge currency risks by entering into forward exchange contracts. But one should be aware of two likely problems with long-term financial hedges: liquidity and cash flow.

Many financial

derivatives markets are very deep, thus the Japanese manufacturer may find forward contracts in sufficient sizes to cover exchange-rate risks for the entire economic life of its plant; on the other hand, commodity derivatives are still relatively thin and it is very unlikely that the chemical company could find an OTC option to cover its risk over more than a few months. The cash-flow problem is linked to liquidity. Many financial derivatives are liquid only over a relatively short term.

When used to cover long-term exposures, positions in short-term

derivatives are stacked up and rolled over. At every rollover, expiring contracts must be settled; between rollovers, margins must be posted. If unlucky, the hedger may accumulate large realised losses on the short-term contracts against unrealised gains on the initial exposure. The cash-flow problem thus created may prove fatal. The textbook case is MGRM, the US subsidiary of the German company Metallgesellshaft. In 1993 MGRM had accumulated positions on 154 million barrels of crude oil futures on the New York Mercantile Exchange (NYMEX) to hedge longterm supply contracts of crude oil at fixed prices it had agreed with its customers. Unfortunately for MGRM, crude oil prices started to decline and futures were in continuo (higher prices than spot), so that at each monthly rollover MGRM had to pay for the decline in prices of contracts it had bought a month earlier. By the end of the year the board of Metallgesellschaft decided that



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they could no longer afford to support the losses of their subsidiary. MGRM had made a simplistic calculation resulting in an over-hedge and misjudged the rollover risks and the risk of holding a very large proportion of the futures contracts (which led NYMEX to call for additional margins), but, most importantly, they had underestimated the potential cash-flow problem resulting from hedging 10-year exposures with short-term financial derivatives. Thus, it is generally considered inappropriate to hedge translation risks or long-term transaction risks using derivatives.

Translation risks relate to revaluation of foreign assets (such as

subsidiaries) and liabilities rather than to cash flows.

Traditional hedging strategies with

derivatives would only be applicable if there is a plan to sell these assets or refinance liabilities in a different currency, thus resulting in cash flows. Most firms prefer instead to hedge translation risks by matching assets and liabilities in the same currency. For example, a foreign subsidiary could be funded in the currency of that subsidiary rather than in the home currency. Hedging economic exposures and long-term transaction exposures with derivatives can also be problematic not only because of the mismatch between short-term and long-term cash flows but also because of the uncertainties attached to these future cash flows and the difficulty in predicting exactly how profits will be affected by a possible market movement. In such cases operational solutions such as those mentioned earlier can be preferable. Financial derivatives remain the choice instruments for hedging short-term transaction risks. If the right instruments are not available on exchanges, firms will find many banks willing to offer tailor-made OTC products. But whether market risks in non-financial firms should be hedged at all remains an interesting question. Some argue against hedging as follows: (i)

Short-term uncertainties will tend to average out naturally over the long term.

(ii)

Hedging is costly, and in the long term it only contributes to banks profits.

(iii)

Shareholders and investors know what the risks are; these risks are already priced in the market or washed out by diversification.

(iv)

If competitors do not hedge a certain risk, it would be unsafe to be too much out of line: winning on the hedge might not be as favourable as losing would be damaging.

But others argue in favour of hedging that: (i)

Market risks in non-financial companies serve no useful purpose. They are not chosen with the expectation of deriving a profit. Indeed, there is no risk premium in the pricing of those market risks that are diversifiable (e.g. currency risks).



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(ii)

On the contrary, market risks create uncertainties in the performance of business units and the firm in general. This makes the planning process more difficult, obscures the true profitability of various activities and confuses the reward scheme.

(iii)

If unnecessary risks are eliminated, results become more stable, the debtequity ratio can be increased and tax benefits can be reaped. 45

(iv)

Reduction in risk can also make the firm more attractive to other stakeholders such as lenders, trade creditors, customers and employees (especially if they hold executive stock and have poorly diversified portfolios). Greater stability of earnings may mean that lower interest rates apply, trade terms are more advantageous, etc. This is known as reducing the costs of financial distress.

Different firms may reach different conclusions. Hedging market risks may be less important for large, diversified, internationally active firms than for smaller, more specialised firms. In large firms, market risks may already be well diversified and treasury departments may have the expertise to decide which risks are likely to be beneficial. In small firms, some market risks might be crippling and should be seen by most stakeholders as unnecessary, avoidable gambles, if only a proper hedging strategy were put in place.

III.A.1.8

Summary

There is a pervasive view today that market risk management consists essentially in calculating a value-at-risk. This introduction should help dispel this false impression. Instead, I hope the reader will have realised that market risks, although relatively well understood, are still hidden in many places, and any attempt to assess them is based on a large number of assumptions. And, of course, market risk management does not stop at the assessment phase but should lead to control and mitigation. To start with the risk identification phase, one should not forget that there are market risks hidden in illiquid assets and liabilities that are not evaluated at fair value. It is not because a firm uses accrual accounting that these risks do not exist. That would be an ostrich-like, head-in-thesand attitude.

Fortunately, the wider adoption of the new International Accounting Standards

favouring fair value and hedge accounting, the valuation of contingency claims (e.g. executive share option schemes) and the recognition of assets and liabilities heretofore not affecting reported company profits (e.g. company pension schemes) will help companies pay attention to market risks. A short-term effect of these changes may be to increase the volatility of company 45

Research into the use of derivatives by non-financial corporations suggests that derivatives are more likely to be used by firms with greater leverage. Or, vice versa, firms that reduce their markets risks can afford a higher leverage and a reduce cost of capital.



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returns and make equity investments less attractive, but in the long term it will help better risk management and should result in a more efficient allocation of resources. The risk assessment phase is mathematical but relies on the choice of an objective and coherent set of assumptions. The objective may be to assess the probability of insolvency within a year; this is what banking supervisors are focusing on. But it may also be to assess some risk-adjusted performance measure and improve resource allocation accordingly; or it could be any of a number of other objectives such as developing contingency plans. To each objective corresponds a reasonable set of assumptions, for instance, a choice of time horizon, whether normal or extreme market conditions should be considered, whether portfolios should be assumed to be static or dynamic, whether the business should be regarded as a going concern or whether some assets should be valued on a fire-sale basis, and so on. The risk control/mitigation phase follows logically from a choice of objective. Depending on the business, there may also be a number of constraints, regulatory or otherwise, limiting the level of acceptable market risk; some of these constraints may be biting. Nonetheless, there cannot be any logical control/mitigation strategy without a clear objective. Fortunately, the implementation of hedging and risk control strategies is now a lesser problem because of the existence of a deep, liquid and efficient market in financial derivatives. There are few market risks that cannot be adequately covered when there is a wish to do so. New hedging requirements create new derivatives markets, as we see happening with telecommunications bandwidths and pollution credits, to name just a couple of new commodity derivatives. Market risk management is still a relatively new and growing field of expertise. To operate efficiently, the market risk management function must be independent of risk-taking functions as well as of the accounting and internal auditing functions, must be able to rely on adequate resources, must communicate regularly with risk-taking departments and senior management and, like internal audit, must have reporting lines through a general risk management function up to the board of directors. The quality of risk management directly affects risk-adjusted performance measures and, ultimately, shareholder value. As banking regulators remind us, capital should not be regarded as a substitute for addressing fundamentally inadequate control or risk management processes (BCBS, 2004a, par. 723).

References Alexander, C (2001) Market Models: A Guide to Financial Data Analysis. Chichester: Wiley. BCBS (1996) Amendment to the capital accord to incorporate market risks (January, modified September 1997). Available at http://www.bis.org/publ/bcbs.htm



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BCBS (2004a) International convergence of capital measurements and capital standards (June). Available at http://www.bis.org/publ/bcbs.htm BCBS (2004b) Principles for the management and supervision of interest rate risk (July). Available at http://www.bis.org/publ/bcbs.htm Black, F, and Jones, R (1987) Simplifying portfolio insurance, Journal of Portfolio Management, Fall, 4851. Black, F, and Perold, A F (1992) Theory of constant proportion portfolio insurance, Journal of Economic Dynamics and Control, 16, pp. 403426. Davis, M H A, and Norman, A R (1990) Portfolio selection with transactions costs, Mathematics of Operations Research, 15, pp. 676713. Dybvig, P H (1988) Inefficient dynamic portfolio strategies, or How to throw away a million dollars, Review of Financial Studies, 1, 6788. Hodges, S D and Neuberger, A (1989), Optimal replication of contingent claims under transactions costs,, Review of Futures Markets, 8, pp. 222239.



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III.A.2 Introduction to Value at Risk Models Kevin Dowd and David Rowe 46

III.A.2.1

Introduction

Value at risk (VaR) has been the subject of much criticism in recent years. Many of these criticisms relate to important precautions as to how VaR results should be interpreted as well as limitations on their use. Other criticisms, however, have been more sweeping, in some cases dismissing the entire concept as misdirected and wrong-headed. In that context, it is useful to consider how trading risk limits were determined before VaR became widely accepted. Market risk arises from mismatched positions in a trading book that is marked to market daily based on uncertain movements in prices, rates, volatilities and other relevant market parameters. Market makers cannot operate successfully if they only broker exactly offsetting trades between customers. To be successful, they need to stand ready to execute trades on demand, and this inevitably results in open positions being created that are exposed to loss from adverse market movements. These open positions are hedged in the short run with less than perfect offsets. A common example is a dealer who executes an interest-rate swap with a customer in which he/she receives fixed and pays floating and then hedges by shorting government bonds and investing the proceeds in short-term instruments. There is still basis risk, since the spread between the swap and bond rates may change, but the major exposure to loss from a general rise in rates has been eliminated. In the longer term, the dealer will try to attract offsetting customer trades by shading future quotes to make such offsets attractive to the market. Failing that, the dealer may execute an offsetting swap with another dealer, although this is less desirable since it requires paying away a bid or offer spread instead of earning the spread on a customer deal. Given that running a market-making function inevitably gives rise to market risk, institutions have always imposed restrictions on traders designed to limit the extent of such risk-taking. Until the early 1990s these limits were in the form of restrictions on: the size of net open positions, including delta equivalent exposures to movements in underlying rates and prices; the degree of maturity mismatch in the net position; the permissible amount of negative gamma in option positions; exposure to changes in volatility.

46 Kevin Dowd is Professor of Financial Risk Management at Nottingham University Business School, UK, and David

Rowe is Group Executive Vice President for Risk Management at SunGard Trading and Risk Systems in London.



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These limits imposed a complex array of constraints on trader positions. They were difficult to enforce effectively, not least because risk exposures, such as those based on option gammas, often move quickly in response to market developments. The information and management systems of the time also meant that these limits were enforced piecemeal, with all sorts of inconsistencies and other undesirable results: good risks were often passed over because they ran into arbitrary risk limits, decisions were made with inadequate appreciation of the risks involved, reducing risk in one area seldom allowed greater risk-taking elsewhere, and so forth. Perhaps the most important shortcoming of this old system was the absence of integrated risk management. There was little coherence between the structure or management of the limits and the range of potential losses that they permitted to occur. Senior management committees charged with approving such limits were often at the mercy of technicians, and even the technicians were hard pressed to translate the limits into a consistent measure of risk, let alone provide an effective system of integrated risk management. Gradually a consensus arose that what was of fundamental interest to the institution was the probability distribution of potential losses from traders positions, regardless of the exact structure of those positions. From this realization was born the concept of value at risk. This gave management a much more consistent way of embedding acceptable levels of risk into the formal limits within which traders were required to operate. Naturally, there was still a heavy dependence on market risk technicians to translate market dynamics and the traders positions into estimates of the risks being taken, but VaR did allow a firms management to define limits that reflected a well-considered risk appetite in a way that the pre-existing system did not. In that sense, one of the most important contributions of VaR has been an improvement in the quality of the management of risk at the firm-wide level.

III.A.2.2

Definition of VaR

VaR is an estimate of the loss from a fixed set of trading positions over a fixed time horizon that would be equalled or exceeded with a specified probability. Several details of this definition are worth emphasizing. VaR is an estimate, not a uniquely defined value. In particular, the value of any VaR estimate will depend on the stochastic process that is assumed to drive the random realisations of market data. The structure of the random process has to be identified and the specific parameters of that process must be calibrated. This requires us to resort to historical experience and raises a whole host of issues such as the length of the historical sample to be used and whether more recent events should be weighted more heavily than those further in the past. In essence, the goal is to arrive at the best



76


possible estimate of the stochastic process driving market data over the specific calendar period to which the VaR estimate applies. Moreover, it is also clear that market data are not generated by stable random processes. Differing methods for dealing with the uncertainty surrounding changes in these random processes are at the heart of why VaR estimates are not unique. The trading positions under review are fixed for the period in question. This raises difficult questions when the evaluation period is long enough to make this assumption unrealistic. 47 In this instance it is most common to scale up a VaR estimate for a shorter period on the assumption that market data move independently from day to day. Otherwise, it is necessary to model trades that mature within the specified time horizon and make behavioural assumptions relating to trading strategies during the period. VaR does not address the distribution of potential losses on those rare occasions when the VaR estimate is exceeded. It is never correct to refer to a VaR estimate as the worst-case loss. Analysis of the magnitude of rare but extreme losses must invoke alternative tools such as extreme value theory or simulations guided by historical worst-case market moves. The use of VaR involves two arbitrarily chosen parameters the holding period and the confidence level. The usual holding period is one day or one month, but institutions can also operate on other holding periods (e.g., one quarter or more), depending on their investment and/or reporting horizons. The holding period can also depend on the liquidity of the markets in which an institution operates. Other things being equal, the ideal holding period appropriate in any given market is the length of time it takes to ensure orderly liquidation of positions in that market. The holding period may also be specified by regulation. For example, Basel Accord capital adequacy rules stipulate that internal model estimates used to determine minimum regulatory capital for market risk must reflect a time horizon of two weeks (i.e. 10 business days). The choice of holding period can also depend on other factors: The assumption that the portfolio does not change over the holding period is more easily defended with a shorter holding period. A short holding period is preferable for model validation or backtesting purposes: reliable validation requires a large data set and a large data set requires a short holding period.

47 One common example of this is the requirement to estimate VaR over a 10-day time horizon for purposes of

calculating regulatory capital for market risk under the Basel Capital Accord.



77


The choice of confidence level depends mainly on the purpose to which our risk measures are being put. Thus, a very high confidence level, often as great at 99.97%, is appropriate if we are using risk measures to set capital requirements and wish to achieve a low probability of insolvency or a high credit rating. Indeed, the confidence levels required for these purposes can be higher than those needed to meet regulatory capital requirements. On the other hand, for backtesting and model validation, relatively lower confidence levels are desirable to get a reasonable proportion of excess-loss observations. For limit-setting, most institutions prefer confidence levels low enough that actual losses exceed the corresponding VaR estimate somewhere between two and twelve times per year (implying a daily VaR confidence level of 95% to 99%). This forces policy committees to take the size of the limit seriously, since losses over that limit can occur with a reasonable likelihood. For the above reasons, among others, the best choice for these parameters depends on the context. What is important is that the choices be clear in every context and be thoroughly understood throughout the institution so that limit-setting and other risk-related decisions are made in light of this common understanding.

III.A.2.3

Internal Models for Market Risk Capital

The original Basel Capital Accord was put into effect at the beginning of 1988. It set down rules for calculating minimum regulatory capital for banks based on a simple set of multipliers applied to credit-risky assets. The minimum capital calculation did not reflect risks associated with a banks mark-to-market trading activities, which were still quite small. However, market risk factors were becoming more important constituents of the risk profile of most major money centre banks, and in the 1990s the Basel Accord was amended to reflect banks exposure to market risk. In a significant departure from traditional conventions, the new Amendment also allowed banks to employ their own internal VaR models to calculate their minimum regulatory capital for market risk. This permission was conditional on several requirements: The models and their surrounding technical and organisational infrastructure had to be reviewed and approved by the banks supervisor. The model used for calculating regulatory capital had to be the same one used for dayto-day internal risk management (the so-called use test). The VaR confidence level used in the regulatory capital calculation had to be 99%. The time horizon for the regulatory capital calculation had to be two weeks (i.e. 10 business days).



78


Assuming the above conditions were met, the capital requirement was to be at a level at least 3.0 times the 10-day VaR estimate averaged over the last 60 business days for any reporting period. Supervisors retained the prerogative of applying a larger multiplier if the results of backtesting exercises suggested that internal models were generating insufficiently high VaR estimates. The extended holding period presented some questions. How should positions that matured during this time horizon be handled? What about trades likely to be booked to correct for the erosion of initial hedges due to ageing of the portfolio? The amended Accord also offered banks a solution to many of these problems, which almost all of them adopted. This was to apply the square root of time rule. That is, banks obtained 10-day VaR estimates by multiplying the daily VaRs by 10

3.16228. This procedure effectively says

that if traders took the same level of risk as indicated by the one-day VaR estimate for 10 consecutive days, the 10-day VaR estimate would be 3.16228 times the daily VaR. This rule is based on the formula for the distribution of the sum of random variables, assuming that daily returns are independent of each other (see Chapter III.A.3). Assuming a static portfolio for one day avoids most of the complications described above for longer time horizons. Moreover, in so far as even a one-day static portfolio is unrealistic, the consequences of this assumption will (hopefully!) be evident from the backtests on the VaR model that are described in Section III.A.2.8.

III.A.2.4

Analytical VaR Models

The assumption that holding period returns (i.e. h-day relative changes in value) are normally distributed provides us with a straightforward formula for value at risk. If our h-day returns R are normally distributed with mean

and standard deviation , we write R

N( ,

2)

(III.A.2.1)

as in Section II.E.4.4.1. Now if the portfolio is currently worth S, our h-day VaR at the confidence level 100(1

)% is given by VaR h, = x S

where x is the lower

percentile of the distribution N( ,

probability that R < x = confidence,

(III.A.2.2) 2).

That is, x is the number such that the

(see Section II.E.4.4.3). Since we require a fairly high degree of

is small (normally 0 <

< 0.1). Thus x will typically be negative. In fact, using the

standard normal transformation (II.E.28) we can write Z = (x )/ . In other words, x =Z

+



(III.A.2.3)

79


where Z is the lower percentile of the standard normal distribution. This can be obtained from standard statistical tables or from spreadsheet functions, such as the NORMSINV function in Excel. For instance, typing =NORMSINV(0.05) into Excel gives the value 1.64485 for Z0.05 Putting together (III.A.2.2) and (III.A.2.3), we have derived the following simple analytic formula for VaR that is valid under assumption (III.A.2.1): VaR h,

= (Z

+ )S.

(III.A.2.4)

Estimating VaR at a given probability using the normal distribution is very easy, once we have an estimate of the mean and standard deviation, as the following example shows. Example III.A.2.1: Analytic VaR calculation Suppose we are interested in the normal VaR at the 95% confidence level and a holding period of 1 day and we estimate

and

over this horizon to be 0.005 and 0.02, respectively. Now

(III.A.2.4) tells us that for a portfolio worth $1 million, VaR 1, 0.05 = (0.005

1.64485

0.02) ? $1 million = $27,897.

Note that the higher the confidence level, the greater the VaR. For instance, if we were interested in the corresponding VaR at the 99% confidence level, our VaR would be 48 VaR 1, 0.01 =

(0.005

2.32634

0.02) ? $1 million = $41,527.

Applying the square root of time rule (explained in Chapter III.A.3), the corresponding VaRs over a 10-day holding period are: VaR 10, 0.05 = 10 VaR 1, 0.05 VaR 10, 0.01 = 10 VaR 1, 0.01

3.16228 ? $27,897 = $88,218, 3.16228 ? $41,527 = $131,320.

Analytical approaches provide the simplest and most easily implemented methods to estimate VaR. They rely on parameter estimates based on market data histories that can be obtained from commercial suppliers or gathered internally as part of the daily mark-to-market process. For active markets, vendors such as RiskMetricsTM supply updated estimates of the volatility and correlation parameters themselves. But while simple and practical as rough approximations, analytic VaR estimates also have shortcomings. Perhaps the most important of these is that many parametric VaR applications are based on the assumption that market data changes are normally distributed, and this assumption

48 =NORMSINV(0.01) in Excel gives 2.32634.



80


is seldom correct in practice. Assuming normality when our data are heavy-tailed can lead to major errors in our estimates of VaR. VaR will be underestimated at relatively high confidence levels and overestimated at relatively low confidence levels. Further discussion on this will be given in Chapter III.A.3. But analytical approaches can also be unreliable for other reasons: Market value sensitivities often are not stable as market conditions change. Since VaR is often based on fairly rare, and hence fairly large, changes in market conditions, even modest instability of the value sensitivities can result in major distortions in the VaR estimate. Such distortions are magnified when options are a significant component of the positions being evaluated, since market value sensitivities are especially unstable in that situation. Analytic VaR is particularly inappropriate when there are discontinuous payoffs in the portfolio. This is typical of transactions like range floaters and certain types of barrier options. In summary, analytic approaches provide a reasonable starting point for deriving VaR estimates, but should not be pushed too hard. They may be acceptable on a long-term basis if the risks involved are small relative to a firms total capital or aggregate risk appetite, but as the magnitude of risk increases, and as positions become more complex, and especially more nonlinear, more sophisticated approaches are necessary to provide reliable VaR estimates.

III.A.2.5

Monte Carlo Simulation VaR

Fortunately, many problems that cannot be handled by analytical methods are quite amenable to simulation methods. For example, we might have stochastic processes that exhibit jumps or certain types of heavy tails that do not allow an analytical solution for our VaR, or the values of the instruments in our portfolio might be complicated functions of otherwise straightforward risk factors, as is often the case with exotic options. Alternatively, our portfolio might be a collection of heterogeneous instruments, whose payoffs interact in ways that cannot be handled using analytical methods. And there again, we might have simple positions that can be handled using analytical methods, but are better handled using simulation. A good case in point would be a portfolio of long straddles: these options are simple, but their maximum loss occurs when there is no market movement at all. Although the VaR of such a position can be obtained using analytical methods, we have to be careful which analytical methods we apply. For example, deltagamma methods, which are often used for options VaR, can be very treacherous when applied to such positions because they assume that the maximum loss occurs when underlying variables exhibit large moves. (These methods are discussed in Section III.A.2.7.6.) In such cases, we might prefer to use simulation methods, because we know they are reliable. Copyright ? 2004 The Authors and the Professional Risk Managers International Association


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In these and similar circumstances, the most natural approach is to use Monte Carlo simulation, which is a very powerful method that is tailor-made for complex or difficult problems. The essence of this approach is first to define the problem specify the random processes for the risk factors of the portfolio, the ways in which they affect our portfolio, and so forth and then simulate a large number of possible outcomes based on these assumptions. Each simulation trial leads to a possible profit/loss (P/L). If we simulate enough trials, we can then produce a simulated density for our P/L, and we can read off the VaR as a lower percentile of this density. In the following when the context is clear we drop the notation for the dependence of VaR on holding period h and confidence level 100(1- )%, writing simply VaR for VaR h, . III.A.2.5.1

Methodology

To illustrate, suppose we wish to carry out a Monte Carlo analysis of a stock price S, and we assume that S follows a geometric Brownian motion process: dS/S = dt + dW where

(III.A.2.5)

is its expected (per unit time) rate of return and

is the spot volatility of the stock price.

dW is known as a Wiener process, and can be written as dW = (dt)1/2, where

is a drawing from a

standard normal distribution. Substituting for dW, we get dS/S = dt +

(dt)1/2.

This is the standard stock-price model used in quantitative finance. The (instantaneous) rate of change in the stock price dS/S evolves according to its drift term dt and realisations from the random term . In practice, we would often work with this model in its discrete-form equivalent. If t is some small time increment, we approximate (III.A.2.5) by S/S

t

t

(III.A.2.6)

where S is the change in the stock price over the time interval t, and S/S is its (discretised) rate of change. Note that (III.A.2.6) assumes that the rate of change of the stock price is normally distributed with mean

t and standard deviation

t . Hence our criticisms of analytic VaR with respect to

the normality assumption will also apply to the Monte Carlo VaR methodology, unless we employ an assumption for the underlying dynamics that is more appropriate than the geometric Brownian motions with constant volatility (III.A.2.5). Now suppose that we wish to simulate the stock price over some period of length T. We would usually divide T into a large number N of small time increments t (i.e. we set t = T/N). We Copyright ? 2004 The Authors and the Professional Risk Managers International Association


82


take a starting value of S, say S(0), and draw a random value of

to update S using (III.A.2.6);

this gives the change in the stock price over the first time increment, and we repeat the process again and again until we have changes in the stock price over all N increments. At this point, we have simulated the path of the stock price over the whole period T. We can then repeat the exercise many times and produce as many simulated price paths as we wish. Some illustrative simulated price paths are shown in Figure III.A.2.1. We assume here that the starting value of our stock price, S(0), equals 1, so each path starts from the 1 on the y-axis. Thereafter the paths typically diverge, moving randomly in accordance with their laws of motion as given in the above equations. Moreover, since

is assumed to be positive, there is a tendency

for the stock prices to drift upwards. The degree of dispersion of the simulated stock prices the extent to which they move away from each other over time is governed by the volatility . The bigger is , the more dispersed the stock prices will be at any point in the simulation. Note, too, that the simulated terminal stock prices will tend to approach the true distribution of terminal stock prices as the number of draws grows larger. Even in this figure, which has only a limited number of paths, we can see that most of the terminal values are clustered around a central value, with relatively few in the tails. If we want to obtain a simulated terminal distribution which is close to the true distribution, all we need to do is carry out a large number of simulation trials. The larger the number of trials, the closer is the simulated terminal distribution to the true terminal distribution.



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Figure III.A.2.1: Some simulated stock price paths

Note: Based on 15 Monte Carlo trials using parameters =0.05, =0.10, T=1 and 30 step increments.

If we wish, we can estimate the VaR of the stock price by simulating a large number of terminal stock prices S(T). We then read the VaR from the histogram of S(T) values so generated. To illustrate, Figure III.A.2.2 shows the histogram of simulated S(T) values from 10,000 simulation trials, using the same stock price parameters as in Figure III.A.2.1. The shape of the histogram is close to a lognormal which it should be, as the stock price is assumed to be lognormally distributed. The figure also shows the 5th percentile of the simulated stock price histogram. This percentile is equal to 0.420, indicating that there is a 5% probability that the initial stock price (of 1) could fall to 0.420 or less over the period, given the parameters assumed. A terminal stock price of 0.420 corresponds to a loss equal to 1 0.420 = 0.580, so we can say that the VaR at the 95% confidence level is 0.580. This example illustrates how easy it is to estimate VaR using Monte Carlo simulation. In addition, Monte Carlo simulation can easily handle problems with more than one random risk factor (see Section II.D.4.2).



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Figure III.A.2.2: Histogram of simulated terminal stock prices

III.A.2.5.2

Applications of Monte Carlo simulation

Monte Carlo methods have many applications in market risk measurement and would be the preferred method in almost any complex risk problem. Examples of such problems include the following, among many other possibilities: We might be dealing with underling risk factors that are badly behaved in some way (e.g. because they jump or show heavy tails) or we might have a mixture of heterogeneous risk factors. For example, we might have credit-related risk factors as well as normal market risk factors, and the credit risk factors cannot be modelled as normal. We might have a portfolio of options. In such cases, the value of the portfolio is a nonlinear (or otherwise difficult) function of underlying risk factors, and might be impossible to handle using analytical methods even if the risk factors are themselves well behaved. We might be dealing with instruments with complicated risk factors, such as mortgages, credit derivatives, and so forth. We might have a portfolio of heterogeneous instruments, the heterogeneity of which prevents us from applying an analytical approach. For example, our portfolio might be a collection of equities, bonds, foreign exchange options, and so forth.



85


III.A.2.5.3

Advantages and Disadvantages of Monte Carlo VaR

Monte Carlo simulation has many advantages over analytical approaches to calculating VaR: It can capture a wider range of market behaviour. It can deal effectively with nonlinear and path dependent payoffs, including the payoffs to very complicated financial instruments. It can capture risk that arises from scenarios that do not involve extreme market moves. Conversely, it can provide detailed insight into the impact of extreme scenarios that lie well out in the tails of the distributions, beyond the usual VaR cutoff. It lends itself easily to evaluating specific scenarios that are deemed worrisome based on geopolitical or other hard-to-quantify considerations. The biggest drawbacks to the Monte Carlo approach to VaR estimation are that it is computer intensive and it requires great care to be sure all the details of the calculation are executed correctly. Non-technicians also find the process of imposing historically consistent characteristics on the scenarios to be quite impenetrable. Hence Monte Carlo VaR estimates are often viewed as coming from a black box whose credibility rests solely on the reputation of the technicians responsible for producing them. Nevertheless, Monte Carlo simulation is the most widely used approach to VaR estimation, and its popularity is likely to grow further as computers become more powerful and simulation software becomes more user-friendly. For large sophisticated trading operations, the only other widely used approach is historical simulation, to which we now turn.

III.A.2.6

Historical Simulation VaR

Historical simulation is a very different approach to VaR estimation. The idea here is that we estimate VaR without making strong assumptions about the distribution of returns. We try to let the data speak for themselves as much as possible and use the recent empirical return distribution not some assumed theoretical distribution to estimate our VaR. This type of approach is based on the underlying assumption that the near future will be sufficiently like the recent past that we can use the data from the recent past to estimate risks over the near future and this assumption may or may not be valid in any given context. III.A.2.6.1

The Basic Method

In applying basic historical simulation, we first construct a hypothetical P/L series for our current portfolio over a specified historical period. This requires a set of historical P/L or return observations on the positions currently held. These P/Ls or returns will be measured over a standard time interval (e.g. a day) and we want a reasonably large set of historical observations



86


over the recent past. Suppose we have a portfolio of n assets, and for each asset i we have the observed return for each of T intervals in our historical sample period. (Our portfolio could equally well include a collection of liabilities and/or instruments such as swaps, but we talk of assets for convenience.) If ri,t is the return on asset i in sub-period t, and if Ai is the amount currently invested in asset i, then the simulated P/L of our current portfolio in sub-period t is: n

(P/L)t

Ai ri ,t . i 1

Calculating this for all t gives us the hypothetical P/L for our current portfolio throughout our historical sample. This series will not be the same as the P/L actually earned on our portfolio in each of those periods because the portfolio actually held in each historical period will virtually never match our current positions. Having obtained our hypothetical P/L data set, we can estimate VaR by plotting the data on a simple histogram and then reading off the appropriate percentile. To illustrate, suppose we have 1000 hypothetical daily observations in our P/L series (approximately four years of data for about 250 business days per year) and we plot the histogram shown in Figure III.A.2.3. If we take our VaR confidence level to be 95%, our VaR is given implicitly by the x-value that cuts off the bottom 5% of worst P/L outcomes from the rest of the distribution. In this particular case, this x-value (or the 5th percentile point of the P/L histogram) is 1.604. The VaR at the 95% probability is the negative of this percentile, and is therefore 1.604. Figure III.A.2.3: Historical simulation VaR



87


III.A.2.6.2

Weighted historical simulation

One of the most important features of basic historical simulation is the way it weights past observations. Our historical simulation P/L series is constructed in a way that gives any observation the same weight on P/L provided it is less than n periods old, and a zero weight if it is older than that. However, a problem with this is that it is hard to justify giving each observation in our sample period the same weight, regardless of age, market volatility, or anything else. For example, it is well known that natural gas prices are usually more volatile in the winter than in the summer, so a raw historical simulation approach that incorporates both summer and winter observations will tend to average the summer and winter P/L values together. As a result, treating all observations as having equal weight will tend to underestimate true risks in the winter, and overestimate them in the summer (see Shimko et al., 1998). This weighting structure also creates the potential for ghost effects we can have a VaR that is unduly high (low) because of a short period of high (low) volatility, and this VaR will continue to be high (low) until n days or so have passed and the observations have fallen out of the sample period. At that point, the VaR will fall (rise) again, but the fall (rise) in VaR is only a ghost effect created by the weighting structure and the length of sample period used. More detailed discussion of this point is left to Chapter III.A.3. We can ameliorate these problems by suitably weighting our observations. In the natural gas case just considered, we might give the winter observations a higher weight than summer observations if we are estimating a winter VaR, and vice versa for a summer VaR. Alternatively, we might believe that newer observations in our sample are more informative than older ones, and in this case we might age-weight our data so that older observations in our historical simulation sample have a smaller weight than more recent ones (see Boudoukh et al., 1998). 49 To implement an age-weighted historical simulation, we begin by ordering our returns, worst return first. We then note the age of each return observation, and calculate a suitable agerelated weight for each return. A good way to do so is to use an exponentially weighted moving average (EWMA). We choose a decay parameter , which indicates how much each observations weight decays from one day to the next. Again, a more detailed discussion of this methodology is left to Chapter III.A.3. A worked example is provided in Table III.A.2.1 and in the Workbook entitled Age-Weighted Historical Simulation. The first column gives the ordered returns, and the second gives the age of the corresponding return observation in days. For illustration only we assume an unrealistically

49 This approach takes account of the loss of information associated with older data and is easy to implement.

However, it can aggravate the problem of limited data in the tails of the distribution.



88


small sample of only 150 observations. Basic historical simulation gives each return a weight of 0.00667, implying cumulative weights of 0.00667, 0.01333, 0.02000, etc. The historical simulation confidence level is 1 minus the cumulative weight plus 0.00667, and the VaR is the negative of the relevant observation. So, for example, in this case the historical simulation VaR at the 95% confidence level is 2.530% of the portfolio size, the point half way between the eighth and ninth worst simulated losses. However, if we apply exponential weighting using

= 0.97, we get the

weights given in the column headed AW weight. These are rather different from the historical simulation equal weights, and give more recent observations higher weights. The cumulative weights given in the next column are 0.02684, 0.04057, and so on. The rest of the analysis then proceeds as before. In this case, the EWMA weighted historical simulation VaR at the 95% confidence level is 2.659% of the portfolio value, falling between the fourth and fifth worst simulated losses. The effect of exponential weighting in this case is to raise the estimated VaR since some of the largest simulated losses occur relatively recently in the sample period. Table III.A.2.1 also shows the same analysis conducted 25 days later. To illustrate the point, we have made the simplifying assumption that the positions are the same as on the first day so that all the simulated historical P/L values for any given calendar day are unchanged. We also assume that there are no large losses in the intervening 25 days, so that the worst simulated losses are the same as those recorded in the analysis at the initial date. Now, however, these observations have aged and their weights are lower, reflecting the observations greater age. Consequently, the cumulative weights are also lower and the impact is to increase the effective confidence level for any given return observation. This, in turn, leads to a lower VaR for any given confidence level. In this case, the VaR at the 95% confidence level is 2.503%, a value interpolated between the ninth and tenth worst simulated losses. By contrast, the historical simulation VaR has remained unchanged because the historical simulation weights are unaltered. Table III.A.2.1: Age-weighted historical simulation Analysis at Initial Date Ordered daily return -3.50% -3.00% -2.80% -2.70% -2.65% -2.60% -2.57% -2.55% -2.51% -2.48% -2.45%

Age 5 27 55 65 30 50 45 10 6 17 24

Basic historical simulation HS weight HS cum. weight cl VaR at chosen cl 0.00667 0.00000 1.00000 3.500% 0.00667 0.00667 0.99333 3.000% 0.00667 0.01333 0.98667 2.800% 0.00667 0.02000 0.98000 2.700% 0.00667 0.02667 0.97333 2.650% 0.00667 0.03333 0.96667 2.600% 0.00667 0.04000 0.96000 2.570% 0.00667 0.04667 0.95333 2.550% 0.00667 0.05333 0.94667 2.510% 0.00667 0.06000 0.94000 2.480% 0.00667 0.06667 0.93333 2.450%



95% VaR

2.530%

89


AW weight 0.02684 0.01373 0.00585 0.00432 0.01253 0.00681 0.00794 0.02305 0.02603 0.01862 0.01505

Age-weighted historical simulation AW cum. weight cl VaR at chosen cl 0.00000 1.00000 3.500% 0.02684 0.97316 3.000% 0.04057 0.95943 2.800% 0.04642 0.95358 2.700% 0.05074 0.94926 2.650% 0.06327 0.93673 2.600% 0.07008 0.92992 2.570% 0.07802 0.92198 2.550% 0.10107 0.89893 2.510% 0.12710 0.87290 2.480% 0.14572 0.85428 2.450%

95% VaR

2.659%

Analysis 25 Days Later Ordered daily return -3.50% -3.00% -2.80% -2.70% -2.65% -2.60% -2.57% -2.55% -2.51% -2.48% -2.45%

AW weight 0.01253 0.00641 0.00273 0.00202 0.00585 0.00318 0.00371 0.01076 0.01216 0.00870 0.00703

Age 30 52 80 90 55 75 70 35 31 42 49

HS weight 0.00667 0.00667 0.00667 0.00667 0.00667 0.00667 0.00667 0.00667 0.00667 0.00667 0.00667

Basic historical simulation HS cum. weight cl VaR at chosen cl 0.00000 1.00000 3.500% 0.00667 0.99333 3.000% 0.01333 0.98667 2.800% 0.02000 0.98000 2.700% 0.02667 0.97333 2.650% 0.03333 0.96667 2.600% 0.04000 0.96000 2.570% 0.04667 0.95333 2.550% 0.05333 0.94667 2.510% 0.06000 0.94000 2.480% 0.06667 0.93333 2.450%

Age-weighted historical simulation AW cum. weight cl VaR at chosen cl 0.00000 1.00000 3.500% 0.01253 0.98747 3.000% 0.01894 0.98106 2.800% 0.02168 0.97832 2.700% 0.02369 0.97631 2.650% 0.02954 0.97046 2.600% 0.03273 0.96727 2.570% 0.03643 0.96357 2.550% 0.04719 0.95281 2.510% 0.05935 0.94065 2.480% 0.06805 0.93195 2.450%

95% VaR

2.530%

95% VaR

2.503%

If we are concerned about changing volatilities, we can also weight our data by contemporaneous volatility estimates. The key idea suggested by Hull and White (1998) is to update return information to take account of changes in volatility. So, for example, if the current volatility in a market is 2% a day, and it was only 1% a day a month ago, then data a month old understate the changes we can expect to see tomorrow. On the other hand, if last months volatility was 1% a day but current volatility is 0.5% a day, month-old data will overstate the changes we can expect tomorrow. Now suppose we are interested in forecasting VaR for day T. Again let ri,t be the historical return in asset i on day t in our historical sample,

i,t

be a forecast of the volatility of the return on asset i



90


for day t, made at the end of day t 1, and

i,T

be our most recent forecast of the volatility of

asset i. We then replace the returns in our data set, ri,t, with volatility-adjusted returns, given by

ri *,t

i ,T

ri ,t .

i ,t

Actual returns in any period t are therefore increased (decreased), depending on whether the current forecast of volatility is greater (smaller) than the estimated volatility for period t. We now calculate the historical simulation P/L as explained in Section III.A.2.6.1, but with r i *,t substituted in place of the original data set ri,t.

The calculations involved are illustrated in Table III.A.2.2 and in the Workbook entitled Volweighted Historical Simulation. Using the same returns as in Table III.A.2.1, this table shows two cases. In the first case, current daily volatility is 1.4%, generally above the range of contemporaneous daily volatilities over the historical dates with the worst losses. Most of the volatility weights are therefore greater than 1.0 and the volatility-adjusted changes in the portfolio are correspondingly greater (in absolute value) than the simulated historical changes. The confidence levels are the same as before. In this case, the effect of the volatility weighting is to increase the effective changes and thus to increase the estimated VaR. In the second case, we have the same contemporaneous volatilities but a current volatility of only 0.8%. The volatility weights are now generally less than 1.0, and the effective or volatility-adjusted changes are correspondingly reduced as is the estimated VaR. Note that the volatility weighting will generally alter the specific historical dates corresponding to the critical confidence level depending on the pattern of contemporaneous volatility. These dates are invariant, however, to the value of current volatility.



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Table III.A.2.2: Volatility weighted vs. equal weighted historical simulation Current volatility generally above contemporaneous historical volatility Basic historical simulation Ordered Daily Return -3.50% -3.00% -2.80% -2.70% -2.65% -2.60% -2.57% -2.55% -2.51% -2.48% -2.45%

Cumulative Weight 0.00000 0.00667 0.01333 0.02000 0.02667 0.03333 0.04000 0.04667 0.05333 0.06000 0.06667

cl 1.00000 0.99333 0.98667 0.98000 0.97333 0.96667 0.96000 0.95333 0.94667 0.94000 0.93333

VaR at Chosen cl 3.50% 3.00% 2.80% 2.70% 2.65% 2.60% 2.57% 2.55% 2.51% 2.48% 2.45%

95% VaR

2.53%

Vol-weighted historical simulation

Daily return -3.50% -2.70% -2.80% -2.51% -2.60% -2.55% -2.48% -2.57% -3.00% -2.45% -2.65%

Contemp Volatility 0.82% 0.80% 0.90% 0.85% 0.95% 1.00% 1.05% 1.10% 1.30% 1.25% 1.50%

Current vol 1.40% 1.40% 1.40% 1.40% 1.40% 1.40% 1.40% 1.40% 1.40% 1.40% 1.40%

Vol weight 1.7073 1.7500 1.5556 1.6471 1.4737 1.4000 1.3333 1.2727 1.0769 1.1200 0.9333

Ordered Voladjusted return -5.98% -4.73% -4.36% -4.13% -3.83% -3.57% -3.31% -3.27% -3.23% -2.74% -2.47%

cl 1.00000 0.99333 0.98667 0.98000 0.97333 0.96667 0.96000 0.95333 0.94667 0.94000 0.93333

VaR at Chosen cl 5.98% 4.73% 4.36% 4.13% 3.83% 3.57% 3.31% 3.27% 3.23% 2.74% 2.47%

95% VaR

3.25%

Current volatility generally below contemporaneous historical volatility Basic historical simulation Ordered Daily Return -3.50% -3.00% -2.80% -2.70% -2.65% -2.60% -2.57% -2.55% -2.51% -2.48% -2.45%

Cumulative Weight 0.00000 0.00667 0.01333 0.02000 0.02667 0.03333 0.04000 0.04667 0.05333 0.06000 0.06667

cl 1.00000 0.99333 0.98667 0.98000 0.97333 0.96667 0.96000 0.95333 0.94667 0.94000 0.93333

VaR at Chosen cl 3.50% 3.00% 2.80% 2.70% 2.65% 2.60% 2.57% 2.55% 2.51% 2.48% 2.45%

95% VaR



2.53%

92


Vol-weighted historical simulation

Daily return -3.50% -2.70% -2.80% -2.51% -2.60% -2.55% -2.48% -2.57% -3.00% -2.45% -2.65%

III.A.2.6.3

Contemp Volatility 0.82% 0.80% 0.90% 0.85% 0.95% 1.00% 1.05% 1.10% 1.30% 1.25% 1.50%

Current vol 0.80% 0.80% 0.80% 0.80% 0.80% 0.80% 0.80% 0.80% 0.80% 0.80% 0.80%

Vol weight 0.97561 1.00000 0.88889 0.94118 0.84211 0.80000 0.76190 0.72727 0.61538 0.64000 0.53333

Ordered Voladjusted return -3.41% -2.70% -2.49% -2.36% -2.19% -2.04% -1.89% -1.87% -1.85% -1.57% -1.41%

cl 1.00000 0.99333 0.98667 0.98000 0.97333 0.96667 0.96000 0.95333 0.94667 0.94000 0.93333

VaR at Chosen cl 3.41% 2.70% 2.49% 2.36% 2.19% 2.04% 1.89% 1.87% 1.85% 1.57% 1.41%

95% VaR

1.86%

Advantages and Disadvantages of Historical Approaches

Historical simulation methods have both advantages and disadvantages. The advantages are: They are intuitive and conceptually simple, providing results that are easy to communicate to senior managers and interested outsiders (e.g. bank supervisors or rating agencies). Dramatic historical events (sometimes irreverently referred to as the markets greatest hits) can be simulated and the results presented individually even when they pre-date the current historical sample. Thus the hypothetical impact of extreme market moves that are strongly remembered by senior management can remain permanently in the information presented, although not directly included in the VaR number. Historical simulation approaches are, in varying degrees, fairly easy to implement on a spreadsheet and can accommodate any type of position, including derivatives positions. They use data that are (often) readily available, either from public sources (e.g. Bloomberg) or from in-house data sets (e.g. collected as a by-product of marking positions to market). Since they do not depend on parametric assumptions about the behaviour of market variables, they can accommodate heavy tails, skewness, and any other non-normal features that can cause problems for parametric approaches, including Monte Carlo simulation. Historical simulation approaches can be modified to allow the influence of observations to be weighted (e.g. by season, age, or volatility). There is a widespread perception among risk practitioners that historical simulation works quite well empirically, although formal evidence on this issue is inevitably mixed. The weaknesses of historical simulation stem from the fact that results are completely dependent on the data set. This can lead to a number of problems:



93


If our data period was unusually quiet (or unusually volatile) and conditions have recently changed, historical simulation will tend to produce VaR estimates that are too low (high) for the risks we are actually facing. Historical simulation approaches have difficulty properly handling shifts that took place during our sample period. For example, if there is a permanent change in exchange-rate risk, it will take time for standard historical simulation VaR estimates to reflect the new conditions. Similarly, historical simulation approaches are sometimes slow to reflect major events, such as the increases in risk associated with sudden market turbulence. Most forms of historical simulation are subject to distortions from ghost effects stemming from updates of the historical sample. In general, historical simulation estimates of VaR make no allowance for plausible events that might occur but did not actually occur in our sample period. There can also be problems associated with the length of our data period. We need a long data period to have a sample size large enough to get risk estimates of acceptable precision. Without this, VaR estimates will fluctuate over time so much that limit-setting and risk-budgeting becomes very difficult. On the other hand, a very long data period can also create its own problems: The longer the data set, the bigger the problem with aged data. The longer the sample period, the longer the period over which results will be distorted by past events that are unlikely to recur, and the longer we will have to wait for ghost effects to disappear. The longer the sample size, the more the news in current market observations is likely to be drowned out by older observations and the less responsive will be our risk estimates to current market conditions. A long sample period can lead to data-collection problems. This is a particular concern with new or emerging market instruments, where long runs of historical data do not exist and are not necessarily easy to proxy. 50 In practice, our main concerns are usually to obtain a long-enough run of historical data. As a broad rule of thumb, many practitioners point to the Basel Committees recommendations for a

50 However, this problem is not unique to the historical simulation approach. Parametric approaches need a reasonable

history if they are to use estimates (rather than just guesstimates or expert judgements) of the relevant parameters assumed to be driving the distributions. For historical simulation VaR it is sometimes possible to synthesize proxy data for markets prior to their existence based on their behaviour over a more recent sample period, but when doing so it is extremely important to avoid overestimating the accuracy of the risk estimates by treating pseudo-data as equivalent to real data.



94


minimum number of observations, requiring at least a years worth of daily observations (i.e. 250 observations, at 250 trading days to the year). However, such a small sample size is far too small to ensure that an historical simulation approach will give accurate and robust results. In addition, as the confidence level rises, with a fixed length sample, the historical simulation VaR estimator is effectively determined by fewer and fewer observations and therefore becomes increasingly sensitive over time to small numbers of observations. For example, at the Basel mandated confidence level of 99%, the historical simulation VaR estimator is determined by the most extreme two or three observations in a one-year sample, and this is hardly sufficient to give us a precise VaR estimate.

III.A.2.7

Mapping Positions to Risk Factors

Portfolio P/L is derived from the P/L of individual positions, and we have assumed up to now that we are able to model the latter directly. However, it is not always possible or even desirable to model each and every position in this manner. In practice, we project our positions onto a relatively small set of risk factors. This process of describing positions in terms of these standard risk factors is known as risk factor mapping. We engage in mapping for three reasons: We might not have enough historical data for some positions. For instance, we might have an emerging market security that has a very short track record or we might have a new type of over-the-counter instrument that has no track record at all. In such circumstances it may be necessary to map our security to some index and the overthe-counter instrument to a comparable instrument for which we do have sufficient data. The dimensionality of our covariance matrix of risk factors may become unworkably large. If we have n different instruments in our portfolio, we would need data on n separate volatilities, one for each instrument, plus data on n(n 1)/2 correlations 51 a total of n(n + 1)/2 pieces of information. As n increases, the number of parameters that need to be estimated grows exponentially, and it becomes increasingly difficult to collect and process the data involved. This problem becomes particularly acute if we treat every individual asset as a separate risk factor. Perhaps the best response to this problem is to map each asset against a market risk factor along capital asset pricing model (CAPM) lines. So, for example, instead of dealing with each of n stocks in a stock portfolio as separate risk factors, we deal with a single stock market factor as represented by a stock market index. A third reason for mapping is that it can greatly reduce the necessary computer time to perform risk simulations. In effect, reducing a highly complex portfolio to a 51 There are, of course, n(n + 1)/2 relevant values in an n ? n symmetric correlation matrix, but we do not have to estimate the values on the main diagonal which are all identically equal to 1.0.



95


consolidated set of risk-equivalent positions in basic risk factors simplifies the problem, allowing simulations to be done faster and with only minimal loss of precision. Naturally, there is a huge variety of different financial instruments, but the task of mapping them and estimating their VaRs can be simplified tremendously by recognising that most instruments can be decomposed into a small number of more basic, primitive instruments. Instead of trying to map and estimate VaR for each specific type of instrument, all we need to do is break down each instrument into its constituent building blocks a process known as reverse engineering to give us an equivalent portfolio of primitive instruments, which we can then map to a limited number of risk factors. There are four main types of basic building blocks. These are: spot foreign exchange positions; equity positions; zero-coupon bonds; futures/forward positions. In this section we will examine the mapping challenges presented by each of these in turn, and compute the normal analytic VaR only, that is to say, we assume the risk factors to which positions are mapped have returns that are normally distributed. The calculation of VaR under more realistic assumptions for risk factor return distributions is discussed in the next chapter of the Handbook. III.A.2.7.1

Mapping Spot Positions

The easiest of the building blocks corresponds to basic spot positions (e.g. holdings of foreign currency instruments whose value is fixed in terms of the foreign currency). These positions are particularly simple to handle where the currencies involved (i.e. our own and the foreign currency) are included as core currencies in our mapping system. 52

We would then already have the exchange-rate volatilities and correlations that we require for the covariance matrix. If the value of our position is A in foreign currency units and the exchange rate (in units of domestic currency per unit of foreign currency) is X, the value of the position in domestic currency units or the mapped position is AX. If we assume A to be a credit-riskless

52 Where currencies are not included as core currencies, we need to proxy them by equivalents in terms of core

currencies. Typically, non-core currencies would be either minor currencies or currencies that are closely tied to some other major currency (e.g. as the Dutch guilder was tied very closely to the German mark before introduction of the euro in both countries). Including closely related currencies as separate core instruments would lead to major collinearity problems; the variancecovariance matrix could fail to be positive definite, etc.



96


instrument bearing an overnight foreign interest rate, its value in units of foreign currency is constant and the only risk to a holder with a different base currency arises from fluctuations in X. In this situation we can calculate VaR analytically in the usual way. For example, if the exchange rate is assumed to be normally distributed with zero mean and standard deviation

X

over the

period concerned, then VaR =

X AX.

(III.A.2.7)

The same approach also applies to other spot positions (e.g. in commodities), provided we have an estimate or proxy for the spot volatilities involved. Example III.A.2.2: Foreign exchange VaR Suppose we have a portfolio worth $1 million, our base currency is the pound, and £0.65 = $1. This means that X = 0.65, A = 1 million (measured in dollars). We estimate the annual volatility of the sterlingdollar exchange rate to be 15%. The daily standard deviation is therefore 0.15/ 250 = 0.009487 and the pound values of the daily VaRs at the 95% and 99% confidence levels are then given by VaR1,0.05 = 1.64485 ? 0.009487 ? 0.65 million = ? 0,143, VaR1,0.01 = 2.32635 ? 0.009487 ? 0.65 million = ? 4,346. III.A.2.7.2

Mapping Equity Positions

The second type of primitive position is equity, and handling equity positions is slightly more involved. Imagine we hold an amount Sk invested in the common equity shares of firm k. If we treat every individual issue of common stock as a distinct risk factor, we can easily run into the problem of estimating a correlation matrix whose dimensions number in the tens of thousands. For example, if we wanted to evaluate the risk for an arbitrary equity portfolio drawn from a pool of 10,000 companies, the number of independent elements in the correlation matrix to be estimated would approach 50 million! It is no wonder that an alternative approach is desirable. In fact, a workable solution to this dilemma is the central concept in the CAPM, which was covered in detail in Chapter I.A.4. The basic assumptions of the CAPM approach imply that the return to the equity of a specific firm k, Rk, is related to the equity market return, Rm, by the following condition: Rk =

k+

kRm

+

k



97


where

k

and

is a firm-specific random element assumed to be uncorrelated with the market. The

k

is a firm-specific constant,

k

is the market-specific component of firm ks equity return

variance of the firms return is then: 2 k

where

2 k

is the total variance of Rk,

2 m

2 k

2 m

2 k ,s

is the variance of the market return Rm and

variance of the firm-specific random element

k

therefore consists of a market-based component

2 k ,s

is the

for company k. The variance of the firms return 2 k

2 m

and a firm-specific component

2 k ,s

.

Assuming the firms equity returns are normally distributed with zero mean, the VaR of an equity position currently valued at xk in the shares of firm k is then: VaR

Z

k

xk .

It is important to recognise that when we aggregate risk across many holdings in a welldiversified portfolio, the main contributor to the total will be the market-based component

2 k

2 m

.

Since the specific risk of each holding is assumed to be uncorrelated both to the market return and to all other specific risk elements, the share of total risk contributed by the specific risk terms falls continuously as the portfolio becomes more diversified and approaches zero when the portfolio approximates the composition of the total market. Look again at the data requirements needed to be ready to estimate VaR based on an arbitrary portfolio drawn from a universe of 10,000 individual stocks. Instead of almost 50 million pairwise correlations we only need the market return volatility, 10,000 market betas for each stock and each stocks specific risk volatility (or more commonly every stocks total return volatility from which, with the market volatility and the individual stock betas, we can derive the specific volatilities). This is 20,001 parameters in all. Compared to a full correlation approach, the CAPM method requires a little more than 0.04% as many parameter values. Estimating just the systematic risk of multi-asset equity portfolios using the CAPM approach reduces to a simple mapping exercise. Assume we have N separate equities holdings with market values of xk for k = 1, , N. Assume the betas of these holdings are the market return volatility is

m.

k

for k = 1, , N and

Then the aggregate systematic VaR of the portfolio is:

VaR

Z

n m

k

xk .

k 1



98


Thus, the systematic VaR is the appropriate critical value times the market volatility times a weighted sum of the market positions, where the weights are the respective betas for each holding. If we define X as the total market value of the portfolio we can modify the above equation slightly to read: n

VaR

ZX

( kx k / X ) .

m

(III.A.2.8)

k 1

In this form, the term in square brackets is the portfolios net beta, or effective market beta. Example III.A.2.3: VaR of equity portfolio Suppose we have a stock market portfolio with five stocks, labelled A, B, C, D and E. The value of the portfolio is $1 million (so X = $1 million) and we have equal investments in each stock (so xA/X = xB/X = = xE/X = 0.20). We are interested in a daily holding period, and the daily stock market standard deviation ( m) is estimated to be 0.025 (hence the annual volatility of the market is approximately 39.5%). The stock market betas are 0.3, and

E

A

= 0.9,

B

= 0.7,

C

= 0.5,

D

=

= 0.1. Substituting the relevant values into equation (III.A.2.8), the VaR at the 95%

confidence level is equal to n

VaR

ZX

m

0.2

k

=1.64485 $1,000,000 0.025 0.2 [0.9+0.7+0.5+0.3+0.1] = $20,561.

k 1

There is no covariance matrix in the example above because all equities are mapped to the same single risk factor, namely, the market index, and this is highly convenient when dealing with equity portfolios. However, this single-factor mapping will underestimate VaR if we hold a relatively undiversified portfolio because it ignores the firm-specific risk, but the underestimation will often be fairly small unless the portfolio is very undiversified. We should also keep in mind that because it assumes a single dominating risk factor it can be unreliable when dealing with portfolios with multiple underlying risk factors (e.g. when there are significant industry concentrations within an equity portfolio). III.A.2.7.3

Mapping Zero-Coupon Bonds

The third type of primitive instrument is a zero-coupon bond (often referred to simply as a zero for short). We will assume for convenience that we are dealing with instruments that have no default risk. In this context it is standard practice to represent prevailing market conditions in terms of a continuously compounded zero-coupon interest-rate curve (sometimes also known as



99


a spot rate curve) across a selected set of future maturity dates. A simplified example of such a curve appears in Table III.A.2.3. 53 Table III.A.2.3

Continuously Compounded Zero Coupon Discount Rates Tenor 3-Mos 1 Year 2 Year 3 Year 5 Year 7 Year

Today 4.5000% 5.0000% 6.0000% 6.5000% 7.5000% 8.0000%

3-Mos

1 Year

2 Year

3 Year

5 Year

7 Year

> > > > > >

In the context of Table III.A.2.3, we can have fixed cash flows maturing on any day out to seven years. Now consider the implications of this for mapping zero-coupon cash flows on arbitrary future dates to a set of fixed grid dates. Given the number of days to the defined grid dates at 3 months, 1, 2, 3, 5, and 7 years, we will interpolate linearly to obtain the effective zero rate on any arbitrary date within this grid. 54 We now want to allocate (or map) a cash flow maturing on an intermediate date to the fixed grid dates in such a way that, to the maximum extent possible, the latter have the same risk characteristics as the original cash flow. What criterion will impose this effective risk equivalence? A common approach is to require the present value impact of a one basis point (0.01%) change in the zero rate at the two surrounding grid points to be the same for the allocated cash flows as it was for the original cash flow. An example will help to illustrate how we can achieve this. Example III.A.2.4: Mapping cash flows for a zero-coupon bond Again using the simplified zero curve shown above, assume we have a risk-free cash flow of 1,000,000 maturing at 2.75 years. We want to create two cash flows, one at two years and one at three years, that have risk-equivalent characteristics to the one cash flow maturing at 2.75 years.

53 In practice, of course, we would have many more grid points at more frequent intervals than is shown in this

simplified illustration. 54 In practice, in developed industrial countries, interest-rate futures are used to calibrate this curve to the market. We

would use an overnight one-day rate as the basis to interpolate out to the next future roll date, which would be no more than three months in the future. It also should be noted that the practice of linear interpolation is not universally preferred because it implies there will be discontinuous changes in the slope of the zero curve at some grid points if the term structure is not uniformly linear over all horizons. To avoid this, some more complex form of interpolation such as cubic splines may be substituted. However, we will assume linear interpolation here to keep the example fairly simple.



100


We begin by recognising that the current interpolated zero rate for 2.75 years is: r2.75 = r2

(3 2.75) + r3

(2.75 2) = 0.0600

0.25 + 0.0650

0.75 = 0.06375.

An increase of one basis point in the two-year rate would result in the 2.75-year rate rising to 0.0601

0.25 + 0.0650

0.75 = 0.063775.

Similarly, an increase of one basis point in the three-year rate would result in the 2.75-year rate rising to 0.600

0.25 + 0.0651

0.75 = 0.063825.

The PV01 (also called the present value of a basis point or PVBP) change in the two-year and three-year rates respectively will be the difference in the present value of the 1,000,000 cash flow after versus before these changes. Thus: PV012 = 1,000,000(e2.75 × 0.063775 e2.75 × 0.063750) = 839,137.04 839,194.73 = 57.69, PV013 = 1,000,000(e2.75 × 0.063825 e2.75 × 0.063750) = 839,021.67 839,194.73 = 173.06. The next step is to find cash flows at two years and three years that have equal PV01 values. To do so, we want to solve the following two equations: C2

e2 × 0.0601 C2

e2 × 0.0600 = 57.69

and C3

e3 × 0.0651 C3

e3 × 0.0650 = 173.06.

We then rearrange slightly to get: C2 = 57.69/(e2 × 0.0601 e2 × 0.0600) = 57.69/0.000377366 = 325,258.99 and C3 = 173.06/(e3 × 0.0651 e3 × 0.0650) = 173.06/0.000246813 = 701,177.56. Thus, the risk sensitivity to changes in the two-year and three-year zero rates of 1,000,000 at 2.75 years is the same as that of two cash flows of 325,258.99 at 2 years and 701,177.56 at 3 years. Put differently, the original cash flow of 1,000,000 at 2.75 years maps to these latter two cash flows, which can therefore be considered its mapped equivalent. 55

55 Obviously this approach to mapping cash flows can be applied in the context of more complex methods for

interpolation of the zero curve. In such cases the mechanics become more complicated, but the core principle of equivalent value sensitivity to small movements in the points that define the zero curve remains the same.



101


Example III.A.2.5: VaR of zero-coupon bond Now suppose we wish to estimate the VaR of a mapped zero-coupon bond. To be more specific, we have a zero-coupon bond with 10 months to maturity, and the nearest reference horizons in our mapping system are three months and 1 year. This means that we need to map our single 10 months horizon cash flow into (nearly) equivalent cash flows at horizons of 3 and 12 months. Now refer to the workbook, VaR of a Mapped Zero Coupon Bond. Given 3-month and 12month zero rates of 4.5% and 5%, we see that the interpolated 10-month zero rate is just under 4.9%. We then carry out the PV01 cash flow mapping and find that a 10-month zero with face value $1m maps to (nearly) equivalent cash flows of $719,217 at 3 months and $654,189 at 12 months. The second sheet of the book then performs the VaR calculation. Assume analysis of the historical data indicates estimated daily volatilities of 1.25% for the three-month rate and 1.00% for the 12-month rate. For the three-month rate this translates into an absolute volatility of 0.0125 ? 4.5% = 0.0563% or 5.63 basis points as a one standard deviation daily change. For the 12-month rate it translates into an absolute volatility of 0.01 ? 5.0% = 0.05% or 5.0 basis points as a one standard deviation daily change. Further assume an estimated correlation between the three-month and 12-month returns of 0.85. The previous worksheet showed that the cash flow mapped to the three-month maturity has a value sensitivity of $17.78 to a one basis point change in the three-month rate, and that the cash flow mapped to the 12-month maturity has a value sensitivity of $62.23 to a one basis point change in the 12-month rate. Based on this information about the value sensitivity to rate changes, the volatility of the rates and their correlation, the portfolio of mapped cash flows has a standard deviation of $399.62. 56 Hence the VaR at the 95% confidence level (taking Z0.05 = 1.645) is 1.645

399.62 = $657.31.

We can confirm this result by deriving the VaR directly from the standard deviation of the 10month zero rate based on the standard deviations of the reference rates, their correlation and their relationship to the 10-month rate.

The derived one-day standard deviation of the 10-

month zero rate equals 4.995 basis points. Since the impact of a one basis point change on the value of the bond is $80.003, the resulting VaR is 1.645 III.A.2.7.4

4.995

80.003 = 657.31.

Mapping Forward/Futures Positions

The fourth building block is a forward/futures position. As explained in Chapter I.B.3, a forward contract is an agreement to buy a particular commodity or asset at a specified future date at a 56 Using the standard formula for the variance of a portfolio (see Section I.A.2.3.2 and/or Section II.D.2.1), the

variance is 399.622 = 159,696 = (17.779

5.625)2 + (62.2253

5.000)2 + 2

0.85

17.779

5.625

62.2253



5.000.

102


price agreed now, with the price being paid when the commodity/asset is delivered, and a futures contract is a standardised forward contract traded on an organised exchange. There are a number of differences between futures and forward contracts, but for our purposes here these differences are seldom important and we can treat the two contracts together and speak of a generic forward/futures contract. To illustrate what is involved for VaR computation, suppose we have a forward/futures position that gives us a daily return that is dependent on the movement of the end-of-day forward/futures price. If we have x contracts each worth F, the value of our position is xF. If the return on each futures contract F is normal with standard deviation

F and zero mean return, the VaR of our

position is

VaR

Z

F

xF .

The main problem in practice is to obtain an estimate of the standard deviation

(III.A.2.9)

F

for the

horizon involved. Typically, we would have estimates for various horizons, and we would use some interpolation method to obtain estimates of interim standard deviations. Section I.C.7.5 presents some examples of VaR calculations for portfolios of commodity futures. III.A.2.7.5

Mapping Complex Positions

Having set out our building blocks, we can now map more complex positions by producing synthetic equivalents for them in terms of positions in our primitive building blocks. (i) Coupon-paying bonds. We can map coupon bonds by regarding them as portfolios of zerocoupon cash flows and mapping each individual cash flow separately to its surrounding grid points. Of course, any one grid point can receive mapped cash flows from one or more actual cash-flow dates on either side of it. These multiple mapped cash flows, and more importantly their associated sensitivities, are aggregated into a single mapped cash flow and associated sensitivity for each grid point. The VaR of our coupon bond is then based on the VaR of its mapped equivalent in zero-coupon cash flows at the fixed grid points. Example III.A.2.6: VaR of Coupon Bond Suppose we have a coupon bond with an original maturity of 4 years and a remaining maturity of 3 years and 10 months. The workbook entitled VaR of a Mapped Coupon Bond illustrates the mapping process and associated VaR calculation. In this example we assume reference points for defining the yield curve at 3 months, 1, 2, 3, and 4 years. As with the zero-coupon bond, the observed daily volatilities for each rate are scaled by the rate levels to derive an absolute daily



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standard deviation of the rate in basis points. We also assume a set of pairwise correlations of the daily changes in the rates. The correlations used in the example are arbitrary, but do reflect the tendency for changes in rates of similar maturities to be more highly correlated than those where maturities differ by a greater amount. In addition, pairs of rates with similar differences in maturities tend to be more highly correlated for rates at longer than at shorter maturities. The worksheet then applies the standard matrix formula (II.D.4) for deriving aggregate volatility to estimate the VaR. Alternately, the mapped cash flows could be treated as the effective positions and analysed by either the Monte Carlo or historical simulation approach. (ii) Forward-rate agreements. A forward-rate agreement (FRA) is an agreement to pay an agreed fixed rate of interest for a specific period starting at a known future date. It is equivalent to a portfolio that is long in a zero-coupon bond maturing at the end of the forward period and short in a zerocoupon bond maturing at the beginning of the forward period. We can therefore map an FRA and estimate its VaR by treating it as a long position in one zero-coupon bond combined with a short position in another zero-coupon bond with a shorter maturity. Here there is one important distinction between FRAs and interest-rate futures. This is that interest-rate futures are settled at the beginning of the forward period, based on the discounted present value of the floating payment determined at that point. FRAs are settled at the end of the forward period at the undiscounted amount of the floating payment that was fixed at the beginning of the period. This means that there is some continuing market risk for an FRA up to the end of its forward period, whereas an interest-rate future is settled at the beginning of the forward period (adjusted for a short settlement delay) and thereafter has no impact on market risk. (iii) Floating-rate instruments. Since a floating-rate note (FRN) reprices to par with every coupon payment, we can think of it as equivalent to a zero-coupon bond whose maturity is equal to the period until the next coupon payment. We can therefore map a floating-rate instrument by treating it as equivalent to a zero-coupon bond that pays its principal plus the current period interest amount on the next coupon date. Example III.A.2.7: VaR of floating-rate note The analysis of an FRN is very similar to that of a zero-coupon bond. The key point here is to appreciate that fixed-income theory tells us that we can price an FRN by treating it as a zero that pays the FRNs current coupon and principal at the FRNs next coupon date. Referring to the workbook VaR of Mapped Floating Rate Note, we see that our FRN has a current coupon rate



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of 5.5%. Given a face value of $1,000,000, this means that we can treat it as a zero that pays $1,027,500 at its FRN coupon date. Given that date is four months hence, this implies that our FRN maps to cash flows of $1,212,992 in three months and $39,406 in 12 months. Given earlier assumptions about rate volatilities and correlations, the portfolio of mapped cash flows has a standard deviation of $185 and therefore a VaR of 1.645

$185 = $304.

(iv) Vanilla interest-rate swaps. A vanilla interest-rate swap (IRS) is equivalent to a portfolio that is long a fixed-coupon bond and short a floating-rate bond, or vice versa, and we already know how to map these instruments. Example III.A.2.8: VaR of interest-rate swap The workbook entitled VaR of Mapped Vanilla Interest Rate Swap illustrates how we deal with this type of instrument. To map an IRS, we first note that for our purposes the swap can be regarded as equivalent to the exchange of a coupon bond for an FRN: one leg of the swap has the cash flows of a coupon bond, and the other the cash flows of an FRN. However, for mapping purposes the FRN is equivalent to a zero, as we have just seen. Suppose, therefore, that we are the fixed-rate receiver on a vanilla IRS, and the fixed-rate leg of the swap has the same features as the coupon bond we have just considered (same notional principal, same term to maturity, etc.). Similarly, the floating-rate leg has the same features as the FRN we have just considered. Then it is as if the cash flows from our swap are as given in Table III.A.2.4. Table III.A.2.4: As if cash flows from vanilla interest-rate swap t (months)

4

10

16

22

28

34

40

46

Fixed rate leg

$30K

$30K

$30K

$30K

$30K

$30K

$30K

$1030K

Floating-rate leg

$1027.5K

Net

$997.5K

$30K

$30K

$30K

$30K

$30K

$30K

$1030K

These are not the actual cash flows from the swap, but for mapping purposes we can consider them as if they were. In Table III.A.2.5 we map these as if cash flows to obtain their (near) equivalents at the same reference points we used for the coupon bond, namely, 3 months, 1, 2, 3, and 4 years. (See the above referenced spreadsheet for the details of the mapping.) Table III.A.2.5: Mapped cash flows from vanilla interest-rate swap t (years)

0.25

1

2

3

4

Mapped Cash Flows

$1,156,000

$16,140

$59,662

$258,256

$843,749

We then derive an analytic estimate of the standard deviation of the portfolios daily change in value. We do this by scaling the value of a one basis point change in each reference rate times



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that rates one standard deviation change and applying the standard correlated aggregation procedure using the assumed correlations for the daily rate changes. The VaR is then the appropriate multiple of this standard deviation estimate.

For the volatility and correlation

assumptions made, the portfolio of mapped cash flows has a standard deviation of $1719, and therefore a VaR (at the 95% confidence level) of 1.645

$1719 = $2827.

(v) Structured notes. These can be regarded as a combination of interest-rate swaps and conventional floating rate notes, which we can already map. (vi) Foreign exchange forwards. A foreign exchange forward is the equivalent of a long position in a foreign currency zero-coupon bond and a short position in a domestic currency zero-coupon bond, or vice versa. Thus it will be sensitive to three market variables, the domestic interest rate, the foreign interest rate and the spot foreign exchange rate. (vii) Commodity, equity and foreign exchange swaps. These can be broken down into some form of forward/futures contract on the one hand, and some other forward/futures contract or bond contract on the other. III.A.2.7.6

Mapping Options: Delta and Delta-Gamma Approaches

The instruments just covered all have in common that their returns or P/L are linear or nearly linear functions of the underlying risk factors. Mapping is then fairly straightforward. However, once we have significant optionality in our positions, their values become nonlinear functions (often highly so) of the underlying risk factors. This nonlinearity can seriously undermine the accuracy of any standard (i.e. linear) mapping procedure. We should be wary of using linear-based mapping systems in the presence of significant optionality. So how do we map option positions? The usual answer is to use a first- or second-order Taylor series approximation. We replace an option position with a surrogate position in an options underlying variables and then use a first- or second-order approximation often known as a delta or delta-gamma approximation to estimate the VaR of the surrogate position. Such methods can be used to estimate the risks of any positions where the value is reasonably approximated by a quadratic function of its underlying risk factor(s). Thus, over a moderate range of variation, they can be applied to option positions that are nonlinear functions of an underlying cash price and to fixed-income instruments that are nonlinear functions of a bond yield. See Sections II.C.3.3, II.C.4 and II.D.2.3.



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Suppose we have a simple European equity call option of value c. The value of this option depends on a variety of factors (the price of the underlying stock, the exercise price of the option, the volatility of the underlying stock price, etc.) but suppose we ignore all factors other than the underlying stock price, and use only the first-order Taylor series approximation of the change in the option value:

c

price respectively, and

S where

c and

S are the changes in the option price and the stock

is the options delta.

If we are dealing with a very short holding period (so we can take

as if it were approximately

constant over that period), the option VaR is approximated by multiplying the underlying VaR by . Hence, if we further assume that S is normally distributed,

VaR where S is the current price and

Z S

(III.A.2.10)

is the standard deviation of returns over the relevant holding

period. The new parameter introduced into the calculation, the option

is also readily available

for any traded option and equation (III.A.2.10) can easily be extended to portfolios of options using the portfolio delta (see Section II.D.2.3.1). Hence the delta approach requires minimal additional data. However, first-order approaches are only reliable when our portfolio is close to linear in the first place, and such methods can be very unreliable when positions have considerable optionality or other nonlinear features. If a first-order approximation is insufficiently accurate, we can try to accommodate nonlinearity by taking a second-order Taylor series (or delta-gamma) approximation: c

S

2

( S )2 .

Readers should refer to Section II.C.4.3 for the mathematical details of delta-gamma approximation and Section II.D.2.3 for further examples of its application. For a long call position, both delta and gamma are positive. In this case, the gamma term contributes an increase in the value of the call option both when the stock price rises and when it falls. If the stock prices rises ( S > 0) the gamma term implies that the call option value rises by more than the delta-equivalent amount. If the stock price falls ( S < 0), the call option value falls by less than the delta-equivalent amount. However, in the case of a short position in the call option, both delta and gamma are negative. Now if S > 0 the option value falls by more than the delta-equivalent amount and if S < 0 the option value will rise (become less negative) by a smaller amount than is implied by the delta term.



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The bottom line is that positive gamma contributes a favourable impact to the value of a position, whether the price of the underlying rises or falls. Conversely, negative gamma contributes an adverse impact to the value of a position, whether the price of the underlying rises or falls. This observation motivates the following as one possible modification to the VaR for an option position to account for the second-order impact of the gamma term: VaR

Z S

2

2

Z S .

(III.A.2.11)

Note that VaR is reduced if gamma is positive and increased if gamma is negative. Example II.A.2.9: Option VaR (delta-gamma approximation) Suppose we wish to estimate the VaR of a European call option. The parameters of the option are: S (underlying price) = X (strike price) = 1, r (risk-free rate) = 5%, option maturity = half a year, and

(annual volatility of underlying) = 25%. We wish to estimate the VaR using a

confidence level of 95% and a holding period of 10 days. One approach is to use the delta approximation (III.A.2.10) VaR

where the sigma term is multiplied by

Z

10/ 250S

10/ 250

1/5 to convert the holding period from one

year (250 business days) to two weeks (10 business days.) To make use of this approximation, we calculate the option delta using the standard formula for the BlackScholes delta (see Table I.A.8.2). This gives us = 0.591. We then input this value and the values of the other parameters into equation (III.A.2.10). This gives the delta approximation for the VaR of a call option as 0.591

1.645

0.25/5 = 0.0486.

If we wish to use a delta-gamma approximation instead, we obtain the option gamma using the standard formula for the BlackScholes gamma (see Table I.A.8.2). This turns out to be 2.198. We then input the relevant parameter values into (III.A.2.11) to get 0.591

1.645

0.25/5 (2.198/2)

[1.645

(0.25/5)] 2 = 0.0412.

Taking account of the positive gamma term therefore reduces our VaR, as we would expect. Now suppose we wish to do the same exercises for a long position in a European put. In this case, the option delta is 0.409, and the gamma remains the same at 2.198. Applying equation (III.A.2.10) then gives us the delta approximation for the VaR as: Copyright ? 2004 The Authors and the Professional Risk Managers International Association


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(0.409)

1.645

0.25/5 = 0.0336.

The corresponding delta-gamma approximation is then found by applying equation (III.A.2.11): (0.409)

1.645

0.25/5 (2.198/2)

[1.645

(0.25/5)]2 = 0.0262.

Again, taking account of the gamma term serves to reduce our VaR. Much the same approach can be used to give us a second-order approximation to the VaR of a bond portfolio, with the first-order term reflecting the bonds duration and the second-order term reflecting its convexity.

III.A.2.8

Backtesting VaR Models

Backtesting involves after-the-fact analysis of the performance of risk estimation models and procedures. In the case of market risk there are two distinct types of backtesting that serve different purposes. These involve comparing ex-ante VaR estimates with ex-post values of (a) actual P/L in the applicable periods and (b) hypothetical P/L assuming positions remained static for the applicable period. The first approach is the one required of banks under the market risk amendment to the Basel Capital Accord and can be thought of as an all-in test. Control of the actual recorded P/L is what risk systems are ultimately designed to achieve. Hence comparing VaR estimates to actual P/L must be a part of any backtesting process. However, one problem with such a test is that there is more than one reason why actual gains and losses may exceed the risk estimates unexpectedly often.

This may occur because of weaknesses in the VaR estimation system, including

(depending on the approach used): inaccurate historical market data and/or parameter estimation; incomplete consolidation of trading positions; inaccurate mapping of trades to risk-equivalent positions in a limited set of primitive securities; incorrect estimation of the portfolio standard deviation or excessive nonlinearities and/or non-normal return distributions resulting in inaccurate VaR estimates using the analytical approach; the generation of Monte Carlo scenarios that fail to match the target characteristics implicit in the estimated parameters; the use of inaccurate or insufficient historical time series to produce historical simulation VaR estimates.



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It also may be caused by gains or losses from intra-day trading that are, by original design, not reflected in any of the three main approaches to VaR estimation. When backtesting reveals weaknesses in the VaR estimation system, it is important to be able to isolate the source of the problem. This is where the second form of backtesting is useful, although not always practiced. The second approach involves comparison of VaR estimates with the hypothetical P/L that would have resulted in the applicable periods if all the trades at the beginning of each period were simply revalued based on end-of-period market prices. This eliminates the impact of day trading that affects the actual P/L, but one must be careful in constructing the hypothetical ex-post P/L results. Assume, for example, that these are based on the same valuation methods used in the VaR estimation process and that these methods are flawed because of one or more of the first three causes noted above. Then the comparison of VaR to these ex-post hypothetical P/L estimates may look acceptable when, in fact, they are both subject to the same flawed calculation methods. This can result in attributing inaccurate VaR estimates to the impact of day trading when the actual problem still lies in the estimation process itself. When conducting the first and more traditional approach to backtesting, it is important to review the official P/L series to establish their relevance to the exercise. Accounting systems are not designed to maintain consistent time series except over fixed reporting periods such as a fiscal quarter. For shorter sub-periods such as a day, which are often the periods of interest for market risk estimation and backtesting, accounting systems attempt to maintain accurate period-to-date information only. Thus, if there is a mistaken entry that results in a large but erroneous daily gain or loss, this will be adjusted by a correcting entry the following day (or later) to bring the fiscalperiod-to-date figures into line. Obviously this will result in misleading daily P/L figures for both the day the mistake was made and the day the correcting entry was booked. It is therefore important to compile the actual P/L to be used for backtesting on a current basis. This allows investigation and documentation of accounting errors and their appropriate treatment in the risk system while the details are still fresh in peoples minds. Failing this, the actual P/L series may be sufficiently flawed to make meaningful comparison impossible. Obviously the key comparison in backtesting is whether the actual or hypothetical P/L series exceeds the corresponding ex-ante VaR estimate (or, more generally, VaR estimates predicated on different confidence levels) with the predicted frequency (or frequencies). In most cases, actual P/L (adjusted for accounting errors and their subsequent correction) tends to produce a lower than expected frequency of observations outside the VaR estimate than is consistent with the probability used in those estimates. This appears to be because day trading allows positions to



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be closed quickly when markets become volatile, thereby reducing actual losses compared to holding a static portfolio for a full 24 hours.

III.A.2.9

Why Financial Markets Are Not Normal

The central limit theorem, sometimes referred to as the law of large numbers, is a fundamental statistical insight with far-reaching consequences. It states that the distributions of sums and averages of random variables exhibit a traditional bell curve or normal distribution even when the individual variables are not normal. While exceptions exist, this holds true for almost any stable random variable found in nature. This gives the normal distribution certain plausibility, and normal distributions are in fact commonly observed in the natural sciences. Thus, it is not surprising that in the early days of modern finance there was some serious debate over whether the distribution of changes in market data departed from a normal distribution in a systematic way. Today the presence of high kurtosis or heavy tails in such distributions is a well-accepted fact. In trying to incorporate such behaviour into market risk analysis, it is important to consider why such persistent departures from the pervasive normal distribution should occur. A key assumption behind the central limit theorem is that the individual observations of random variables going into an average or sum are statistically independent. More often than not this is a reasonably good description of the thousands, or even millions, of individual buy and sell decisions that drive changes in demand and supply on any given day. Since the market clearing price reflects the net balance of these largely independent decisions, it is not surprising that changes in such prices often exhibit a roughly normal distribution. This is, however, not always the case. Consider an example totally unrelated to finance. Suppose you equip the passengers of a singledeck cruise ship with a device that allows you to locate them exactly at any given moment. Then proceed to calculate once every minute the centre of gravity of all these locations with reference to the two-dimensional framework of the ship and plot the resulting distribution. At most times passengers will be in a variety of locations based on their personal preferences, their energy levels, their mood of the moment and the available alternatives. The resulting distribution of their centre of gravity over time will be a cloud of points bunched around the centre of the available passenger areas. We would expect it to exhibit something very close to a bivariate normal distribution. Now assume there is an announcement over the ships loudspeaker that a pod of whales is breaching off the port bow. The consequences are fairly obvious. We would see a sudden outlier in the distribution as passengers rush to find a good viewing spot among the limited



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spaces available. In the immediate aftermath of the announcement, all passengers know several things: There is an opportunity to see something quite unique. The time to see it is limited. There is an ideal location for viewing the phenomenon. Everyone else knows what they know. It is this final point, this mutual self-awareness, that makes for the sudden mad rush to the port bow. Each passenger reacts to the knowledge that speed is of the essence if a good viewing place is to be secured. If the ship was nearly empty, or if only a few people were aware of the opportunity or were likely to take advantage of it (if, say, most passengers were confined to their cabins with sea sickness) the sense of urgency would be greatly reduced. There is a relevant scene in the movie Rogue Trader about Nick Leeson and the Barings debacle. He is awakened by a call at home in the early hours of the morning from another member of the firm. The voice at the other end of the phone says urgently, Turn on CNN!! The TV in the bedroom flickers to life showing scenes of the Kobe earthquake. The voice at the other end of the phone says, This is just going to kill the market!!! In effect this is much like the announcement on the ship but on a global basis. Observers around the world are suddenly focused on a common crystallising event with obviously directional implications for the market. In addition, everyone knows that everyone else knows. Suddenly the millions of decisions that drive the market are no longer randomly independent. Rather they are subject to a common shared perception. The core structural assumptions that underpin a normal distribution have temporarily broken down and we see a sudden extreme observation. Various statistical methods are used to try to build such behaviour into distributions of risk factor returns. But what these approaches cannot do is predict in advance when such events will occur. Thoughtful consideration of such potential scenarios, especially those that present special threats given existing open positions in the book, is therefore an essential component of effective market risk management. Such analysis remains in the realm of experience and seasoned judgement that no amount of advanced analytical technique can replace. This topic will be discussed in detail in Chapter III.A.4.

III.A.2.10

Summary

In this chapter we have explained the underlying methodology for the three basic VaR model approaches analytic, historical simulation and Monte Carlo simulation. A main focus of this Copyright ? 2004 The Authors and the Professional Risk Managers International Association


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chapter has been the analytic VaR models that are only valid when portfolio values are linearly related to the underlying risk factors and portfolio returns are normally distributed. We have also considered simple analytic formulae for VaR based on delta-gamma approximation to simple portfolios of standard European options. However, in reality things are not that straightforward. Portfolio returns are not normally distributed and, as is evident from Chapter I.B.9, options portfolios typically contain products with many underlying risk factors and various exotic features. The next chapter of the Handbook will consider how the basic VaR methodology that we have introduced here may be extended to more realistic assumptions about the products traded and the behaviour of asset returns.

References Boudoukh, J, Richardson, M, and Whitelaw, R (1998) The best of both worlds: a hybrid approach to calculating value at risk, Risk, Vol. 11(5), pp. 6467. Hull, J, and White, A (1998) Incorporating volatility updating into the historical simulation method for value-at-risk, Journal of Risk, Vol. 1 (Fall), pp. 519. Shimko, D B, Humphreys, B, and Pant, V (1998) Hysterical simulation, Risk, Vol. 11(6), p. 47.



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III.A.3: Advanced Value at Risk Models Carol Alexander and Elizabeth Sheedy 57, 58

III.A.3.1

Introduction

The previous chapter introduced the three basic VaR models: the analytic model and two simulation models, one that is based on historical observations and another that uses a covariance matrix to generate correlated scenarios by Monte Carlo (MC) simulation. The fundamental assumptions applied in the basic forms of each model can be summarised as in Table III.A.3.1. A number of variations on these assumptions are in common use and some of these will be reviewed later in this chapter. Table III.A.3.1: Basic VaR model assumptions Analytic

Historical

Monte Carlo

VaR

VaR

VaR

Risk Factor/Asset Distributions

Normal

No Assumption

Normal

P&L Distribution

Analytic

Empirical

Empirical

(Normal)

(Historical)

(Simulated)

Requires Covariance Matrix?

Yes

No

Yes

Risk Factor/Asset Returns i.i.d.?59

Yes

Yes

Yes

Clearly the analytic and the MC VaR models are very similar. Indeed, if one were to apply the basic MC VaR model above to a linear portfolio i.e. one without options, for which the portfolio value is a linear function of the underlying risk factors the VaR estimate should be similar to the analytic VaR model estimate. If it were different, that would be because not enough simulations were used and a small sample error had been introduced. But of course, one would not apply MC VaR to a linear portfolio; this method is quite computationally intensive, and would only be applied to portfolios containing options. The Excel spreadsheet SimpleVaR.xls examines the VaR of a simple portfolio ($1000 a point on the Johannesburg Top40 index, actually). From the single historical price series in the spreadsheet we compute the 1% 10-day VaR as: $878,233 according to the analytic VaR model,

57 Carol Alexander is Chair of Risk Management and Director of Research, ISMA Centre, Business School, University

of Reading, UK. Elizabeth Sheedy is Associate Professor, Applied Finance Centre, Macquarie University, Australia. 58 Many thanks to Kevin Dowd and David Rowe for their careful editing of this chapter. 59 i.i.d. stands for independent and identically distributed.



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$839,829 according to the historical VaR model, something close to $878,233 (hopefully) according to the MC VaR model. The MC VaR model result changes every time we perform the simulation. In this spreadsheet only 1000 simulations are used so the result can vary a lot each time we press F9 to generate new simulations. However, if we used 100,000+ simulations, the MC VaR result would be $878,233 (or very close to that) as in the analytic VaR model. The main problem with the basic analytic and MC VaR models is that they are both based on the very restrictive assumptions that risk factor (or asset) returns are i.i.d. normally distributed. Most practitioners are aware that the assumption of normality is violated in reality, with returns often exhibiting skewness and leptokurtosis. This has led to the popularity of the historical simulation method. The historical simulation method, which can be applied to any type of portfolio, can give results quite different from the other two methods, even for an extremely simple portfolio as shown above. If the portfolio P&Ls are non-normal, then the historical VaR should be more accurate, assuming the sample contains enough data points to estimate the 1% lower percentile of the historical distribution with sufficient accuracy. But this requires a very large amount of historical data, at least 1000 days.60 Even if the data were available, the portfolio price series is computed over these 1000 or so days using the current portfolio weights, and this may not be plausible (would you have traded the same portfolio 5 years ago, when the market was in all probability quite different?). And even if the current weights are plausible, we still have to assume that returns are i.i.d. and use the square root of time rule to compute the 10-day VaR (see Section III.A.3.2). 61 The use of a covariance matrix in the analytic and the MC VaR models has both advantages and limitations. One advantage is that the covariance matrix will not normally be based on a very large amount of historical data. Regulators, for some reason, require at least one year of historical data to be employed when computing regulatory capital using a VaR model (see Section III.A.3.4), but for internal purposes less than a year of data may be used as, for instance, in Section III.A.3.4.1. Hence the assumption of current portfolio weights is not as problematic as it is with the historical VaR model. However, an important limitation is that the use of a single covariance matrix assumes that all co-variations between risk factors are linear; but they are not

60 Regulators recommend 35 years of daily data, i.e. 7501250 observations.. 61 You cannot use 10-day returns instead, since these would have to be non-overlapping, and one would need a history

spanning decades to obtain enough data!



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linear. It is well known that the extreme variations in major risk factors can be more highly correlated than the normal variations. We shall address this issue in Section III.A.3.5.3. Thus, all three of the approaches are defective in different ways when the basic assumptions are used. The nub of the problem is this: how can VaR be estimated using assumptions that are consistent with the stylised facts we observe in financial markets? Most of this chapter is directed at this problem, which is fundamentally concerned with model risk. Section III.A.3.2 examines the standard distributional assumptions more closely and the ways in which they are violated. It concludes, somewhat counter-intuitively, that the most pressing problem for those modelling market risks is not heavy tails, but volatility clustering. Accordingly, Section III.A.3.3 examines two different approaches to modelling volatility clustering: EWMA and GARCH. These models are then applied to the problem of VaR estimation in Section III.A.3.4. Section III.A.3.5 examines some other solutions to the problem of heavy tails in VaR estimation: Students t, EVT and normal mixtures. The remaining sections of the chapter address some other advanced topics in VaR modelling. It is often desirable to decompose VaR into components for purposes of limit setting and performance measurement.

These decomposition techniques are considered in

Section III.A.3.6. Section III.A.3.7 examines principal component analysis, an important tool for analysing bond and futures portfolios. Section III.A.3.8 concludes the chapter.

III.A.3.2

Standard Distributional Assumptions

When measuring risk, for example in a VaR calculation, it is often necessary to make some assumption about the distribution of portfolio returns. The most widespread choice is the assumption that returns are i.i.d. normal. That is, each return is an independent realisation from the same normal distribution (see Section II.E.4.4). Normality implies that the distribution can be completely described with only two parameters: the mean and the variance. Yet many financial analysts are sceptical about the assumption of normality. The skewness and (excess) kurtosis should equal zero, but this is rarely the case. To illustrate this point, let us consider daily returns for the USD/JPY for the period from January 1996 to July 2004 as illustrated in Figure III.A.3.1 and described in Table III.A.3.2. Table III.A.3.2 highlights the stylised facts or characteristics commonly observed in financial returns: The mean of the daily returns is very close to zero. Risk is expressed in two ways standard deviation per day and volatility, which is the annualised standard deviation. If we assume returns are i.i.d. then the square root of time rule applies. This rule is explained in Section II.B.5.3. It states that the standard deviation



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of h-day returns is h

standard deviation of one-day returns; 62 or equivalently, that

volatility is constant. 63 Note that throughout this Handbook we adopt the standard convention of quoting volatility on an annual basis. The skewness (see Section II.B.5.5) is estimated using the Excel function SKEW and is found to be negative. The excess kurtosis (see Section II.B.5.6) is estimated using the Excel function KURT and is found to be positive, indicating that the distribution has heavy tails relative to the normal. Figure III.A.3.1: USD/JPY returns, Jan. 1996 to July 2004 0.04

0.02

0

-0.02

-0.04

-0.06

-0.08

The implication of negative skewness and positive excess kurtosis is that the true probability of a large negative return is greater than that predicted under the normal distribution. This finding has potentially grave consequences for the measurement of VaR at high confidence levels. If VaR is calculated under the assumption of normality, yet the actual data have heavy tails, then VaR will understate the true risk of a disastrous outcome. Consequently, capital reserves may be an insufficient buffer to withstand disaster at the desired confidence level.

62 To be precise, the rule is based on log returns, defined as ln(P /P ) where P is the price at time t. Log returns have t+h t t the nice property that the sum of h consecutive one-day log returns is the h-day log return. Also, if h is small, it is easy to show that log returns are very close indeed to the absolute return (Pt+h Pt )/Pt. 63 To see this, let denote the standard deviation of one-day returns. Then the volatility based on 1-day returns = 250 Under the square root of time rule, the standard deviation of h-day returns is h, and since there are 250/h time periods of length h days per annum, the volatility based on h-day returns = h (250/h) which also equals 250 So in Table III.A.3.2, volatility = 0.00718 250 = 0.1135.



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Table III.A.3.2

Analysis of USD/JPY returns, Jan. 1996 to July 2004

Number of observations

2161

Mean return

2.60047 10

5

Standard deviation per day

0.718%

Volatility

11.35%

Skewness

0.8255

Excess kurtosis No. of observations below the lower bound of 99% confidence interval

6.241 48 vs. 22 expected

Under the normal distribution, the lower bound of the 99% one-tailed confidence interval is defined by the mean less 2.33 standard deviations. In this case the lower bound equals 2.33 × 0.00718 = 0.0167, so we should expect that only 1% of the return observations will be lower than this figure. With 2161 observations, only 22 returns (1% of 2161) should lie in this lower tail. In fact, examination of these data reveals that 48 returns are below the lower bound. This kind of analysis is often performed to demonstrate that finance data violate the assumption of normality.

The problem with this analysis, however, is that it incorrectly assumes that

volatility has remained constant for the entire sample period of 8.5 years! Reviewing Figure III.A.3.1, we can see that the volatility of USD/JPY returns has varied considerably, with the period of greatest volatility being 1998. This was a time when most financial markets exhibited high risk following the Russian crisis. In 1998 we see that oscillations about the mean were much greater than at other times. High-volatility periods are characterised by large returns, both positive and negative. Note that it is the absolute size of the return that is important rather than direction. On one day at the height of the Russian crisis, the US dollar depreciated by more than 6% against the Japanese yen. The largest positive return (in excess of 3%) occurred during the same period of high volatility. Of the 48 observations in the lower tail of the confidence interval, 18 fall in 1998, a period of very high volatility. This leads us to an alternative way of understanding the data: the large number of observations that appear in the lower tail is a result of changes in volatility throughout the sample period. Sometimes, as in 1998, volatility is much higher and so the confidence interval is correspondingly wider. Instead of a lower bound of 0.0167, the lower bound would be very much lower (for example, 0.0335 if volatility were to double). If we take account of the increase in volatility it is likely we will find that the number of observations in the tail is close to 1% as expected. In other words, the main problem with financial data is not skewness/kurtosis (often referred to as heavy tails), but the fact that volatility is changing over time. In short, if we



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take account of these changes in volatility then the assumption of normality may actually be quite a reasonable one. This brings us to a further discussion of i.i.d. returns. 64 If returns are identically distributed, then the parameters of the distribution (mean and variance) should be constant over time. This assumption is clearly violated since the variance parameter is changing.

If returns are

independent, then yesterdays return will have no bearing on todays return. Tossing coins is a good example of independent outcomes. Even if I toss 10 heads in a row, the probability of heads on the next toss is still 50% (assuming a fair coin!). In other words, the next toss is unaffected by what has gone before. However, the empirical analysis of financial returns shows that the assumption of independence is unsupported: the size (but not the direction) of yesterdays return does have implications for todays return. A large return yesterday (in either direction) is likely to be followed by another large return in either direction. This concept is commonly referred to as the heat wave effect or volatility clustering. We can test for volatility clustering by examining the autocorrelation in squared returns. Squared returns are used because squaring removes the sign of the return we can focus purely on its magnitude rather than its direction. Financial data often generally exhibit significant positive autocorrelation in squared returns, in which case the data are not i.i.d. Volatility clustering is arguably the most important empirical characteristic of financial data. The most useful financial models take account of this fact. Indeed, the world of finance research was for ever changed in the 1980s when volatility clustering was first identified by Robert Engle and his associates. In 2003 Robert Engle won (jointly) the Nobel Prize for Economics, largely because of his groundbreaking contribution in this area. To summarise this section, we can say that traditional financial models have often assumed that returns are i.i.d. normal. In reality we observe changes in volatility and volatility clustering. We will show in the following sections that these characteristics can be modelled successfully. Having taken account of volatility clustering, the issue of heavy tails becomes less significant.

64 See Section II.B.5.3 for further discussion of this concept.



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III.A.3.3

Models of Volatility Clustering

If we believe that todays volatility is positively correlated with yesterdays volatility, then it is appropriate to estimate conditional volatility, that is, volatility that is conditional on the recent past. We will discuss two methods for achieving this:

exponentially weighted moving average

(EWMA) and generalised autoregressive conditional heteroscedasticity (GARCH). The latter is more difficult to implement but offers some potential advantages. The application of these methods to VaR models will be discussed in Section III.A.3.4. III.A.3.3.1

Exponentially Weighted Moving Average

The EWMA method for estimating volatility was popularised by the RiskMetrics Group. Todays estimate of variance is: t2

rt 2 1

1

t2 1 ,

(III.A.3.1)

where is a smoothing constant and rt1 is the most recent days return. 65 Notice that since is positive, todays variance will be positively correlated with yesterdays variance, so we see that EWMA captures the idea of volatility clustering. The parameter, , may also be referred to as the persistence parameter. The higher the value of , the more will high variance tend to persist after a market shock. The EWMA variance also reacts immediately to market shocks.

If

yesterdays return is large, in either direction, the variance will increase through the first term on the right-hand side of (III.A.3.1). The greater is 1 , the greater will be the size of the reaction to a return shock. In Section III.A.3.3.2 we shall see that this type of reaction to, and persistence following, a market shock is the characteristic of a GARCH model. In fact, the EWMA can be thought of as a very simple GARCH model. However, whilst GARCH volatility models are based on firm statistical foundations, EWMA volatility models are not, for the following reasons: There is no proper statistical estimation procedure for the smoothing constant: the user simply assumes some value for . According to RiskMetrics, a value for

of

around 0.94 is generally appropriate when analysing daily data. 66

65 This can be calculated in Excel by typing (III.A.3.1) in the formula bar, or by using the exponential smoothing

analysis tool. Note that technically, in (III.A.3.1) the volatility estimate depends on the entire historical data set rather than a limited past history. A pragmatic alternative calculation that is much easier to replicate for the auditors is to truncate the historical sample at n past observations, where n is defined as the point at which n drops below some critical value C. For instance, if = 0.94 and C = 0.002 then n = 100 observations. The EWMA on n observations can be written as a finite sum: 2t

n

1 t 1

t 2 t

r

n

/

1

t

t 1

66 See Section 5.3.2 of RiskMetrics Technical Document, 1996, available at: www.riskmetrics.com/techdoc.html#rmtd



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In portfolio models, where the variance is calculated using a covariance matrix, some portfolios can have negative variance unless this matrix is positive semi-definite (see Section II.D.3.2). For this reason one cannot form an EWMA covariance matrix using different values for

for different assets: in fact the same value of

must be used for

all variances and covariance in the matrix. More details are given in Section III.A.3.4.1. The EWMA variance estimate is converted to volatility by taking the square root, to obtain the standard deviation, and then applying the square root of time rule: EWMA volatility estimate at time t for a horizon of h days = t h

250/ h .

But, as explained in Section III.A.3.2, this is equivalent to a constant volatility assumption. Hence the EWMA model is not really appropriate for estimating the market evolution over time horizons longer than a few days. Figure III.A.3.2: EWMA volatility USD/JPY returns, Jan. 1996 to July 2004 40% 35% 30% 25% 20% 15% 10% 5% 0%

Figure III.A.3.2 highlights the variability of conditional volatility. Here the EWMA model has been applied with

= 0.94. Recall from the previous section that unconditional volatility for this

same time period was calculated as 11.35% pa. Following the market shocks of 1998, conditional volatility reaches a maximum of 35%. In the quieter periods of 1996 and early 2004, conditional volatility briefly dips below 5% pa. Having estimated conditional volatility in this way, we can then standardise the daily returns. That is, each daily return is divided by the relevant standard deviation. We do this to try to make



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the returns comparable. Prior to standardisation, each return comes from a distribution having a different volatility and is therefore not strictly comparable. We then repeat the analysis of Table III.A.3.2, but this time using the standardised returns (see Table III.A.3.3). Table III.A.3.3

Analysis of standardised USD/JPY returns, (Feb. 1996 to July 2004

Number of observations

2140

Skewness

0.1974

Excess kurtosis

0.2727

No. of observations below the lower bound of 99% confidence interval

29 vs. 21 expected

Having standardised returns on conditional volatility, we find that the problems noted earlier are much diminished. That is, skewness and excess kurtosis are now much closer to zero. The number of observations in the lower tail is much closer to that expected under normality. This suggests that the issue of heavy tails, often noted by finance analysts, can be partly explained by volatility clustering. III.A.3.3.2

Generalised Autoregressive Conditional Heteroscedasticity Models

GARCH models are similar to EWMA in that both focus on the issue of volatility clustering. The word heteroscedasticity is Greek for different scale, so GARCH models are concerned with the process by which the scale of returns, or volatility, is changing. GARCH models are generalised in the sense that they can be varied, almost infinitely, to take account of the factors specific to a particular market. What all GARCH models share, however, is a positive correlation between risk yesterday and risk today; that is, an autoregressive structure in risk. The simplest GARCH model consists of two equations that are estimated together. The first is the conditional mean equation and the second is for the conditional variance: rt

c

t

2 t

2 t 1

0, , The parameters

,

2 t 1

(III.A.3.2)

0. and

of the GARCH model are usually estimated using maximum

likelihood estimation (MLE). This involves using an iterative method to find the maximum values of the likelihood function. In normal GARCH models, i.e. when the distribution of on all information up to time t is normal with variance

2 t

t

conditional

, the likelihood function is a

multivariate normal distribution on the model parameters (see Section II.E.4.7).



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In its simplest form (shown here) the conditional mean equation merely adjusts for the mean (c), leaving an unexpected return t. The conditional variance equation appears very similar to the EWMA equation where

stands in place of ,

stands in place of 1 , and an extra constant

term ( ) is also included. Perhaps the most important difference between EWMA and GARCH is the fact that in GARCH there is no constraint that the sum of the coefficients ( + ) should equal one. It is possible that data are best explained when the estimated coefficients do add to one, 67 but in this case the GARCH forecasts behave just like those from a constant volatility model. If the sum

+

is less than one (the more usual case) then volatility is said to be mean-

reverting and the rate of mean reversion is inversely related to this sum. This means that the variance will, in the absence of a market shock, tend towards its steady-state variance defined by 2

1

.

(III.A.3.3)

The GARCH volatility forecasts then behave like the volatility term structures we observe in implied volatilities, where implied volatilities of long-term options do not vary as much as the implied volatilities of short-term options. From (III.A.3.3) we can see that if

1 (as in

EWMA) then the denominator equals zero so the steady-state variance is undefined. In this case variance is not mean-reverting and is assumed constant as we project forwards in time. Example III.A.3.1: GARCH model for spot USD/JPY We estimate a GARCH(1,1) model for daily log returns from January 1996 to July 2004. The conditional variance equation is estimated using a maximum likelihood technique: 2 t

3.78 10

7

0.03684

2 t 1

0.95571

2 t 1

.

The constant term ( 3.78 10 7 ) is statistically significant but very small. Conditional variance for the USD/JPY is quite persistent (with a persistence coefficient of 0.955710) and not particularly reactive (reaction coefficient of 0.03684) compared with some other markets. This means that the initial reaction to new information will be more muted, but its effect relatively long-lasting. As the sum of these two coefficients is less than unity, we can say that conditional variance is mean-reverting to a steady-state variance defined by (III.A.3.3).

Substituting the estimated

parameters into this equation, the steady-state variance is 0.0000505794, equivalent to an annualised volatility of 11.24%. This is close to the sample volatility of 11.35% p.a. in Table III.A.3.2.

67 When this happens, the model is called an integrated GARCH (IGARCH) model.

commonly used for modelling currency returns.

The IGARCH is most



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Figure III.A.3.3: GARCH vs. EWMA volatility USD/JPY returns 40% 35%

EWMA GARCH

30% 25% 20% 15% 10% 5% 0%

Figure III.A.3.3 shows the conditional volatility resulting from the GARCH model. The two series, EWMA and GARCH, track each other closely, so the figure emphasises the similarity between the two approaches. We have argued that volatility clustering is the main cause of the apparent heavy tails observed in financial data. If GARCH modelling is successful, then it should account for these heavy tails, leaving residuals that are i.i.d. normal. Is this in fact the case? It can be tested by examining the series of standardised returns. To obtain the standardised returns we estimate a GARCH model and from this create a series of daily conditional standard deviation estimates. We then divide each daily return by the relevant conditional standard deviation to obtain a standardised return, which we then test for normality. If this series is normally distributed then we can conclude that volatility clustering fully explains the extreme moves.

If not, then further action may be

necessary to adjust for the extreme moves (or for the heavy tails of the distribution). For example, it may be necessary to use a more complex GARCH specification. For instance, we might use an asymmetric specification for the conditional variance that can account for a leverage effect in equities, and/or a conditional distribution for

t

that is non-normal. There is a

huge body of research literature on GARCH models, but this is beyond the scope of the PRM exam.



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III.A.3.4

Volatility Clustering and VaR

Volatility clustering has important implications for VaR. Consider the situation in which new and unexpected information comes to the market, as it did when Russia defaulted on its debt in 1998, causing a large reaction in market prices. Our knowledge of volatility clustering tells us that this market shock is likely to be followed by large returns (in either direction) for some time. Ideally, our VaR measure will increase significantly, sending the appropriate signal to risk managers either to reduce risk through hedging or to ensure that capital is adequate to withstand the higher risk environment. Figure III.A.3.4: USD/JPY volatility in 1998 35%

30%

EWMA 1-year Volatility

25%

20%

15%

10%

Figure III.A.3.4 compares two volatility measures for USD/JPY returns for calendar year 1998. The EWMA measure of conditional volatility reacts swiftly as the crisis unfolds in mid-1998. The other series shows unconditional volatility calculated using rolling one-year samples. Since each day in the sample is equally weighted, and has a small weight of only 1/250, this measure is very slow to react to market shocks. The choice of volatility measure will have major implications for the VaR measure and consequently for capital adequacy. By failing properly to take account of volatility clustering, the risk is that financial institutions will take unduly large risks (or will hold insufficient capital) in periods of market crisis. In addition, they will hold too much expensive capital at other times. Backtests of such models will reveal clusters of exceptions where the actual losses exceed VaR. Unfortunately, the timing of exceptions is often ignored in traditional backtesting procedures which focus only on the number of exceptions (see Section III.A.2.8). Both regulators and risk



126


managers are concerned about the timing and size of exceptions. A series of large losses in quick succession is potentially far more serious for solvency than smaller losses spread over time, even if the total combined losses are equal. For reasons that remain unclear to the authors, the Basel regulations relating to market risk currently require financial institutions to measure volatility using at least one year of data. This regulation encourages the use of volatility measures that react very slowly to new information such as the one illustrated above. In our view, this is exactly the wrong way to proceed. 68 III.A.3.4.1

VaR using EWMA

Volatility clustering can be relatively easily incorporated into VaR measures using the EWMA approach. There are at least three ways that this can be done: Historical simulation using volatility weighted data. This method is explained in detail in Section III.A.2.6.2.

Historical returns are standardised using conditional volatility

estimates calculated using EWMA. This approach has a number of attractions, especially for option-affected portfolios. Historical simulation makes no assumption about the distribution of historical returns apart from independence. Thus it is attractive if it is feared that the standardisation process has not entirely eliminated the heavy tails evident in raw returns. MC simulation using EWMA. Returns could be simulated under the assumption of normality, but using a covariance matrix created using EWMA. Again, this method is most relevant for option-affected portfolios. Analytical VaR using EWMA.

We will explain this method here, building on the

discussion in Section III.A.2.4. Of these three methods, the last two make use of the assumption that returns are conditionally normally distributed. In Section III.A.3.3 we explained that normality is a much more reasonable assumption to make if we properly account for changes in volatility. One way to do this is to calculate the covariance matrix using the EWMA measures of variance and covariance. This is preferable to the standard measures of unconditional variance and covariance, which are slow to respond to new information when estimated using long samples. 69 Equation (III.A.3.1) shows the EWMA equation for variance. The analogous equation for covariance between assets 1 and 2 is:

68 See the discussion of historical observation period in Basel Committee Amendment to the Capital Accord to Incorporate Market Risks, January 1996, at p. 44. 69 See Sections II.B.5 and II.B.6 for descriptions of these standard measures.



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12,t

1

r1,t 1r2,t

1

12,t 1 ,

where r1,t1 and r2,t1 are yesterdays returns for assets 1 and 2, respectively. Note that when constructing a large covariance matrix it is always important to ensure that it is positive semidefinite, otherwise portfolio volatility may not be defined (see Section II.D.3.2). If we use a different value of

for each variance and covariance term, the matrix will not necessarily be

positive semi-definite. RiskMetrics gets around this problem by using the same value for (being 0.94 in the case of daily data) throughout the covariance matrix. When calculating VaR we need to forecast portfolio variance for the horizon of interest, say, 10 business days. Unlike GARCH, EWMA is a non-stationary model of variance. That is, under EWMA, variance is not mean-reverting. This means that the EWMA forecast of any future variance (or covariance) is the same as the estimate made today. Equivalently, as explained in Section III.A.3.3.1, the average volatility over any forecast horizon is a constant, equal to the volatility estimated today. Once the covariance matrix has been defined, it can then be used for VaR calculations using either: the analytical method (appropriate for simple linear portfolios) or MC simulation (best for option-affected portfolios). In the analytical method (See Section II.A.2.4) the h-day VaR estimate at the significance level

is

given by: VaR ,h = Z P , where Z is the standard normal

(III.A.3.4)

critical value, P is the current value of the portfolio and

is the

forecast of the standard deviation of the h-day portfolio return. The standard deviation in (III.A.3.4) is computed using a forecast covariance matrix of h-day returns as follows (see Section II.D.2.1): (i)

Representation at the asset level, w ' Vw where w = (w1, , wn) is the portfolio weights vector and V is the h-day forecast of the covariance matrix of asset returns. 70

70

Equivalently, VaR in each asset.

,h

=Z

P&L

Z

p ' Vp , where p = P w is the vector of nominal amounts invested



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(ii)

Representation at the risk factor level, 'V where

= ( 1, ,

n)

is the portfolio sensitivity vector and V is the h-day forecast of

the covariance matrix of risk factor returns. 71 To calculate analytic VaR using EWMA we simply calculate todays portfolio variance as above. To convert to a 10-day horizon we simply multiply the one-day VaR by 10; or we use a 10-day covariance matrix for V. In either case, the results will be the same. The 10-day matrix can be obtained, using the square root of time rule, by multiplying every element of the one-day covariance matrix by 10. But note that these two conversion methods assume both constant volatility and no serial correlation in the forward projection of volatility, which is one reason why the EWMA VaR estimate does not fully reflect volatility clustering. By contrast, for non-linear portfolios the MC VaR method uses a covariance matrix, and you should be sure to use the 10-day covariance matrix directly in the simulations. You will get an incorrect result if you simulate the one-day VaR and multiply the result by 10 Example III.A.3.2: Analytical method with EWMA for two-asset portfolio Consider a simple portfolio with $1m invested in asset 1 and $2m invested in asset 2. Suppose the EWMA model estimated on daily returns for asset 1 gives a variance estimate of 0.01, for asset 2 returns the EWMA variance is 0.005 and the EWMA covariance is 0.002. What is the 5% 10-day VaR?

We have p = (1, 2) and V is the matrix

0.01 0.002 . So the P&L volatility = 0.002 0.005

0.038 = $0.19m and the 5% one-day VaR is 1.645 VaR = 0.3207

p ' Vp =

0.195 = $0.3207m. Thus the 5% 10-day

10 = $1.014m. 72

This is high, because the assets have a very high variance (and covariance). For instance, a daily variance of 0.01 corresponds to an annual volatility of 2.5 = 158%! In fact, the 1% 10-day VaR for this portfolio is 2.33

0.195

10 = $1.43m, so almost half the amount invested would be

required for risk capital to cover this position! However, the example illustrates an important 71 Equivalently VaR , = Z h factor in nominal terms.

P&L

Z

p ' Vp , where p = P

is the vector of sensitivities to each risk

72 Of course, the same result would be obtained if V is the given matrix but with each element multiplied by 10 and we calculated Z p ' Vp .



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weakness in the use of the analytic VaR method for long-only positions. Being based on the assumption that portfolio P&L is normally distributed, there is a chance, however small, that the estimated VaR will be more than the total investment. For a long-only position, this clearly is a nonsensical result. Example III.A.3.3: Analytical method for portfolio mapped to two risk factors Suppose a US investor buys $2m of shares in a portfolio of UK (FTSE100) shares, and the portfolio beta is 1.5. Suppose the FTSE100 and GBP/USD volatilities are 15% and 20% respectively (with corresponding variances of 0.152 = 0.0225 and 0.202 = 0.04) and their correlation is 0.3. What is the 1% 10-day risk factor VaR in US dollars?

We have two risk factors (FTSE and GBP/USD) with p = (3, 2) . Note that the $2m exposure to the equity portfolio described above is equivalent in risk terms to a $3m exposure to the FTSE index since beta is 1.5. The one-day variances are: 0.0225/250 = 0.00009 for the FTSE and 0.04/250 = 0.00016 for GBP/USD. With a correlation of 0.3, the one-day covariance is (0.3 0.15 0.2)/250 = 0.000036. Hence, 73 p Vp = 3 2

90

36

3

36 160

2

and the 1% 10-day VaR is therefore 2.32634

10

5

342 428

3 2

10 5 = 0.01882

10 0.01882 = $0.31845m.

In concluding this section, we can say that VaR calculations, whether analytical or simulationbased, can be greatly improved by taking account of volatility clustering. In this regard EWMA is far superior to the approach, unfortunately encouraged by regulators, which employs unconditional volatility based on large samples of data. III.A.3.4.2

VaR and GARCH

Alternatively, the issue of volatility clustering may be incorporated into VaR estimates using the more sophisticated GARCH approach. While GARCH is undoubtedly more challenging to implement than some other methods, it also has some distinct advantages. For example, because GARCH is a more general model it can explain the characteristics of the data more precisely to

73 Note that the one-day covariance matrix is

0.0009 0.00036 0.00036 0.0016

90 36 36 160

0.00009

0.000036

0.000036

0.00016

so the 10-day covariance matrix is

10 5 .



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ensure that all evidence of non-normality and dependence is removed from the standardised returns. A distinct advantage of GARCH over EWMA is that GARCH is a mean-reverting model of volatility. This fits better with the stylised facts observed in the market; we regularly observe that volatility will tend toward a mean or steady-state value after a period of unusually high or low risk. If we are forecasting volatility over, say, the next 10 days, then it could be helpful to take account of this mean reversion feature. We cannot do this with EWMA volatility models because they use the square root of time rule, as explained above. Example III.A.3.4: Forecasting volatility with GARCH Suppose that we have estimated a GARCH model for a stock index such that the mean return is zero and the conditional variance equation is as follows: 2 t

5.0 10

6

0.07

2 t 1

0.88

2 t 1

In this case the steady-state variance (using (III.A.3.3)) is equal to 0.0001, equivalent to daily volatility of 1% and annual volatility of 1%

250 = 15.81% p.a. Suppose that we are estimating

VaR at a time when the market has recently been unusually quiet. The current estimate of unconditional annual volatility (using, say, one year of daily data) stands at 13.0% and todays estimate of conditional daily volatility is 0.007746, equivalent to annual volatility of 0.7746% 250 = 12.25%. Assume that today new and unexpected economic data hit the market, causing a large return of +0.04 or 4%. To put this in perspective, a return of 4% in one day is a greater than 5 standard deviation event (using unconditional volatility). To forecast variance tomorrow we proceed as follows, substituting the appropriate values for todays shock (0.04) and todays variance (0.00006): 2 t 1

5.0 10

6

0.07 0.04 2

0.88 0.00006

0.0001698

Notice the large size of forecast variance (0.0001698 vs. 0.00006) as variance reacts to todays market shock. When forecasting variance on the subsequent day (and thereafter) we do not know the return shock so variance is forecast as: 2 t 2

5.0 10

6

(0.07 0.88) 0.0001698 0.0001663

and similarly for the days afterwards. Table III.A.3.4 shows the daily variance forecasts for each day in the 10-day horizon.



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While tomorrows forecast volatility is significantly higher than todays, thereafter the forecast volatility falls slightly. The GARCH model tells us that in the absence of any further shock, volatility will gradually revert towards the steady-state volatility. We obtain a forecast of volatility over the next 10 days by summing the 10 daily variances, multiplying by 250/10 and taking the square root. This gives 19.7% pa. In contrast, the unconditional volatility forecast for the next 10 days is approximately 13% todays large return having only a marginal impact due to its small weighting of 1/250. Table III.A.3.4 Forecasting Volatility with GARCH Day

Variance T+1 T+2 T+3 T+4 T+5 T+6 T+7 T+8 T+9 T+10 10-day horizon i.e. T+1 to T+10

0.0001698 0.0001663 0.0001630 0.0001598 0.0001569 0.0001540 0.0001513 0.0001487 0.0001463 0.0001440

Equivalent volatility % pa 20.6% 20.4% 20.2% 20.0% 19.8% 19.6% 19.4% 19.3% 19.1% 19.0%

Sum = 0.0015601

19.7%

The significance of this for the VaR estimate is obvious; using the GARCH conditional volatility forecast will significantly boost the VaR, signalling to risk managers that risk should either be substantially reduced or capital increased to withstand the new high-risk environment. This example also illustrates the point that the square root of time rule is inappropriate in a world of volatility clustering. The GARCH model tells us that in this situation, volatility is likely to increase initially and then gradually decline over the 10-day horizon.

To assume constant

volatility for 10 days is not suitable. For the professional risk manager calculating VaR, the task of implementing GARCH techniques might appear daunting. How might she actually go about it? There are a number of possibilities: Analytical VaR. As explained in Section III.A.3.4.1, but this time using a covariance matrix based on GARCH variancecovariance forecasts over the risk horizon. This is a convenient solution for linear portfolios. Historical simulation using volatility weighted data. As explained in Section III.A.3.4.1, but this time standardising returns using GARCH volatility estimates. This could be done quite simply, using a univariate GARCH model for each asset/factor. This method makes no assumption about the distribution of standardised returns apart from independence.



132


Monte Carlo simulation. A GARCH covariance matrix forecast could be used to simulate returns going forward. This would allow for changes in volatility, as well as the underlying, over the risk horizon an important advantage for portfolios containing options. Professional risk managers are generally analysing investment or trading portfolios containing multiple assets. To use some of the GARCH methods listed above it is necessary to evaluate an entire covariance matrix. This can present substantial implementation problems. The difficulty compounds as the number of assets or risk factors grows, particularly as we must ensure that the covariance matrix is positive semi-definite. Various approaches have been suggested in the academic literature, and these are surveyed by Bauwens et al. (2003). One very simple alternative is to use GARCH volatilities but assume a constant correlation between assets or risk factors, as proposed by Bollerslev (1990). Variance terms in the matrix are estimated using the simple GARCH(1,1) specification as shown in (III.A.3.2). The covariance terms in the matrix are formulated as follows: ij ,t 1

where

ij

ij

i ,t 1

j ,t 1

is the sample correlation of mean-corrected returns. Ease of estimation is achieved by

constraining the correlation to be constant over the estimation period and by ignoring crossmarket effects in volatility. The benefits of the GARCH approach to VaR estimation have recently been illustrated by Berkowitz and OBrien (2002).

They have shown that a simple reduced-form GARCH

implementation produces regulatory capital estimates that are an improvement on the methods used by some large commercial banks. The reduced-form GARCH approach applies a univariate GARCH model directly to portfolio P&L data, thus avoiding the need for a large covariance matrix. As in the standard historical simulation approach described in Chapter III.A.2, an artificial history of daily P&Ls is created using the current portfolio weights. But instead of taking a lower percentile of the empirical distribution of these P&Ls for the VaR estimate, they apply a GARCH model to the portfolio returns series and use formula (III.A.3.4). They find that the VaR estimates based on this type of GARCH model are more sensitive to changes in volatility; they allow for more risk-taking (or less capital) when volatility is low and less risk-taking (or more capital) when volatility is high. These excellent results are achieved without adversely affecting the number of exceptions in backtests (see Section III.A.2.8 for a description of VaR model backtests); in fact the size of the maximum exception is reduced. In summary, GARCH techniques have the potential to greatly enhance our modelling of market risk and to ensure that appropriate capital buffers are in place. Copyright ? 2004 The Authors and the Professional Risk Managers International Association


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III.A.3.5

Alternative Solutions to Non-normality

Here we consider some approaches to estimating VaR in the face of non-normality that differ from those considered in Section III.A.3.4 in that they do not take account of volatility clustering. We limit ourselves to discussing only three possibilities, although many others exist. III.A.3.5.1

VaR with the Students distribution

The Students t distribution is often proposed as a possible candidate for describing financial returns because of its heavy tails. It is actually a poor candidate because it assumes returns are i.i.d. Under the Students t distribution each days return is assumed independent of the previous days return; we know this is unlikely to be the case because of the heat wave effect. The implication of this is that the t distribution will tend to underestimate VaR in periods of market crisis the time when risk measurement is most crucial and overestimate VaR when market conditions are quiet. Backtests will reveal clusters of exceptions where actual losses exceed VaR. Nevertheless the Students t distribution remains quite popular with some professional risk managers. Statistical background is provided in Section II.E.4.6. The standard Students t has only one parameter, , the degrees of freedom. The distribution was originally designed for working with small samples where the degrees of freedom are one less than the sample size. As approaches infinity, the distribution converges to the normal. Under the standard Students t distribution: the mean is equal to zero, the variance is equal to

2

,

the skewness is equal to zero and the (raw) kurtosis is equal to

3

2 . 4

In VaR applications we will be working with large data sets and attempting artificially to select the parameter

to fit the shape of the tails of the distribution (that is, to match the sample

kurtosis). 74 Since the observed variance will not be equal to /

2 it will be necessary to scale

the variance. 75 It will generally not be necessary to scale the mean as mean returns (at daily or

74 It is also possible to estimate the degrees of freedom parameter using maximum likelihood techniques. 75 This is analogous to the way in which we adapt the standard normal distribution. The standard normal has variance

and standard deviation of one. When we use data having an observed standard deviation different from one, we divide the observed standard deviation by unity.



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higher frequencies) are close to zero. Dowd (2002) explains how to adapt the standard Students t distribution for VaR calculations (for a single asset) as follows: (a)

Select the degrees of freedom parameter by matching it to the sample kurtosis such that: ( Raw )Kurtosis

(b)

3

2 . 4

(III.A.3.5)

The empirical variance should be scaled by: 2

(c)

.

(III.A.3.6)

Select the appropriate critical point from the t distribution, based on the desired level of probability (e.g. 0.01) and the degrees of freedom selected in (a).

(d)

Proceed with VaR calculation using, for instance, the analytical method but the MC VaR model could also be applied.

Example III.A.3.5: VaR with Students t For purposes of comparison we use the USD/JPY data described in Section III.A.3.2. The returns have empirical excess kurtosis of 6.241, equivalent to raw kurtosis of 9.241. Using equation (III.A.3.3) and solving for gives a value of 4.961. The empirical variance is scaled using equation (III.A.3.4) to give adjusted daily variance of: 0.00718 2

4.961 2 4.961

3.07693 10 5 ,

being equivalent to daily volatility of 0.005547. The critical value of the t distribution can be found in Excel using the TINV function. 76 At the 99% level of confidence and with 5 degrees of freedom, the critical value of the t distribution is 3.36 (the comparable critical value of the normal distribution is 2.33). We use this information to calculate VaR using the analytical method for a position held long $1m. We choose a 10-day holding period and ignore expected returns. We compare VaR for the Students t with VaR from the more familiar normal distribution. Note that the square root of time rule is appropriate here as returns are assumed i.i.d.: Students t VaR = 3.36

10 0.005547 $1,000,000 $58, 938

76 Note that the Excel TINV function assumes a two-tailed probability distribution, whereas VaR calculations

generally apply a one-tailed probability distribution. When using TINV, take care to double the probability parameter to get the appropriate value for a one-tailed distribution. If you are interested in probability of 0.01 (i.e. one-tailed 99% confidence), then you should instead use a probability of 0.02. Note also that the Excel TINV function assumes that is an integer. In this example we use =TINV(0.02,5). If we instead used =TINV(0.02,4.961) Excel would round 4.961 down to 4, returning a value of 3.75.



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Normal VaR = 2.33

10 0.00718 $1,000,000 $52,903

The Students t VaR estimate is 11% higher than that calculated using the normal distribution. This example illustrates how the heavier-tailed Students t distribution will tend to give larger VaR estimates than the normal. Applying the Students t approach will tend to result in greater capital requirements over time (or reductions in the amount of risk taken). Potentially these larger VaR estimates afford financial institutions greater protection from extreme variations as they will respond either by reducing risk or increasing capital. Given the existence of volatility clustering, however, even an 11% increase in VaR will not be sufficient protection in times of market crisis.

Returning to Figure III.A.3.4, we see that in 1998,

conditional volatility more than tripled, reaching a peak of 35% p.a.! III.A.3.5.2

VaR with Extreme Value Theory

Some VaR applications focus on the tail behaviour of financial returns distributions. For instance, economic capital is often estimated at an extremely high percentile, such as 99.97% for a AA company (see Section III.0.2.3). For another example, when conducting stress tests the possible behaviour of portfolio returns is pushed to an extreme limit (see Chapter III.A.4). Several large banks have been developing risk capital models based on extreme value theory (EVT). In these models, it is not VaR but an associated risk metric called conditional VaR also called expected shortfall, expected tail loss or tail VaR by various authors that is estimated. Whereas VaR is the cut-off point, above which the largest losses occur, the associated conditional VaR is the average of these largest losses. For instance, the 1% VaR based on 1000 P&Ls is the 10th largest loss; the 1% conditional VaR is the average of these 10 largest losses. Though not admissible under the Basel rules for the computation of regulatory capital, conditional VaR measures are commonly favoured for the computation of economic capital. That is because conditional VaR is sub-additive, i.e. the sum of the component conditional VaRs can never be less than the total conditional VaR (see Section III.A.3.6). Of course, any risk metric that is not sub-additive is not a good risk metric. The incentive for holding portfolios just dissolves if it is better to assess risk on individual positions and simply add up the total! When VaR is modelled using normal distributions for the risk factor returns VaR is always sub-additive. However, when the VaR model is extended to allow for heavy-tailed risk factor distributions then sub-additivity, in theory, need not hold. However, in practice market VaR is almost always found to be sub-additive; by contrast, there can be real problems when VaR is applied to credit risk, due to lack of sub-additivity.



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Extreme value distributions, as their name suggests, examine the distribution of the extreme values of a random variable, which is typically assumed to be i.i.d. These extreme returns (or exceptional losses) are extracted from the data and an extreme value distribution may be fitted to these values. There are two approaches. Either one models the maximal and minimal values in a sample using the generalised extreme value distribution (GEV) or one models the excesses over a predefined threshold using the generalised Pareto distribution (GPD). For instance, suppose the underlying time series consists of hourly observations on a portfolio P&L. On each day, we record the maximal loss and subsequently fit these daily data using a GEV distribution. Alternatively, we forget about the time spacing of the data, but record only those P&Ls that exceed a certain loss threshold, regardless of when this happens. Then we would fit the data using the GPD. EVT models are often fitted using some form of maximum likelihood technique so the data requirements can be substantial. Very large data sets are crucial for EVT models as we are concerned only with the tail of the distribution and we require many extreme data points for robust estimation of parameters. The assumption that the observations are i.i.d. over such long sample periods is not very realistic in financial markets except, perhaps, when they have already been standardised using a volatility clustering model, as explained above. Nevertheless many banks do apply GEV distributions for intra-day VaR estimates. The GPD may be used to estimate conditional VaR; in fact there is a simple formula for this once the density has been fitted. However, the conditional VaR is quite sensitive to the choice of threshold loss, which must first be defined. For instance, the threshold could be set at the VaR that is estimated using a standard VaR model and then the GPD density can be fitted to obtain a more precise estimate of the conditional VaR. III.A.3.5.3

VaR with Normal Mixtures

A normal mixture density function is a sum of normal density functions. For example, a mixture of only two normal densities f1(x) =

x;

1

,

2 1

and f2(x) =

x;

2

,

2 2

is the density

function: 77 g(x) = pf1(x) + (1 p)f2(x). The parameter p can be thought of as the probability that observation x is governed by density f1(x). In effect there are two regimes for x, one where x has mean

77That is, g ( x )

p 2

2 1

1/ 2

exp

1 2

x

2 1

/

2 1

1

p

2

2 2

1/ 2

exp

1

and variance

1 2

x

2 2

/

2 1

2 2

and another

Note that there

is only one random variable so it would be misleading to call the densities f1 and f2 independent.



137


where x has mean

2

and variance

2 2

In the case where x denotes the return on a portfolio, one

can naturally identify these two regimes as a high volatility (or even, for an equity portfolio, a crash market) regime with a low probability and the rest of the time a regime that governs ordinary, everyday market circumstances. Consider a mixture of two zero-mean normal components, i.e.

1

=

2

= 0. In this case the

variance is just NM(2) variance = p

2 1

+ (1 p)

2 1

(III.A.3.7)

The skewness is zero and the kurtosis is given by: NM(2) kurtosis = 3

p p

4 1 2 1

(1

p)

(1 p )

4 2 2 2

2

.

(III.A.3.8)

The kurtosis is always greater than 3, so the mixture of two zero-mean normal densities always has a higher peak and heavier tails than the normal density of the same variance. For instance, Figure III.A.3.5 shows four densities: three zero-mean normal densities with volatility 5%, 10% (shown in grey) and 7.906% (shown in red); and a normal mixture density, shown in black, which is a mixture of the first two normal densities with probability weight of 0.5 on each of the grey normal densities. From formula (III.A.3.7), the variance of this mixture is 0.5

52 + 0.5

102 = 62.5. Since 7.906

= 62.5, the mixture has the same volatility as the red normal curve. However, it has kurtosis of 4.87 (substitute p = 0.5,

1

= 5 and

1

= 10 into formula (III.A.3.7)). In other words, it has an

excess kurtosis of 1.87, which is significantly greater than zero, the excess kurtosis of the equivalent normal (red) curve. More generally, taking several components of different means and variances in the mixture can lead to almost any shape for the density. Maclachlan and Peel (2000) provide pictures of many interesting examples. The parameters of a normal mixture density function can be estimated using historical data. In this case the best approach is to employ the expectationmaximisation (EM) algorithm. The idea, as always, is to choose the parameters to maximise the likelihood of the data. But the EM algorithm differs from standard MLE in that the EM algorithm allows for some hidden variables in the data that we cannot observe, so that we can only maximise the



138


expected value of the likelihood function and not the likelihood function itself. 78 Alternatively, the parameters of a simple (e.g. two-component) normal mixture can be chosen in a scenario analysis of portfolio risk as, for instance, in Example III.A.3.7 below. Figure III.A.3.5: A normal mixture density

0.08

0.06

0.04

0.02

0 f1(x)

f2(x)

Mixture

Normal

There is no explicit formula for estimating VaR under the assumption that portfolio returns (or, equivalently, P&L) follow a normal mixture density. However, there is an implicit formula, so the problem is akin to that of implying volatility from the market price of an option (see Section II.G.1). That is, we can apply the Excel Goal Seek (see Section II.G.1.4) or Solver (see Section II.G.2.4) methods to back out the normal mixture VaR. To see how, suppose for the moment that we have a normal distribution for the P&L of a linear portfolio. Then the analytic formula for VaR follows directly from the definition of VaR. That is, by definition, Prob(P&L < VaR ) = . So if P&L has a normal distribution with mean

and standard deviation , we have

Prob (Z < [VaR ]/ ) = , where Z is a standard normal variable. Hence [VaR ]/ = Z , the

critical value of Z, and

rearranging this gives our analytic formula for normal VaR: VaR = Z

.

(III.A.3.9)

78 The EM algorithm is far beyond the scope of the PRM syllabus, but interested readers should consult the book by

Maclachlan and Peel (2000) which deals almost exclusively with this approach.



139


Using exactly the same type of argument we can derive the normal mixture VaR, but this time it is given by an implicit formula. For instance, with only two zero-mean components in the mixture (so there are only three parameters p,

1

and

2,

the standard deviations now being those

of h-day P&L in each regime) we know everything except VaR in the identity: p Prob(Z < VaR / 1) + (1 p)Prob(Z < VaR / 2) = .

(III.A.3.10)

Hence the h-day VaR can be backed out from (III.A.3.10) using an iterative approximation method such as Goal Seek or Solver. Example III.A.3.6: Scenario VaR using normal mixtures A risk manager assumes there is a small chance, say 1 in 100, that the market will crash, in which case the expected portfolio return over a 10-day period is 50% with an (annualised) volatility around this mean return of 100%. However, at the moment in ordinary market circumstances there are steady positive returns of 10% per annum with a volatility of 20%. His portfolio is currently valued at $2m. What is his 10-day VaR? To answer this we extend (III.A.3.10) to the non-zero-mean case, and rephrase it in terms of the means and standard deviations of returns in the two regimes, rather than P&L . This gives: p Prob (Z < [VaR

1]/P 1)

+ (1 p)Prob(Z < [VaR

where p is the probability of regime 1 (i.e. 1/100 = 0.01), 0.5),

1

1

2

P 2) = ,

is the 10-day return in regime 1 (i.e.

= 10-day standard deviation in regime 1 (i.e. 1/ 25 = 0.2),

regime 2 (i.e. 0.1/25 = 0.004),

2

(III.A.3.11)

2

is the 10-day return in

is the 10-day standard deviation in regime 2 (i.e. 0.2/ 25 = 0.04)

and P is the current portfolio value ($2m). Using the Excel spreadsheet NMVaR.xls with Solver (or Goal Seek) 79 applied to cell B21 each time we change the significance level, we obtain the NM VaR figures in the first row of the Table III.A.3.5. Table III.A.3.5: NM VaR vs. normal VaR Significance level

10%

5%

1%

NM VaR ($)

157,480

500,003

1,012,632

Equivalent normal VaR ($)

281,845

335,437

435,965

Ordinary normal VaR ($)

94,524

123,588

178,107

79 To apply Goal Seek, click Tools Goal Seek Set cell B21 to value 0 by changing cell B18.



140


The two normal VaR figures are calculated using equation (III.A.3.9), or equivalently VaR = [Z where

and

]P,

are the returns standard deviation and mean over the holding period. The

ordinary normal VaR figures are computed using the second (more likely) distribution of 10-day mean and standard deviation of: = 0.004 (i.e. a 10% annual return), = 0.04 (i.e. a 20% annual volatility). That is, we ignore the possibility of a market crash in the ordinary normal VaR. For the equivalent normal VaR we use (III.A.3.7) to obtain an equivalent standard deviation and similarly the equivalent mean is p

1

+ (1 p)

2

These adjust the ordinary market circumstances mean and standard deviation to take account of the possibility of a crash, but after that the VaR is computed using the normal assumption for portfolio returns. From the results in Table III.A.3.5 it is clear that ignoring the possibility of a crash can seriously underestimate the VaR. Even if one were always to assume a normal distribution, the VaR will be almost three times larger when the standard deviation and mean are adjusted to account for the possibility of a crash. It is the relationship between the NM VaR and the equivalent normal VaR that is really interesting. Our example exhibits some typical features of NM VaR: For low significance levels (e.g. 10% or 20%), the normal assumption can seriously overestimate VaR (in this example, it was about twice the size of the NM VaR). For higher significance levels (e.g. 5% or 1%), the normal assumption can seriously underestimate VaR (in this example, it was about half the size of the NM VaR). The significance level at which the NM VaR becomes greater than the normal VaR based on an equivalent volatility/mean depends on the degree of excess kurtosis in the data. When the excess kurtosis is relatively small it may be that the 5% (or even 1%) VaR is actually smaller under the NM assumption. In fact, when the parameters are estimated using actual historical observations on the portfolio returns it is common to find that the normal mixture VaR is less than the normal VaR at the 5%, and even at the 1% level. Finally, it should be noted that, although we have not described the generalisation of the MC VaR model to the normal mixture case, this is a simple (one- or two-line) extension to the simulation code. We outline the method for the two-component zero-mean case:



141


Define two risk factor covariance matrices V1 and V2 and associated probability weights (p, 1 p) in the mixture (either estimated from historical data using the EM algorithm, or assumed in a scenario analysis). Break each of the 10,000 or so simulations into two steps: (a) draw from a Bernoulli variable with probability p; (b) if the result is success use V1 to generate the correlated risk factors in the simulation, else use V2. In summary, the normality assumption is not necessary for the analytic and Monte Carlo VaR methodologies. This section has shown how the analytic and MC VaR methods can be used with a normal mixture distribution for the portfolio returns. Such a distribution is better able to capture the skewness and excess kurtosis that we commonly observe in portfolios of most types of financial assets. Hence, if the parameters of the normal mixture distribution are estimated from historical data, the resulting VaR estimate will reflect the actual properties of the data more accurately than the normal VaR. Another very useful application of normal mixture VaR is to probabilistic scenario analysis, where the portfolio returns are generated by a high-volatility (or crash) component with a low probability and, the rest of the time, by another component that applies to the ordinary market circumstances.

III.A.3.6

Decomposition of VaR

As explained in Chapter III.0, VaR models form the basis of internal economic capital allocation and limit setting. Hence, firms need to aggregate VaR over different risk types and over different business activities and, likewise, to disaggregate VaR into different components. Disaggregation of risk is used in risk management for setting limits, assessing new investments, hedging and performance measurement. It allows risk managers to understand the drivers of risk in their portfolio. A number of different rules for disaggregating risk are considered in this section. Their common theme is that each rule is based on the analytic VaR formula; that is, the rule is derived using the rules for the variance operator (see Section II.E.3.4). However, it is common practice, for instance in the allocation of economic capital, to apply these rules to all VaR estimates, irrespective of the model used to compute them. But not all these rules for VaR decomposition are appropriate for historical or MC VaR estimates. Here the VaR corresponds to a percentile, but percentiles do not satisfy simple rules, like the variance operator. Hence the usefulness of each rule varies, depending on the intended application.



142


III.A.3.6.1

Stand-alone Capital

Suppose a line manager operating on a VaR-based risk limit wants to assign separate VaR limits to the equity and foreign exchange desks so that aggregate losses only exceed the aggregate VaR limit an appropriately small proportion of the time. But since risk limits do not correspond to real capital, it is not necessary for the two limits to add up to his overall VaR limit. On the contrary, in theory it could even be that the VaR limit for the equity desk, say, exceeds the overall risk limit! We shall see why, and how, in this section. In Example III.A.3.3 we considered a simple portfolio that has been mapped to two risk factors, an equity index and an exchange rate. The total 1% 10-day VaR due to both risk factors was estimated as $319,643. However, the VaR due to equity risk alone was: Equity VaR

= 2.33

10-day standard deviation FTSE

= 2.33

0.15

= 2.33

0.2 ?

$3m

10/ 250 ? 3 = $209,700.

Similarly, FX VaR

10/ 250

2 = $186,400.

Hence, Equity VaR + FX VaR = $396,100 > $319,643 = Total VaR. In the normal analytic VaR model VaR follows the same rules as the standard deviation operator. 80 In this case it is easy to show that VaR is sub-additive in the sense that: Total VaR

Sum of component VaRs

with equality if, and only if, all the correlations in the covariance matrix V are one. We shall state the complete rule for the decomposition into two components in (II.A.3.12) below. It shows that the total VaR is only equal to the sum of the component VaRs if there is perfect correlation between the components. But in the above example the risk factors had a correlation of 0.3, which is much less than one. So the total VaR was much less than the sum of the two component VaRs. In fact, had the correlation been large and negative, the total VaR might actually have been less than either of the component VaRs. The fact that VaR aggregates take account of correlations, and that these are typically far less than one, is one of the reasons why banks favour VaR over traditional risk measures, such as duration for a bond portfolio, or the Greeks for an options portfolio, or beta for an equity portfolio. These traditional risk measures ignore the benefits of diversification that apply in a

80 But only when the risk factor returns are assumed to be normal: as mentioned already in Section III.A.3.5.2, if risk factor returns are heavy-tailed then VaR need not be sub-additive.



143


portfolio exposed to multiple risk factors. In contrast, VaR accounts not only for the risks due to the risk factors themselves, but also for the less than perfect correlation of risk factors when aggregating the risk. Also, VaR is a risk measure with consistent dimensions across markets and therefore allows greater consistency in setting policy across products and in evaluating the relationship of risk and return. III.A.3.6.1.1

Decomposing non-linear portfolios

Decomposition of risk becomes more complex for portfolios with non-linearities. VaR measures for such portfolios are much more likely to violate the criterion of sub-additivity. For this reason VaR is criticised as being a poor risk metric. It is possible to construct extreme cases where individual portfolios containing, say, short, well out-of-the-money digital options have a very low or even zero VaR at, say, the 95th percentile. However, if two similar portfolios are combined together, the diversified portfolio has higher VaR than each of the components.

In this

anomalous situation, stand-alone capital is not appropriate for limit setting. 81 Indeed, a strong argument could be made here to avoid VaR and to use conditional VaR instead, for limit-setting purposes. Of course typically, non-linear portfolios will be analysed using the simulation approaches. Disaggregation of VaR (or conditional VaR) can be undertaken in the context of simulation by restricting the risk factor scenarios in different ways. For instance, total VaR can be disaggregated into an equity VaR component, corresponding to a lower percentile of a simulated portfolio returns distribution with only the equity risk factors changing, and an FX VaR component, corresponding to a lower percentile of a simulated portfolio returns distribution with only the foreign exchange rates changing. Since percentiles do not obey simple rules (except that a percentile is invariant under a monotonic transformation of variables) there is, in this case, no simple rule that relates the sum of equity VaR and FX VaR to the total VaR. III.A.3.6.1.2

Specific vs. Systematic Risk

Another way of disaggregating risk is to decompose VaR into its systematic and specific components. That is, those risks that apply to the market/factor generally and those that arise from lack of diversification or deviations from the market portfolio. A VaR model can be used to assess the specific risks of a portfolio i.e. the risks that are not captured by the risk factor mapping.

81 Each separate portfolio could be within limit yet the business overall could be in breach of limits when considered

on a diversified basis.



144


Example III.A.3.7: Specific VaR To obtain the beta of 1.5 for our portfolio of UK equities in Example III.A.3.3 we re-created an artificial price history of the portfolio using the current portfolio weights, and regressed the time series of returns to this portfolio on the FTSE index returns. This gave the Excel output shown in Table III.A.3.6. Table III.A.3.6: A CAPM regression Regression Statistics

Coefficients

Standard Error

t Stat

R Square

0.7284

Intercept

-0.00016

0.00045

-0.34991

Standard Error

0.00802

FTSE

1.50312

0.02876

52.26426

The standard error is the standard deviation of the model residuals (see Section II.F.6). Since the returns are observed daily, the 10-day standard deviation of the residuals in this model is 0.00802 Hence, the 1% 10-day specific VaR is 2.33

10 = 0.02536 0.02536

$2m = $118,184.

Of course, having re-created an artificial price history of the portfolio using the current portfolio weights, we did not need to estimate a factor model in order to calculate the total VaR. We could have simply obtained the daily standard deviation of the re-created portfolio returns and used formula (III.A.3.4). In this case, we would have a daily standard deviation 0.01636, giving a direct estimate of 1% 10-day total VaR as: Total VaR = 2.33

0.01636

10 ? $2m = $241,117.

However, the 1% 10-day systematic VaR (in this case, the equity risk factor VaR) is $209,700. Adding this to the specific VaR of $118,184 we thus obtain: Systematic VaR + Specific VaR = $327,884. Clearly, simply adding systematic VaR and specific VaR is a very conservative way to estimate the total VaR, that is, direct estimation of total VaR will normally give a result that is considerably lower than the sum of systematic and specific VaR. III.A.3.6.1.3

Sub-additivity

To understand why this is so, we do some simple algebra showing that the analytic VaR (for linear positions) obeys the following sub-additive rule for any decomposition of total VaR into two components, VaR1 and VaR2 where the component risks have correlation : Total VaR2 = VaR 12 + VaR 22 + 2 VaR1 VaR2. Copyright ? 2004 The Authors and the Professional Risk Managers International Association


(III.A.3.12) 145


Note that expression (III.A.3.12) simplifies when the correlation is one or zero: If

Total VaR2 = VaR 12 + VaR 22 + 2 VaR1 VaR2 = (VaR1 + VaR2)2 and

= 1:

Total VaR = VaR1 + VaR2. If

Total VaR2 = VaR 12 + VaR 22 and

= 0:

Total VaR = ( VaR 12 + VaR 22 ).

(III.A.3.13)

Now recall that if the factor model is capturing most of the variation in the portfolio then specific risks and systematic risks should be uncorrelated (see Section II.F.2). In that case, the Total VaR will be closer to the square root of the sum of the squared component VaRs than to the simple sum of the two VaRs. In other words, simply adding specific VaR and systematic VaR is not a good way to estimate Total VaR if the systematic and specific components are (more or less) uncorrelated. When disaggregating VaR into different components, it is sometimes assumed that all component correlations lie between zero and one. In this case the two ways of calculating total VaR (as a straightforward sum of VaRs, or as the square root of the sum of the squared VaRs) provide approximate upper and lower bounds for the total VaR, respectively. III.A.3.6.2

Incremental VaR

Incremental VaR (IVaR) is a measure of how portfolio risk changes if the portfolio itself is changed in some way. It is ideal for assessing the effect of a hedge or a new investment decision on a traders VaR limit, or for assessing how total business risk would be affected by the sale/purchase of a business unit. There are two ways of proceeding: (a) The before and after approach. Here we measure the VaR under the proposed change, compare it to the current VaR and take the difference. This is the best approach if the proposed change to the portfolio is significant. (b) The approximation approach. We can find an approximate IVaR by first calculating the DelVaR vector (see below). This vector is then multiplied by another vector containing the proposed changes in positions for each asset/risk factor.

Like all approximations based on partial

derivatives, it is suitable only for examining small changes to the portfolio composition. Under the analytical method 82 the DelVar vector is calculated as follows: 82 With simulation methods a DelVaR vector could also be constructed, but with considerably greater difficulty. Each

element of the vector would be determined by assessing the change in total VaR (according to simulation) for a oneunit change in the portfolio holdings of the relevant asset.



146


VpZ

DelVaR

p'Vp

0.5

,

(III.A.3.14)

where V is the covariance matrix, p is the position vector and Z is the relevant critical value of the normal distribution. Note that the denominator of this expression is simply the standard deviation of the portfolios P&L. The DelVaR vector will contain an element for each asset/risk factor in the covariance matrix (the first element will contain information relating to the risk of the first asset/risk factor, and so forth). Example III.A.3.8 Approximate IVaR We continue with the example first presented in Example III.A.3.3. The portfolio consists of an exposure to the FTSE index and exposure to the exchange rate since the investor is US$ based. Note that the standard deviation of the portfolios P&L over a one-day horizon is equal to $0.043382m (being 0.001882). Hence, for the base position where the portfolio is unchanged, equation (III.A.3.14) becomes: 0.00009 0.000036 0.000036 0.00016

Vp

DelVaR

3 2

0.000342 , 0.000428

0.000342 2.33 0.043382

0.018368

0.000428 2.33 0.043382

0.022987

.

Now suppose that we consider hedging half of the currency exposure so that the exposure to the exchange rate is reduced to only $1m, rather than the current $2m. We can construct a vector of portfolio changes where the first element (the change in the exposure to FTSE) is equal to zero, and the second element (the change in the exposure to currency) is 1. The incremental VaR can be approximated as follows: IVaR

0

1

0.018368 0.022987

0.022987.

In other words, the hedging decision will reduce 1% one-day VaR by approximately $22,987.

As the proposed change to the portfolio is quite large in this case, the approximation method is unlikely to be accurate and the before and after method would be preferable. With the before and after method, we know from Example III.A.3.3 that the one-day VaR of the original position is 2.33

0.001882 m$ = $101,080.



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With the currency hedge, the 1% one-day VaR is $80,241 because p Vp = 3 1

90

36

3

36 160 1

10

6

306 268

3 1

10 6 = 0.001186,

0.001186 m$ = $0.08241m. Hence, according to the exact method, the IVaR is

and 2.33

101,080 80,241 = $20,839, which is less than the IVAR given by the approximation. In other cases the actual IVaR will exceed the approximation, depending on the level of correlation between assets. III.A.3.6.3

Marginal Capital

Sometimes referred to as component VaR (CVaR), this method of decomposition is useful for gaining a better understanding the drivers of risk within a portfolio. Sometimes it is also used for performance measurement. Unlike the previous methods, marginal capital is additive; the sum of each of the component VaRs will be equal to the total VaR. 83 We take the DelVaR vector discussed above and multiply each element by the corresponding position for each asset/risk factor. The result is a set of CVaRs for each asset. The sum of these CVaRs will be equal to portfolio VaR. Since the analysis is performed using the DelVaR vector of partial derivatives with respect to asset weights, it is only relevant for the current portfolio weightings. Significant changes to the portfolio will change the risk contributions of the various assets, necessitating new analysis based on the revised DelVaR vector. Example III.A.3.9: Marginal capital We continue to analyse the portfolio first introduced in Example III.A.3.3. The DelVaR vector, calculated in Example III.A.3.8 is used here: DelVaR

0.018368 . 0.022987

The CVaR for equities is equal to the position ($3m) multiplied by 0.018368, or $55,104. The CVaR for foreign exchange is equal to the position ($2m) multiplied by 0.022987, or $45,974. Note that they sum to $101,078, which is approximately equal to $101,080, the 1% one-day VaR (the error is due to rounding in the DelVaR vector). CVaR is used to assign a proportion of the total risk that can be attributed to each component. For instance, in this example the equity exposure is contributing 55,105/101,078 or around 55% of 83 This relationship is assured for the analytical method and if the portfolio is comprised of simple linear positions. It

may not hold for other cases.



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the risk in the portfolio, while currency contributes around 45%. This technique can be used to evaluate the risk of an asset (or asset class) in the context of a diversified portfolio. In contrast, the stand-alone capital (in Section III.A.3.6.1) measures the risk of an asset class in isolation. Either can be used for performance measurement purposes, although stand-alone capital is most commonly used as the risk measure in a risk-adjusted performance measure context. For instance, we would use stand-alone capital to compare the performance of the equity and foreign exchange trading desks because it is generally argued that the diversification benefit should not enter into the analysis. That is, the team managing foreign exchange risks presumably has no say in the way the portfolio of businesses is constructed (whether or not there is an equity desk, and the relative sizes of those businesses). They should therefore be neither rewarded nor penalised for the diversification of the overall portfolio of businesses. Instead, performance measures should be centred only on the issues over which they have direct control, being foreign exchange risks in this example.

III.A.3.7

Principal Component Analysis

Principal component analysis (PCA) is a statistical tool that decomposes a positive semi-definite matrix into its principal components. 84 For instance, Section II.D.5.5 shows how PCA on an n

n

covariance matrix is used to write the portfolio variance as a sum of n positive terms that become progressively smaller, with the first principal component explaining the largest part of the variation in the system represented by the covariance matrix and so on. The nth principal component explains the least variation indeed, the variation captured by the lower-order principal components is commonly ignored, because it just picks up the noisy variation that we would prefer to ignore. PCA applied to a covariance matrix or a correlation matrix has many applications to financial risk management. It is particularly effective in highly correlated systems such as term structures, i.e. yield curves, or futures prices or even implied volatilities. Here only few components are needed to explain almost all of the variation. In this respect PCA is a useful technique for reducing dimensions. By retaining only the first few principal components enough, say, to explain 95% of the variation observed historically it cuts out the noise for the subsequent analysis.

84 Positive definiteness is discussed in detail in Section II.D.5.4.



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The other great advantage of PCA is that the principal components are uncorrelated with each other. This means, for instance, that one can perform a simple scenario analysis on each of the main principal components separately, leaving the other components fixed. The change translates into a meaningful scenario, i.e. one that could be observed historically. For instance, when applying PCA to a covariance matrix of zero-coupon yields of different maturities, 85 a scenario for the change in the first principal component generally will mimic an almost parallel shift in the entire yield curve. Without PCA one would need to take care that only correlated scenarios are used for instance, if the scenario specifies that the one-month interest rate increases by 100 bps, one could not have the three-month interest rate decreasing by 200 bps in the same yield curve scenario. Correlated scenarios are generated using the Cholesky decomposition of a covariance matrix see Section II.D.4.2. This problem is compounded if simulations are extended more than a short time into the future using this method, as it is difficult to prevent generating implausible shapes in the resulting yield curves. III.A.3.7.1

PCA in Action

From 4 January 1993 until 20 November 2003 we have daily closing New York Mercantile Exchange (NYMEX) futures prices, from 2 to 12 months out, on West Texas Intermediate (WTI) light sweet crude oil and on natural gas. Some of these are shown in Figure III.A.3.6. The system of futures returns is clearly very highly correlated indeed there are perhaps just one or two independent sources of information driving the whole system of futures prices. The correlation matrix of the returns to futures from 2 to 12 months out, based on the entire sample from 4 January 1993 until 20 November 2003, is shown in Table III.A.3.7. This exhibits the pattern that is typical of term structures, with correlation decreasing as the maturity difference increases. The correlations are so high that almost all the variation can be attributed to two or perhaps three components. An analysis of eigenvalues tells us how many components we need. The eigenvalues 1, 2, ,

n

of an n

n correlation matrix sum to n, and the amount of variation

captured by the ith principal component is i/n. So in our example, with n = 11, if the largest eigenvalue is, say, 10, then the first principal component explains 10/11 = 90.9% of the variation.

85 More precisely, to the covariance matrix of the daily (or weekly or monthly) changes in yields.



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Figure III.A.3.6: A highly collinear system Crude Oil Futures Prices 40 35 30

2mth

4mth

6mth 10mth

8mth 12mth

25 20 15 10

Table III.A.3.7: Correlation matrix of returns m2

m3

m4

m5

m6

m7

m8

m9

m10

m11

m2 m3 m4 m5 m6 m7 m8 m9 m10 m11

1 0.993 0.984 0.974 0.963 0.951 0.939 0.927 0.915 0.901

1 0.997 0.990 0.981 0.972 0.962 0.951 0.940 0.928

1 0.998 0.993 0.986 0.978 0.969 0.960 0.949

1 0.998 0.994 0.988 0.981 0.974 0.965

1 0.998 0.995 0.990 0.984 0.977

1 0.999 0.996 0.991 0.986

1 0.999 0.996 0.992

1 0.999 0.996

1 0.999

1

m12

0.888

0.916

0.938

0.956

0.969

0.979

0.987

0.992

0.996

0.999

m12

1

Table III.A.3.8 shows the eigenvalues and the first three eigenvectors of this matrix (see Section II.D.5 for an explanation of these). The first three eigenvalues show that the first principal component explains 97.5% of the total variation, the second component explains a further 2.2% (since 0.245/11 = 0.022) and the third component only explains a tiny amount, 0.145% of the movements in the futures prices in our sample.



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Table III.A.3.8: Eigenvectors and eigenvalues Eigenvalues

Future

1st Eigenvector

2nd Eigenvector

3rd Eigenvector

10.732 0.245 0.016 0.004 0.002 0.001 0.000 0.000 0.000 0.000

m2 m3 m4 m5 m6 m7 m8 m9 m10 m11

0.293 0.299 0.302 0.304 0.305 0.305 0.304 0.303 0.302 0.300

0.537 0.412 0.280 0.161 0.054 -0.044 -0.132 -0.211 -0.282 -0.350

0.623 0.078 -0.225 -0.348 -0.353 -0.276 -0.162 -0.028 0.115 0.247

0.000

m12

0.298

-0.411

0.364

The first three eigenvectors in Table III.A.3.8 are used to compute the first three principal components as: PC1 = 0.293 m2 + 0.299 m3 + .. + 0.298 m12,

(III.A.3.12a)

PC2 = 0.537 m2 + 0.412 m3 + .. 0.411 m12,

(III.A.3.12b)

PC3 = 0.623 m2 + 0.078 m3 + .. + 0.364 m12,

(III.A.3.12c)

where m2, , m12 denote the (normalised) returns to the futures of different maturities from 2 to 12 months out. Since the eigenvectors are always orthogonal (see Section II.D.5.2), equations (III.A.3.12) can be rewritten as: m2 = 0.293 PC1 + 0.537 PC2 + 0.623 PC3, m3 = 0.299 PC1 + 0.412 PC2 + 0.078 PC3, .. .. m12 = 0.298 PC1 0.411 PC2 + 0.364 PC3. This is known as the principal component representation of the system. Since the coefficients on PC1 are all approximately the same, when PC1 moves (holding the other components fixed) the term structure of futures prices will shift in (almost) parallel fashion. For this reason PC1 is often called the trend component when PCA is applied to term structures. When PC2 increases the term structure of futures prices will shift up at the short end but down at the long end so PC2 is called the tilt component. And when PC3 increases the term structure shifts up at both ends but looking again at Table III.A.3.8 the medium term futures prices will go down. Hence PC3 is called the curvature component.



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In this example the curvature component is not important it corresponds to much less than 1% of the movements normally found in futures prices. But in other examples interest rates, for instance the curvature component can be more important. And when PCA is applied to other types of systems, such as equities or currencies, we normally need more than three components to model the system with sufficient accuracy. Of course, there will be no intuitive interpretation of the components as there is for term structures because we cannot order a system of equities or currencies in any sensible way although the first principal component will normally capture the common trend (if the system is sufficiently correlated that there is a common trend!). III.A.3.7.2

VaR with PCA

PCA is commonly used in the analytic VaR method for cash flows, and the MC simulation VaR method for interest-rate options portfolios. Perhaps the most important input in both these approaches is the covariance matrix of risk factor returns in this case, the covariance matrix of changes in a zero-coupon yield curve. Typically this yield curve will have more than 10 different maturities, so the dimension of the risk factor space is large. But, as we discussed earlier, largedimensional covariance matrices are very difficult to estimate using GARCH models. And EWMA matrices normally assume the same value for the smoothing constant for all the risk factors. This may seem fine if the risk factors are just one yield curve, but for international fixed income portfolios the risk factors consist of many yield curves. In order to take proper account of (less than) perfect correlation between these risk factors when estimating the total VaR, we need to use a covariance matrix of the whole system i.e. all yields of all maturities in all countries. Then in the EWMA matrix it is very unrealistic to apply the same value for the smoothing constant for all the risk factors. And direct estimation of the matrix using a multivariate GARCH model will be out of the question because the likelihood surface will not be well defined. The solution is to apply PCA to the entire system of risk factors, retaining enough components in the principal component representation to explain most (say, 95%) of the variation. Then, because they are uncorrelated, it is only the variance of each component that matters their covariances are zero. 86 So one can treat each component separately, estimating and forecasting only its variance using either GARCH or EWMA. Note that with this method we do not have to use the same smoothing constant in each EWMA; and even if we do use the same smoothing constant, the final risk factor covariance matrix will not have the same effective smoothing constant for all risk factors, as happens in the RiskMetrics methodology. Nor do we need to 86 Note that it is only the unconditional covariances that are zero, because PCA is based on an unconditional covariance matrix, which does not have time-varying covariances. Hence the application of a time-varying (i.e. EWMA or GARCH) model to variances, whilst assuming correlations are constant, does entail a strong assumption.



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constrain the GARCH models in any particular way, as in most multivariate GARCH models. The large risk factor covariance matrix that we obtain in the end will always be positive semidefinite. Full details are given in Alexander (2001).

III.A.3.8

Summary

The crucial question for any VaR model is whether the VaR estimate provides a good indication of the true risk in financial markets. Any deviation between the VaR estimate and this truth represents a model risk. Such a risk is potentially disastrous since it could cause a financial institution to have insufficient capital given the risks taken, making it more vulnerable than it should be to insolvency. Concern about this kind of model risk is well justified because financial markets are prone to periods of high volatility in which extreme market movements can occur. Many analysts have noted the heavy tails of empirical distributions relative to the standard i.i.d. normal assumptions. These are a symptom of the clusters of volatility that occur periodically. Simple VaR models as explained in the previous chapter are all flawed to some extent; either their assumptions are too simple and/or they suffer from lack of data. This chapter has explored various methods for improving on VaR models so that they are more consistent with the behaviour observed empirically in financial markets. We have argued that the most crucial issue for analysts to address in this regard is volatility clustering. It is now well established that volatility in financial markets exhibits elevated values over certain periods before reverting to lower levels. The variation in volatility explains most of the extreme moves observed in financial markets. We focused on models that incorporate the volatility clustering idea (EWMA and GARCH) and considered their practical application to VaR estimation. We also examined some alternative approaches to the issue of extreme returns that do not incorporate volatility clustering (Students t, EVT and normal mixtures). Of course, it should always be remembered that model risk can never entirely be eliminated. The advanced models presented here have their own problems, and these should be well understood by the user. Every model is a simplification of the reality it represents, and should be used with caution. The pervasiveness of model risk is a key reason why stress tests (covered in the next chapter) are useful. This chapter also completed the discussion of VaR for market risk by examining two other, more advanced topics: risk decomposition and principal component analysis. We have described the ways in which total VaR can be disaggregated into different components, for capital allocation on a stand-alone and on a marginal basis, and how traders can use an incremental VaR analysis



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to evaluate the effect of a proposed trade on their VaR limit. Finally, we have explained how PCA can greatly simplify VaR estimation when there are multiple risk factors that are highly correlated. For instance, the VaR for any portfolio of bonds or loans that are mapped to riskequivalent cash flows (so that the risk factors are a term structure of interest rates) should be calculated using PCA. Similarly, commodity portfolios with exposures to futures prices at different maturities are best represented using VaR with PCA than by a direct VaR analysis. Not only does this simplify the VaR estimation itself, but the use of PCA greatly facilitates the application of stress tests and scenario analysis to these types of portfolios, as we shall see in the next chapter.

References Alexander, C (2001) Market Models: A Guide to Financial Data Analysis. Chichester: Wiley. Bauwens, L, Laurent, S, and Rombouts, J (2003) Multivariate GARCH models: a survey. To appear in Journal of Applied Econometrics. Available at http://www.core.ucl.ac.be/econometrics/Bauwens/Papers/papers.htm Berkowitz, J, and OBrien, J (2002) How accurate are value-at-risk models at commercial banks? The Journal of Finance, LVII, pp. 10931111. Bollerslev, T (1990) Modelling the coherence in short-run nominal exchange rates: A multivariate Generalised ARCH model, Review of Economics and Statistics, 72, pp. 498505. Dowd, K (2002) Measuring Market Risk. Chichester: Wiley. Maclachlan, G, and Peel, D (2000) Finite Mixture Models. New York: Wiley.



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III.A.4 Stress Testing Barry Schachter 87

III.A.4.1

Introduction

The previous chapters have introduced value-at-risk (VaR) as a risk measure and discussed methods to deal with some of its shortcomings. It is common belief that VaR does not provide a complete picture of portfolio risk and that stress testing is a means of addressing that (at least in part). While most of this chapter is devoted to questions related to the construction of stress tests, it is important to step back and formulate some ideas about why we need to have a chapter on stress testing. By now the need for stress testing of portfolios of financial instruments is taken for granted. We find stress testing on every list of risk management best practices. Stress testing has long since been solemnised by regulators. If this chapter were to be simply a review of stress-testing techniques, then I could immediately proceed by providing a typology and discussion of the various techniques that have been presented in the literature on the topic. I think, however, that an appreciation of the usefulness of stress testing requires that I first step back a bit and ask a couple of basic questions. The first question is by what historical process we have arrived at this general acceptance of the need for stress testing. The second question is by what conceptual framework we have concluded that stress testing is needed. My definition of stress testing is as follows: Stress testing (str s' t st' ng) n. 1. A method for the quantification of potential future extreme, adverse outcomes in a portfolio of financial instruments. 2. A palliative for the anxiety that is experienced by managers with significant risk exposures. The key words in this definition are quantification, potential future, extreme and palliative. I am taking a very broad view of stress testing here to include just about any method for attaching a monetary figure (i.e., a quantity) to potential losses from extreme events that is not specifically VaR. It is important to note that quantitative estimates are not necessarily statistical estimates. In fact, most stress-testing methods, unlike VaR, are not statistical measures of risk. That is, no probabilities are attached to and no confidence intervals estimated for the adverse outcomes. It is a great challenge in stress testing, whatever the method employed, from historical scenario analysis

87 Chief Risk Officer, Balyasny Asset Management, LLC. Thanks go to Steve Allen and Carl Batlin for reviewing and

commenting on a draft of this chapter. Any remaining errors or omissions are my own. Parts of this chapter draw on Schachter (1998a, 1998b, 2000a, 2000b).



157


(reliving a past market crisis) to factor push (shifting a market rate to see the impact on a portfolio), to effectively gauge potential future losses. A palliative is something that reduces pain, but does not cure an underlying problem. To a large extent, this is the role that stress tests have served to date, which is somewhat unsatisfying. The risk management profession is still seeking consensus on theoretically and practically consistent ways to integrate stress testing into decision making. Uses vary widely from a simple informational tool to a formal element of limit setting and capital allocation. The last point above deserves emphasis. Even if we agree what stress testing is, we still need to agree (or at least understand) its purpose. Stress testing must fit into some wider context, or have some point. We should be able to establish how stress testing provides some incremental value as either a means to achieving some goal (e.g., the optimal allocation of assets in a portfolio) in other words, it is a direct input in some decision function or a way of measuring the movement towards some goal in other words, it is a benchmark or feedback mechanism for modifying or improving a decision. If we cannot show either of these things, then we cannot know how to make rational economic decisions which use stress-test results, and we cannot articulate why we should be performing stress tests. It is common belief that stress testing has incremental value because VaR is a sufficient statistic for risk under only a restricted set of conditions. For instance, in a world where all financial asset returns are jointly normally distributed and portfolios are passive, the magnitude of a loss of any probability is a scalar multiple of the VaR. The loss on a portfolio corresponding to a one standard deviation move in the portfolio value is equal to the 99% VaR divided by 2.33. To know the VaR is to know everything about risk, and stress tests are not needed. However, if asset returns are nonlinear functions of underlying market risks, or market risks are not distributed according to some distribution that is stable under addition, then to know VaR is to have only an incomplete picture of portfolio risk. Stating it more colloquially, while VaR models provide a notion of a bad portfolio return, they do not convey just how bad bad can get. Even if we agree that stress testing has incremental value, then the context for extracting that value in decision making still needs to be shown. This chapter, focusing mainly on methods for stress testing, does not settle this issue.

III.A.4.2

Historical Context

In the classic 1967 film, The Graduate, Benjamin Braddock (Dustin Hoffman) returns home after graduation from university, and receives the following sage (and frightening) career advice from a family friend:



158


MR MCGUIRE: I just want to say one word to you. Just one word. BEN: Yes sir. MR MCGUIRE: Are you listening? BEN: Yes I am. MR MCGUIRE: Plastics. BEN: Exactly how do you mean? MR MCGUIRE: Theres a great future in plastics. Think about it. Will you think about it? Perhaps Mr McGuire should have allowed for a second word, too. Derivatives. Or perhaps he should have substituted derivatives and omitted plastics entirely. By the end of 1967, work was already well advanced in the development of a preference-free option pricing formula, based on the work of Sprenkle, Boness, Samuelson, and Thorp.

Merton, Black and Scholes were

eventually to become household names (well, almost) after their revolutionary results were published in 1973. It is widely acknowledged that financial markets were forever changed by wide adoption of derivatives as a fundamental tool of risk allocation, as recognition of the importance of arbitrage-free option pricing spread; see MacKenzies (2003) history. Alas, every boon is also a bane. In the crash of 1987, portfolio insurance strategies, based on option replication arguments, were given much of the blame. Following the various studies of the crash, regulatory authorities prescribed additional regulation over the function of equity markets, in the form of circuit breakers, to mitigate future risk. Against this backdrop the Basel Committee on Banking Supervision, in 1988, ushered in a new era of international cooperation in international financial regulation.

And the Committees attention immediately turned to

derivatives, motivated by a concern for the soundness and stability of the international financial system. Also in 1988, the Chicago Mercantile Exchange adopted a system for setting daily margin requirements, the Standard Portfolio Analysis System (SPAN? , based on a type of stress test discussed below, scenario analysis. SPAN has since been adopted by many derivatives exchanges. In the 1990s a series of corporate financial collapses occurred which were associated (in one way or another) with derivatives usage. Some of the names are familiar, such as Orange County (see Jorion, 1995), Procter and Gamble, Gibson Greetings (see Overdahl and Schachter, 1995) and Barings.

Others are less familiar, such as Metallgesellschaft (see Culp and Miller, 1999),

Sumitomo (Gilbert, 1996), Daiwa Bank, and SK Securities (see Gay et al., 1999). As these events unfolded, regulatory authorities sought ways to enhance internal risk controls. In the series of reports and rule makings that resulted, recommendations for best practice frequently



159


included a reference to the importance of the role of stress testing for identifying otherwise hidden risks.

It is these recommendations that have pushed stress testing to a relatively

prominent position among risk management tools. Recommendation 6 of the G-30 report (Global Derivatives Study Group, 1993) states: Dealers should regularly perform simulations to determine how their portfolios would perform under stress conditions. Simulations of improbable market environments are important in risk analysis because many assumptions that are valid for normal markets may no longer hold true in abnormal markets. These simulations should reflect both historical events and future possibilities. Stress scenarios should include not only abnormally large market swings but also periods of prolonged inactivity. The tests should consider the effect of price changes on the mid-market value of the portfolio, as well as changes in the assumptions about the adjustments to mid-market (such as the impact that decreased liquidity would have on close-out costs). Dealers should evaluate the results of stress tests and develop contingency plans accordingly. The US Comptroller of the Currency in Banking Circular 277 (Comptroller of the Currency, 1993) states: National banks ... systems also should facilitate stress testing and enable management to assess the potential impact of various changes in market factors on earnings and capital. The bank should evaluate risk exposures under various scenarios that represent a broad range of potential market movements and corresponding price behaviors and that consider historical and recent market trends. The Basel Committee on Banking Supervision (1994) states: Analysing stress situations, including combinations of market events that could affect the banking organisation, is also an important aspect of risk measurement. Sound risk measurement practices include identifying possible events or changes in market behaviour that could have unfavourable effects on the institution and assessing the ability of the institution to withstand them. These analyses should consider not only the likelihood of adverse events, reflecting their probability, but also worst-case scenarios. Ideally, such worst-case analysis should be conducted on an institution-wide basis by taking into account the effect of unusual changes in prices or volatilities, market illiquidity or the default of a large counterparty across both the derivatives and cash trading portfolios and the loan and funding portfolios. The Derivatives Policy Group (1995) included stress testing among the necessary risk measurement tools of derivatives dealers. Specifically, they state, Mechanisms should be in place



160


to measure market risk consistent with established risk measurement guidelines. These procedures should ... provide the information necessary to conduct stress testing. Concurrent with the implementation of the use of VaR models for the calculation of regulatory market risk capital as permitted by the EU Capital Adequacy Directive, banks were required to conduct a routine and rigorous programme of stress testing (Securities and Futures Authority, 1995). The Basel Committee on Banking Supervision (1996) eventually made stress testing a prerequisite for banks to be eligible for the internal models approach to market risk capital. The requirement had previously been laid out by the Committee in an earlier release (1995). More specifically, the Basel Committee on Banking Supervision (1995) states: Banks that use the internal models approach for meeting market risk capital requirements must have in place a rigorous and comprehensive stress testing program. Stress testing to identify events or influences that could greatly impact banks is a key component of a banks assessment of its capital position. Banks stress scenarios need to cover a range of factors that can create extraordinary losses or gains in trading portfolios, or make the control of risk in those portfolios very difficult. These factors include low-probability events in all major types of risks, including the various components of market, credit, and operational risks. Stress scenarios need to shed light on the impact of such events on positions that display both linear and non-linear price characteristics (i.e. options and instruments that have options-like characteristics). Banks stress tests should be both of a quantitative and qualitative nature, incorporating both market risk and liquidity aspects of market disturbances. Quantitative criteria should identify plausible stress scenarios to which banks could be exposed. Qualitative criteria should emphasize that two major goals of stress testing are to evaluate the capacity of the banks capital to absorb potential large losses and to identify steps the bank can take to reduce its risk and conserve capital. This assessment is integral to setting and evaluating the banks management strategy and the results of stress testing should be routinely communicated to senior management and, periodically, to the banks board of directors. Banks should combine the use of supervisory stress scenarios with stress tests developed by banks themselves to reflect their specific risk characteristics. Specifically, supervisory authorities may ask banks to provide information on stress testing in three broad areas, which are discussed in turn below. (a) Supervisory scenarios requiring no simulations by the bank Banks should have information on the largest losses experienced during the reporting period available for supervisory review. This loss information could be compared to the level of capital that results from a banks internal measurement system. For example, it could provide



161


supervisory authorities with a picture of how many days of peak day losses would have been covered by a given value-at-risk estimate. (b) Scenarios requiring a simulation by the bank Banks should subject their portfolios to a series of simulated stress scenarios and provide supervisory authorities with the results. These scenarios could include testing the current portfolio against past periods of significant disturbance, for example, the 1987 equity crash, the ERM crises of 1992 and 1993 or the fall in bond markets in the first quarter of 1994, incorporating both the large price movements and the sharp reduction in liquidity associated with these events. A second type of scenario would evaluate the sensitivity of the banks market risk exposure to changes in the assumptions about volatilities and correlations. Applying this test would require an evaluation of the historical range of variation for volatilities and correlations and evaluation of the banks current positions against the extreme values of the historical range. Due consideration should be given to the sharp variation that at times has occurred in a matter of days in periods of significant market disturbance. The 1987 equity crash, the suspension of the ERM, or the fall in bond markets in the first quarter of 1994, for example, all involved correlations within risk factors approaching the extreme values of 1 or 1 for several days at the height of the disturbance. (c) Scenarios developed by the bank itself to capture the specific characteristics of its portfolio In addition to the scenarios prescribed by supervisory authorities under (a) and (b) above, a bank should also develop its own stress tests which it identifies as most adverse based on the characteristics of its portfolio (e.g. problems in a key region of the world combined with a sharp move in oil prices). Banks should provide supervisory authorities with a description of the methodology used to identify and carry out the scenarios, as well as with a description of the results derived from these scenarios. The results should be reviewed periodically by senior management and should be reflected in the policies and limits set by management and the board of directors. Moreover, if the testing reveals particular vulnerability to a given set of circumstances, the national authorities would expect the bank to take prompt steps to manage those risks appropriately (e.g. by hedging against that outcome or reducing the size of its exposures). Another round of calls for stress testing, including recommendations for stress tests of counterparty credit exposures, followed the LTCM and liquidity crises of 1998. See for example, the report of the Presidents Working Group on Financial Markets (1999). The Basel Committee provided further impetus to the application of stress testing to credit in 2002 as part of the new Basel Capital Accord (Basel II), stating (Basel Committee on Banking Supervision (2002): Banks adopting an [internal ratings based] approach to credit risk will be required to perform a meaningfully conservative credit risk stress test of their own design with the aim of estimating the



162


extent to which their IRB capital requirements could increase during such a stress scenario. Banks and supervisors will use the results of such stress tests as a means of ensuring that banks hold a sufficient capital buffer under Pillar Two of the new Accord. Credit stress testing is discussed in the Credit Risk section of the PRM Handbook.

III.A.4.3

Conceptual Context

As noted in the Introduction, the apparent presumption in all this emphasis on stress testing is that using stress tests will lead to better decisions with respect to risk taking, either as an integral component of the decision makers objective function, or as a tool measuring the distance from some goal. What is not at all clear, however, is what are the mechanisms at work here. 88 To put this in the form of a question, if we develop a system of stress testing, what are we then actually supposed to do with the stress-test results anyway? Berkowitz (1999) takes up this question. 89 It is presumed that stress tests are needed because there is something lacking in the VaR derived from the firms risk model. By risk model Berkowitz means the computational process by which VaR is ultimately derived. Stress tests, if they address this lack, could be thought of as a way of enhancing the assessment of portfolio risk, specifically by providing a way of correcting VaR. To achieve this correction, stress tests should be incorporated into the risk model. He argues that there is no reason to support the modelling of stress tests as a thing separate and apart from the normal process used to evaluate risk, for then there is no internally consistent way to put stress results to use. (Of course, one could still view and use stress-test results in any arbitrary way.) It is easiest to see how this approach to employing stress tests would work by thinking about the historical simulation approach to VaR estimation (see Chapter III.A.2). In (the basic application of) historical simulation, each day of history is assigned an equal probability in the forecast portfolio return distribution. A set of stress scenarios may then be added to the history, each scenario weighing equally with each day of history. The resulting corrected simulation is then used in evaluating the usual desired percentile for the calculation of VaR, now the corrected VaR. The risk model employed by Algorithmics, the enterprise-wide risk management software company, follows a similar approach. This is a powerful and attractive argument for placing stress tests on a sound footing for use in risk management. The key prerequisite for accepting this approach is a willingness to assign to

88 See Shaw (1997) for a critique of stress tests. 89 See also Zangari (1997a, 1997b) and Cherubini and Della Longa (1999), who apply the BlackLitterman/Bayesian

approach to incorporate stress scenarios into the VaR framework.



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each stress-test scenario a subjective probability (it is not necessary to assume equal probabilities). While putting stress testing on a solid foundation, this approach is not a panacea for the risk manager. Even were this approach to be adopted, the risk manager would still be left to deal with the anxiety and doubt over including inappropriate scenarios and excluding overlooked scenarios.

Interestingly, too, the Basel Committees risk-based capital rules may create a

disincentive for banks to adopt this integrated approach in favour of a fuzzier application of stress testing. Integrating stress tests with the basic risk model will result in an increase in measured VaR, which will result in an increase in regulatory capital. To the extent that capital requirements are a binding constraint on the bank (or nearly so), the bank will incur some real incremental economic costs using this approach. Is that it, then? Is that the only application of stress testing that is sensible? Well, no, even if it is the most theoretically comforting (i.e., internally consistent) application. As a risk yardstick, stress tests also provide information to decision makers about risk taking in relation to risk appetite. For this reason some institutions set stress-test limits at the enterprise level. Others use stress-test results to measure capital usage or assess capital charges.

Less formally, many

institutions rely on stress-tests results as a means for the decision maker to perform an intuitive check on his or her comfort with the set of risks in the portfolio. In this sense, stress tests can provide a reality check or a way of assessing model risk (e.g., the sensitivity of valuation to parameter estimates or inputs) where complex models are used in risk assessment. This use of stress testing is long-established practice in the credit world. Regulators also view stress tests as a way to assess their own comfort level with the risks being run individually by institutions for which they are responsible, and more recently also to check their comfort level with the systemic risks implied by the collective positions of the same institutions.

III.A.4.4

Stress Testing in Practice

Most of the information available on current practice was obtained from a survey of 43 large financial institutions conducted by a task force of G-10 central banks established by the Committee on the Global Financial System (2001). The results were also summarised by Fender et al. (2001) and Fender and Gibson (2001a, 2001b). The survey asked risk managers to list the most important stress scenarios used firm-wide. The task force identified nine stress-scenario themes: four themes related to asset class (e.g., stress tests on commodity indices); four themes related to geographic region (e.g., stress tests on exposures to emerging markets).



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The most common scenarios were the 1987 stock market crash, widening of credit spreads, and various versions of a hypothetical stock market crash. Among interest-rate themed scenarios, the bond market crash of 1994 was the most common. In the emerging markets theme, an Asian crisis was the most common. US dollar weakness/strength scenarios were the most common among the remaining geographic themed scenarios. Commodity themed scenarios focused most often on a potential Middle East crisis.

Stress tests, other than those centred on foreign

exchange, tended to reflect the predominance of long exposures in the respondents portfolios (e.g., institutions conducted more spread widening than narrowing scenarios). Banks are not terribly keen to disclose much about their stress-testing approach. Below are produced extracts from the 2003 annual reports of the three largest US banks (by assets). Citigroups report states: Stress testing is performed on trading portfolios on a regular basis ... on individual trading portfolios, as well as on aggregations of portfolios and businesses, as appropriate. It is the responsibility of independent market risk management, in conjunction with the businesses, to develop stress scenarios, review the output of periodic stress testing exercises, and utilize the information to make judgments as to the ongoing appropriateness of exposure levels and limits. Bank of America states: [S]tress scenarios are run regularly against the trading portfolio to verify that, even under extreme market moves, we will preserve our capital; to determine the effects of significant historical events; and to determine the effects of specific, extreme hypothetical, but plausible events. The results of the stress scenarios are calculated daily and reported to senior management .

JP Morgan Chase, which typically provides the most comprehensive disclosures, states: Stress testing ... is used for monitoring limits, cross business risk measurement and economic capital allocation. ... The Firm conducts economic value stress tests for both its trading and its nontrading activities, using the same scenarios for both. The Firm stress tests its portfolios at least once a month using multiple scenarios. Several macroeconomic event-related scenarios are evaluated across the Firm, with shocks to roughly 10,000 market prices specified for each scenario. Additional scenarios focus on the risks predominant in individual business segments and include scenarios that focus on the potential for adverse moves in complex portfolios. Scenarios are continually reviewed and updated to reflect changes in the Firms risk profile and economic events. Stress-test results, trends and explanations are provided each month to the



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Firms senior management and to the lines of business, to help them better measure and manage risks and to understand event risk-sensitive positions. The Firms stress-test methodology assumes that, during an actual stress event, no management action would be taken to change the risk profile of portfolios. This assumption captures the decreased liquidity that often occurs with abnormal markets and results, in the Firms view, in a conservative stress-test result. [S]tresstest losses are calculated at varying dates each month. The following table represents the worstcase potential economic value stress-test loss (pre-tax) in the Firms trading portfolio as predicted by stress-test scenarios: Trading Economic-Value Stress-Test Loss Results Pre-Tax as of or for the year ended December 31 Loss in USD $m 2003

2002(A)

AVE.

MIN.

MAX.

DEC. 4

AVE.

MIN.

MAX.

DEC. 5

508

255

888

436

405

103

715

219

(A) Amounts have been revised to reflect the reclassification of certain mortgage banking positions from the trading portfolio to the non-trading portfolio.

The potential stress-test loss as of December 4, 2003, is the result of the Equity Market Collapse stress scenario, which is broadly modeled on the events of October 1987. Under this scenario, global equity markets suffer a sharp reversal after a long sustained rally; equity prices decline globally; volatilities for equities, interest rates and credit products increase dramatically for short maturities and less so for longer maturities; sovereign bond yields decline moderately; and swap spreads and credit spreads widen moderately.

III.A.4.5

Approaches to Stress Testing: An Overview

The definition and practice of stress testing encompass several different techniques (see Table III.A.4.1 for a high level overview). The choice of which test to employ in a particular institution and situation is driven by several factors. Regulatory requirements are important, of course. Also important is the cost, in time and resources, needed to generate stress-test results. The choice also reflects the specific needs of the users. These needs depend on the complexity of the portfolio, the frequency with which it is traded, the liquidity of the instruments in the portfolio, the volatility of the markets in which the instruments are traded, and the strategies employed. 90

90 The discussion that follows pertains most directly to stress testing for traded instruments. Stress testing either the

banking book or a non-financial firms exposures presents additional challenges and needs. Approaches specific to



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Table III.A.4.1: Typology of stress tests Approach Historical scenarios

Description Replay crisis event

Pros It actually happened that way

Cons Proxy shocks may be numerous No probabilistic interpretation No guarantee of worst case

Hypothetical scenarios

Algorithmic

1. Covariance matrix

1. Relatively easy

1. Empirical support mixed

2. Create event

2. Very flexible

2. No guarantee of worst case

3. Sensitivity analysis

3. Can be detailed

3. Limited risk information

1. Factor push

1. Minimal qualitative elements

1. No guarantee of worst case

2. Maximum loss

2. Identifies worst case in feasible set (maybe)

1. Ignores correlations 2. Assumes data from normal periods are relevant 2. Computationally intensive

Most of the regulatory attention has focused on stress testing at the portfolio level. For the regulators it is the aggregated impact of stressed market environments that poses risks that interest them. For some time international organisations have pursued the idea of aggregating the results of standardised stress scenarios for individual financial firms into financial sector stress tests to attempt to estimate the economy-wide impact of crisis events (e.g., the Committee on the Global Financial System of the Bank for International Settlements, and the Financial Sector Assessment Program of the IMF and World Bank). Stress testing of a sort is widely practised at the individual trading desk, trader and position levels, too. To the extent that trading desks already perform their own position-by-position stress testing, we have to ask why we cannot merely aggregate those results for the purpose of examining exposure at the portfolio level.

Desk-level stress tests are (as they should be) highly focused on the specific risks being

run at the desk. Perhaps much of the information generated is not relevant at the aggregate portfolio level. Equally important, risks that are deemed to be of second order at the trading desk, and hence are subject to little or no stress testing, may be important (at the margin at least) when viewed in the context of risks being taken at other desks. Also, desk-level tests are used for evaluating and actively managing risk position by position (or at least strategy by strategy), possibly in near real time.

Portfolio stress testing serves a more strategic purpose in the

those needs have been developed (e.g., net interest income stress tests and economic value added stress tests), but they are outside the scope of this discussion.



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identification and control of event-related risk, and the most useful metrics for the one purpose need not also serve best for the other. Desk-level stress tests tend to be more of the factor-push variety, whereas portfolio stress tests tend to employ the scenario approach. Irrespective of whether the stress-testing programme is intended for desk level or portfolio level risk management, useful stress tests require full revaluation for all positions with nonlinear or discontinuous payouts. 91 Perhaps it is foolish to state the obvious. However, full revaluation can entail significant computational and time costs, and it is tempting to trade off accuracy for speed. But, as the raison dêtre of stress testing is to explore portfolio losses in crises where nonlinearities and discontinuities may expose hidden risks, approximate revaluation approaches should be strictly limited. It may be necessary to develop alternative revaluation strategies to overcome technology constraints that limit the employment of full revaluation for other risk management purposes. The only exception to this requirement is for some desk-level stress testing. Here stress testing of sensitivity-based risk exposure information (deltas, vegas, etc.) is useful for tactical portfolio management decisions as well as being perhaps the only practical way of delivering stress results on demand or in real time.

III.A.4.6

Historical Scenarios

Historical scenarios, as a variety of stress testing, seek to quantify potential losses based on reenacting a particular historical market event of significance. Scenario shocks that determine the impact on portfolio valuation are taken from observed historical events in the financial markets. This is in contrast to stress tests where shocks are based, for example, on specified changes in the covariance matrix of asset returns, or on shifting prices or rates by an arbitrary number of standard deviations.

We are all empiricists when we say, it is reasonable, because it actually

happened. Historical scenario stress testing is required by the Basel Committee, prescribing that scenarios could 92 include testing the current portfolio against past periods of significant disturbance, for example, the 1987 equity crash, the ERM crises of 1992 and 1993 or the fall in bond markets in the first quarter of 1994, incorporating both the large price movements and the sharp reduction in liquidity associated with these events (Basel Committee on Banking Supervision, 1996). It sounds so easy. Designate a period in history with a suitable crisis environment and find out what would happen to the current portfolio if that crisis were replicated today. However, as is the case with so many things, it really is necessary to sweat the details. I will consider the

91 Discontinuities can present a challenge for stress tests of all types, except perhaps in principle maximum loss. It

may be prudent to construct stress tests with the points of discontinuity specifically in mind. 92 With regulators, to say could is to say do this unless you have some compelling reason not to.



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elements of historical stress testing in turn, namely, choice of a historical period and specifying shocks. III.A.4.6.1

Choosing Event Periods

The first question in choosing a historical period for stress testing is which periods to choose. A historical event may be defined in one of two ways. In the first, the event is defined relative to a well-known crisis period, such as the Asian crisis of 1997. In the second, the event is defined by examining the historical record of moves in market risk factors relative to some user-defined threshold level of shocks. The second approach will no doubt also turn up events that correspond to most well-known crises, but may identify other event periods as well, depending on the particular risk factors whose histories are being scanned for large movements. The former approach is more prevalent. Some potential scenarios were mentioned in previous sections.

Common candidates for

historical stress tests include the following: the US stock market crash of 1987, the European exchange-rate mechanism crisis of 1992, the US bond market sell-off of 1994, the Mexican peso crisis of 1994, the so-called Asian crisis of 1997, the Russian default of 1998, and the LTCM and liquidity crises of 1998. The attacks of 11 September 2001 also constitute a candidate. 93 Davis (2003) creates a typology of financial crises. Systematic characterisation of market crises may be useful both for ensuring that the set of events employed in a stress-testing programme contains a reasonable portion of the spectrum of possible crises, and for guiding the construction of hypothetical scenarios (discussed later). The next question is how many events should be part of the stress-testing programme. No matter how many historical periods are selected, it is not possible to guarantee that the prospective worst-case scenario is covered. It can be hoped, however, that a judicious selection of scenarios will provide indicative information about areas of vulnerability, especially in identifying risks that may not be obvious from other risk measurements, such as VaR. Ideally, the risk manager will vary the number and variety of historical scenarios evaluated through time depending on both the changing composition of the portfolio and the changing economic environment. The first thing one realises when looking at a historical event candidate for a stress scenario, say the liquidity crisis of 1998, is that the start and end dates of the event are not always obvious. For any particular market rate, the period of interest is likely to be unambiguous. However, the

93 See, for example, Malz and Mina (2001), Hickman and Jameson (2001) and Jorion (2002).



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more complex and varied the instruments in the portfolio to be stressed, the more difficult is the problem of identifying the start and end date for the stress event. Two approaches are possible. Define the event interval, making sure that the interval selected encompasses all (or essentially all) of the significant moves in individual market rates. Then use as the shock magnitude for each risk factor the greatest change in that factor (e.g., peak-to-trough) found within the interval, regardless of the start or end date. The advantage of this is that the scenario will entail the largest possible moves in each risk factor.

The

disadvantage is that the shocks, when taken together, may make no economic sense. Define the event interval such that it comes as close as is possible to capturing exactly the greatest moves in the factors of most interest. That is, the peaks (or troughs) occur at the start date and the troughs (or peaks) occur at the end date. Then use as the shock magnitude for each risk factor the change in that factor from the start date to the end date. The ideal event window cannot be achieved in practice. That is the disadvantage of this.

The advantage is that the scenario has the potential to be economically

meaningful. The second approach is preferred, as a key element in establishing the plausibility of a scenario is that the shocks, taken together, must be sensible. Historical scenarios rarely play out within a single trading day. Given international market linkages observed through contagion and feedback effects, even an event sharply focused in time is likely to engender after-shocks that continue for a few days. More commonly, an event will develop over a period of a few days or even weeks, as in the liquidity crisis of 1998. As a result the specification of historical events raises questions about how the passage of time is affecting the test results. Two areas of concern come to mind: trading or hedging out risks, and modelling the effect of time passing on expiration, maturity and carry (e.g., for fixed income or options positions). No matter how illiquid a market, there is a price at which a portfolio manager can trade out of a position. Since those costs may be prohibitive, it is common to assume in historical stress scenarios that positions cannot be traded or hedged no matter how long the interval of calendar time spanned by the scenario. It is common to argue (at least, I have argued) that this assumption, while extreme, approximates a worst-case scenario in which illiquidity is extreme. Still, it stretches the plausibility requirement to apply this assumption to a very extended stress period (as traders will readily argue). Another question that arises with historical scenarios is how to model the impact of the passage of time on instruments in the portfolio whose values depend on time. Instruments that are affected include futures and forwards, bonds and options. It is possible to argue that the scenario telescopes the historical record into a single trading



170


day, thereby making it unnecessary to deal with the passage of time, and this is the prevalent approach.

However, by assuming that time is telescoped, the effects of

illiquidity on the portfolio are muddled somewhat, and the plausibility of such assumed one-day moves in rates can be questioned. If explicitly allowing for the passage of time, the expiration of options and the cash flows from payouts, if any, need to be incorporated into the scenario. Similarly, bond coupons and repo payments should be considered. Allowance must be made, too, for rolling of forwards and futures. III.A.4.6.2

Specifying Shock Factors

A fundamental element in specifying shocks for a historical scenario is the choice of relative (or proportional) versus absolute (or additive) shocks. Consider the following example. Suppose that the GBP/USD exchange rate is currently 1.8. Also assume that during the period of historical interest the rate moved from 1.5 at the beginning of the event to 1.75 at the conclusion of the event. The absolute change in the exchange rate during the event was 0.25, and the relative change was 16.7% (appreciation of the GBP). A decision must be made whether to calculate the shocked exchange rate as 1.8 + 0.25 = 2.05 or 1.8 ? 1.167 = 2.1. This issue arises in VaR modelling as well, but because the changes in market risk factors are generally larger in stress testing, the effects can be more dramatic. If the levels of market rates are similar between the beginning of the historical event and the current stress-test as of date, then the choice is less important. However, this will not generally be the case. Relative shocks are generally preferred for a couple of reasons. Firstly, when applying a relative shock one will not inadvertently cause a rate to change sign. Secondly, a relative shock (generally) corresponds directly to the rate of return on a portfolio, which is (generally) thought to be the parameter of interest in an individuals utility function. Nevertheless, relative shocks are not always appropriate. Some market spreads can be either positive or negative. To maintain that property in a stress scenario, absolute shocks need to be applied. Applying relative shocks to interest rates can be a problem as well. For example, a relative interest-rate shock that is derived from a historical period when interest rates are very low may imply unrealistic moves in rates when they are at higher levels. As a general rule, then, most shocks should be relative. However, shocks to interest rates generally should be absolute. Shocks to volatility should generally be relative, in part because volatilities cannot be negative. Exceptions to these rules should be made on a rate-by-rate basis where it is appropriate to do so. An issue arises when the portfolios fixed-income instruments are priced from both zero curves and par curves, perhaps as a result of the way different trading desks prefer to view their risks, or when different portfolios are marked using different back-office systems.

In this case,

consistency between the historical shocks for par curves and zero curves must be imposed.



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Applying historical interest-rate shocks to current yield curves requires special attention as well. Years of research (and probably quite a few doctoral theses) have shown that typically three factors can explain about 95% of the movements in yield curves, commonly interpreted as level, slope and curvature. Ideally, it would be the shocks arising from those three key yield curve factors that are used in scenario construction, perhaps through principal components analysis (PCA) as described in Frye (1997). If historical (absolute) changes at various points on the yield curve are applied point-by-point to the current yield curve, the problem that can arise is that the shocked yield curve can take on some very implausible shapes. It is a good idea, at the very least, to monitor the look of shocked curves as part of the stress-test process. PCA is useful for generating realistic shocked yield curves in a tractable manner.

Firstly,

dimensions are reduced so that only the three key risk factors need to be shocked. Secondly, these factors are orthogonal, so they can be shocked independently and the shocked result will be a realistic curve. This technique is equally important when the scenario specifies shocks to any term structure, such as the term structure of volatility, a term structure of commodity futures of different maturities and a term structure of foreign exchange rates. If PCA is not applied it may be necessary to modify historical shocks that give rise to implausible curve shapes, perhaps by applying the Cholesky matrix (derived from the historical covariance matrix) to the independent shocks, thus making them correlated (see Section II.D.4). III.A.4.6.3

Missing Shock Factors

In many instances it is not possible to refer to the historical record for a particular instrument when specifying stress shocks. The more distant in the past is the historical scenario, the more likely this is to be a problem. In some cases instruments in the current portfolio simply were not traded during the historical period. For example, default swaps were not traded at the time of the Mexican peso crisis. In other cases, even if instruments were traded, there is no reliable source for historical data; either the data are bad or there are no data. It is necessary to have a policy for dealing with missing shock factors. A simple rule to follow is not to leave any current positions without an assigned historical shock unless a zero shock is the best guess at what that shock would have been. If positions in particular instruments are deemed to be immaterial at the time of the specification of the scenario, the situation may subsequently change, and the specification of the scenario would then need to be revisited.

If stress scenarios become part of the risk limit structure or capital

allocation or performance evaluation processes, leaving positions without shocks not only results



172


in an inferior decision-making process but also creates incentives for strategic behaviour that may not be desirable from the perspective of risk appetite. Assumptions about correlations play a big role in following the leave-no-position-unshocked rule, as the best guess is usually obtained from examining historical shocks of instruments thought to be highly correlated with the position in need of a shock assignment. There are two basic approaches to ensuring this: employing proxies or using interpolation. In the case of proxies, a shock is assigned from another instrument. Equities are a good example. With equities, mergers, spin-offs, and changes in business focus may make this existing historical record irrelevant, at least if plausibility is to be maintained. In this case it may be prudent to at least ensure that changes in a companys industry classification are noted.

Then a policy decision can be made whether to ignore the history and

instead proxy shocks for such equities to a historical industry-specific shock factor. When assigning proxies, it is useful to employ more rather than fewer proxies (equity sector proxies, rather than just a single market proxy), in order to obtain a betterarticulated result. More sophisticated data filling approaches than discussed here are possible as well, of course. Interpolation (or extrapolation) may be appropriate in the case of fixed-income instruments. For example, swap and forward foreign exchange markets tend to both become more liquid and extend over time as comfort increases in assessing the longerterm risks. This filling in and out of the term structure is especially noticeable in emerging markets. In part because of the correlation between instruments of different tenors and in part because of the structure of implied forward rates, it is reasonable to interpolate rate shocks (usually linearly) from available historical shocks at adjacent points in the term structure. It should be clear that even when using historical scenarios for stress testing, scenario creation is not a once- only event. Not only should new scenarios be constantly under consideration for development, but even existing scenarios need to be constantly re-evaluated and sometimes tweaked to maintain their usefulness. This can be tedious and unexciting, so it is a good idea to establish a policy to formally review stress scenarios periodically to assist in establishing a good discipline.



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III.A.4.7

Hypothetical Scenarios

History does not conveniently present the risk manager with a template for every plausible future market crisis (though the sample size of crises does keep increasing with time). For this reason, it may be desirable to create a hypothetical economic scenario as a stress test.

Ideally, a

hypothetical scenario is based on a structural model of the global financial markets (perhaps with a real or physical goods and services component, too), in which the specification of a parsimonious set of market shocks provided as inputs to the model will result in a complete specification of responses in all markets. Well, in most cases that is not going to happen. 94 Still, it is good to keep that ideal in mind when constructing an economic scenario, because it is very easy to make a bad scenario by ignoring cause, effect and co-determination in economic relationships. III.A.4.7.1

Modifying the Covariance Matrix

Some argue that the key feature of a stress event is embedded in the behaviour of asset correlations. The intuition is strong. In a crisis, investors may make fewer distinctions among assets and issue blanket buy or sell orders in a flight to safety that tends to drive whole classes of assets in the same direction. Or interconnections between market participants may manifest in a crisis where the actions of one agent create the need for other agents to take similar actions. Or interconnections between markets may manifest when an agent, faced with a liquidity crisis in one market, attempts to liquidate positions in other markets, precipitating a liquidity crisis throughout the system. A stress test can be constructed from a modified covariance matrix in several ways. For example, if it is assumed that asset returns are jointly normally distributed, then the stress portfolio valuation can be obtained by computing the monetary value of a one standard deviation change in the portfolio value (using the modified covariance matrix) and scaling up the result by the desired number of standard deviations (a common multiplier is 4). As with any intuition, it is prudent to test it against the available data. Boyer et al. (1999) show that careless data mining can lead one to conclude incorrectly that correlations are different in stressful markets. However, they acknowledge that there is evidence that correlations do change. Taking Boyer et al.s points into account, Kim and Finger (2000) conduct an empirical study from which they conclude that that data provide considerable support for the existence of a separate stressful market environment with distinct asset correlations. In particular, they propose that observed asset returns are generated by a mixture of normal return-generating processes, one for an ordinary market environment and one for a stressful market environment.

Nevertheless,

94 However, that is exactly the approach taken by UK regulatory authorities who employed a macroeconomic model in

their experimentation with macroeconomic stress tests. Hoggarth and Whitley (2003) present a very interesting discussion of the issues involved.



174


Loretan and English (2001) argue that the empirical evidence might not support correlation breakdown, but rather be the residual of time-varying volatility. Similarly, Forbes and Rigobon (2002) argue that, taking into account the apparent relation between correlation and volatility, they are unable to find evidence of changes in correlations during the 1997 Asian crisis, the 1994 Mexican peso crisis, or the 1987 US stock market crash. Then again, Dungey and Zhumabekova (2001) and Corsetti et al. (2002) say that the results in Forbes and Rigobon (2002) may be overstated, first because the number of crisis periods in the sample is small, and second because their econometric specification is too restrictive. In sum, despite the intuition, the empirical evidence is not uniformly supportive of the notion that correlations increase in crisis situations. Still, the force of intuition is strong, and it is common to construct stress scenarios with increases in correlation.

Note, however, that

increased correlation does not in itself guarantee to stress a portfolio. It is simple to construct thought experiments in which the VaR of a portfolio will decline with increases in correlation. It is not advisable to change correlations by arbitrarily setting selected correlations to 0, 1 or 1, because implausible stressed portfolio returns can result (specifically, covariance matrices that are not positive semi-definite, meaning that some portfolios could have negative variance!). As a result, any stress scenario that involves causing correlations to differ from the relationships embedded in the historical data that were used to estimated them must follow certain rules. When changing the correlation between two risk factors, it is important to understand that the correlations can only change in reality if the underlying returns on the two factors change relative to each other (possibly in such a way that the average return and the variance of the two do not change). If those underlying returns change, then by implication every correlation between each of those two risk factors and the remaining risk factors may change, too. Thinking in terms of the correlation matrix, if we want to change the value of one correlation, which means changing the value in two cells of the matrix (from the symmetry property), then all the correlations in the rows and columns intersecting those two cells can change as well. Generally, many possible pairs of altered return vectors will yield any desired correlation (given average returns and variances). By specifying the method to (explicitly or implicitly) adjust the return vectors, it will be possible to determine the corresponding induced changes to the other affected correlations that are necessary to maintain consistency. Finger (1997) proposes modifying the return vectors of the risk factors whose correlations are to be modified such that the return on each factor on any day, t, is a linear combination of the historically observed return of that factor on day t and the average of the returns on the affected



175


market factors on day t. The transformed return vectors will need to be rescaled if the original individual variances are to be unchanged. Note that shocking individual asset variances can be incorporated into this method as well. Further, an adjustment may also be made to ensure the mean returns are unchanged, but this is not necessary if the mean returns are ignored (i.e., assumed to equal zero) for purposes of the risk calculations. Table III.A.4.2 contains statistics derived from the logarithms of the daily changes in closing prices (in New York) for IBM, GE and MSFT for one year ending on 20 April 2004. The upper triangular (red) entries are correlations; 95 the remaining entries are variances and covariances (all annualised from the daily data). Table III.A.4.2: Estimated correlations and annualised variances and covariances IBM

GE

MSFT

IBM

0.038

0.411

0.558

GE

0.016

0.041

0.474

MSFT

0.025

0.022

0.054

In this example, the correlation between IBM and MSFT will be increased to 0.850 (from 0.558) using Fingers methodology. Table III.A.4.3 illustrates the operations that are performed on the IBM and MSFT return vectors. Table III.A.4.3: Modification of returns on a representative date Date Return

Modified Return

R(IBM) 0.005117

4

R(IBM mod )

June 2003

R( MSFT)

The parameter

0.0004

MSFTmod

Normalised Return

0.005117 0.0004 (1 2 0.005117 0.0004 (1 2

) 0.005117

R(IBM mod )

) 0.0004

R( MSFTmod )

determines the modified correlation between the two return series. For any

= 1, then the two time series will be perfectly correlated. A simple numerical search

(e.g., with Solver in MS Excel) can be used to identify the value of

that results in a correlation

equal to the target level of 0.850. For the data used here, the desired correlation is obtained with = 0.4625. Table III.A.4.4 shows the final modified correlations and covariances for these three equities.

95 For example, Corr(IBM, GE) = Cov(IBM, GE)/{Var(IBM) ? Var(GE)}1/2 = 0.016/{0.038 ? 0.041}1/2 = 0.411.



MSFT MSFTmod

choice of , returns for every date are modified as illustrated for the representative date in the table. If

IBM IBM mod

176


As before, the entries above the diagonal are correlations, and the remaining entries are variances and covariances. As should be expected, given the methodology, the correlations between GE and both IBM and MSFT have been affected by changing the correlation between IBM and MSFT.

Table III.A.4.4: Modified correlations and annualised variances and covariances IBM

GE

MSFT

IBM

0.038

0.470

0.850

GE

0.018

0.041

0.498

MSFT

0.038

0.023

0.054

When several instrument pairs are chosen to have their correlations fixed at a level different from the historical correlation, this approach in general requires computing a separate

for each of

those pairs of instruments. In this case, because of the interdependencies among the risk factors, it will be necessary, in general again, to solve simultaneously for the set of s that together yield the desired set of correlations. More general techniques for modifying correlations in a consistent manner have been suggested; see, for example, Kupiec (1998), Rebonato and J鋍kel (1999), Higham (2002), or Turkay et al. (2003). These approaches focus explicitly on eliminating the negative eigenvalues of the stressed correlation matrix, and seek to identify the consistent matrix that is closest to the stressed matrix (according to some metric). III.A.4.7.2

Specifying Factor Shocks (to create an event)

Rather than creating a stress test through modification of the covariance matrix, it is possible to create a hypothetical scenario simply by specifying hypothetical shocks to the market factors. Without an economic model, it is a daunting task to attempt to describe a coherent set of hypothetical shocks encompassing every market risk factor.

Thus, even for a hypothetical

economic scenario, actual historical behaviour of market prices can provide useful guidance for the specification of plausible shocks. Another element in specifying shocks is to specify which no-arbitrage relationships are to hold in the scenario. Arbitrage is a term that is used very loosely, so much so that a variation has come into use, pure arbitrage. A pure arbitrage is an opportunity for a riskless and certain profit. Pure arbitrage is achieved through the implementation of a self-financing replicating portfolio (or exact hedging strategy). Speaking somewhat loosely, instruments that are (in principle) tied by



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this type of arbitrage relationship include simple European options and their underlying assets, forwards/futures and the corresponding cash market instrument, and relative exchange rates. The scenario designer must decide which relationships are to be fixed in the scenario. In part, this decision is helped by observing what happened in the historical record. In the case of the relationship between futures and cash equities, it has been observed that the parity relationship did not hold continuously through the 1987 market crash. Nevertheless, if desired, the futures cash relationship can be enforced in scenario construction simply by defining the futures price to be fairly valued relative to the shocked index level, forgoing the implementation of a separate historical shock for the futures. Currency traders sometimes will make large bets on the actions of governments that either have pegged or are managing the float of their exchange rate. For this reason, it is important when constructing a hypothetical economic scenario to make a conscious decision about what to do with pegged currencies. Depending on the scenario, it may be appropriate to assume that certain pegs are broken. The shocks that are assumed (including spill-over effects), then, may depend on historical cases when other currency pegs were broken, or on expectations of the future exchange rate implied in non-deliverable currency forwards. An example of a hypothetical economic scenario is the commodity themed Middle East crisis scenario common among banks, as noted in Section III.A.4.2. It may be assumed in such a scenario that the outbreak of war results in a disruption of oil production. Some spike in crude prices and volatility must then be assumed. The impact on related energy products prices is then estimated. However, the impacts do not end there, as such an event is likely to modify investor expectations of future inflationary impacts, lead to possible shortages in certain markets, and bring about pre-emptive central bank responses, etc., all of which will affect asset prices and volatilities. For all of these effects factor shocks must be assumed in a way that creates a plausible and coherent picture of the impacts on various markets. These shocks to market prices and rates are then applied to portfolio positions to evaluate potential exposure to the hypothetical event. III.A.4.7.3

Systemic Events and Stress-Testing Liquidity

Another hypothetical economic scenario that is of great interest is a systemic liquidity event. The stress tests discussed above do not probe vulnerabilities arising from the interrelationships among institutions. The voluntary withdrawal of a derivatives dealer from the market is a commonly cited event of this type; see, for example, Greenspan (2003) and Jeffery (2003). Since the LTCM and liquidity crises of 1998, regulators and some money centre banks have shown increasing interest in the both immediate and the follow-on effects of such an event. The focus



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of this interest is on the mechanisms that tie together institutions and provide channels for contagion and feedback. Borio (2000) writes that, for a proper understanding of liquidity under severe stress, the interaction of basic order imbalances with cash liquidity constraints and counterparty risk needs to be explained. Leverage and risk management play a key role. It also suggests that some factors that may contribute to liquidity in normal times can actually make it more vulnerable under stress. Consider the following features of the financial system, all of which are generally regarded as improving stability and efficiency in normal markets. Firstly, collateral requirements for over-thecounter derivatives and performance bonds on exchange-traded derivative transactions reduce the likelihood of default in normal times. However, an organisation that trades these may plan to have liquidity sufficient for normal daily mark-to-market contingencies and long-term average liquidity needs, but may still find itself unable to post the collateral as required in a significant market event. Failure to do so requires the counterparty absorb a portion of the loss. This loss may, in turn, force the counterparty to fail to make required payments on its obligations, creating a contagion of defaults. Secondly, risk management policies, such as risk limits, will moderate risk taking to tolerable levels in normal times. However, where a market event causes a sharp increase in measured risk, traders may choose (or be directed) to exit positions to avoid triggering risk limits. The resulting trades may contribute further volatility, and further knock-on effects. This likelihood is greater if large market participants measure and limit risk taking in similar ways, or risk managers at large firms just respond to market events in similar ways. Then these individual responses will be exacerbated at the level of the financial system as a whole. Thirdly, position transparency and risk disclosures, as well as advances in information technology, generally promote efficient price discovery and fair market pricing. However, they may also contribute to herd behaviour, in which large market players have similar trades, and in which market participants react similarly and at simultaneously in response to an event. This behaviour can exacerbate the initial effects of an event. Fourthly, a linchpin of efficient functioning of the over-the-counter derivatives market is the ability of dealers to control portfolio risk exposures through the construction of synthetic hedges (replicating the desired set of risk characteristics through another portfolio of instruments). Illiquid sectors of the market rely on the instruments traded in the more liquid sectors to create hedges cheaply. A reduction in liquidity in the liquid sectors can significantly affect the prices in the illiquid sectors. Thus pricing is interrelated across the range of derivatives products. If a



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large dealer were to pull back from trading, the resulting reduction in market liquidity would create losses at other institutions, as markets repriced to reflect the reduced liquidity. In some instances, less liquid instruments might become uneconomical to trade, forcing institutions to exit those positions, causing further price impacts in a cascading effect. The data needs for an institution to construct such a stress test are great. But, as with any stresstesting method, value-added information is still possible even though the reality falls short of the ideal. Firstly, an institution must first incorporate counterparty information in its risk database. Secondly, an institution should be able to identify the impact on required collateral of a proposed set of systemic shocks. This may be especially difficult for institutions with many types of instruments held with a prime broker.

Prime brokers will take a portfolio view of their

counterparty risk and the algorithm they apply for determining collateral may not be transparent. Thirdly, an institution should estimate the distribution of positions across the financial system. For example, the institution should ask whether dealer A is the only dealer making a market in certain instruments in the institutions portfolio; who are the others and what is the market share of each; and whether the market is one-sided (e.g., the dealer is long in comparison to most counterparties and is primarily relying on hedges to control overall risk) or two-way (the dealers book has a balance of long and short positions). For the instruments that the institution uses in its own hedging, it should identify the other major institutions that either make markets in those instruments or heavily employ those instruments and estimate market share. Since much of this information is not internal to the institution, some estimation will be necessary. Some potentially useful sources of data for this type of stress test are BIS statistical summaries, reports of derivatives activities of banks published by the US Comptroller of the Currency, and the Commodities Futures Trading Commission commitments of traders reports. As is the case with data, specifying shocks can be a significant challenge as well. Some historical guidance is available by reviewing the behaviour of markets and spreads in periods of illiquidity, such as October 1998. It may be possible to use the institutions own trading data to estimate market impact functions in certain instruments. An impact function measures the cost of trading as a function of position size, given other parameters that describe the trading environment (e.g., volume and volatility). The institutions own pricing models may be used to estimate the price impact on positions of a wider bidask spread. Ultimately, it will be necessary to rely heavily on intelligent guesstimates of possible shocks. Supranational organisations that have an interest in the financial stability of the global economy are embracing a similar approach to stress testing, the most prominent bit being the Financial Stability Assessment Program. The umbrella organisation for these efforts is the Financial



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Stability Forum (www.fsforum.org), with the actual work being undertaken by the International Monetary Fund (www.imf.org) and the World Bank (www.worldbank.org) in conjunction with local country supervisory authorities. See, for example, Blaschke et al. (2001) and International Monetary Fund (2003) for an overview of this effort. III.A.4.7.4

Sensitivity Analysis

Sometimes it may be desirable to create simple, somewhat artificial portfolio shocks, in a method referred to as sensitivity analysis. In this type of stress test at most a few risk factors are shocked and correlation is typically ignored. These are easy to implement, but they only provide a partial picture, and must be accompanied by a lot of judgement on the part of the risk manager. Examples of sensitivity analysis are a parallel shift of the yield curve, or a 10% drop in equity prices. The Derivatives Policy Group (1995) report contains recommendations for a parallel yield curve shift of 100 basis points, and for curve steepening and flattening of 25 basis points. 96 Since the biggest losses do not always correspond to the largest moves in factors, it is common in scenario analysis to create a ladder of shocks, in which price impacts are calculated for intermediate values of the risk factors. Design of these sensitivity ladders requires attention to two issues: granularity and range. The range of shocks should be wide enough to encompass both likely and unlikely (but plausible) moves in the market factors. The recommendations of the Derivatives Policy Group noted above might have represented an adequate range in 1995, but would probably be considered inadequate in 20032004, when interest-rate volatility was very high. Similarly, the range selected should take into account the time horizon for the analysis, with the range increasing with the horizon (except, perhaps, for very strongly mean-reverting risk factors). For these reasons, it makes sense to use the volatility (or perhaps, empirical percentiles of the percentage change) of the factor to determine the range. The granularity of the analysis refers to the distance between the rungs of the ladder, or more exactly, the increment chosen for the change in the risk factor. Granularity should reflect the nature of the portfolio. A portfolio whose valuation function is linear in the risk factor can have a larger increment than a highly nonlinear portfolio with some instruments whose payouts might be discontinuous in the risk factor.

96 Implementing even these simplistic hypothetical market moves contains some hidden issues that can have a big

impact on the results. For example, to implement changes in curve slope, a point on the curve must be chosen around which to rotate the curve, and a second point on the curve must be chosen from which to measure the amount of steepening or flattening. These points should reflect the market environment and the manner in which the curve is traded by market participants. A fair place to start might be to take the two-year point as the point of rotation and the 10-year point as the point to measure the change in basis points.



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III.A.4.7.5

Hybrid Methods

Kupiec (1998) proposes a methodology that is a particular hybrid of covariance matrix manipulation and economic scenarios. His approach can also be applied to the problem of missing historical data in specifying shocks to be used in a historical scenario. In his approach, which he calls stress VaR, the risk manager can specify partial what if scenarios and use the VaR structure to specify the most likely values for the remaining factors in the system. Assume that the risk manager wishes to specify the shocks to a subset of the market risk factors. Using the covariance matrix of factor returns and this subset of fixed shocks, it is then possible to compute a conditional mean vector and a conditional covariance matrix for the remaining risk factors. Using the resulting conditional distribution of factor returns (the factors with fixed shocks have means given by the shocks, zero variance, and hence zero correlations), a conditional, stress VaR can then be calculated. Kupiec goes on to demonstrate how this approach can be generalised to the case in which selected variances and covariances are given prespecified shocks as well. Assuming that market risk factors are jointly normally distributed makes the approach very tractable. Using the unconditional covariance matrix as the starting point for this approach is potentially very limiting, if it is true that in crisis periods there is a structural change in the relationships among market risk factors. However, in applying this approach, it is not necessary to create the conditional return distribution from the same covariances as are used in an unstressed VaR. Any covariance matrix may be taken as a starting point, such as the historical covariance matrix modified in the manner of Finger (1997) discussed above. Alternatively, Kim and Finger (2000) employ Kupiecs method after first estimating a stress-environment covariance matrix. They assume that returns are drawn from a mixture of two normals, one the stress environment and the other a normal environment, calculating the stress VaR using the estimated parameters of the stress-environment return distribution.

Perhaps more simply, the stress VaR could be

calculated using the covariance matrix estimated from a particular crisis period.

III.A.4.8

Algorithmic Approaches to Stress Testing

One of the deficiencies of historical and hypothetical stress scenarios is that the user has only a fuzzy level of confidence that the full extent of the potential badness for the portfolio has been exposed. A more systematic approach might yield a greater level of comfort. The goal is to create a search algorithm to identify the worst outcome for the portfolio within some defined feasible set; that is to say, an optimisation is performed. The key issues with such approaches are the following:



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Are the relationships (e.g., correlations or other measures of interdependence) used in the optimisation relevant for identifying worst-case scenarios? Is the algorithm capable of identifying the globally worst-case outcome in the feasible set? Does the feasible set include implausible outcomes, and are those outcomes represented disproportionately in the optimal results? Two approaches are discussed below, namely, factor-push and maximum loss. III.A.4.8.1

Factor-Push Stress Tests

This type of stress test is so named because it involves pushing each individual market risk factor in the direction that results in a loss for the portfolio. Construction is straightforward. 1.

A push magnitude, m, is selected; it can be stated as a number of standard deviations. For example, each market risk factor, r, may be pushed four standard deviations. The magnitude chosen is, ultimately, subjective. It may be chosen with reference to (an average of) observed movements in market prices during some significant historical event. Or it may be chosen to correspond to some quantile of returns based on an assumed distribution (or perhaps an empirical distribution quantile). If you are setting the push magnitude using standard deviations or quantiles, the period over which these are estimated must be chosen as well. If you are using unconditional estimates, then one year of daily return history is good, if available.

When using a methodology for

estimating volatility conditionally, such as GARCH, then it is good to use as much return history as there is reliable data. 2.

The portfolio, P, is revalued twice by applying shocks, s, to a single market risk factor: once applying a shock s+ = (+1 ? m), and once applying a shock s = (1 × m).

3.

The two portfolio revaluations are compared and the shock resulting in the lower portfolio value is adopted for the stress test.

4.

Steps 2 and 3 are repeated for each of the N market risk factors affecting the portfolio.

5.

The portfolio is revalued once more, this time simultaneously applying the shocks selected in the prior steps for each risk factor.

Consider the following example. A portfolio consists of a long position of 1000 shares in IBM and a short position of 1700 shares in GM. On 23 February 2004 the closing prices of the two equities were $95.96 and $47.51, respectively. Their respective daily standard deviations of return (based on one year of closing prices) were 0.005955 and 0.006972, respectively. Set the push magnitude to 6 (more on this below). The shock for IBM will be +1, because the position is long and the return is linear in the price change. The shock for GE will be 1, by analogous reasoning. The factor-push portfolio stress loss is then $6807.09.



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The selection of the push magnitude is subjective and ignores correlations. Note that the largest one-day move downward in IBM in the five years ending 23 February 2004 is 9.31%, or about 15.63 times the estimated standard deviation of IBM. The third largest move downward is still 7.78 times the standard deviation. The largest one-day move downward in GE in the same period is 6.31%, or 9.06 times the estimated standard deviation of GE. The third largest move down in GE is 5.35 times the standard deviation. If we assume, as in a historical simulation, that any of the one-day IBMGE return pairs observed over that five year period were equally possible when looking forward one day, then the potential worst-case portfolio loss would be $22,175.00, much bigger than the factor-push loss. Of course, the factor-push loss can be made arbitrarily large by increasing the push magnitude. However, the goal is not to create an arbitrarily large loss, but rather to estimate a plausible, if unlikely, loss. Small perturbations of the push magnitudes, if applied to individual positions, can result in greater estimated losses. For example, if the magnitude applied to IBM is reduced to 5.99 and the magnitude applied to GE is increased to 6.01, the factor-push loss will increase (slightly) because GE has the greater estimated standard deviation. This fact highlights the implicit assumption here that the marginal return distributions have the same shape and thus the marginal probabilities of the moves are equal. It may be better to select push-magnitudes position by position, based on the volatility of each factor. A simple thought experiment also illustrates that the factor-push loss need not be greater than the losses from all unambiguously smaller moves, if some of the portfolio positions have returns that are nonlinear functions of the market risk factors. A long option straddle has its least payout associated with small moves in the underlying market factor. Since the factor-push method only searches among large moves for the worst outcomes, the factor-push method will be less useful in such cases. This type of stress test has a variety of other drawbacks as well. Most importantly, these stress tests ignore correlations among risk factors. As a result, the specification of the factor changes employed in the stress test may imply highly implausible market dynamics.

For example,

neighbouring tenor points of a given yield curve are generally very highly positively correlated. Consider a situation where the push factor is 4 standard deviations, and the stress shock at the three-month point on a given yield curve is 1, while at the adjacent six-month point the shock is +1. Such a scenario fails the laugh test. Also, cross effects in instruments whose values are



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affected by multiple risk factors are ignored. For example, for an option on an equity, the option delta is itself related to the volatility of the equity. Thus the correct choice of shock, +1 or 1, for the equity price is dependent on the choice of shock for the volatility. Choosing the push factor based on the standard deviation, or even the VaR, may defeat the purpose of the test. If it is the case that extreme events are, in effect, observations from a distribution that is distinct from that which is experienced most of the time, there may be no obvious choice of push factor for the portfolio that maps from the estimated standard deviations to a stress environment. The particular set of shocks that are chosen can change with every run of the stress test. This makes it more difficult to communicate intuition about the nature of the shocks that generated the observed stress-test result. III.A.4.8.2

Maximum Loss

A maximum loss scenario is defined by the set of changes in market risk factors that results in the greatest portfolio loss, subject to some feasibility constraint on the allowable changes in market risk factors (see Studer, 1995, 1997). The constraint is necessary because potential portfolio losses need not be bounded, and thus, absent a constraint, the result may lack plausibility. Note that the maximum loss scenario is dependent on the structure of the portfolio, and not related to any particular economic environment, except as an environment is reflected in the statistical description of market factor returns employed in the analysis. Thus it is well adapted to answer a question such as just how bad can things plausibly get. 97 One natural method of constraining the set of feasible risk factors is to consider only those scenarios that have a likelihood in excess of some small probability. Being able to associate a probability with a stress-test loss is potentially very useful. To proceed down this path is it necessary to be able to state joint probabilities of specific sets of risk factor changes, which in turn requires that some statements be made about the co-movements of risk factors.

Those

statements will be based on either historical or simulated data and can be conditional (e.g., stressed) or unconditional (normal). Note that, in general, to apply this approach, it is necessary to search through the entire space of risk factor changes with joint probability less than or equal to the plausibility constraint. With positions whose values are nonlinear functions of risk factors in the portfolio, the maximum loss need not occur at the limits of the allowable risk factor changes. Breuer and Krenn (2000) suggest Monte Carlo simulation (or some quasi-random method) as a natural method to use in

97 Boudoukh et al. (1995) propose a risk measure they call the worst-case scenario (WCS). More akin to conditional VaR than stress testing, WCS is the expected value of the distribution of maximum loss over some time horizon, for example, the expected worst-day portfolio P&L over a month. It is also related to the extreme-value theory method discussed below.



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this regard. A na飗e approach would be to run the Monte Carlo for a predetermined sample size, throw out any draw of (an n-tuple of) factor returns with a corresponding theoretical probability less than the plausibility constraint, and pick from the remaining sampled (n-tuples of) factor returns the one with the largest associated portfolio loss. Practically, however, too many simulated risk factor vectors would be required to attain a high level of confidence that the maximum loss had been identified using this approach. Quasirandom methods may ensure a more efficient sampling of the feasible return space. However, since a complex portfolio may have many local return minima and possibly many discontinuities in the portfolio return function, even quasi-random methods may not make the search for that maximum loss tractable. To employ this approach in a practical manner, it may be necessary to develop a smart search algorithm that, based on the qualities of the positions in the portfolio, makes guesses as to which parts of the return space can be ignored. 98

III.A.4.9

Extreme-Value Theory as a Stress-Testing Method

Extreme-value theory (EVT) is based on limit laws which apply to the extreme observations in a sample. These laws allow parametric estimation of high quantiles of loss (negative return) distributions without making any (substantial) assumptions about the shape of the return distribution as a whole. The application to stress testing is immediate (sort of). Firstly, it must be assumed that the historical data employed in the estimation of the tail are representative (i.e., are drawn from the distribution relevant for examining the stress event). Second, the size of the historical data sample must be large enough to yield enough tail observations to get good estimates of the tail distributions parameters. Two flavours of EVT have been employed in looking at risk measurement in finance. The first is called the block maxima approach. An example is the set of yearly maxima of negative daily returns on the S&P500 over some period. The second method is the peak-over-threshold approach, illustrated by the greatest z (i.e., some natural number of) negative daily returns on the S&P500 over the same period. It is apparent that the two methods have somewhat different definitions of extreme events. The block maxima approach may be better suited to estimation of stress losses and the peak-over-threshold approach may be better suited to VaR estimation. If we take the block to be a year, then we can use this approach to find the loss that is expected to be exceeded once in every k years. The details of this approach are beyond the scope of this chapter. For an application of the approach to stress testing, see Cotter (2000).

98 Gonzalez-Rivera (2003) develops an interesting related approach.



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III.A.4.10


The main points of this chapter are as follows: Stress testing is perceived as being a useful supplement to value-at-risk because value-at-risk does not convey complete information about the risk in a portfolio. Stress tests should be part of an rational economic approach to decision making. However, in practice, most uses of stress results are somewhat ad hoc. Useful stress tests must represent plausible, if unlikely, factor changes. Creating useful stress tests requires detailed consideration of the potential behaviour of the market risk factors, individually and jointly, and an understanding of structural relationships among risk factors (e.g., no arbitrage requirements and spread relationships). Stress-test methods generally fall into three categories, historical scenarios, hypothetical scenarios, and algorithms. Historical scenarios seek to re-create a particular economic environment from the past, Hypothetical scenarios can represent a complete, but not yet experienced, economic story or simply a set of ad hoc factor movements. Algorithms attempt to systematically identify the set of factor changes (within some bounds) that give the worst-case portfolio loss. In the case of the factor-push method, the result may not represent a plausible economic story. There is no best or right type of stress test. The context in which the results will be used should determine the approach to be taken in stress testing.

Further Reading Most of the papers listed here and in the References following may be found on www.GloriaMundi.org Basel Committee on Banking Supervision (1999) Recommendations for public disclosure of trading and derivatives activities of banks and securities firms. Mimeo (October). Bouy? E (2002) Multivariate extremes at work for portfolio risk management. Working Paper (January). Bouy? E, Durrleman, V, Nikeghbali, A, Riboulet, G, and Roncalli, T (2000) Copulas for finance: A reading guide and some applications. Working Paper (July). Bouy? E, Durrleman, V, Nikeghbali, A, Riboulet, G, and Roncalli, T (2001) Copulas: An open field for risk management. Working Paper (March). ih醟, Martin (2004a) Designing stress tests for the Czech banking system. Czech National Bank, mimeo (March). ih醟, Martin (2004b) Stress testing: A review of key concepts. Czech National Bank, mimeo (February).



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Costinot, A, Riboulet, G, and Roncalli, T (2000) Stress testing et th閛rie des valeurs extr阭 es: Une vision quantifi閑 du risque extr阭 e. Working paper, Credit Lyonnais (September). Hong Kong Monetary Authority (2003) Stress testing. (February).

Supervisory Policy Manual IC-5

International Association of Insurance Supervisors (2003) Stress testing by insurers. Guidance paper (October). Jouanin, J-F, Riboulet, G, and Roncalli, T (2004) Financial applications of copula functions. In G Szego (ed.), Risk Measures for the 21st Century. New York: Wiley. Oesterreichische Nationalbank (1999) Guidelines on market risk, Volume 5: Stress testing. Mimeo. Wee, L-S, and Lee, J (1999) Integrating stress testing with risk management. Bank Accounting & Finance (Spring), pp. 719.

References Basel Committee on Banking Supervision (1994) Risk management guidelines for derivatives. Mimeo (July). Basel Committee on Banking Supervision (1995) An internal model-based approach to market risk capital requirements. Mimeo (April). Basel Committee on Banking Supervision (1996) Amendment to the capital accord to incorporate market risks. Mimeo (January). Basel Committee on Banking Supervision (2002) Basel Committee reaches agreement on new capital accord issues. Press release (10 July). Berkowitz, J (1999) A coherent framework for stress testing. Journal of Risk, 2 (Winter), pp. 1 11. Blaschke, W, Jones, M T, Majnoni, G, and Martinez Peria, S (2001) Stress testing of financial systems: An overview of issues, methodologies and FSAP experiences. Mimeo (June). Borio, C (2000) Market liquidity and stress: Selected issues and policy implications. BIS Quarterly Review (November), pp. 3851. Boudoukh, J, Richardson, M, and Whitelaw, R (1995) Expect the worst. Risk, 8(9), pp. 100101. Boyer, B, Gibson, M, and Loretan, M (1999) Pitfalls in tests for changes in correlations. Working paper, Federal Reserve Board. Breuer, T, and Krenn, G (2000) Identifying stress test scenarios. Working paper. Cherubini, U, and Della Longa, G (1999) Stress testing techniques and value-at-risk measures: A unified approach. Rivista di Matematica per le Scienze Economiche e Sociali, 22(1/2), pp. 7799. Committee on the Global Financial System (2001) A survey of stress tests and current practice at major financial institutions. Mimeo (April). Comptroller of the Currency (1993) derivatives. Mimeo (October).

Banking Circular 277: Risk management of financial



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Corsetti, G, Pericoli, M, and Sbracia, M (2002) Some contagion, some interdependence. Working paper, Yale University. Cotter, J (2000) Crash and boom statistics for global equity markets. Working paper, University College Dublin. Culp, C and Miller, M (eds) (1999) Corporate Hedging in Theory and Practice: Lessons from Metallgesellschaft. London: Risk Publications. Davis, E P (2003) Towards a typology for systematic financial instability. Working paper, Brunel University (November). Derivatives Policy Group (1995) Framework for voluntary oversight. Mimeo (March). Dungey, M, and Zhumabekova, D (2001) Testing for contagion using correlations. Working paper, Australian National University. Fender, I, and Gibson, M (2001a) The BIS census on stress tests. Risk, 14(5), pp. 5052. Fender, I, and Gibson, M (2001b) Stress testing in practice: A survey of 43 major financial institutions. BIS Quarterly Review (June), pp. 5862. Fender, I, Gibson, M, and Mosser, P (2001) An international survey of stress tests. Current Issues in Economics and Finance, 7(10), pp. 16. Finger, C (1997) A methodology to stress correlations. RiskMetrics Monitor, 3-12. Forbes, K, and Rigobon, R (2002) No contagion, only interdependence. Journal of Finance, 43. Frye, J (1997) Principals of risk: Finding value-at-risk through factor-based interest rate scenarios. In S Grayling (ed.), Understanding and Applying Value at Risk. London: Risk Books. Gay, G, Kim, J, and Nam, J (1999) The case of the SK Securities and J.P. Morgan swap: Lessons in VaR frailty. Derivatives Quarterly, 5(Spring), 1326. Gilbert, C L (1996) Manipulation of metals futures: Lessons from Sumitomo. Working paper, University of London (November). Global Derivatives Study Group (1993) Group of Thirty

Derivatives: Practices and Principles. Washington, DC:

Gonzalez-Rivera, G (2003) Value in stress: A coherent approach to stress testing. Journal of Fixed Income (September), pp. 718. Greenspan, A (2003) Corporate governance. Speech at the Conference on Bank Structure and Competition (May). Hickman, A, and Jameson, R (2001) Benchmarking the US attack crisis. ERisk.com (September). Higham, N J (2002) Computing the nearest correlation matrix a problem from finance. IMA Journal of Numerical Analysis, 22, pp. 329343. Hoggarth, G, and Whitley, J (2003) Assessing the strength of UK banks through macroeconomic stress tests. Financial Stability Review, 3(6), pp. 91103.



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International Monetary Fund (2003) Analytical tools of the FSAP. Mimeo (February). Jeffery, C (2003) The ultimate stress test: Modeling the next liquidity crisis. Risk (November). Jorion, P (1995) Big Bets Gone Bad. Amsterdam: Elsevier Science. Jorion, P (2002) Risk management in the aftermath of September 11. Working paper, University of California-Irvine (April). Kim, J, and Finger, C (2000) A stress test to incorporate correlation breakdown. Journal of Risk, 2(1). Kupiec, Paul (1998) Stress testing in a value at risk framework. Journal of Derivatives, 6, pp. 724. Loretan, M, and English, W (2001) Evaluating correlation breakdown during periods of market volatility. Working paper. Malz, A, and Mina, J (2001) Risk management in the aftermath of the terrorist attack. Working paper, RiskMetrics Group (September). MacKenzie, D (2003) An equation and its words: Bricolage, exemplars, disunity and perfomativity in financial economics. Working paper. Overdahl, J and Schachter, B (1995) Derivatives regulation and financial management: Lessons from Gibsons Greetings. Financial Management, 24, pp. 6878. Presidents Working Group on Financial Markets (1999) Hedge funds, leverage, and the lessons of Long Term Capital Management. Mimeo. Rebonato, R, and J鋍kel, P (1999) The most general methodology to create a valid correlation matrix for risk management and option pricing purposes. Journal of Risk, 2 (2). Schachter, B (1998a) The value of stress testing in market risk management. Derivatives Risk Management (March). Schachter, B (1998b) Move over VaR. The Financial Survey (May), 1214. Schachter, B (2000a) How well can stress testing complement VaR? In, T. Haight, Derivatives Risk Management. (Arlington, Virginia) A. S. Pratt & Sons. Schachter, B (2000b) Stringent stress tests. Risk, 13, (December), pp. S22S24. Securities and Futures Authority Limited (1995) Value-at-risk models. Board Notice 254, 26 May. Shaw, J (1997) Beyond VaR and stress testing. In VaR: Understanding and Applying Value-at-Risk. New York: Risk Publications, pp. 211224. Studer, G (1995) Value at risk and maximum loss optimization. Working paper, ETHZ (December). Studer, G (1997) Maximum loss for measurement of market risk. Doctoral thesis, Swiss Federal Institute of Technology. Turkay, S, Epperlein, E, and Nicos, C (2003) Correlation stress testing for value-at-risk. Journal of Risk, 5(4), pp. 7589.



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Zangari, P (1997a) Exploratory stress-scenario analysis with applications to EMU. RiskMetrics Monitor, Special Edition, pp. 3154. Zangari, P (1997b) Catering for an event. Risk (July), pp. 3436.



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III.B.1 Credit Risk Management Dr. J鰎g Behrens 99

III.B.1.1

Introduction

After a decade of success in the field of market risk modelling mainly driven by advances in derivative pricing and computer technology, the past couple of years have seen a clear shift in interest towards credit risk modelling. Although a significant part of this shift is attributable to regulatory initiatives such as Basel II, an important contribution comes from the demand for more sophisticated credit products and from better management of credit risk. Interestingly enough, these two worlds, pricing on the one hand and risk management on the other, are now converging as credit risk at some institutions is actively managed using credit derivatives. As a result a wealth of new models for pricing and risk management has been developed. Many of these state-of-the-art credit risk concepts are explained in Chapters III.B.2 to III.B.6. In fact so many methods have been developed and so many papers are being published, that one is tempted to ask to what degree the wealth of new information has reached the central figure in this context: the credit risk manager! While the business of publishing the latest developments in glamorous magazines and discussing it at conferences has reached a high point, discussion about the extent to which these developments actually find their way into daily credit risk management seems to be absent. The aim of this chapter therefore is to shed some light on this topic. Are credit risk officers taking advantage of recent advances in credit risk methods? Do the next generation of credit officers need to be rocket scientists? Maybe we are just observing the exploitation of rather academic theories that do not really add a lot of value to practical credit risk management? In order to answer these questions, let us examine the daily work of a credit risk manager and find out which areas have benefited from recent credit developments, and where is there further need for improvement. This chapter will focus on some general topics of relevance for a particular credit officer (Mr Noloss) who has just been appointed chief credit risk officer of Universal Bank. Mr Noloss is so named to highlight the traditional defensive role of the credit officer: to avoid credit losses as much as possible and take corrective action when losses are incurred. While this role is changing, Mr Noloss, like many of his peers, is primarily concerned with protecting his firm from

99 Partner, Global Financial Services Risk Management, Ernst & Young.



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significant credit losses. From his individual situation as a practitioner we hope to derive conclusions of general relevance. Before looking at a typical day in the working life of Mr Noloss, let us browse through his job description in order to better understand his responsibilities: 1. Ownership of credit risk function and framework. This includes responsibility for the management of credit-related work groups such as the credit portfolio group, credit modelling team, credit risk policy and reporting teams. The portfolio group supports the credit officer by complementing a typically rather transaction-focused view with monitoring expected and unexpected losses of the credit book, reviewing provisions, etc. The credit modelling team reviews the banks credit risk methodologies and needs to be independent of the business. 2. Compilation and responsibility for credit risk reporting, including information on limit excesses, counterparty ratings, exposures, concentrations, etc. 100 Supervision of credit data quality process and delivery of all critical credit risk information to various stakeholders and board level. 3. Representative for credit risk, dealing with external credit bodies such as rating agencies and regulators. In particular, ownership of all credit risk-related aspects of Basel II. 4. Ownership of credit processes, including limit setting, provisioning, credit stress and scenario testing and calculating capital requirements. These topics will be discussed in more detail below. 5. Benchmarking of performance of credit risk functions within business units according to group credit policy. Such a list looks good on paper, but how does it translate into practice? This obviously depends on the individual situation. But in the spirit of this chapter, let us just look at a typical day in the life of our credit officer Mr Noloss.

III.B.1.2

A Credit To-Do List

Clearly there are critical tasks and other activities that cover the day. In order not to get sidetracked by too many other issues such as administration, Mr Noloss has produced a to-do list to remind him of the key tasks he wants to focus on (Table III.B.1.1). This also allows a deputy to smoothly take over the work whenever Mr Noloss is not around.

100 We will introduce terminology and show components of a detailed credit risk report below.



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Table III.B.1.1: To-do list Review strategic credit positions: o

Any changes to largest exposures (net of collateral)?

o

How about changes to counterparty ratings?

o

Any significant credits to be approved by chief credit officer or board?

Credit limits and provisions: o

Any limit excesses?

o

Limits to be reviewed?

o

Provisions still up to date?

o

All concentrations within limits?

Credit exposure: o

All exposures covered and correctly mapped?

o

Any wrong-way positions?

Credit reporting: o

All significant risks covered in credit report?

o

Report distributed to all relevant parties?

o

Any significant credits that must be discussed at top management/board level?

Stress and scenario analysis: o

Any surprises from stress and scenario analysis at portfolio or global level?

o

Anything not covered by current set of scenarios?

Provisions: o

Any past or anticipated changes in general loss provisions?

o

Any changes to specific provisions?

Documentation: o

Full documentation in place for all transactions?

o

Break clauses and rating triggers fully recognised?

Credit protection: o

Credit protection utilised and understood?

o

Any further possibility to exploit credit protection?



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Clearly this list is not exhaustive. In particular, it does not address topics with an inherently long time horizon. But let us stick with the daily critical tasks for the moment. III.B.1.2.1

Review of Strategic Credit Positions

Probably the most critical question Mr Noloss asks himself when arriving at his office is Are there any recent or overnight changes in critical exposures? This would generally be most important in the corporate book or in some of the structured transactions, where credit lines could fill quickly in turbulent market conditions. In such scenarios collateral 101 could significantly lose value within hours and diversification could be low. Mr Noloss has a couple of these XXL size positions with individual corporates and funds. Some of these are collateralised and a significant move in market value would probably trigger a margin call. Not too long ago collateralised lending at Universal Bank was based on a simple table of haircuts reflecting collateral volatility and mismatch of each collateral type. Table III.B.1.2 shows collateral types and corresponding haircuts. If a client pledges US government bonds, a haircut of 5% would be applied to the lending value, i.e. the client would receive 95 units of cash for collateral worth 100 units. The haircut is low because the value of government bonds is considered to be quite stable. As can be seen in the table, the haircut increases (i.e. lending value decreases) for other, more volatile collateral types. Table III.B.1.2: Haircuts for collateralised lending at Universal Bank Collateral type

Restrictions

Hair-cut

Government bond

EUR, USD, CHF or GBP

5%

Blue chip equity

SPI or DAX

20%

Corporate bond

Investment grade

10%

Other equity

Listed

50%

Clearly such tables cannot easily capture collateral of a complex nature such as derivatives, and would therefore either need to be very conservative (i.e. non-competitive in the market) or updated very often, which is not desirable. Also sometimes a client would want to pledge collateral of similar type, for example a securitised paper with AAA rating not covered by the table, but in spirit an investment grade bond. Universal Bank therefore has accepted some of these, thereby generating a mismatch of collateral that was difficult to manage. A natural evolution of collateralised lending is to implement a dynamic collateral system that is, basically, a

101 Collateral refers to the assets used to secure a loan or other credit-sensitive transaction. The lender has recourse to

these assets if the borrower defaults on its contracted obligations. See Section III.B.3.4.2 for further discussion.



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value-at-risk system for a collateral portfolio. This was a big achievement for Mr Noloss. The approach not only is dynamic as it daily reflects changes in collateral value, but also accounts for correlations between the underlying risk factors. Non-collateralised positions require even closer attention, as defaults would translate into a bigger loss to the lending bank. This is why Mr Noloss carefully monitors major credit lines 102, in particular credit line utilisation. In some cases, the line will be fully drawn in bad times, prior to default. For obvious reasons in credit card portfolios one even observes line utilisation of more than 100% at the time of default, that is, when credit lines are cut. So high credit line utilisation might eventually indicate an increased default probability. This is a classic case of wrong-way exposure that we will discuss a little later: in the event of default the credit exposure is always a bit bigger than usual. III.B.1.2.2

Credit Limits and Provisions

While the management of credit lines can be regarded as straightforward, the mechanics of defining appropriate limits is not! The first question is What shall we limit against? A traditional variable is the notional amount, which could limit gross exposure against an individual counterparty that is, the total outstanding exposure must not exceed a certain notional amount. However, this measure does not really capture risk, as we have not differentiated different products within this limit. An alternative limiting variable could be expected loss, that is, the credit line remains open provided expected loss does not exceed a certain level. But there are other dimensions to limits, such as the term structure of credit: should limits vary with the term of the borrowing (or other credit-sensitive transaction)? Some people argue that long-term credit exposure is riskier than short-term exposure as the uncertainty in exposure and default probability increase with time, therefore credit lines should be a decreasing function of term-to-maturity. However, this has implications for business incentives. For example, a currency forward that allows a counterparty to exchange one currency against another at a predefined rate has a potential credit exposure profile that increases with time: the longer the time to maturity, the bigger the potential credit exposure. 103 Other products have different potential exposure profiles. A limit that is decreasing with term-to-maturity therefore hardly provides incentives to do long-term currency forwards, and indeed there might not be any room for any new business at all.

102 A credit line defines the total amount that a borrower (or credit card user) may borrow. The term is used

synonymously with credit limit.

103 See the discussion of exposure profiles in Section III.B.3.3.



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In addition to daily position monitoring, a critical review of loss provisions 104 will be at the heart of a credit officers job. Every now and then yesterdays strategic position turns into tomorrows legacy position, leading to significant write-offs (and even the departure of key personnel). So our Mr Noloss will study closely Universal Banks so-called leveraged finance book 105 and check what happens to those wonderful spreads earned some time ago, and carefully monitor provisions that might have to be built up in order to cover increases in expected loss. These days the task of provisioning is made more difficult by the complexity of many credit risks. Mr Noloss might find transactions that do not appear to be exposed to credit risk at all, but in fact hide a ticking credit bomb. Didnt Universal Bank invest in that famous hedge fund on which it is now writing (and hedging) put options? In order to dynamically delta-hedge the short option position, they have to sell or buy further into the fund. That sounds like market risk, but what happens if the fund value goes down and Universal Bank needs to buy even more shares in the fund to hedge its short put position exactly at a time when the credit quality of the fund becomes doubtful? An example of wrong-way exposure (explained in the next section) and in this case even the pure credit risk component which perhaps could be interpreted as a hidden straight loan might not have been fully recognised. III.B.1.2.3

Credit Exposure

This brings us to the problem of exposure measurement: that is, how large are potential losses due to credit risk? While the theory of exposure measurement seems to be fully explored, several problems often remain in practice. The biggest ones simply have to do with complexity. Not only does Universal Bank comprise numerous businesses, as its name suggests it also operates in various countries and runs different IT systems. Some of the more challenging issues around exposure measurement in practice are the following: Full exposure coverage and aggregation: The collection of all relevant exposures seems to be trivial but in practice one often finds a special spreadsheet for that recent complex structure that does not yet feed into the official exposure system. Some banks apply materiality rules of some kind and allow minor exposures to be ignored in the credit aggregation process. However, these exposures might grow and at some stage exceed global counterparty limits. Other problems with exposure measurement arise when aggregating exposures of different businesses. For example, a private client might pledge 104 Provisions for bad debts are made when it becomes apparent that debtors will not repay their debts, or will repay

them only in part. An amount is set aside out of profits in the accounts to reflect this diminution in the value of assets. A corresponding amount is normally deducted from the total assets in the balance sheet. See the discussion in Section III.B.1.2.6. 105 The term leveraged finance is often used for direct investments in sub-investment grade exposures where the credit quality outlook is believed to be significantly better than the implied rating.



198


stock in company ABC as collateral while at the same time company ABC is a major corporate client with significant borrowings. The total exposure to ABC might remain undetected until a default event, where the realised loss seems to be significantly larger than a fully utilised credit line. Globally consistent definition and usage of risk factors and risk measures: As Universal Bank has undergone several mergers and acquisitions, several of its exposure systems are built around inconsistent risk definitions and measures. For example, some systems use a 95% confidence level to define potential credit exposure while others use a 99% confidence level. Even worse, other systems might not know the concept of potential exposure at all but deliver expected exposure only. In addition, some traders start using new concepts such as negative credit exposure. In contrast to the classical definition of credit exposure that is floored at zero, this approach would recognise a loan received by another bank as negative exposure, as one could offset this negative exposure with another loan resulting in a net zero credit exposure. Rather than offsetting such negative credit exposure with the original counterparty, the trading arm of Universal Bank investigates opportunities to package up negative credit exposure for other clients to exploit new revenue streams. This problem highlights a key function of the credit risk manager: He must respond to new developments and judge if these are in line with the banks overall policy, or if required initiate a change in that policy! Correct capture of credit derivative exposure: Classical exposure systems sometimes have difficulty differentiating between normal counterparty exposure and exposure from credit derivatives, which could be offsetting rather than additive especially in the case of bought credit protection. For example, if a firm buys protection to lower credit exposure, many systems still compute credit exposure of the credit derivative and add both exposures rather than recognising the offsetting effect as they cannot correctly reflect counterparty (initial exposure) and issuer (bought protection). Correct treatment of wrong-way exposure: Despite much discussion in journals and at conferences, wrong-way exposure is still often not adequately accounted for. Wrongway exposure basically is exposure that goes against you when it hurts most, mainly caused by ignoring positive correlation between default probability and credit exposure (see Winters, 1999). A drastic example is a long put position on counterparty X issued by X: if the share price of X drops dramatically, the put option would gain significantly in value and therefore credit exposure would go up. However, will X be able to pay in particular if the share price falls even further? While this problem is fully addressed in



199


various academic papers and books (see, for example, Jarrow and Turnbull, 1996), this often remains a problem in practice. The best solutions truly integrate market and credit risk, thereby recognising exposuredefault correlations. As a first step in this direction one often finds banks addressing such problems with specific scenarios in their stress testing exercises (see below). Other problems deal with the complexity of netting agreements 106 (in particular when ISDA agreements are not in place), the mapping of complex exposures onto market risk factors in the simulation of potential credit exposure, or difficulties with forecasting variables such as prepayment in mortgages that can lead to large uncertainties for credit exposure modelling. III.B.1.2.4

Credit Risk Reporting

One of Mr Nolosss key tasks is the collection and reporting of credit information. Not only must he report the most relevant credit risk information to top management (usually the chief risk officer or chief financial officer) and the board as a basis for strategic decisions, this information should also be the basis for his own action plan! So what are the most relevant pieces of credit information one would want to look at? A good starting point are the largest individual counterparty exposures. How did overnight market moves affect these positions? Is there any news in the press about any of those firms? Are there rumours regarding mergers that could affect any of these positions or eventually merge two positions into a single, even bigger one? Large wrong-way exposures could be highlighted as well as large exposures with significant cash flows within the next week (e.g. amortisations or pay-offs). A good credit report would list these largest individual exposures by business type and country. Other credit exposure would be reported against various variables such as sector/industry, risk factor and by country. Most likely these reports would include information on limits and utilisation as well. Several firms also combine information on credit exposure and counterparty ratings: how much exposure is AAA, AA, or how much exposure is investment grade or below? Here the focus is on the overall portfolio rather than on individual credits. Credit risk concentrations are of major concern: does the firm fully recognise credit risk concentrations in certain sectors, risk factors, countries, etc.? Will concentration limits 106 A netting agreement allows to offset exposures between counterparties, for example exposure to firm A through a

Currency Forward bought from firm B, with exposure to firm B through an Interest Rate Swap bought from firm A. See Section III.B.3.4.1 for a full discussion.



200


still be appropriate after adverse market movements? Can credit risk be further diversified and business be extended without substantially increasing concentration risks? Changes of credit risk variables are of major importance, and good credit reports do highlight relevant changes such that they immediately catch the eye. Clearly there is no single answer to the question of what information should be in such a report. Mr Noloss believes that less is more and produces an executive summary report fitting the most critical information onto two or three pages, and complements it with a second report with everything else for those interested in the detail.

Example III.B.1.1: Largest single credit exposures A key component of a good credit report is a list of the largest individual credit exposures. This information often covers exposures before and after netting (gross and net exposure) and can be combined with information about counterparty credit quality. Stress net is the anticipated exposure measured under stressful conditions, as explained in Section III.B.1.2.5. In the table below CP1 refers to counterparty 1 and so on and numbers are in units of currency. Ten largest individual exposures

CP1 CP2 CP3 CP4 CP5 CP6 CP7 CP8 CP9 CP10

rating AA AA AAA BBB BBB BBB AAA AA AA AAA

investment grade gross net 3400 3354 1500 1416 1200 1030 980 909 950 843 780 737 645 617 632 512 500 462 450 366

stress net 3390 1470 1140 970 933 770 639 601 480 430

CP11 CP12 CP13 CP14 CP15 CP16 CP17 CP18 CP19 CP20

rating B BB B B B C BB B C C

sub-investment grade gross net 950 894 800 676 750 609 670 607 650 582 500 451 401 378 303 279 230 210 150 144

stress net 932 725 655 652 593 496 390 298 211 149



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Example III.B.1.2: Exposures by product category Information about individual exposures is complemented by the aggregated view usually listed by product type. Again the information can be combined with credit quality information, in this case by splitting aggregated exposures into investment grade (rating BBB or better) and lower. Total exposure by risk type investment grade sub-investment gross net gross 20.2 15.4 5.4 12.4 10.1 3.2 8.8 8.2 3.1

Trading Banking Cash Management

grade net 1.5 1.2 1.8

Example III.B.1.3: Time evolution of credit exposure Many firms report time evolution of credit exposure. This allows monitoring of trends that can be useful to trigger timely corrective action. Again it is good practice to combine this information for various periods with credit quality information as shown below. Grade BB

Grade B

Grade C

Grade D

$ 7.0 $ 6.0

0.6

0.8

2.5

2.4 1.4

$ 5.0 $ 4.0

0.6

$ 3.0

0.6 1.1

1.2

$ 2.0

0 .8

1.2

1.6

1.4

1.4

1.4

P6

P7

P8

$ 1.0

0.7

0.7

0.7

2.2

2.4

2.4

1.4

1.4

1.5

1.4

0 .9

0 .7

0 .7

P9

P 10

P 11

P 12

$ 0.0

Example III.B.1.4: Arrow plots Another good way of highlighting recent changes is an arrow plot, that indicates a shift in risk parameters by arrows. Universal Bank uses two-dimensional plots to visualise monthly risk parameter changes. The graph below shows the change in expected loss (x-axis) and extreme tail risk (y-axis) for the biggest five counterparty exposures this month (solid squares) compared to last month (empty squares) without actually using an arrow. Moves of 5 largest exposures 19.000 14.000 9.000 4.000 1.000

1.500

2.000

2.500

3.000

3.500

EL

CP1: 200 M

CP2: 193 M

CP4: 175 M

CP5: 159 M

CP3: 184 M

Example III.B.1.5: Watch list Some transactions or counterparties might require special monitoring. A good credit report combines the most relevant information into one table or graphic. The table below shows the Copyright ? 2004 The Authors and the Professional Risk Managers International Association


202


largest sub-investment grade exposures bundled with information about rating outlook and cost of rating triggers in the event of a downgrade. The cost of rating triggers is explained in Section III.B.1.2.7. CP11 CP12 CP13 CP14 CP15 CP16 CP17 CP18 CP19 CP20

Rating watch list rating B BB B B B C BB B C C

(sub-investment grade) outlook exposure stable 894 pos 676 neg 609 neg 607 pos 582 neg 451 pos 378 pos 279 stable 210 stable 144

cost of triggers 90 13 54 44 70 14 24 9 79 15

Clearly there are other important components of credit risk reporting that cannot be discussed here. In practice the difficulty is often not so much to produce all the pieces of a credit report (though this might be a challenge for some very complex firms) as to select the most relevant reports and implement contingency plans in case anything unexpected happens.

III.B.1.2.5

Stress and Scenario Analysis

This topic received a big push with recent Basel II pillar 2 requirements but has been at the heart of credit risk management for a long time. Why is this? Firstly, because classical what-if analysis was the only way to estimate large potential losses before statistical credit (portfolio) models 107 were developed. But stress and scenario analysis also allows one to address shortcomings in the firms risk models, for example through specific scenarios when tail event probabilities are underestimated by the firms standard models, or when dependencies between probability of default and exposure measurement are not correctly captured by standard risk management models. Stress and scenario analysis can also significantly expand the coverage of risk models, for example by inclusion of additional risk types such as business or strategic risk. Clearly there is no general guideline covering all aspects of stress and scenario testing, but in order to provide some practical hints let us once more look to Mr Noloss, who has produced a simple but pragmatic stress testing schedule: Historical stress scenarios: Once every quarter Universal Bank runs a selection of historical crisis scenarios, such as the Asian crisis, the Russia crisis and the oil crisis. For these events historical market data are used for the re-evaluation of all traded positions and books, and the impact of credit risk is quantified and reported against stress limits. In the event that any of these stress limits are exceeded, the position will be reduced immediately. Most recently Universal Bank has also started to perform a unified risk 107 These models are explained in Chapter III.B.5.



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stress test where joint moves in different risk factors are recognised. In particular, a combined credit and market risk stress test highlights problems such as wrong-way exposures, as a shock to market risk factors would directly impact counterparty default probabilities. Potential stress scenarios: In addition to historical scenarios, Mr Noloss has defined a number of hypothetical scenarios that have not (yet?) been realised but are considered to have a decent chance of materialising. These scenarios are defined and reviewed by the credit committee, which Mr Noloss chairs, once a month. Testing the models: Another method of stress testing concerns the risk models themselves. Leveraging experience with derivative pricing models, Universal Bank carries out annual testing of the performance of its credit risk models by stressing input parameters and verifying their performance in such events. The benefit of this analysis that Mr Noloss performs jointly with Universal Banks independent modelling team, is an increased understanding of model behaviour. Does the model behave intuitively? If not, is there a problem with the model, or did Mr Noloss fail to recognise a specific tail risk? Needless to say, the results of all these exercises are reflected in the standard reports. III.B.1.2.6

Provisioning

Textbooks usually introduce economic capital as the difference between unexpected loss (typically the 99th percentile of a loss distribution) and expected loss (typically the average loss). As expected losses as the word suggests are anticipated, they would have to be priced into the product, while unexpected losses would be covered by risk capital. 108 But sometimes so many firms argue expected losses cannot be priced in, that is, they cannot be directly be charged to customers, either because of competitive pressure or for other reasons. So the next best thing is to set up provisions to recognise expected losses internally and avoid the possibility that such losses would harm future profit and loss statements. In practice, provisioning takes place at two levels. General loss provisions are usually made at portfolio level in line with portfolio expected loss (or the difference between loss charges priced into the product and portfolio expected loss). However, for some large transactions, individual loss provisions might include other factors. For example, liquidity reserves are booked for transactions where market prices can be observed for low volumes only and prices are expected

108 See Chapter III.0 for a discussion of economic capital generally and Chapter III.B.6 for a detailed discussion of

capital for credit risk.



204


to drop significantly when large volumes are sold. Individual provisions could also cover model risk, that is, in cases where market prices are observable only for similar instruments typically for non-standard derivative instruments. As expected losses vary over time, for example due to changes to exposure and default probabilities, reserves need to be monitored and adjusted on a regular basis. Typically general loss provisions are reviewed monthly or after specific events, such as big market movements. In addition, individual loss provisions are also reviewed after rating changes of the relevant counterparties. Triggered by market moves or individual loss events, provisions are then adjusted and losses or gains booked accordingly. Mr Noloss has a list of specific transactions with individual provisions that he monitors frequently. Do any individual rating changes justify an adjustment of these provisions? Are correlation assumptions for joint obligor default still realistic? Is individual credit protection really offsetting the amount of risk that it was supposed to cover? For a long time Universal Bank has used long-term average default probabilities for the computation of provisions. Recently, however, it has begun to compute provisions using observable default information taken from market prices such as CDS spreads. Once sufficient data are available, it will compute provisions entirely based on market information. III.B.1.2.7

Documentation

For many reasons, credit documentation plays a crucial in credit risk management. Let us start with an obvious example, which is particularly relevant for credit derivatives: when exactly has a counterparty defaulted? It looks as though Basel II has finally arrived at a useful default definition, but it must be crystal clear whether a party selling protection has to pay or not. As for traditional derivatives, standardisation is a good way to resolve problems with formulation and interpretation of contract details, and ISDA agreements are certainly a very good example of a solution. Break clauses on one or both sides of a contract allow a firm to terminate a transaction contingent on a predefined event, typically at a predefined point in time. In practice, a break clause is often used to renegotiate the terms and conditions of a transaction and then continue with a modified contract. For example, a break clause could give a corporate the right to prepay a loan after 2 years if interest rates have decreased by more than 2 per cent which in this case is an embedded interest-rate option. On the other hand a break clause could give the lending bank the right to force a corporate to prepay a loan in case of a merger with another counterparty. This is a powerful tool for a credit officer as it allows him to limit maximum credit exposure to an individual counterparty.



205


Another feature requiring careful documentation is a rating trigger. This contractual feature provides protection against additional credit risk due to a downgrading of the counterparty. By accepting rating triggers within their contracts, some firms have generously granted their counterparties implicit credit options without adequately measuring or pricing them. A better way would be to map a rating trigger into the corresponding credit protection product, for example a put option triggered by counterparty rating. The discovery of rating triggers written for free can come as a bad surprise, in particular as a rating trigger, by nature, hits at times when the firm most likely already has problems (i.e. a downgrade)! Rating triggers are often implemented as a break clause. For example, pension funds might be restricted to dealing with counterparties holding a very good credit rating. Should the rating of the counterparty deteriorate, a break clause linked to a downgrade below AAA or AA would allow the pension fund to withdraw investments without incurring additional costs at time of the trigger event.

III.B.1.2.8

Credit Protection

Sometime Mr Noloss is faced with the temptation of allowing his traders to continue trading with counterparties that already have their credit lines filled. These are the days when arbitrage opportunities seem to be obvious and it would be a real shame to pass them up. Of course all standard options to increase credit lines or to reduce credit exposure have already been explored: in light of the good credit outlook credit lines have already been increased twice by the credit committee and short-term exposure has been repackaged to long-term exposure using specially designed swaps in order to fully utilise the entire term structure of credit lines. To help with such cases Universal Bank has hired Dr Quant, a former rocket scientist with brilliant ideas. Why not transfer some of the credit risk to another counterparty for which there is still plenty of credit line available? So Dr Quant designs credit instruments, or sometimes simply identifies credit derivatives readily available in the market, that Universal Bank can buy to protect itself: if the original counterparty defaults, it will be the new counterparty that pays the outstanding amount. Of course Dr Quant has realised that sometimes it is even more profitable to assume credit exposure synthetically, that is, by directly taking on credit exposure through credit derivatives. However, in other cases Dr Quants ideas are just too expensive to implement, in particular when credit derivatives for credit protection (as in the first example) cannot be found in the market. Also the exact amount of protection for future points in time is difficult to estimate. Dr Quant has decided to consider expected exposure as the relevant figure, but does this really provide full credit protection in the future? Most certainly not, as the expected exposure is simply a correct



206


probabilistic measure today good enough for pricing and reserving today but will need adjustment if any of its constituents (default probability, exposures, loss given default) changes. Finally Dr Quant comes up with an even better idea: why not consider all counterparty exposures within one portfolio jointly, securitise them 109 and then sell them to other counterparties? In this way there should be a large diversification effect and credit lines could be significantly reduced. So now Dr Quant is working jointly with Mr Noloss and the portfolio team of Universal Bank to identify opportunities. Mr Noloss is ensuring that this process is in line with Universal Banks overall credit risk framework, that it fits the global credit risk strategy and that any new risks in particular, with regard to limit management and reporting are fully taken into account. So credit derivatives and securitisation are instruments/techniques that do actively link trading and risk management. Whereas in bygone days traders would have done their business and an independent risk control unit in particular, the credit officer would have either approved or blocked the business, we nowadays see joint teams exploiting opportunities to integrate both aspects. Good business links returns to risks, while good risk management links risks to returns.

III.B.1.3

Other Tasks

We have mainly focused on the credit officers more frequent tasks. Clearly there are other tasks with an inherently long time horizon. Some of these were listed in the job description at the beginning of this chapter. In the following we highlight a few tasks that are usually addressed only annually but then sometimes lead to very controversial discussions. Review of credit strategy: We previously discussed the review of strategic positions that clearly is a frequent key task for any credit officer. However, a more fundamental question often remains unanswered, perhaps even unasked: Do we really want to be in that business? Are we fooling ourselves when we believe we can exploit what is called arbitrage today? In some firms the duration of employment of those people who propose and eventually benefit from delicate transactions is typically far shorter than that of a credit officer. Mr Noloss therefore has a reputation for asking many nasty questions, but he certainly also gets the respect he deserves. Review of models: Backtesting of ratings and models for loss given default is not only required by Basel II. Mr Noloss is keen to ensure that model calibration is done seriously so that credit granting is based on the best information available and

109 See Section III.B.6.6 for a discussion of securitisation.



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provisions are computed with realistic default probabilities and loss given default values. At Universal Bank backtesting is performed by an independent modelling team that also approves any new models before they go into production. Review of credit process: Another topic that is usually triggered by events for example, a change in regulatory requirements, a large unforeseen loss, or when a new chief credit officer assumes his role is the review of the credit process itself. This is a good opportunity to clean up legacy positions, for example to better integrate new businesses into the existing credit framework. These sometimes start with a complex spreadsheet trade on a remote desk rather than with a formal approval of the new business committee and might require a dedicated infrastructure including appropriate management information systems. Review of stress testing: Credit officers will also from time to time meet with economists to hear about changes in the global economical outlook and adapt their stress scenarios accordingly. Clearly the frequency of such meetings varies with the volatility of the business.

III.B.1.4

Conclusions

Let us return to our initial questions: Are credit risk officers taking advantage of recent advances in credit risk methods? And does the next generation of credit officers need to be rocket scientists? Looking at a day in the life of a credit officer, we observe many of the credit officers classical tasks and rather few forays into the hot areas in credit risk such as credit modelling or portfolio theory. For those who like to be critical, we could argue that a credit officer usually plays a rather defensive role. He monitors developments that could lead to credit losses and ensures that corrective action is taken, that is he helps to avoid execution of bad (i.e. too credit risky) transactions. In the simplest terms, the traditional role of the credit officer is to control the business units and avoid losses. By contrast, the head of the business unit has the job of optimising profits. But is a credit officer necessarily restricted to a defensive role to protect or can he also be someone who actively makes suggestions, identifies opportunities and provides insights? Can he proactively support the business to better generate value for the firm? The answer is a clear yes, and some firms have started to significantly expand the credit officers role and in fact integrate the two functions.



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This is best highlighted by the use of credit derivatives. Credit officers have begun actively using these instruments to manage credit risk. Another development is the formation of portfolio teams comprising business people and credit risk managers who jointly explore opportunities to do the right trades trades that optimize return versus risk. What used to be a two-stage process has been integrated into one. In addition to joining the business team and providing it with a better perspective on risk, the new credit officer has the role of independent oversight of credit risk which, at least for banks, under the Basel regulations, is an essential qualitative requirement. Whereas in earlier times traders would have done their business and a credit officer would have either approved or blocked the business, we nowadays see joint teams exploiting opportunities to integrate both aspects. Good business links returns to risks, while good risk management links risks to returns. So it seems that the new credit officer, unlike his traditional predecessor, can directly benefit from many advances in credit risk management. The benefits of a portfolio view including up-to-date credit methodologies are so obvious, that it is only a matter of time until the typical credit officer not only reads about recent advances in credit risk but perhaps even writes about them. So we will not be surprised to learn of Mr Nolosss replacement, on his retirement, by Mr Raroc, 110 who has something of a reputation for his outspoken views on risk and return

References Winters, W (1999) Wrong way exposure, Risk, July. Jarrow, R, and Turnbull, S (1996) Derivative Securities. Cincinnati, OH: South Western College Publishing, pp. 574580.

110 See Section III.0.5 for discussion of RAROC (risk-adjusted return on capital).



209




210


III.B.2 Foundations of Credit Risk Modelling Philipp Sch鰊bucher 111

III.B.2.1

Introduction

This chapter introduces the three basic components of a credit loss: the exposure, the default probability and the recovery rate. We define each of these as random processes, that is to say, their future values are not known. Instead, their values are represented as a probability distribution of a random variable. The credit loss distribution is then defined as a product of these three distributions. The credit loss distribution for a large portfolio of credits can become quite complex and we shall see in Chapter III.B.5 that many advanced techniques are available for portfolio modelling. In this chapter we merely aim to provide an introduction to this distribution. Section III.B.2.2 makes the definition of default risk precise, and Sections III.B.2.3 and III.B.2.4 introduce the three basic processes and the credit loss distribution. Section III.B.2.5 makes a careful distinction between expected and unexpected loss, and Section III.B.2.6 gives a detailed discussion of recovery rates for a portfolio of credits. Section III.B.2.7 summarizes and concludes.

III.B.2.2

What is Default Risk?

Default risk is the risk that a counterparty does not honour his obligations. Such an obligation may be a payment obligation, but it is also a default if a supplier does not deliver the parts he promised to deliver, or if a contractor does not render the services that he promised. In this wide context, default risk is everywhere, and there is no transaction that does not involve the risk that one of the two parties involved does not deliver. For a financial institution the largest and most important component of default risk refers to payment obligations such as loans, bonds, and payments arising from over-the-counter derivatives transactions. This risk of a payment default, in particular when it refers to loans and bonds, is called credit risk. We distinguish: Default: An obligation is not honoured. Payment default: An obligor does not make a payment when it is due. This can be: o

Repudiation: Refusal to accept a claim as valid.

111 D-MATH, ETH Z黵ich, R鋗 istrasse 101, 8092 Z黵ich



211


o

Moratorium: Declaration to stop all payments for some period of time. Usually only sovereigns can afford to do this.

o

Credit default: Payment default on borrowed money (loans and bonds).

Insolvency: Inability to pay (even if only temporary). Bankruptcy: The start of a formal legal procedure to ensure fair treatment of all creditors of a defaulted obligor. For instance, an obligor may be: in default but not in payment default (e.g. if he defaults on a non-financial obligation), in payment default but not insolvent (e.g. if he could pay, but chooses not to), and in default but not in bankruptcy (e.g. if the bankruptcy procedures have not been started (yet) or if there are no bankruptcy procedures, e.g. if the obligor is a sovereign). Strictly speaking, default risk is tied to the obligation it refers to, and a priori there is no reason why an obligor should not default on one obligation and honour another. Fortunately, in most cases we can rely on the existence of a functioning legal system which ensures that such selective defaults are not possible. The creditor of the defaulted obligation has the right to go to a court which (eventually) will force the obligor to honour his obligation (if this is possible) or if the obligor is generally unable to do so instigate a formal bankruptcy procedure against him. The aim of this procedure is an orderly and fair settlement of the creditors claims and possibly also other social priorities such as the preservation of jobs. The details of the bankruptcy procedure depend on the applicable local bankruptcy law and will vary across countries. Thus, the existence of a bankruptcy code allows us to speak of the credit risk of an obligor and not just of an individual obligation, which we will do in the following.

III.B.2.3

Exposure, Default and Recovery Processes

To analyse the components of default risk in more detail we now need to introduce some notation. We consider a set of I obligors indexed with i = 1, , I, and we call

i

the time of

default of the i th obligor. The following three processes describe the credit risk of obligor i: Ni(t): The default indicator process. At time t, Ni(t) takes the value one if the default of obligor i has occurred by that time, and zero if the obligor is still alive at time t. Obviously knowledge of the full path of the default indicator process is equivalent to knowledge of the exact time of default of the obligor, but frequently we only have partial knowledge (i.e. the obligor has not defaulted so far), or we are only interested in a partial event (i.e. default before the maturity of a loan).



212


Ei(t): The exposure process. The exposure at default (EAD) to obligor i at time t is the total amount of the payment obligations of obligor i at time t which would enter the bankruptcy proceedings if a default occurred at time t. The exposure process will be covered in detail in Chapter III.B.3. Li(t): The loss given default (LGD) of obligor i, given default at time t. The LGD usually takes values between zero and one, and Ri(t) = 1 Li(t) is known as the recovery rate of the obligor. The LGD is often less than one, reflecting the fact that some proportion of the exposure at default may be recovered in bankruptcy proceedings. We will discuss recovery rates in more detail in Section III.B.2.6 . Let pi(T) be the individual probability of default (PD) of obligor i until some time horizon T. For instance, if the one-year probability of default is pi(1) = 0.01%, this means that there is a 1/10,000 chance that the obligor will default at some point in time over the next year. Using these processes, the losses due to defaults of obligor i before time T can be represented as follows: Default loss = Default arrival ? EAD ? LGD or, in our mathematical notation where denotes the time of default of the obligor, Di(T) = Ni(T) Ei( ) ? Li( ).

(III.B.2.1)

For a fixed time-horizon T, the default indicator Ni(T) is a binary (0/1) variable which allows one to capture the default arrival risk, that is, the risk whether a default occurs at all. A fixed timehorizon (typically one year) is a common point of view in credit risk management, but in many cases this is not sufficient and the timing risk of defaults must also be considered.

III.B.2.4

The Credit Loss Distribution

For a bank, individual credit defaults of obligors are not unusual events and although painful and inconvenient these events are part of the normal course of business. If the exposure is not too large it can be buffered using normal operating cash flows. But when multiple defaults occur simultaneously (or within a short time span) this can threaten the existence of a financial institution. Thus, a major task of the credit risk manager is to measure and control the risk of losses from a whole portfolio of credits. For instance, suppose the credit losses are correlated, so that a default from one obligor makes default of other obligors more likely. Then there is a concentration risk in the portfolio that needs to be managed.



213


Using the definition of individual default losses (III.B.2.1), the portfolio loss D(T) at time T is defined as the sum of the individual credit losses Di(T), summed over all obligors i = 1, , I: I

D( T )

I

Di ( T ) i 1

N i ( T ) Ei ( ) L i ( ).

(III.B.2.2)

i 1

The individual default losses Di(T) are unknown in advance. Thus the full portfolio loss D(T) is a random variable. The portfolios credit loss distribution is the probability distribution of this random variable (see Chapter II.E). F ( x ) : P[ D( T ) x ].

(III.B.2.3)

An important ingredient of the distribution of D(T) is the dependency between the individual losses. In Chapter III.B.4 some popular models are presented which show how default correlation can be modelled. Here we only mention two important points: (a) Assuming independence between individual default losses almost always leads to a gross underestimation of the portfolios credit risk. (b) The default correlation parameters typically have a strong influence on the tail of the loss distribution and thus on the value at risk (VaR). Figure III.B.2.1: Density function of the portfolio loss of a typical loan portfolio. Mean loss (expected loss), 99% VaR, and 99.9% VaR are shown as vertical lines Mean Loss

14.00%

99% VaR

99.9% VaR

12.00%

10.00%

8.00%

6.00%

4.00%

2.00%

0.00% 0

50

100

150

200

250

300

350

400

Loss



214


Figure III.B.2.1 shows the density function of the credit loss distribution of a typical credit portfolio using a hypothetical portfolio of 100 obligors with 10m exposure each (i.e. 1bn total portfolio volume). We have used a CreditMetrics type of model (see Section III.B.5.3) with individually varying unconditional default probabilities, 50% assumed recovery rate and a constant asset correlation of 20%. Credit portfolio loss distributions have several features that distinguish them from the profit and loss distributions (or returns distributions) from market variables such as equities, interest rates or FX markets: The distribution is not symmetrical. The upside is limited (the best possible case is a credit loss of zero), while the downside can become extremely large. The distribution is highly skewed. Most of the probability mass is concentrated around the low loss events (e.g. losses between 0 and 50 in Figure III.B.2.1). These are also the events that we are most likely to observe in historical data sets. In the example of Figure III.B.2.1, losses will be less than 60m in 80% of the cases. The distribution has heavy tails. This means that the probability of large losses decreases very slowly, and VaR quantiles are quite far out in the tail of the distribution. This can also be seen from Figure III.B.2.1.

III.B.2.5

Expected and Unexpected Loss

The standard scenario which people intuitively expect to happen when they consider the default risk of an obligor is no default, that is, no loss. This is indeed the most likely scenario if we consider each obligor individually (unless we consider an obligor of extremely low credit quality). This scenario is also still used quite frequently for accounting purposes: a loan or bond is booked at its notional value (essentially assuming zero loss), and only if it gets into distress is it depreciated. In some institutions return on capital is still (incorrectly) calculated this way. Unfortunately, this is one of the cases wheren naive intuition can lead us astray: the standard scenario is not the mathematical expectation of the loss on the individual obligor. If we assume that exposure Ei and loss given default Li are known and constant, the expected loss is E[ Di ( T )]

pi ( T ) Ei Li

0,

(III.B.2.4)

where pi(T) is the default probability of the obligor.



215


At the level of individual obligors in isolation, the concept of expected loss may be counterintuitive at first because we will never observe a realisation of the expected loss: either the obligor survives (then the realised loss will be zero) or the obligor defaults (then we will have a realised loss which is much larger than the expected loss). There is a related trick question. Next time you go out, offer to buy your friend a drink if the next person entering the bar does not have an above average number of legs. Of course your friend will have to buy you a drink if the person does indeed have an above average number of legs. You will win the bet if the next person has two legs: the average number of legs per person in the population must be slightly less than two because there are some unfortunate people who have lost one or both legs (but there are no people with more than two legs). The same idea applies to credit obligors: most obligors will perform better than expected (they will not default), but there are some who perform significantly worse than expected. But nobody will perform exactly as expected. Typically, the expected loss is small (because pi will be small) but it is positive. These small errors will accumulate when we consider a portfolio of many obligors. In a portfolio of 1000 obligors we may no longer assume that none of these obligors will default. Even if each of the obligors has a default probability of only 1%, we will have to expect 10 defaults. In Figure III.B.2.1 the level of the expected loss of the portfolio is shown by the first (leftmost) vertical line; it is at a level of about 35m, that is, 3.5% of the portfolios notional amount. Expected loss is an important concept when it comes to performance measurement, in particular in connection with risk-adjusted return on capital (RAROC) calculations (see Chapter III.0). When a loans expected gain (in terms of excess earnings over funding and administration costs) is not sufficient to cover the expected loss on this loan, then the transaction should not be undertaken. Suffering the expected loss (in particular, on a portfolio) is not bad luck: it is what you should expect to happen. Consequently, the expected loss should be covered from the portfolios earnings. It should not require capital reserves or the intervention of risk management. The expected loss on a portfolio is the sum of the expected losses of the individual obligors: 112 E[ D( T )]

I

E[ Di ( T )].

(III.B.2.5)

i 1

112 This follows from the property of the expectation operator: E(X + Y) = E(X) + E(Y) see Chapter II.E.



216


However, this simple summation property will not hold for the unexpected loss! Unexpected loss is usually defined with respect to a VaR quantile and the probability distribution of the portfolios loss. Let us assume that D99% is the portfolios 99% VaR quantile, that is, P[ D( T ) D99% ] 99%.

Then the unexpected loss of the portfolio at a VaR quantile of 99% is defined as the difference between the 99% quantile level and the expected loss of the portfolio:

UEL D99%

E[ D(T )]

(III.B.2.6)

If another risk measure such as conditional VaR is used in place of VaR, then (III.B.2.6) is easily extended. We define unexpected loss in these situations by replacing D99% with the general risk measure. More details on some alternative risk measures are given in Section III.A.3.5.2. The term unexpected loss may be confusing at first, because it does not concern losses that were unexpected, but only something like a worst-case scenario. Intuitively, one might define unexpected loss as the amount by which the portfolios credit loss turns out, in the end, to exceed the originally expected loss: max{D( T ) E[ D( T )], 0}.

(III.B.2.7)

Here, we will use unexpected loss in the sense of equation (III.B.2.6), and not in the sense of (III.B.2.7). In Figure III.B.2.1, the 99% and 99.9% VaR loss quantiles were shown with two vertical lines intersecting the tail of the loss distribution. The 99% VaR level is at approximately 160m, which yields an unexpected loss of 160m

35m = 125m. The 99.9% VaR level is at approximately

220m, with an unexpected loss of 185m. It is no coincidence that, even at a very high VaR quantile of 99.9%, the unexpected loss is still much less than the maximum possible loss of 1bn that is suffered when the total portfolio defaults with zero recovery. This effect stems from the partial diversification, which is still present in the portfolio despite an asset correlation of 20%. As opposed to expected loss, the unexpected loss is not additive in the exposures. If, for example, we assume that with zero recovery and unit exposure each obligor defaults with a probability of 3%, then each obligors individual 99% VaR will be 1, its total exposure. But the 99% VaR of a large portfolio of such obligors will not be the total exposure of the portfolio (unless we have the extreme case of perfect dependency between all obligors).



217


The unexpected loss is frequently used to determine the capital reserves that have to be held against the credit risk of the portfolio. It is not economically viable to hold full reserves against total loss of the portfolio, but reserving against unexpected loss at a sufficiently high quantile is viable and effective if it is done centrally under exploitation of all diversification effects. Thus, coverage of the (linear) expected loss is in the domain of responsibility of the business lines, but the management of the (highly non-linear) unexpected loss is usually a task for a centralised risk management department. The risk management department then makes appropriate risk charges to the business lines. A stylised capital allocation procedure is as follows: 1. Fix a VaR quantile for credit losses (usually 99% or 99.9%). This quantile should reflect the institutions desired survival probability due to credit losses (this probability can be derived from its targeted credit rating). This is a management decision that has to be made at the top level. 2. Determine the portfolios expected loss. 3. Determine the unexpected loss of the portfolio according to (III.B.2.6). 4. Allocate risk capital to the portfolio to the amount of the unexpected loss. 5. Split up the portfolios risk capital over the individual components of the portfolio according to their risk capital contributions. Losses in the portfolio up to the amount of the expected loss will have to be borne by the individual business lines (because these losses are economic losses), but any losses that exceed the expected loss will hit the risk capital reserves. Should these reserves not suffice to cover all losses, the bank itself will have to default. But by setting the original VaR level, the probability of this event can be controlled.

III.B.2.6

Recovery Rates

As we can see from equations (III.B.2.1) and (III.B.2.4), the recovery rate (or the LGD) of an obligor is as important in determining default losses or expected default losses as the default probability. Nevertheless, research into recovery rates has been neglected for a long time, and most focus has been put on default events and default probabilities. This is partly due to the fact that data on recovery rates are much more fragmented and unreliable than data on default events. While we may expect that obligors will do their best to avoid bankruptcy in almost any legal environment (which will make default arrivals largely independent of the legal framework), recovery rates are closely tied to the legal bankruptcy procedure which is entered upon the default of the obligor.



218


In a typical bankruptcy procedure all creditors register their legal claim amounts with the bankruptcy court. There is a well-defined procedure to determine these legal claim amounts which does not necessarily reflect the market value of the claim: for example, for loans or bonds only the notional amount (and any interest payments which are currently due) is considered, but not the actual market value of the bond or the value of the future coupon payments. These values may be significant if interest rates have moved since the issuance of the bond. According to the ISDA standard definitions, the legal claim amount for over-the-counter derivatives is usually the current replacement value of the contract, where it is assumed that the counterparty has the same rating as the defaulted counterpartys pre-default rating. Claims are then grouped by priority class (collateralised, senior, junior, etc.). At this point, the bankruptcy procedures start to diverge significantly: some procedures aim to find a way to restructure the bankrupt obligor and to enable him to become profitable again (e.g. Chapter 11 in the USA), while others aim is to liquidate the obligors business and use the proceeds to pay off the debts (e.g. Chapter 7 bankruptcy in the USA). The final outcome for the creditors usually depends first on the choice of bankruptcy procedure, and then on complicated negotiations between many parties. We will have to ignore these effects here, and just warn the reader that, because of the strong dependency on procedural details, recovery rates are not easily comparable across countries. Generally, the outcome of any bankruptcy procedure will be a settlement in which creditors of the same legal claim amount and the same priority class will be treated identically, with secured creditors having the first claim on their collateral and the rest of the firms assets, unsecured creditors come next, and then stockholders have the last claim and may not receive anything. If the creditors claims are settled in cash, the definition of the recovery rate is relatively straightforward: if each dollar of legal claim amount receives a 40 cent cash settlement, then the recovery rate is 40% and the LGD is 1

40% = 60%. The only problem is to decide whether the

payment should be discounted back to the actual date of default (in particular, for large obligors, bankruptcy procedures can easily take several years, and the usual answer is yes), and whether legal costs should be subtracted from it (again yes). Unfortunately, cash settlements tend to be rare. Much more frequently (in particular, if the obligor is restructured and not liquidated) the settlement is partly cash, and partly some other type of security such as equity, preferred equity or restructured debt of the reorganised obligor. In this case, determining the value of the settlement is often close to impossible, in particular if



219


the obligor was de-listed from the stock exchange in the course of the bankruptcy. For these reasons, recovery rates are generally defined without reference to the final settlement. Definition 1 (Market Value Recovery). The recovery rate is the market value per unit of legal claim amount of defaulted debt, some short time (e.g. 1 or 3 months) after default. This definition also coincides with the way recovery rates are determined in credit default swaps with cash settlement (see Chapter I.B.6). It generally applies to larger obligors (e.g. obligors rated by public rating agencies). When smaller obligors are concerned (e.g. retail obligors and/or small and medium-sized enterprises) there will be no market price for distressed debt and we will have to attach a direct valuation to the final default settlement, and define the recovery rate as: follows Definition 2 (Settlement Value Recovery). The recovery rate is the value of the default settlement per unit of legal claim, discounted back to the date of default and after subtracting legal and administrative costs. According to the discussion above, we expect the following factors to directly influence the recovery rates of defaulted debt: collateral; the legal priority class of the claim; the legislature in which the bankruptcy takes place (the UK tends to be a rather creditorfriendly legislature, while France and the USA are more obligor-friendly and thus have lower recovery rates). Some other, less easily observed factors turned out to be significant in empirical investigations (see, for example, Altman et al., 2001; Gupton et al., 2001; Van de Castle et al., 2001; Renault and Scaillet, 2004): The industry group of the obligor. Financial institutions tend to have significantly different (higher) recovery rates than industrial obligors. The less capital intensive the obligors business, the less substance there is to liquidate in the event of a bankruptcy. For instance, dotcom companies tended to have recoveries close to zero. The obligors rating prior to default. An obligor who has spent much time close to default usually has fewer assets to liquidate to pay off the creditors than an obligor who defaulted quite suddenly from a high rating class.



220


The average rating of the other obligors in the industry group, and the business cycle. This affects the liquidation value of the obligors business and/or the value and viability of a restructured firm. Recoveries tend to be lower in recessions and in industry groups that are in cyclical downswings or which have large overcapacities. Despite these empirical findings, it turns out that it is virtually impossible to predict a recovery rate of an obligor with much certainty. The margins of error are very large indeed. Table III.B.2.1 shows estimated recovery rates and their standard errors for US corporate debt of different seniority classes. It can be seen that the average standard error is often of the same order of magnitude as the mean of the recovery rate. Table III.B.2.1: Recovery rates by seniority of claim Seniority

Observations

Mean (%)

Standard deviation (%)

82

56.31

23.61

Senior unsecured

225

46.74

25.57

Subordinated

174

35.35

24.64

Junior subordinated

142

35.03

22.09

Total

623

42.15

25.42

Senior secured

Source: Renault and Scaillet (2004) Data set: S&P, 19811999, US Corporates From a risk management point of view the large prediction errors in the recovery rates would not be too serious if we could at least hope that our estimation errors will cancel out on average over several defaults. Unfortunately, the systematic dependence of recoveries on the business cycle destroys this hope. Recoveries depend on a common factor and thus they will not diversify away. In particular, we will get hit twice in a recession: first, because there are more defaults than usual; and second, because we will have lower than average recovery rates. Thus, we should stress the recovery-rate assumptions of our credit risk models when we consider recession scenarios. This is confirmed in Table III.B.2.2, which shows average recovery rates for different phases of the business cycle. In the years 19822000 (a proxy for a long-term average) recoveries tended to be much higher than in the recent (2001 and 2002) downswing, where also the default incidence has increased significantly. For example, recoveries on senior unsecured debt dropped from a long-term average of 43.8% to 35.5% and 34.0% in the 2001/02 recession.



221


Table III.B.2.2: Average recovery rates (%) of defaulted debt at different periods in time Asset class

19822002 19822000 2001 2002

Secured bank loans

61.6

67.3

64.0

51.0

Equipment trust

40.2

65.9

NA

38.2

Senior secured

53.1

52.1

57.5

48.7

Senior unsecured

37.4

43.8

35.5

34.0

Senior subordinated

32.0

34.6

20.5

26.6

Subordinated

30.4

31.9

15.8

24.4

Junior subordinated

23.6

22.5

NA

NA

All bonds

37.2

39.1

34.7

34.3

Source: Moodys KMV (2003) A popular mathematical model for random recovery rates is the beta distribution. The beta distribution is a distribution for random variables with values in [0, 1] which has the density f ( x ) c x a (1 x )b ,

(III.B.2.8)

where a and b are the two parameters of the distribution, and c is a normalisation constant. By choosing different values for a and b, a large variety of shapes for the recovery distribution can be reached. One can directly fit the parameters of the beta distribution to the mean and the variance of the dataset using the following formulae: 2

a Here,

is the mean of the data, and

(1

) 2

and b

)2

(1 2

.

is its standard deviation. In Figure III.B.2.2, the beta

distributions fitted to the data in Table III.B.2.1 are plotted. Recovery-rate distributions for senior, senior unsecured, subordinated, junior subordinated, and all debt issues are shown. Quite clearly, higher seniority classes tend to have higher average recoveries, but there is a large overlap between the distributions of the various seniority classes. This is caused by the relatively large variance of the values reported for each class. This can happen even if priority rules of the seniority classes are observed for each obligor individually.



222


Figure III.B.2.2: Beta distributions fitted to the recovery data of Table III.B.2.1 2.5

2

1.5

1

0.5

0 0%

10%

20%

30%

Senior Secured

III.B.2.7

40% Senior unsecured

50%

60%

Subordinated

70% Junior Subordinated

80%

90%

100%

All

Conclusion

Modelling credit risk has become an essential tool for modern risk management within a financial institution. Such models are used to determine both expected loss (important for pricing loans and other assets or contracts) and unexpected loss (necessary for assessing the appropriate size of the capital buffer). The fundamental idea presented in this chapter is that probability of default, exposure amount and loss given default (or conversely, recovery rates) combine to give the credit loss distribution for a portfolio of assets. This distribution is typically skewed, with a small probability of large losses. Recovery rates are one of the three crucial elements in determining the credit loss distribution. They will vary according to seniority, level of security, the economic cycle and local bankruptcy laws, amongst other things. Recovery rates are particularly difficult to estimate in advance, having large standard deviation and dependence on the business cycle.

References Altman, E I, Resti, A, and Sirone, A (2001) Analyzing and explaining default recovery rates. Report submitted to the ISDA, Stern School of Business, New York University, December. Gupton, G M, Gates, D, and Carty, L V (2001) Bank-loan loss given default, in Enterprise Credit Risk Using Mark-to-Future. Algorithmics Publications, pp. 6992. Available from www.algorithmics.com Copyright ? 2004 The Authors and the Professional Risk Managers International Association


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Renault, O, and Scaillet, O (2004) On the way to recovery: A nonparametric bias-free estimation of recovery rate densities. Journal of Banking and Finance, 28 (to appear). Van de Castle, K, Keisman, D, and Yang, R (2001) Suddenly structure mattered: insights into recoveries from defaulted debt, in Enterprise Credit Risk Using Mark-to-Future. Algorithmics Publications, pp. 6168.



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III.B.3 Credit Exposure Philipp Sch鰊bucher 113

III.B.3.1

Introduction

Chapter III.B.2 established the three components of credit loss upon default of an obligor: exposure amount, loss given default (or, conversely, recovery rates) and probability of default. Successful credit risk management and modelling will require an understanding of all three components. Accordingly, this chapter explores in more detail the exposure amount, which may also be referred to as credit exposure or exposure at default. The exposure at time t is the amount that we would lose if the obligor defaulted at time t with zero recovery. Section III.B.3.2 will distinguish between pre-settlement and settlement risks, as the management techniques vary depending on whether default occurs prior to or at the time of settlement. Section III.B.3.3 explains exposure profiles, that is, how exposures vary over time for the various asset and transaction types. Finally, Section III.B.3.4 discusses some techniques for reducing credit exposures (risk mitigation techniques) and Section III.B.3.5 concludes. The exposure is closely related to the recovery rate in the sense that it is usually identified with the legal claim amount or the book value of the asset. But this identification is not quite correct: We should also recognise that we might stand to lose value which is not recognised as a legal claim amount (see the discussion in Section III.B.2.6), for example large future coupon payments. Thus, the correct definition of exposure is the market price or the replacement value of the claim. In practice many simplifications are used when exposure amounts are estimated. This is partly justified by the fact that any accuracy that is gained by a more accurate measurement of exposure will probably be swamped by the large uncertainty surrounding the recovery rates of the obligors. Exposure and recovery enter the loss at default multiplicatively. Thus if we decrease the relative error in the exposure measurement by 5%, it will not help much in improving the loss estimation if on the other hand we face a relative error of 20% or more in the recovery rate. Credit exposure may arise from several sources:

113 D-MATH, ETH Z黵ich, R鋗 istrasse 101, 8092 Z黵ich, Switzerland.



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Direct, fixed exposures. These arise from lending to the obligor or from investment in bonds issued by the obligor (which is another kind of lending). This is the most straightforward type of exposure. Commitments. Although they frequently have zero current exposure, committed lines of credit constitute large potential exposure because they will usually be drawn should the obligor get into financial difficulties. The bank may have covenants that allow a termination of the lending facility in the event of an adverse change of the obligors credit quality, but the borrower usually has an information advantage and can draw at least parts of the line of credit before his financial problems become known to the bank. This raises the question how the potential exposure embedded in a line of credit should be measured. A common pragmatic solution is to assume that the obligor will have drawn a certain fraction of the line of credit if he should default. Thus, we consider a fixed fraction of a committed line of credit as exposure at default, even if it is not drawn at the moment. Variable exposures. These arise mainly from over-the-counter (OTC) transactions in derivatives. As they are exposed to (uncertain) moves in the interest rates, fixed-coupon loans and bonds could also be considered variable exposures, but they are usually considered fixed exposures. Futures contracts are generally ignored for the purposes of credit assessment as their institutional features are designed to effectively eliminate credit exposure. The futures clearing mechanism interposes the clearing house as the ultimate counterparty to all trades. The clearing house in turn manages its credit exposure by trading on a fully collateralised basis through the system of initial and daily margin calls (see Section I.C.6.3.1). While it is theoretically possible that a clearing house might default on its obligations as counterparty, it has never actually happened and is generally regarded as a remote possibility. For OTC transactions, current exposure is defined as the current replacement value of the relevant derivative contract (after taking netting into account). Difficulties arise when it comes to future exposures because this involves the projection of the future value of the derivative conditional on the occurrence of a default while recognising the effects of any netting agreements which may be in place.

The future value of the derivative depends, in turn, on market

movements which cannot be accurately predicted in advance. For pragmatic reasons, most credit portfolio risk management models currently map future exposures at default into loan equivalent exposures. In this method, the random future exposure of an OTC derivatives transaction is mapped into a non-random exposure (possibly time-



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dependent). For credit risk assessment, the derivatives transaction is then treated as a loan with this exposure.

III.B.3.2

Pre-settlement versus Settlement Risk

In Section III.B.3.3 on exposure profiles we shall see that it is not unusual in OTC transactions for very large cash flows to change hands at the settlement of the transaction, even if the net value of the transaction is much smaller than these cash flows. Thus it makes sense to distinguish between the risks that arise specifically at the settlement of these transactions, the settlement risk, and the pre-settlement risk of the transaction before its maturity. The techniques used for managing each of these risks vary. III.B.3.2.1

Pre-settlement Risk

This is the risk that the counterparty to a transaction (e.g. an OTC derivative transaction) defaults at a date before the maturity (settlement) of the transaction. Here, the existence of early termination clauses is crucial in the reduction of the exposure due to pre-settlement risk. The right of early termination is the credit risk equivalent of cutting your losses by closing out a lossmaking market exposure.

Usually, these clauses are incorporated in the master agreement

between the counterparties. They involve triggers due to failure to perform on this or a related contract, rating downgrade (usually to a class below investment grade), bankruptcy. Should one of these events occur, the contract is terminated and settled immediately, with final payment being the replacement value of the contract (i.e. the value of an otherwise identical contract with a non-defaulted counterparty). If the replacement value of the contract is positive to the defaulted counterparty, it will receive the final payment. Otherwise, the defaulted counterparty will have to make the final payment to the non-defaulted counterparty. In the latter case the defaulted counterparty may default on this final payment, too, which means that the non-defaulted counterparty will have to enter the bankruptcy proceedings with a legal claim amount of the final payment. Thus, the pre-settlement exposure is E(t ) = max{0; Replacement value at time t}.

(III.B.3.1)

Thus, the exposure for pre-settlement risk is only the replacement value of the derivative, and only if this value is positive to us. The exposure of the settlement risk on the other hand is roughly equivalent to the gross exposure of the transaction.



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III.B.3.2.2

Settlement Risk

Many financial transactions between two counterparties involve two simultaneous payments or deliveries for example, counterparty A delivers a bond and counterparty B delivers the purchase price for this bond (in a straightforward cash-trading transaction), or one counterparty delivers USD and the other counterparty delivers EUR (in a spot FX transaction). Similar final payments are made at the maturity of FX swaps (exchange of principal) or at the maturity of forward contracts. Settlement risk arises only at the final settlement of a transaction if there are timing differences between the two payments of the transaction. For example, an FX transaction may involve a payment in EUR by bank A, which is made at 9am in London, and a payment in USD by bank B at the same day, which must be made in New York. Because New York is five hours behind London, this payment will be made later than the EUR payment. Should bank B default after receiving bank As payment, but before making its own payment, bank A will have to try to recover its claim in the bankruptcy court. A famous example of settlement risk is the case of the German bank Herstatt, which on 26 June 1974 had taken sizeable foreign currency receipts in Europe but went bankrupt (it was shut down for insufficient capital by the German office for banking supervision) at the end of the German business day before it settled its USD payments in New York. The effect of settlement risk is that bank A has a very large exposure (the total notional amount of the transaction) but only for a very short period of time (a few hours). Compared to presettlement risk, the difference in exposure size can be very large. This risk can be mitigated by improving the clearing and settlement mechanisms, netting agreements to minimise cash flows (in particular, the cash settlement of price differences instead of physical delivery), or the introduction of a central clearing house which takes both sides payments in escrow. Other risk management tools for both settlement and pre-settlement risks are explained in Section III.B.3.4.

III.B.3.3

Exposure Profiles

III.B.3.3.1

Exposure Profiles of Standard Debt Obligations

Standard debt contracts usually have fairly straightforward exposure profiles. The simplest case is a bullet bond or a bullet loan with a fixed notional amount paid off at maturity of the loan (see Chapter I.B.2). The top panel of Figure III.B.3.1 shows the exposure profile over time of a tenyear, 5% coupon loan where we have assumed constant interest rates of 5%.



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Figure III.B.3.1: Bond exposure profiles Exposure profile for a ten-year fixed-coupon bond/loan with 100 notional amount, bullet principal repayment at maturity and an annual, fixed coupon of 5%. 120

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Exposure profile for a ten-year amortising loan with 100 notional amount, and annual payments of 12.95. Interest rates are constant at 5% . 120

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Whenever a payment is received, the exposure drops by the payment amount. This causes the characteristic sawtooth pattern in Figure III.B.3.1 and also in all other exposure profiles for assets with intermediate payoffs. Between payment dates the exposure increases smoothly. This reflects the increase in the time-value of the outstanding payments. For the coupon bond in the top part of Figure III.B.3.1, the largest payment is the final repayment of principal (with the final coupon), thus the exposure profile remains largely constant with a large drop in the exposure at maturity. A common approximation is to set the exposure constant until maturity, that is, to ignore the sawtooth pattern caused by the coupon payments Copyright ? 2004 The Authors and the Professional Risk Managers International Association


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and to use some average of the exposure level. Essentially this is equivalent to assuming that defaults only occur in the middle between two coupon payment dates. Not all loans have the full principal repayment at maturity of the transaction. Amortising loans, for example, spread the principal repayments over the life of the transaction in such a way that the total payment amount is the same at all payment dates. This yields the downward sloping exposure profile in the bottom part of Figure III.B.3.1. The exposure profile of amortising loans is slightly concave because, initially, less of the annual payments goes towards principal repayments than at later dates. III.B.3.3.2

Exposure Profiles of Derivatives

OTC derivative contracts such swaps, forward contracts or FX transactions have several special features that complicate the calculation of the corresponding exposure amounts. Derivatives often have an initial value of zero (or close to zero). This means that current exposure is a very bad measure of future exposure, which can vary dramatically with market movements. We must consider exposure profiles over time and cannot set exposure to a constant value as we did for fixed coupon bonds. These exposure profiles are not only time-dependent but also stochastic. Here, assumptions must be made in order to decide how multiple possible realisations of the exposure are to be captured by a single number. At future dates, an OTC derivative can have a positive or a negative value. But by equation (III.B.3.1) the exposure is floored at zero. By definition, exposure cannot become negative. The exposure is the amount that we would lose if the obligor defaulted. So, for instance, with an interest-rate swap, at each payment date either we owe the other party money or the other party owes us money. In the first case our exposure to default of the other party is zero; in the latter case it is positive. Thus we have to cope with an inherent nonlinearity here. The volatility of the underlying asset will also enter the calculation as it defines the range of likely future outcomes for the asset and its derivative. Derivatives usually involve payments by both contract parties. Thus, it makes a difference whether the two payments are netted (then the exposure is only to the net value of the payments), or whether both payment streams are considered in isolation (in which case we are exposed to the full payment stream of our counterparty). For the current exposure of a derivative with replacement value D(t) at time t, the starting point for the calculation of derivatives exposures is E(t ) = max{0; D(t)}.



(III.B.3.2)

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If we assume that D(t) can be determined at any time t, there is no inherent difficulty in equation (III.B.3.2) yet. The difficulties arise when we consider future exposures because exposure is a random variable and, since we usually measure credit risk over a long time horizon (such as one or five years), derivatives exposures may change significantly over the time horizon. Put another way, for a future point in time T > t we cannot predict D(T) with certainty given information at time t, so the exposure will be stochastic. The exposure measurement problem is now to find a curve E(t, T) which for all relevant T > t maps the distribution of the spot exposure E(T) to a single number: the forward-looking exposure E(t, T) given information at time t. Viewed as a function of time horizon T, the forward-looking exposure E(t, T) is called the exposure profile of the transaction. Figure III.B.3.2 illustrates this using the example of an interest-rate swap with a 5% swap rate from the point of view of the receiver of the floating rate. The surface plot shows the current exposure (i.e. the positive part of the value of the replacement value of the swap) at different points in time and for different possible levels of the interest rates. Unfortunately, seeing this from t = 0, we cannot predict future levels of interest rates, we can only make statements over the probability distribution of the interest rates at different time horizons. Figure III.B.3.2 shows the density of the interest rates at t = 2 and t = 10, and equal colours of the mesh correspond to equal quantiles of the interest-rate distribution. Clearly, uncertainty increases over time, so the density for t = 10 is wider than the density for t = 2. Because the underlying interest rate is random, the exposure is random, too: it could take any positive value. The bold blue line in Figure III.B.3.2 represents the 90% quantile exposure; that is, at any time in the future, the exposure will be below the blue line with 90% probability. The simplification is now to assume that this 90% point will be the realisation of the exposure, in the hope that this assumption while clearly wrong will at least be conservative.



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Figure III.B.3.2: Interest-rate swap exposure profile Exposure distribution and exposure profile for a ten-year interest-rate swap at 5%

Assigning a fixed number to the (stochastic) exposure at time T is a simplification, similar to assigning a single risk measure such as value at risk (VaR) to the full distribution of a random variable. But this simplification allows us to map the complex derivative contract to a loanequivalent exposure amount, so we can view the derivative as a loan with an admittedly rather strange amortisation schedule. Nevertheless, this clever mapping allows us to integrate derivatives contracts into a risk management system which otherwise could only accept loans. An obvious candidate for an exposure measure is the expected exposure E(t, T) = Et [(D(t))+],

(III.B.3.3)

where (x)+ = max{x, 0} is the common shorthand notation for the positive part of x and Et[穄 denotes the conditional expectation given information up to time t. Expected exposure has the advantage of being comparatively easy to evaluate. Essentially the problem is reduced to the calculation of the value of a European call option on D(T) see Section I.B.5.1. In many cases, this calculation can be done quite easily. Expected exposure has the additional advantage of being coherent in the sense of Artzner et al. (1999). That is, expected exposure is: non-negative: any derivative with a non-negative replacement value will generate a nonnegative exposure number;



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homogeneous: a positive scaling of the derivative position will result in the same positive scaling of the exposure measure; and subadditive: if you add two (or more) derivatives, the joint expected exposure is less than the sum of the individual exposures. Despite its nice properties, it is frequently felt that the expected exposure is not conservative enough because in many cases the actual exposure at default will be larger than the expected exposure. An alternative, yielding higher exposure values, is to use a quantile-based exposure measure. These measures are defined very much like VaR levels: the p-quantile exposure at time T is the level below which the exposure will remain with probability p. We denote this by qp*, so: Pt (D(T )

q *p ) = p

and

E(t, T) = q *p .

(III.B.3.4)

Frequently, the price of the derivative is a monotonic function of the underlying asset and the distribution of the underlying asset is assumed known (e.g. it is lognormal). In this case the pquantile exposure of the derivative can be calculated by: determining the corresponding quantile in the bad direction of the underlying asset, and calculating the value of the derivative at this level of the underlying asset. This is usually simpler than calculating the expected exposure at the same level. These calculations have much in common with the calculation of risk measures in market risk management: expected exposure is roughly equivalent to expected shortfall (see Section III.A.3.5.2) and quantile-based exposure measurement in very similar to VaR-based measures of market risk (see Chapter III.A.2). There are some practical difficulties, in that for credit exposure measurement these numbers must be calculated at several time horizons, and for longer time horizons, than the ones commonly used in market risk. Figure III.B.3.3 shows the exposure profile of an interest-rate swap contract (see Chapter I.B.4). The typical feature of most swap contracts is that they are initially entered at zero exposure: both sides of the swap have the same value. The potential exposure increases as we look further forward in time. This so-called diffusion effect is caused by the randomness of the underlying interest rate: the range of possible outcomes increases with time. But as time proceeds, the number and total notional amount of the remaining payments decrease, so that eventually the future exposure has to decrease back to zero. From a statistical point of view we can say that the worst time for the counterparty to default is around seven years into this ten-year swap. The potential for large credit losses at this point exceeds the potential for large credit losses at any earlier time because the range of likely market movements is greater. A large and favourable



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market movement could mean the loss of a very valuable asset should the counterparty default at this time. Beyond the seven-year point, the amortisation effect dominates over the diffusion effect, so potential losses are reduced. Figure III.B.3.3: 95% exposure profile of an interest-rate swap 160

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Figure III.B.3.4: 95% exposure profile of an FX swap 120

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Figure III.B.3.4 shows the typical exposure profile of an FX swap. An FX swap differs from an interest-rate swap in that there are no interim cashflows and therefore no amortisation of risk. The exposure profile reflects only the diffusion effect; the greater the passage of time, the greater the potential for a large favourable exchange rate movement. If the counterparty defaults when the swap is in profit, then a significant asset may be lost. The exposure profile tells us that statistically, the worst possibly time for default is towards maturity when the potential mark-to-



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market value is greatest. Note also that an FX swap, unlike an interest-rate swap, also has a final exchange of principal. Consequently the settlement risk at maturity is significantly greater. In Table III.B.3.1 we present a numerical example to illustrate the calculation of the exposure levels of a fixed-for-floating interest-rate swap from the point of view of the floating-rate receiver with a fixed swap rate of 5% and a notional of 100. We assume that the interest rates follow a lognormal random walk with drift zero and volatility 5%, and that the term structure of interest rates remains flat at all times. Table III.B.3.1(a) shows the calculation of current exposure for one simulated path of the interest rates (shown in the column floating). In years 1 and 2 interest rates rose above the fixed rate of 5% so that the swap has a positive value to us. The value of the swap is calculated by calculating the value of an annuity that pays a fixed payment of $1 for the remaining life of the swap, and then multiplying it by the difference between the current interest rate and 5%. (This calculation yields the value of the net payments if we were to enter an offsetting swap.) We see that the value of the swap turns negative from year 3 onwards. Thus, the current exposure is zero in those years, and it is only positive when the swap also has a positive value. Unfortunately, to carry out the calculations in Table III.B.3.1(a) we need to know the future development of the interest rates, so it cannot be done at time t = 0 to generate an exposure profile. The calculations to generate an exposure profile are shown in Table III.B.3.1(b). For the floating-rate receiver, the exposure is worst (highest) whenever interest rates are high. So, in order to calculate a 95% quantile exposure, we need to calculate the upper 95% quantiles of the interest rates; that is, for each year we calculate those levels of interest rates which are not exceeded with 95% probability. These levels we can calculate already at time t = 0; they are shown in the second column. The rest of the exposure calculation now proceeds as for the calculation of current exposure: we calculate the values of the annuities for these interest rates, and then the value of the swap. As the value of the swap is always positive to us, this is also the exposure in these scenarios. Again, we see the characteristic shape of a hump-shaped exposure profile similar to the one in Figure III.B.3.3.



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Table III.B.3.1: Exposure calculations for an interest-rate swap (a) Current exposure (simulated interest-rate path) Time

Floating

Fixed

0 1 2 3 4 5 6 7 8 9 10

5.00% 5.30% 5.17% 4.61% 4.55% 4.74% 4.61% 4.13% 3.99% 3.88% 4.06%

5% 5% 5% 5% 5% 5% 5% 5% 5% 5% 5%

Value of annuity 7.77 7.06 6.47 5.91 5.19 4.40 3.61 2.79 1.90 0.97 0.00

Value of swap 0.00 2.12 1.12 -2.31 -2.35 -1.13 -1.40 -2.42 -1.92 -1.09 0.00

Current exposure 0.00 2.12 1.12 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

(b) 95% quantile exposure profile Time 0 1 2 3 4 5 6 7 8 9 10

III.B.3.4

Floating: upper 95% quantile 5.00% 5.42% 6.08% 6.98% 8.19% 9.78% 11.87% 14.63% 18.27% 23.13% 29.62%

Fixed

Value of annuity

5% 5% 5% 5% 5% 5% 5% 5% 5% 5% 5%

7.77 7.03 6.24 5.44 4.64 3.86 3.09 2.34 1.60 0.84 0.00

Exposure (value of swap at 95% quantile) 0.00 2.96 6.71 10.77 14.80 18.44 21.23 22.53 21.24 15.27 0.00

Mitigation of Exposures

Whenever possible, both counterparties to an OTC derivatives transaction should engage in exposure-minimising agreements. This is a winwin situation for both sides because with welldesigned agreements, counterparty exposure (and thus credit risk) can be significantly reduced without needing economic capital and without large additional costs to the counterparties. In this section we consider the problem of aggregating all exposures to a particular counterparty, where netting and other mitigation agreements may be in place. Note that for the calculation of the total exposure profile with respect to a counterparty we have to consider a whole portfolio of derivatives transactions. This usually means that we will no longer be able to calculate expected exposures with closed-form solutions for European options on the underlying transaction, nor will we be able to identify the critical values for quantile-



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based exposures, as these values will now be a surface in a multidimensional risk space. Thus, the calculation of exposure profiles is reduced to Monte Carlo simulations or other numerical methods (see Chapter II.G). III.B.3.4.1

Netting Agreements

To illustrate the magnitude of the benefits of netting agreements, one need only look at the latest OTC derivatives market statistics released by the BIS in Table III.B.3.2. The presence of netting agreements reduces the gross credit exposure of all outstanding OTC derivatives contracts to 28% of the total market value of these contracts. This risk-reduction effect can be even stronger when the two counterparties are involved in a large number of transactions for example, if they are both market makers in certain OTC markets, or for a very active trader/hedge fund with his prime broker. If the transactions are hedged transactions (i.e. delta-hedged derivatives positions) then netting can eliminate almost all exposure. Table III.B.3.2: The effect of netting on global OTC derivatives exposure (USD bn) Total outstanding notional amounts

197,177

Total market value of outstanding contracts

6,987

Gross credit exposure after netting

1,986

Source: BIS market statistics, End of December 2003.

A necessary requirement before netting can be applied is the existence of a legally watertight netting agreement, which is usually embedded in a master agreement between the two counterparties. 114 A master agreement is a contract that sets the framework under which the counterparties can undertake derivatives transactions. Each individual transaction is economically independent, but it is governed by the same master agreement. Legally, all transactions are part of this one master contract. Thus, the master agreement can specify rules that apply across several transactions. The form of netting advocated by ISDA is called bilateral close-out netting: Once a credit event occurs on one of the counterparties, all transactions under the netting agreement are closed out. Closing out means that the current market value (replacement value) of all transactions is determined, then these numbers are netted, and then the net amount becomes immediately due.

114 The International Swaps and Derivatives Association (ISDA) was and is engaged in proposing and promoting

legislation to enable netting agreements, and such legislation has been adopted in most OECD countries. More legal background information can be found on the ISDAs web site (www.isda.org); see also Werlen and Flanagan (2002).



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Suppose counterparty A has entered swaps with counterparty B with current market values (to A) of +34, 97, +45 and +5. Then the net value of these four transactions is 13. If A defaulted, B would have a claim of 13 against A with which it would go to the bankruptcy court. At an assumed recovery rate of 40%, B would suffer a loss of 7.8. If counterparty B defaulted, then A would pay the net value 13 to B and would not have any further obligations. In this case, A does not have any immediate credit exposure to B, thanks to the existence of the netting agreement. If, on the other hand, there is no bilateral close-out netting agreement, then B would have claims of 97 against A which B would have to try and recover in bankruptcy court, while B would still have to perform on the other three contracts with a total market value of 84. Thus, Bs loss would be significantly larger: at 40% recovery B would lose 58.2. The situation without netting is also bad to A: while A had no net exposure with netting, without netting A does have significant losses. A would have to perform on the 97 swap but would have to try and recover some of the total value of 84 of the other three contracts. The credit losses to A would be 50.4. The reason for the efficacy of netting is that it prevents cherry-picking by the administrator of the defaulted counterparty. In many legislations, the administrator (receiver/bankruptcy court) can decide only to enforce contracts which are beneficial to the defaulted entity while sending all other contracts and obligations to the bankruptcy courts. The introduction of a master agreement ensures that all swap transactions are viewed as one contract, which can only be accepted or rejected as whole. From a risk managers point of view, the existence of a netting agreement with a certain counterparty allows him to aggregate all exposures with that counterparty and to consider only the net exposure that arises from these transactions. He is able to calculate an exposure profile with respect to the counterparty and not just with respect to an individual transaction. III.B.3.4.2

Collateral

Apart from netting agreements, collateral is another popular instrument to mitigate counterparty risk, in particular if one counterparty (e.g. a hedge fund) has a significantly more likely to default than the other (e.g. an investment bank). The International Swaps and Derivatives Association carries out regular surveys on the use of collateral and estimates that at the beginning of 2004, around USD 1017bn of collateral were used in OTC derivatives transactions. Comparing this to the USD 1986bn of credit exposure after netting found by the BIS, we see that about half of the non-netted credit exposure is managed by collateral. Another, less obvious form of collateralisation arises when derivatives are embedded into bonds or notes, as for example in credit-linked notes or equity-linked notes. By buying the note, the investor implicitly posts full



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collateral for all possible exposures that may arise from the derivative which is embedded in the note. This explains why this is a very popular structure for derivatives transactions with retail customers. Like netting, collateral also requires a corresponding legal document (again the ISDA publishes template contracts on this topic), but here the legal difficulties are less severe as collateral is a very old means of mitigating credit risk. Generally, as in netting agreements, the idea is to reduce credit risk by reducing exposure. In the case of collateral, the credit risk of the counterparty is enhanced by the credit risk of the collateral. The lender only suffers a loss if both counterparty and collateral default. Let us assume that hedge fund A and investment bank B want to enter an OTC derivatives transaction such as an interest-rate swap. To mitigate As credit risk, the parties enter a collateral agreement which specifies which assets may be delivered by A as collateral, and under which conditions B may ask for collateral, including the amount of the collateral. Typical collateral assets are cash (used in about 70% of cases) and government securities (about 15% of cases), but other assets are possible (e.g. equities). As the collateral may decrease in market value just when the counterparty defaults, non-cash collateral is not counted at its full face value but at less; that is, it is reduced by a haircut factor depending on the volatility of the collateral and its correlation with the underlying exposure. Over the life of the swap, A will have to deliver the required amount of collateral assets to B. The assets are legally still the property of A but they are under administration by B. If the collateral becomes insufficient (e.g. because of market movements or because of changes in Bs credit rating), B issues a margin call asking A to post additional collateral. If there is excess collateral, A may remove the collateral from the collateral account. If at the end of the transaction A has not defaulted, all unused collateral is returned to A. If on the other hand counterparty A misses a payment of the swap at some point, then B is entitled to sell some of the collateral assets to make the payment to himself. If an early termination event occurs because of a default of A, then B is allowed to sell the full collateral in the market and use the proceeds to cover the replacement value of the contract. If there are any remaining proceeds from the sale of the collateral, these are returned to A. If the collateral was not sufficient to cover the replacement value of the derivative, the remaining value will have to be recovered in the usual way.



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For exposure measurement, the value of the collateral must be deducted from the replacement value at risk at every time horizon. If the value of the collateral itself is random this introduces an additional level of complexity, but for the most common case of cash collateral we can simply subtract the collateral amount from the exposure value. Collateral management is the non-trivial task of keeping track of both the collateral a business has to post, and the collateral it should receive. The institution must ensure that it can always provide sufficient collateral to support its transactions, even if adverse market movements or a credit downgrade suddenly increase the collateral requirements. Thus, a collateral manager also has to monitor the credit quality of his own institution very closely. There are many famous examples of the failure of collateral management policy, including the downfall of the Long-Term Capital Management hedge fund. III.B.3.4.3

Other Counterparty Risk Mitigation Instruments

Limits. Counterparty exposure limits are not a mitigation instrument. Exposure limits are counterparty risk management devices that are used to avoid undue counterparty risk concentrations with respect to any particular counterparty. They are also used to avoid exposure to potential counterparties that are deemed to be insufficiently creditworthy; in such cases a limit is not granted, preventing any trades or lending activity which might result in credit exposure. In most financial institutions the credit committee is responsible for determining whether the institution

is

willing

to

expose

itself

counterparty/borrower, and to what extent.

to

performance

risk

for

any

particular

The existence/size of a limit will either be

determined on the basis of its own analysis, or can effectively be outsourced by relying on ratings from ratings agencies. Non-financial corporations also establish counterparty limits, but are more likely to rely on ratings agencies for credit assessments. Separate limits should apply for settlement and pre-settlement risk. Termination rights/credit puts. One or both counterparties may reserve the right to terminate the transaction should the credit risk of the other party worsen significantly. This allows the recovery of the exposure before the actual default event at a higher (often full) recovery value. Establishing third-party guarantees: A traditional and popular way to mitigate credit risk is to establish a guarantee for the exposure from a third party. This is very similar to collateralisation but its effect is to replace the default probability (as opposed to the exposure) of the counterparty with the combined default probability of the counterparty and the guarantor.



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Credit derivatives.

Counterparty exposures can also be managed with credit derivatives (see

Chapter I.B.6). Although it is technically possible to specify a credit derivative which pays off exactly the exposure at default of a specific counterparty, this is usually not done in practice for two reasons: first, this would entail a disclosure of the transactions that have been made with the counterparty; and second, because of the unusual nature of the payoff, the credit protection would be rather expensive. Nevertheless if one is willing to accept some residual exposure to the counterparty, a significant part of the counterparty exposure can be laid off using single-name credit default swap contracts. All counterparty risk mitigation instruments necessarily introduce legal risk and documentation risk. Collateral calls may not be made or fulfilled in time, and netting agreements and guarantees may be legally challenged. These risks are hard to quantify and must be carefully monitored. Furthermore, these risk mitigation instruments also have a cost in terms of administration costs, which must be justified by the reduction in risk. More details on credit risk management instruments can be found in Chapter III.B.6.

References Artzner, P, Delbaen, F, Eber, J-M, and Heath, D (1999) Coherent measures of risk, Mathematical Finance, 9(3), pp. 203228. Werlen, T J, and Flanagan, SM (2002) The 2002 Model Netting Act: A solution for insolvency uncertainty, Butterworths Journal of International Banking and Finance Law, April, pp. 154164.



241




242


III.B.4 Default and Credit Migration Philipp J. Sch鰊bucher 115 Having covered recovery rates and exposures in the previous chapters, this chapter discusses the modelling and measurement of the third determinant of default loss: the probabilities of default of individual companies (and, by extension, sovereign nations). After setting up a framework to describe and analyse default probabilities we will consider three different credit risk assessment methods: agency ratings, internal ratings and market-implied default probabilities. Finally, we compare the three approaches and discuss their differences.

III.B.4.1

Default Probabilities and Term Structures of Default

Rates In this section we introduce the terminology for describing default probabilities. The starting point of any representation of default probabilities is the one-period default probability depicted in Figure III.B.4.1. Figure III.B.4.1: A one-step default tree

Starting from node S0 at time T0, there are two possible outcomes at time T1 in Figure III.B.4.1: default (node D1) which is reached with the default probability p1; and survival (node S1) which is reached with the survival probability, 1 p1. Apart from the direct specification of the default probability p1 (or the survival probability 1 p1), we could also specify the odds of default. The odds of an event are defined as the ratio of the probability of the event (i.e. the default probability) to the probability that the event does not occur (i.e. the survival probability):

115 D-Math, ETH Zurich, Switzerland.



243


H1 :

p1 . 1 p1

(III.B.4.1)

Hence, given the odds of an event, one can recover the probability of the event as: p1

H1 . 1 H1

This definition differs only slightly from the odds that are routinely quoted by bookmakers on sports (and other) events. In fact, bookmakers usually quote 1/H1 and not H1. We can interpret H1 to be the odds of a fair bet on the event, in that if you bet $1 on the event that the obligor defaults then:

you lose your $1 if the obligor survives, or

you get 1/H1 if the obligor does indeed default.

The expected payoff of this bet is 1

p

p

, so the bet is fair.

From a modelling point of view the advantage of using odds is mostly of a technical nature: probabilities are restricted to lie between 0 and 1, while odds can take any value between 0 and infinity. For small values (such as for short-horizon default probabilities), odds and probabilities are almost equal, with odds being slightly larger. For example, if the default probability is p1 = 2%, then the odds of default are H1 = 2.0408%. Figure III.B.4.2: A schematic representation of default and survival over time

A complication in the representation of default probabilities arises when several points in time are considered. Figure III.B.4.2 shows a simple representation of default and survival over time in a model with three periods ([T0, T1], [T1, T2] and [T2, T3]) Over each period [Ti1, Ti], the obligor can default (with probability pi) or survive (with probability 1 pi).



244


By considering the individual periods in isolation, we can define local default probabilities (and local odds of default) as before. So, we can still analyse the situation in period [T1, T2] with default probability p2 and odds of default H2 = p2/(1 p2). But these quantities are now conditional default probabilities, that is, they are conditional upon survival until T1. They are only valid if we have reached the node S1. If a default has already occurred in the first period, then p2 and H2 have no meaning. These conditional default probabilities are also often called marginal default probabilities. If we now want to know the cumulative default probability until T2, that is, the probability of default at any point in time over [T0, T2], then we must take several possible paths across the tree into account, which all can lead to a default during the period [T0, T2]:

the obligor survives the first period ( S0

S ) and defaults in the second period S1

D ,

and

the obligor defaults in the first period ( S0

D ).

The first scenario has probability (1 p1 ) p2 and the second scenario has probability p1, so that the total default probability over [T0, T2] equals (1 p1 ) p2

p1

p2

p1

p1 p2 .

If we project further into the future it quickly becomes more convenient to consider cumulative survival probabilities because for survival we only have to consider one path across the tree. The cumulative survival probability over [T0, T3], for example, is given by the product of the marginal survival probabilities over the individual periods which are spanned by the interval [T0, T3], that is, P[Survival over [T0, T3]]= 1

p1 1

p2 1

p3

P T0 , T3 .

(III.B.4.2)

Clearly, the cumulative default probability over [T0, T3] equals one minus the cumulative survival probability over [T0, T3]. The general formula for the probability of Si

S j , that is, of going

from survival in Ti to survival in T j with i < j, is P ( Ti , T j ) (1 pi 1 )(1 pi 2 ) (1 p j ). Starting from the tree in Figure III.B.4.2, we can specify default probabilities (or odds of default) for future time periods, that is, we can specify a term structure of default probabilities. In many cases it may be desirable to increase the resolution of the term structure by inserting additional points in time, for example, going from yearly to quarterly, monthly or even daily intervals as this



245


will allow us to place the nodes of our tree exactly on the payment dates of the bonds or loans of the obligor under consideration. As the length of the time periods gets smaller, local default probabilities and odds of default should also decrease. For example, suppose we have an obligor with a 10% default probability over one year. To be consistent with this when moving to monthly time periods, we should assume that the monthly default probability is 0.874%. The constraint becomes stronger as we take smaller steps. For instance, when calculating the daily default probability that is equivalent to a 10% one-year default probability, even a small overestimation of the daily probability can lead to a one-year default probability that is much too large. In summary, to avoid strange limiting behaviour as the time period decreases we must reduce the term default probabilities in a coordinated way. A common assumption that always achieves a stable balance when time-steps are reduced is that the odds of default over a small time interval [Ti, Ti+1] are approximately proportional to the length of the time interval. Let us denote the length of the time interval by

= Ti+1 Ti. Then the

hypothesis may be written: Hi

r

i

where the proportionality factor is denoted by interval gets smaller and smaller (i.e., as of default get smaller, too.

T

Ti

i

,

(III.B.4.3)

Ti . Now, if we take the limit as the time

0 ), we leave

Ti

unchanged. But the odds H i

is the default probability per unit of time, evaluated at the

gridpoint which corresponds to time T. This quantity is known as the default rate, the default intensity or the default hazard rate at time T; it is the instantaneous probability of default in a continuous time setting. Suppose that, somehow, we are able to specify a default hazard rate function (t) for all t

0.

Armed with this function we can calculate survival and default probabilities from 0 until a given time horizon T. If we then let the interval length

go to zero, it can be shown that eventually the

survival probability will be P (0, T ) exp

T 0

t dt ,

(III.B.4.4)

and the probability of a default between time 0 and time T will thus be 1 P(0, T).



246


Conversely, suppose we are given a term structure of survival probabilities, that is to say, a set of probabilities P (0, T ) for different time horizons T. Then we could compute the corresponding default intensity function by differentiation. In fact: (T )

T

ln P (0, T ).

(III.B.4.5)

So, there is a correspondence between the term structure of survival probabilities (or default probabilities) and the default hazard rate function. We may, for instance, try to estimate a term structure of default probabilities using one of the methods described later in the chapter. Whatever the shape of this term structure and market-implied term structures can have quite strange shapes it is usually possible to find a default hazard rate function that reproduces this shape. 116 An important special case arises when the default hazard rate is constant, an assumption that is at the foundations of the CreditRisk+ model (see Chapter III.B.5). In this case, let us write the hazard rate as

0

Then by equation (III.B.4.4) the survival probabilities are just: P ( t ) exp

0

T .

(III.B.4.6)

This gives a simple and effective way to interpolate default probabilities between different dates. For example, suppose we choose survival probability is P (1) exp{

0

= 10.53%. Since by (III.B.4.6), 0

0

ln( P (1)) , the one-year

T } 90% so the one-year default probability is 10%. Also,

assumption (III.B.4.3) gives the six-month default probability as 5.13%. Similarly, over one month the default probability will be 0.87% and over one day it will be 0.029%.

III.B.4.2

Credit Ratings

The goal of any credit rating system is the accurate assessment of the credit risk of the obligor. Credit ratings are one of the most important tools to assess the likelihood that an obligor defaults and their use is actively encouraged by the new capital adequacy rules for credit risk proposed by the Basel Committee on Banking Supervision. The aim of a credit rating procedure is the accurate classification of obligors according to their credit quality, usually by specifying an estimate of the obligors default probability or by giving them a letter rating.

116The

only exception are term structures of survival probabilities which have jumps or which drop to

zero. Copyright ? 2004 The Authors and the Professional Risk Managers International Association


247


Large public rating agencies such as Standard and Poors, Moodys and Fitch classify obligors into a number of rating classes. 117 For example, Standard and Poors use the classes AAA, AA, A, BBB, BB, B, CCC, D, with the interpretation that lower classes (such as CCC) carry a higher risk of default over a given time horizon than higher classes (AAA or AA).

A classification into a finite set of classes is not necessarily the only output of a credit rating system. Many other systems directly produce estimates of the default probability of an obligor which can vary on a continuous scale, essentially from 0% to 100%. Such systems are typically based upon a quantitative and statistical model of the default risk of the obligor. Examples are KMVs expected default frequencies (see Chapter III.B.5) or Kamakuras default probabilities; most proprietary internal ratings models also fall into this class. III.B.4.2.1

Measuring Rating Accuracy

There are many different approaches to the problem of determining a credit rating, and it is important to have a methodology to judge the accuracy of the output of the rating procedure. Unfortunately, a credit rating can be correct but still unlucky (it correctly classifies obligors, but some good ones default and the bad ones do not). Conversely, it can be incorrect, but lucky. Distinguishing between these two possibilities can be difficult.

When classifying rating systems one might be tempted to say that the best rating system is the one that produces the true default probability. Unfortunately, the concept of the true default probability is very elusive: it depends on the information that is available, and it has no direct relation to observable quantities.

Imagine we had access to an ideal rating system, let us call it Crystal Ball. Crystal Ball only needs two classes to accurately classify all obligors: D and S. An obligor is classified as D if it will default, and as S if it will survive, and our Crystal Ball can do this without error. Having Crystal Ball, we no longer need default probabilities. There are only two trivial values for default probabilities: 0% (for the S class) and 100% (for the D class). Essentially, these are the only true default probabilities. In reality we do not have a Crystal Ball, so we specify default probabilities

117 Moodys bought KMV and formed Moodys KMV in 2003. Today, Moodys KMV provides both classical Moodys letter ratings and the expected default frequency (EDF) ratings of KMV. As these two rating systems are fundamentally different, we will refer to Moodys ratings when the letter ratings of Moodys KMV are meant, and we will refer to KMV ratings for the EDF equity-based default risk measures calculated according to the KMV methodology. Copyright ? 2004 The Authors and the Professional Risk Managers International Association


248


and/or intermediate rating classes to measure the degree of our confidence about the fact that the obligor truly belongs to class S or class D.

A common method to compare different rating models is to focus on their ability to accurately rank obligors according to their credit quality. This method ignores any possible concrete values that are given for the default probabilities, which can be an advantage if the system does not directly give us such probabilities but only qualitative rankings like the letter ratings of the public rating agencies.

Figure III.B.4.3 shows a cumulative accuracy plot (CAP) with which the accuracy of the risk ranking of rating models can be compared. The plot is constructed as follows. First, the obligors are ranked according to the default risk that the model assigns to them, starting with those with the highest default risk. The rank numbers will form the x-axis of the plot. The CAP now shows, for each number/percentile x, what percentage y% of the actually defaulted obligors can be found among the x% worst obligors according to the models ranking. Figure III.B.4.3: A cumulative accuracy plot



249


For example, the marked point of the blue CAP is found as follows. The basic data set contained 100 obligors, of which 10 defaulted during the next year. We now use our credit model to classify these 100 obligors based upon the data available at the beginning of the year and rank the obligors in order of decreasing credit risk (according to the models prediction). Next to that, we also keep track of the actual default/survival behaviour of the obligors. The marked value in the graph is at an x-value of 19, which means that we must consider the 19 worst obligors according to the model. Out of these 19 obligors, the 5th, 6th, 7th, 13th and 16th actually did default, so that we have captured 50% (5 out of 10) of the actual defaulters in these 19 worst obligors. Thus, the y-value at x = 19 is y = 50%. The same calculation is done for every x between 0 and 100. Clearly, any CAP plot must start at 0% (an empty set of zero obligors cannot contain any defaulters) and it must end at 100% (if you have all obligors, you also have all defaulters).

The CAP of the Crystal Ball model is the yellow line, and shows the best possible default prediction accuracy. The ten worst obligors of the Crystal Ball model are also the ten obligors that default in reality. Thus, the accuracy profile shows a very steep increase up to 100% explained (correctly classified) obligors already at the 10th ranked obligor. Beyond that, the CAP for the Crystal Ball model remains flat because there is nothing more to explain.

The other extreme is a random model in which the obligors are ranked completely at random, without any reference to their actual credit quality. With such an approach, we can expect to find 50% of the actual defaulters in the 50% (randomly chosen) worst obligors, 10% of the actual defaulters in the 10% worst, etc., because any randomly chosen subset of x obligors will on average contain 10x/100 defaulters. Thus, the proportion of actual defaulters should be proportional to the proportion of the number of obligors selected, and this purely random credit risk ranking produces the diagonal line shown in pink in Figure III.B.4.3.

A good credit risk model will exhibit a CAP profile that is as close to the yellow line of the perfect model and as far away from the pink line of the purely random model as possible. It will concentrate the defaulters at the high risk scores at the beginning of the ranking, so that the slope of its CAP will be steep initially. The closer the CAP is to the yellow line, the better it is. The dark blue line in Figure III.B.4.3 shows the cumulative accuracy profile of an average default prediction model. If we compare two different rating models and the CAP of one model is consistently above the CAP of the other, then this is a sign of superior classification ability for the first model and it will be considered the better model.

A CAP gives a very good visual impression of the predictive performance of a credit risk model. Nevertheless, it is often useful to reduce the CAP to a single number, the accuracy ratio (AR). The Copyright ? 2004 The Authors and the Professional Risk Managers International Association


250


AR is the ratio of the area between the CAP of the credit risk model under consideration and the random models CAP (the diagonal), and the area between the perfect prediction models CAP (Crystal Ball) and the diagonal. (Here the x-axis is now also a percentage scale.)

The area between the ideal model and the random model is easily found to be (1 D)/2, where D is the fraction of defaulted obligors. Thus the accuracy ratio of any given model with CAP function CAP(x) is: 1

AR

0

CAP( x )dx 1 2

(1 D )

1 2

(III.B.4.7)

.

An accuracy ratio close to 1 means that the model is almost as good as the ideal model, while an accuracy ratio of 0 means that it is as bad as a purely random classification. A negative accuracy ratio is possible: it means that the model does an even worse job than a purely random credit ranking and that the models ranking should be inverted.

The CAP and the AR only measure the models ability to rank obligors, they say nothing about the models ability to give correct values for the probability of default. This is an advantage when we want to compare models that do not necessarily give us numerical probabilities of default at all (like the letter ratings of public rating agencies) or if we want to use the model as a support tool for yes/no loan decisions.

Yet, theoretically, a model which gives completely absurd values for the default probabilities may still have an almost perfect-looking CAP. As an extreme example, consider a model that assigns a default probability of 50.1% to the first ten obligors (the ones who do default in the end) and a default probability of 49.9% to the other 90 obligors. This model correctly ranks the obligors, thus it has a perfect CAP, but the assignment of almost 50% default probability to all obligors is strongly invalidated by the actual experience of only 10 defaults: 10 or fewer defaults out of 100 obligors would have a probability of essentially zero ( 10

17

) if the individual default

probabilities were indeed at 50%. 118 Despite this extreme example, in practical applications we may expect a model that does a good job in ranking the obligors also to provide good estimates for the default probabilities. We simply have to check that aspect of the model also.

118This is the value if defaults are independent. But even with reasonable default correlation, we will be able to

strongly reject the hypothesis that the default probabilities equal 50%.



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III.B.4.3

Agency Ratings

III.B.4.3.1

Methodology

The core business of public rating agencies such as Standard and Poors, Moodys and Fitch is the analysis of issuers of debt instruments, in terms of their credit quality and their ability to pay their investors. This analysis is summarised in a rating classification into one of several rating classes: Aaa to C for Moodys, and AAA to C for Standard and Poors and Fitch. Agency rating classifications are publicly available and are an important factor driving the decisions of many potential investors in these bonds.

Issuing a bond (instead of a loan) has the advantage that the issuer is able to borrow many small amounts from a large number of investors simultaneously. These investors will also be able to trade the bond on a secondary market. Unfortunately, a thorough analysis of an issuers credit quality is usually too expensive compared to the small investment amount of most individual investors. This might prevent them from investing in the bond in the first place, and the issuance of the bond may fail. The secondary market for the bond will also suffer from the same problem. But if the credit research is carried out by an independent and trustworthy agency which acts for all bond investors and which makes the findings of the research public, a duplication of research effort is avoided. Investors can invest many small amounts in many different bond issues and thus diversify their risk without spending unfeasible amounts on research. Hence a recognised public credit rating is almost indispensable for issuance in modern bond markets. Indeed, the cost of the rating is usually paid for by the issuer of the bond.

Because of their crucial position as one of the most important sources of information to investors, rating agencies often gain access to information that may be unavailable to ordinary investors. This information includes such things as direct interviews with the issuing firms management, internal planning, research and budgeting numbers, and the accounting data of the rated firm. These data are summarised by a credit analyst who then possibly with the aid of proprietary statistical models assigns a rating to the issuer and the bond issue. Naturally, rating agencies are rather secretive about the precise methodology used and the weightings of the factors that influence a rating. There is a portion of human judgement involved, but a surprisingly large fraction of the credit ratings can be reproduced using purely statistical models.

Public agency ratings are revised and updated at regular intervals to reflect changes in the credit quality of the rated obligor. Here, rating agencies try to strike a balance between rating stability and rating accuracy. As the rating is meant to represent a long-term view, the effects of general business cycle variations should average out. This rating through the cycle policy is not



252


uncontroversial because it can lead to delays in rating adjustments. It will also cause empirical difficulties when we try to back out estimates for default probabilities from agency ratings.

The Standard and Poors ratings from AA to CCC shown in Table III.B.4.1 may be modified by the addition of a plus or minus sign to show relative standing within the major rating categories. Obligors rated BB, B, CCC, and CC are regarded as speculative-grade investments, obligors rated BBB and better are regarded as investment-grade.

Besides the classical, fundamental ratings, quantitative credit rating providers have more recently entered the market using proprietary quantitative statistical models to produce an output which can be more directly interpreted as a probability of default. Examples are KMVs expected default frequencies and Kamakuras default probability estimates. These ratings typically are point-in-time ratings that do not attempt to smoothen business cycle effects.



253


Table III.B.4.1 Standard and Poors long-term issuer credit ratings definitions

An obligor rated AAA has extremely strong capacity to meet its financial commitments. AAA is the highest issuer credit rating assigned by Standard and Poors.

An obligor rated AA has very strong capacity to meet its financial commitments. It differs from the highest-rated obligors only in small degree.

An obligor rated A has strong capacity to meet its financial commitments but is somewhat more susceptible to the adverse effects of changes in circumstances and economic conditions than obligors in higher-rated categories.

An obligor rated BBB has adequate capacity to meet its financial commitments. However, adverse economic conditions or changing circumstances are more likely to lead to a weakened capacity of the obligor to meet its financial commitments.

An obligor rated BB is less vulnerable in the near term than other lower-rated obligors. However, it faces major ongoing uncertainties and exposure to adverse business, financial, or economic conditions which could lead to the obligors inadequate capacity to meet its financial commitments.

An obligor rated B is more vulnerable than the obligors rated BB, but the obligor currently has the capacity to meet its financial commitments. Adverse business, financial, or economic conditions will likely impair the obligors capacity or willingness to meet its financial commitments.

An obligor rated CCC is currently vulnerable, and is dependent upon favourable business, financial, and economic conditions to meet its financial commitments.

An obligor rated CC is currently highly vulnerable. Source: Standard & Poors.

III.B.4.3.2

Transition Matrices, Default Probabilities and Credit Migration

We have seen that public rating agencies only use letter ratings in order to classify obligors according to their credit quality. This may be adequate to make relative comparisons and maybe intuitive buy/hold/sell investment decisions, but in order to be able to use these ratings in a quantitative risk management system we need to map the letter ratings to numbers, to default probabilities. Essentially, we face the problem of backing out what the rating actually means in terms of default probability; the verbal definitions as they are given in Table III.B.4.1 are not sufficient for this.



254


Table III.B.4.2: Standard and Poors one-year average rating transition frequencies, 19811991 (percentages) AAA

AA A

BBB

BB

B

CCC

D

AAA 89.10

9.63

0.78

0.19

0.30

0

0

0

AA

0.86

90.10

7.47

0.99

0.29

0.29

0

0

A

0.09

2.91

88.94

6.49

1.01

0.45

0

0.09

BBB

0.06

0.43

6.56

84.27

6.44

1.60

0.18

0.45

BB

0.04

0.22

0.79

7.19

77.64 10.43

1.27

2.41

B

0

0.19

0.31

0.66

5.17

82.46

4.35

6.85

CCC

0

0

1.16

1.16

2.03

7.54

64.93 23.19

D

0

0

0

0

0

0

0

100

Rating agencies may be secretive about their methodology, but fortunately they publish a lot of historical data about both rating transitions and defaults of rated obligors. A typical summary of such data published by Standard & Poors is the transition matrix shown in Table III.B.4.2; similar transition matrices are also regularly published by other agencies (see, for example, Hamilton et al., 2002). In Table III.B.4.2 we have eliminated the not rated class and added the D rating class for defaulted obligors in the last row, using the assumption that a defaulted obligor remains in the default class with 100% probability.

Each row of a transition matrix gives the historical rating transition frequencies for the obligors of the corresponding rating class. For example, according to the BBB row of Table III.B.4.2, on average 0.06% of all BBB-rated obligors were upgraded to AAA in the course of one year, 0.43% were upgraded to AA, 6.56% were upgraded to A, 84.27% did not change their rating, 6.44% were downgraded to BB, 1.60% were downgraded to B, 0.18% were downgraded to CCC, and 0.45% defaulted.

A first glance at Table III.B.4.2 already confirms some stylised facts about rating transitions. First, default frequencies decrease with increasing rating classes, thus the ratings do indeed reflect an ordering according to default probability. Second, the frequencies of unchanged ratings (i.e. the values on the diagonal) are very high; this is an indicator of the rating stability aimed for by the agencies.

119 The no rating category has been eliminated.



255


As they stand, the numbers given in Table III.B.4.2 are not probabilities, they are the realised frequencies of the transitions over the observation period. Thus, a zero entry (as in the AA

D cell)

does not mean that AA-rated obligors have a zero probability of default, it only means that no AA-rated obligor defaulted in the years 19811991. Generally, small transition frequencies are usually based upon a very small number of observations, so that we have to expect inconsistencies like zero probabilities. Also, realised frequencies can be non-monotonic. For instance, in Table III.B.4.2 the CCC-rated obligors had a higher upgrade frequency to A than BBrated obligors: 1.16% for C-rated obligors as opposed to only 0.79% for B-rated obligors! Clearly, transition frequencies can only be considered as noisy estimates of the true transition probabilities.

In order to use the information given in the transition matrix to calculate transition probabilities we represent the transition table as a matrix P ( Pij ) , 1 i , j

k , where the entry Pij in the ith

row and the jth column represents the transition probability from class i to class j over the time interval, and k is the total number of rating classes (including default). Having added the default class D row ensures that the transition probability matrix is actually a square matrix. Now, to calculate transition probabilities, we need the following two assumptions:

Time-invariance. The probability of a transition from rating class i at time t to rating class j at time T > t does not depend on the calendar dates t and T but only depends on time via the length T t of the time interval.

Markov property. Besides the length of the time interval, the probability of a transition from rating class i to rating class j only depends on the rating class that we come from (class i ) and the rating class that we go to (class j ), and on no other external variables.

The assumption of time-invariance is not an uncommon assumption in statistical analysis. Essentially it says that the future will be like the past. Although common, it is very restrictive as it rules out phenomena such as business cycle effects. Empirically, downgrades are very much more likely in recessions than in boom phases. The second assumption, the Markov property, is also restrictive. It essentially rules out the use of any information beyond the current rating class all we need to know in order to determine future transition probabilities is the current rating class. Besides disallowing other explanatory variables beyond the rating itself, the Markov property also implies that the history of the obligors rating is irrelevant as long as the current rating is known. This is in contradiction to empirically observed rating momentum. Recently downgraded obligors are much more likely to be downgraded again than other obligors of the same rating which have already been in that rating class for a long time. However, the Markov property implies an absence of rating momentum.



256


Both assumptions thus contradict empirical observation. In practice, the inaccuracies of these assumptions have to be weighed against the significant simplifications that they allow: under time-invariance and the Markov property we can speak of the one-year transition probability matrix P from which we can calculate the two-year and longer-horizon transition probability matrices. For example, the two-period transition probability matrix is reached as follows. The probability of going from rating class A to rating class BB over two periods equals the sum of the following probabilities:

p( A

p

AAA ) ( AAA

B)

, the probability of going from A to AAA in the first year times the

probability of going from AAA to BB in the second year,

+ p( A

p

AA ) ( AA

B)

, the probability of going from A to AA in the first year times the

probability of going from AA to BB in the second year,

+

+ p( A

D)

p( D

B)

, the probability of going from A to D in the first year times the

probability of going from D to BB in the second year. Essentially, we need to sum over all possible paths that the rating can take from A at t = 0 to BB at t = 2. Mathematically, pij( 2)

K n 1

pin pnj and the two-year transition matrix is therefore given by P( 2)

P P

that is, the matrix product of the one-year transition matrix with itself. Note that we need timeinvariance so that the transition probabilities in the second year are the same as the ones in the first year, and the Markov property is used when we calculate the transition probabilities for the second year (i.e. we just multiply the transition probabilities, irrespective of whether we multiply two downgrade probabilities, or a downgrade and an upgrade probability). Similarly, the transition matrix over n years is obtained from the one-year transition matrix by multiplication n times with itself: P( n )

Pn .

(III.B.4.8)

In summary, assuming time-invariance and the Markov property means that the one-year transition data given by the rating agencies can be used to calculate the transition probabilities (and default probabilities) for all time horizons, by a simple matrix multiplication.



257


If transition probabilities are sought for shorter time horizons than the one-year horizon given by the original transition matrix P, one may try to find a short-term (say, monthly) transition matrix A, that is consistent with P, or in other words a 1/n-period transition matrix such that A n

P.

In many cases the problem of finding such a short-term transition matrix usually does not possess an exact solution, but for most transition matrices found in practice highly accurate approximate solutions can be found numerically.

III.B.4.4

Credit Scoring and Internal Rating Models

Because the large public credit rating agencies have traditionally concentrated on larger, US-based corporate bond issuers, the set of obligors covered by public credit rating agencies is only a fraction (albeit an important one) of all obligors that make up a banks credit portfolio. Important missing categories of obligors are small and medium-sized businesses, many larger European, Japanese and Asian obligors, and of course the whole retail portfolio.

In order to assess the credit risk of such obligors, various statistical methods have been developed. Generally, when it comes to statistical models of default prediction, the choice of the inputs seems to be more important than the choice of the particular methodology, although both are frequently intertwined. Important variables that have been found to drive the default behaviour of obligors include:

balance-sheet data capturing the indebtedness of the obligor;

profits and free cash flows capturing the ability to pay;

the riskiness (volatility) of the business;

(if available) market data, such as the firms market capitalisation;

macro-economic data capturing the effects of the business environment on the obligor.

III.B.4.4.1

Credit Scoring

One of the earliest published credit scoring model goes back to Altman (1968), and was further developed in Altman et al. (1977). Credit scoring models usually rely on accounting ratios like the ones listed in Table III.B.4.3.



258


Table III.B.4.3: Key accounting ratios Some key accounting ratios, and the averages of these ratios over the firms in the data set used by Altman (1968). Bankrupt firms defaulted within one year. Accounting Ratio

Bankrupt

Non-bankrupt

Working Capital / Total Assets

X1

6.10%

41.40%

Retained Earnings / Total Assets

X2

62.10%

35.50%

Earnings before Interest and Taxes/ Total Assets

X3

31.80%

15.30%

Market Capitalisation / Debt

X4

40.10%

247.70%

Sales / Total Assets

X5

150%

190%

Altman proposes to calculate for each obligor the score function Z(i )

1.2 X

i

1.4 X

i

3.3 X

i

0.6 X

i

1.0 X

i

(III.B.4.9)

the value of which is the so-called Z-score of the ith obligor. The variables X 1( i ) ,..., X 5( i ) are the values of the accounting ratios for the ith obligor. Table III.B.4.3 shows the definitions of these ratios and their averages in Altmans data set. Altman proposes to use the Z-score to make loan decisions. If the score function has a value less than the cutoff score of 1.8 then the obligor is likely to default and a loan is to be denied. The score can also be used to rank obligors: a higher score is a sign of better credit quality.

The variables that are used to explain the default behaviour of the obligor are similar for most internal credit scoring models. Indeed, accounting ratios are typically included to measure such things as: indebtedness (for instance, using the ratio of market capitalisation to debt), cash flow available for debt service (for instance, via earnings before interest and taxes), profitability (for instance, using the retained earnings to total asset ratio), as well as numbers to measure earnings stability and the debt service load.

The scoring weights (1.2, 1.4, 3.3, 0.6, 1.0) and the cutoff level of 1.8 in (III.B.4.9) were estimated using a statistical method. The training set lists the values of each of the variables Xi for a set of obligors and, of course, the information whether the obligors eventually defaulted or whether they survived. When choosing the training set it is important to take care that it is diverse, that it



259


is representative of the real world, and that it contains a sufficient number of defaulted obligors. In particular, one has to be very careful if only internal data of a bank is used to estimate a scoring model because this data set will only contain obligors who have already been pre-selected according to the existing lending criteria. If, for example, the existing criteria are very strict regarding earnings before interest and taxes (EBIT), then the internal data will contain very few obligors with bad EBIT numbers and the scoring model may mistakenly conclude that EBIT was not relevant to default prediction. Figure III.B.4.4: An example of credit scoring Defaulted obligors are shown as red points, obligors which survived as blue points. The green line shows the points where the score is exactly at the cutoff level.

The scoring weights and the cutoff level in (III.B.4.9) are chosen to maximise the number of correctly classified obligors. Figure III.B.4.4shows the principle for the case with only two accounting ratios. The red cloud of points are the defaulted obligors in the training set, the blue cloud are the surviving obligors. It is not surprising that we have more and more survivors, the further we go into the northeast direction of higher earnings and lower indebtedness. For a given cutoff level, the score weights define a line in this graph which consists of the points that have a score of exactly the cutoff score. In Figure III.B.4.4, the green line shows this line for the optimal cutoff. Any points above that line will have a higher score than the cutoff (and thus will be accepted), any points below it will have a lower score (and will be rejected).



260


III.B.4.4.2

Estimation of the Probability of Default

The two most common models for the estimation of default probabilities are logit and probit models. In the probit model, it is assumed that the default probability of obligor i can be written as: pi

where

X

i N

X Ni

(III.B.4.10)

is the cumulative normal distribution function,

n

are parameters that are estimated

statistically, and X n( i ) are given accounting ratios and other explanatory variables for obligor i, with n = 1, , N.

The structure of equation (III.B.4.10) is very similar to a scoring model like (III.B.4.9). Essentially, we are mapping from the Z-scores to default probabilities by using the cumulative normal distribution function. However, note that the scores have a different interpretation in the two models: In the probit model (III.B.4.10) high scores are mapped to high default probabilities. In a scoring model such as (III.B.4.9) high scores are mapped to low default risk.

Thus high scores are good in the scoring model, but bad in the probit model. The estimated parameters of the two models are therefore quite different, even when the same explanatory variables used in the two models.

One advantage of probit models is that we can tell a story about how defaults occur in this setup. Let us define the credit index for the ith obligor as X

where

(i )

i N

X Ni

i

(III.B.4.11)

is a standard normally distributed noise component. If we assume that obligor i

defaults if his credit index drops below zero, then the default probability of obligor i is exactly equal to the pi given by the probit model. Thus, there is a natural link from probit models to credit portfolio models like CreditMetrics (see Chapter III.B.5).

The logit model differs from the probit model only in that it does not use the cumulative normal distribution function to map from scores to default probabilities but uses a different function, the logistic transformation. Here, the default probability of obligor i is set equal to



261


pi

1 exp{

1 Xi

N

X Ni

(III.B.4.12)

The logit model is slightly easier to estimate than the probit, but generally results from logit and probit models do not differ much. III.B.4.4.3

Other Methods to Determine the Probability of Default

Besides the rating models presented above, a number of other models are used to assess the credit quality of individual obligors. The most important of these is the KMV model. This is explained in detail in Section III.B.5.6.

In a gross simplification, the KMV model can also be viewed as a scoring model where the accounting ratio is the distance-to-default. But, this approach is special because the distance-todefault contains both market information (the market capitalisation of the firm) and volatility information. Both market and volatility information are usually not found in classical accounting ratios. From a purely practical point of view, the large historical database underlying the model is valuable, in particular the empirical transformation that KMV uses to reach a default probability estimate for a given distance-to-default.

Another class of potentially powerful models for credit classification are neural networks and other expert systems from artificial intelligence research. Generally, such models have been found to be equally powerful compared to good statistical models (like logit or probit models), but their acceptance in practice has been hindered by their complex nature, which is essentially a black box to the credit officer.

III.B.4.5

Market-Implied Default Probabilities

The credit default tree presented in Figure III.B.4.2 is a perfectly adequate model to price simple credit-sensitive instruments such as corporate bonds or credit default swaps (CDSs), provided that the necessary conditional default probabilities are already specified. In this section we will turn the model around. Instead of determining prices for given set of parameters, we now determine parameters for the prices of a set of benchmark securities, the calibration securities. The default probabilities that are reached in this calibration exercise are called market-implied default probabilities.

This approach rests on several assumptions. First, we assume that there is information to be found in the prices. The prices of the defaultable bonds/CDSs must be meaningful, that is to say, liquid and not unduly affected by external factors beyond default risk (such as taxes). Also, the peculiarities of different markets enable the participants to incorporate their information into the



262


prices to different degrees. For example, while it is easy to short credit risk in the CDS market, it can be difficult to short defaultable bonds. Therefore, calibration securities should be taken from markets with similar conditions. Second, it is necessary that all calibration instruments are subject to the same type of credit risk, that is, they reference the same obligor (and have the same default definition in the case of CDSs), and they have the same recovery rate and seniority class in default. We also need to know the value of the recovery rate (or at least its average).

Third, we use the risk-neutral pricing paradigm, that is, we price a security by taking the discounted expected value of its payoffs. Thus, the probabilities that we will reach are pricing or martingale measure probabilities. It is important to realise that these are different from historical probabilities because they are loaded with risk premia. This issue will be discussed in the next subsection.

Finally, we assume that movements of risk-free interest rates and defaults are independent. This is mostly a technical assumption, which significantly simplifies the analysis. In normal situations this correlation has only a second-order effect on the resulting default probabilities. III.B.4.5.1

Pricing the Calibration Securities

As a running example in this section we consider the problem of calibrating a term structure of default probabilities to the bond prices shown in Table III.B.4.4. Table III.B.4.4: Calibration securities Prices of five corporate bonds issued by Daimler Chrysler NA Holding. Trade date: 17 November 2003, effective date 19 November 2003. Coupons are paid annually, currency is USD, notional is 100. No. Dirty Price Coupon Maturity 1 105.46 4.5 03-01-2005 2 106 5.75 23-06-2005 3 105.27 4.62 10-03-2006 4 100.84 3.75 02-10-2006 5 109.46 5.62 16-01-2007 We can represent the prices of the calibration securities with two elementary types of (hypothetical) building-block securities: defaultable zero-coupon bonds which only pay in survival, and a recovery security which pays at the time of default. Specifically, we denote by B (0, T ) the value at time t = 0 of receiving $1 at time T in survival (and nothing if default occurs



263


before T), and by E(0, T ) the value at time t = 0 of receiving $1 at default, if the default occurs before time T.

The value of a defaultable coupon bond with coupon payment dates Tk, k = 1, , K, coupon amount c and recovery rate R can now be written as 120 V Bond

cB (0, T1 ) cB (0, T2 )

(1 c )B (0, TK ) R E(0, TK ).

(III.B.4.13)

Here, the first summation represents the value of the promised coupon payments and the final repayment of principal, and the final term represents the value of the recovery received at default, with R denoting the recovery rate.

For example, the first bond in Table III.B.4.4 has coupon payments of c = $2.25 at T1 = 3 January 2004 and T2 = 3 January 2005, and the principal repayment of $100 at T2. Furthermore, we assume that the recovery rate is 40%; thus we will have an additional payoff of R = 40 at default, if a default occurs before T2 .

Similarly, we can represent the prices of the other four bonds as discounted values of the principal, coupons and recovery cash flows. For all bonds together, we have promised cash flows on the following payment dates: 3 January 2005 16 January 2005, 10 March 2005, 23 June 2005, 2 October 2005, 19 November 2005, and 3 January 2006.

These are the maturities of the

defaultable zero-coupon bonds that we need to represent the cash flows of our calibration securities.

120We ignore day count conventions and other technical adjustments. Notional amounts are normalised to 1.



264


Figure III.B.4.5: The term structure of risk-free interest rates USD forward Libor rates on 17 November 2003. 6.00%

5.00%

4.00%

3.00%

2.00%

1.00%

0.00% 0

1

2

3

4

5

6

7

8

9

10

Maturity

The value of a protection-buyer position in a CDS on the reference credit with CDS rate s and the same payment dates Tk, k = 1, ... , K, is: V CDS

s B(0, T1 ) ... B(0, TN )

(1 R ) E

TK

(III.B.4.14)

Here, the first sum represents the value of the fee stream (a liability to the protection buyer), and the last term represents the value of receiving 1 R at default of the obligor. The market CDS rate is chosen such that the value of the CDS position is zero: s

(1 R )

E(0, TK ) . B(0, T1 ) ... B(0, TN )

(III.B.4.15)

¯ (0, T) and E(0, T), we Having reduced the pricing problems to the problem of finding prices for B now have to represent these prices in terms of the survival probabilities defined in Section III.B.4.1. The price of a defaultable zero coupon bond is easily seen to be B (0, T ) B(0, T ) P (0, T ) ,

(III.B.4.16)

where B(0,T) is the price of a default-free zero-coupon bond with maturity T, and P(0,T) the survival probability until T. It can also be shown that: E 0, T

p1B 0, T1

p 2 P 0, T1 B 0, T2

...

pk P 0, Tk

1

B 0, Tk .



(III.B.4.17)

265


Here, pk P (0, Tk 1 ) represents the probability of surviving until time Tk1 (which is P (0, Tk 1 ) ), and then defaulting in the period [Tk1, Tk] (which occurs with probability pk). This takes us to the default branch at time Tk. Then, after discounting with B(0, Tk ) , we reach the formula above. Example III.B.4.2 In the example given above, we use the USD forward Libor curve as the risk-free interest rates. This yields the prices of the default-free zero-coupon bonds B(0, T ) at the payment dates of the calibration bonds. In equations (III.B.4.7) and (III.B.4.8) we also need default and survival probabilities for several different time horizons. Here we made the assumption that these probabilities are given by a constant default-intensity model as in Section III.B.4.1, that is, that P ( T ) exp{

0

T } , where

0

is a parameter that we need to find. Finally, we assume a common

recovery rate of 40%.

Given all these assumptions, the only remaining degree of freedom that we can use to match model prices to market prices is the default intensity that a value of

0

0.

By using Excels Solver routine we find

= 1.0029% minimises the squared pricing errors of the bonds model prices

relative to the market prices. The results are presented in Table III.B.4.5. The default payoffs row shows the values of the potential recovery payoffs. The survival payoffs row shows the values of the promised coupon and principal payments of the bonds. The sum of these two yields the model price of the bond. Table III.B.4.5: Model prices of the calibration securities The model prices of the bonds of Table III.B.4.4 under the assumption of a constant default intensity 0 = 1.0029%, a recovery rate of 40% and using the default-free interest rates shown in Figure III.B.4.5. Coupon Maturity Market Price Default Payoff Survival Payoffs Model Price

4.5 5.75 03-01-2005 23-06-2005 105.46 106 Calibration Model Prices 0.4482 105.0228 105.4710

0.6289 105.3767 106.0056

4.62 10-03-2006 105.27

3.75 02-10-2006 100.84

5.62 16-01-2007 109.46

0.8966 104.5247 105.4212

1.1020 99.2955 100.3975

1.2055 108.5792 109.7847

We have introduced a variety of ways to represent default probabilities in Section III.B.4.1. Similarly, there are several ways to represent the prices of defaultable securities. A particularly simple representation can be reached for the fair CDS rate (III.B.4.15) if the odds of default Hk are used: s

(1 R )

wH

wK HK



(III.B.4.18) 266


That is, the CDS rate s equals the loss on default (1 R) times a weighted average of the odds of default Hk where the weights of the average are wk

B(0, Tk ) . B(0, T1 ) ... B(0, TK )

Clearly, these weights are non-negative and sum to 1.

In the particularly simple case of constant odds of default (i.e. if all Hk take the same value H) we have s =(1 R)H.

(III.B.4.19)

Hence, if we equate the odds of default with the default hazard rate (a very accurate approximation) we can say that the CDS rate equals the loss given default times the default hazard rate. III.B.4.5.2

Calculating implied default probabilities

Backing out an implied default hazard rate from a single CDS quote is very straightforward, given equation (III.B.4.19). If we observe a CDS spread s in the market, then the corresponding implied default hazard rate is reached by solving equation (III.B.4.19) for the hazard rate: H

s /(1 R ).

Example III.B.4.3 In the case of Daimler-Chrysler, the quote for a CDS with five years maturity on 17 November 2003 was 108.13bp. According to the formula given above, the implied default hazard rate is 1.8% (at an assumed recovery rate of 40%). It is not unusual that implied default intensities from CDSs are higher than the corresponding implied default intensities from bond prices. This effect is partly due to the fact that the embedded delivery option of a CDS makes the recovery rate with a CDS smaller than the recovery rate of a bond, and partly caused by market imperfections in the bond markets.

In a general situation, we may have CDS quotes for several maturities or we may have market prices for different bonds with different maturities and different coupons. In order to find a set of implied default probabilities that is consistent with these prices and that simultaneously looks sensible, a numerical optimisation routine must be used. This procedure is quite similar to the bootstrapping procedures used to back out term structures of interest rates from default-free bond prices. More details can be found in Sch nbucher (2003). Generally, apart from introducing Copyright ? 2004 The Authors and the Professional Risk Managers International Association


267


a time-dependence in the default probabilities, the results are qualitatively similar to the simple result for the single CDS. That is, credit spreads are approximately equal to the default hazard rate times the loss given default.

III.B.4.6

Credit Rating and Credit Spreads

In order to compare implied default probabilities and historical default probabilities, it is instructive to study Figure III.B.4.6. In particular, let us take the year 1997. For that year, the dark blue line shows an average credit spread for US Baa-rated corporate bonds of 0.65%, which will yield an implied default hazard rate of 1.08% (assuming a recovery rate of 40%). This spread is measured as spread over Aaa-rated corporate bonds and not as spread over treasuries (in order to avoid tax and liquidity premia, which would be present in the treasury market). 121 The spread is a compensation for the default risk that we bear if we invest in Baa-rated bonds, so it is interesting to see if this compensation was adequate. If the realised default hazard rate equals the implied default hazard rate, then an investment in Baa-rated bonds will just break even. This is what we would expect to happen with risk-neutral investors and in the absence of market imperfections. We now investigate how the implied default intensities compare, historically, to actual default rates, and whether the compensation through credit spread was adequate for the credit risk incurred. In fact, it turns out that the compensation is much more than adequate. Imagine we had bought the total Baa-rated corporate bonds market (or an equally weighted fraction of it) in the year 1997 and held it to maturity let us call this portfolio Baa97. The value of the yellow line in 1997 tells us the rate of defaults that we would have suffered over one year, that is, 19971998. Clearly, this rate of defaults is almost zero! So, what if we had to hold the Baa97 portfolio for three years starting in 1997? 122 The blue line tells us that over 19972000 the annual default rate of the Baa97 bond portfolio was only 0.3%. This is still much less than the implied default rate, and even less than the spread. Even over a five-year horizon starting 1997 the historical default rate is only about 0.55% p.a.(see the value of the yellow line in 1997), much below the implied default rate. This situation repeats consistently without exception over the whole period from 1976 to 1997 (1997 is the last year in which we could calculate a five-year forward-looking default rate).

121Choosing Aaa instead of risk-free only makes our estimate more conservative. 122We want to hold until maturity in order to eliminate effects due to market price movements.



268


Hence, investing in Baa portfolios would always and without exception have outperformed an investment in the risk-free reference portfolio (which here is Aaa-rated bonds). Figure III.B.4.6: Implied default probabilities vs. historical default frequencies This figure shows, for the pool of Baa-rated US corporate bonds: implied default rates (black, for 40% recovery), credit spreads (dark blue, measured as spreads over Aaa, not as spreads over treasury), the default rate of the pool over the next year (yellow), over the next three years (blue), and over the next five years (green). 4.00

3.50

3.00

2.50

2.00

1.50

1.00

0.50

0.00 1975

1980

1985 ImplDefIntens

1990 1Y forw

3Y forw

1995 5Y forw

2000

Baa Spread

Other studies also indicate that there seems to be a significant discrepancy between implied default rates and historical default rates. Typically, implied default rates tend to be larger by a factor of 23. This situation has been termed the spread premium puzzle: why are spreads so much higher than seems to be justified by their actual credit risk component? Or is this an arbitrage opportunity? Several explanations have been put forward. First, there is risk aversion. An investment in Baa97 will lose money if a recession comes, but a recession is exactly the situation in which the average investor needs money. Therefore, the investor will demand a higher return in order to be compensated for this wrong-sided risk. Secondly, maybe the actual default risk was much higher than historical incidence, we just did not experience the truly bad scenario, but the possibility of this bad scenario was nevertheless priced into the spreads. Copyright ? 2004 The Authors and the Professional Risk Managers International Association


269


Third, maybe there were tax effects at work that made the Baa-rated bonds unattractive, or liquidity premia were demanded for investments in Baa97. Unfortunately, none of these explanations can fully explain the effect. Risk aversion is certainly present, but risk premia can never lead to a shift in spreads that amounts to a virtual arbitrage opportunity: that would be inconsistent with rational investor behaviour because even the most risk-averse investor could have picked up a seemingly risk-free excess return here. The second explanation concerning the unrealised Armageddon scenario also cannot explain the size of the effect. If this mysterious extremely bad scenario were to explain a sizeable proportion of the spreads, then it must also have a probability that is not too small compared to the individual default probability of an obligor. But if this is the case, why has this scenario never occurred so far? The liquidity argument is certainly valid if spreads over treasuries are considered. But here we are considering spreads over Aaa corporates which should have similar liquidity problems to Baarated bonds. Finally, the tax argument again does not hold for spreads over Aaa, it only has the possibility of being relevant if spreads over treasuries are considered, because treasuries are exempt from state taxes in the USA. Besides, the spread premium puzzle is also observed in markets which are not affected by US tax rules, for example in the markets for Eurobonds or for bonds of non-US issuers. So the spread-premium puzzle remains a puzzle. Maybe it is indeed a market imperfection. But maybe it has already disappeared: since mid-2003 spreads have decreased significantly compared to the spreads used in Figure III.B.4.6. It is quite possible that the market has finally reached a level where implied and actual default risk are approximately equal, or at least in a more realistic relationship to each other. Of course, we will only know this for sure after it is too late to make an investment decision.

III.B.4.7

Summary

Credit rating and the estimation and measurement of default probabilities are a classical problem of credit analysis, and one that never seems to be perfectly solvable. The classical solution to this problem is to rely on the rating assessment of an external rating agency essentially, this is reliance on expert advice. We have seen how these rating classifications can be translated into concrete numbers for default and survival probabilities over different time horizons.



270


Recent advances in computing power and (more importantly) the increasing availability of the necessary data in electronic form have made it possible to estimate default probabilities on a purely statistical basis. In many cases, such quantitative approaches are able to compete successfully with agency ratings, and frequently they are the only option when no agency rating is available.

Finally, we discussed methods to imply default probabilities from observed market prices of traded credit-sensitive instruments such as bonds and credit default swaps. While these methods are indispensable to assess the risk compensation that one should get for the type of credit risk under consideration, spread-implied probabilities usually differ significantly and systematically from statistical and historical default rates. This credit-spread puzzle remains an open question to date. Nevertheless, because they are systematically above historical default rates, spread-implied probabilities may be useful as very conservative estimates of default probabilities.

In practice, implied probabilities are the correct probabilities to use for pricing applications, because these probabilities already contain the risk premia that are paid for the credit risk contained in the calibration securities. Risk management, capital allocation and value-at-risk calculations, on the other hand, require historical probabilities because here the preferences and risk aversion can be added later on.

References Altman, E (1968) Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. Journal of Finance, 23(4), pp. 589609. Altman, E, Haldeman, R, and Narayanan, P (1977) Zeta analysis: a new model to identify bankruptcy risk of corporations. Journal of Banking and Finance, 1, pp. 3154. Hamilton, D T, Cantor, R, and Ou, S (2002) Default and recovery rates of corporate bond issuers. Special comment, Moodys Investor Service Global Credit Research, February. Sch鰊bucher, P J (2003) Credit Derivatives Pricing Models. Chichester: Wiley.



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272


III.B.5 Portfolio Models of Credit Loss Michel Crouhy, Dan Galai and Robert Mark 123 This chapter describes the main approaches to the modelling of credit risk in a portfolio context (credit value-at-risk), i.e. the credit migration approach, the contingent claim or structural approach, and the actuarial approach. It reviews the assumptions of the credit portfolio models and the pros and cons of each approach. Finally, it discusses the relationship between credit value-at-risk, economic capital and regulatory capital.

III.B.5.1

Introduction

In this chapter we review the main approaches to modelling credit risk. For each approach we explain the basic logic behind it, describe the data required and evaluate its strengths and weaknesses. The interested reader can find a more detailed description of the approaches in Crouhy et al. (2001). A bank should be concerned with the estimation of the risk of default of a specific creditor, since this is the basis for pricing a loan and charging the borrower with the appropriate interest rate. But, at the same time, the bank should be looking at the quality of its loan portfolio as a whole, since the stability of the bank depends to a large extent on the performance of its portfolio, and on the size of credit-related losses in the portfolio in a given period. Portfolio analysis may in turn affect the pricing of individual loans and the lending decision as each assets contribution to portfolio risk must be considered. Modelling credit risk and pricing risky loans or bonds is a complicated task. The factors that affect credit risk are many and varied. Some factors are exogenous or economy-wide, such as the level of interest rates and the growth rate of the economy. Other factors are endogenous, such as the business risk of the firm, its capital structure, and the flexibility of its production technology. A major consideration is whether to evaluate credit risk as a discrete event, and concentrate only on the potential default event, or whether to analyse the dynamics of the debt value and the associated credit spread, and to estimate its risk over the whole time interval to its maturity. Another important issue is the data sources that are available in order to assess credit risk. Can the analyst rely on accounting data, or are these too stale and subject to manipulation? To what extent are market data available, and then, to what extent are the markets efficient enough to convey reliable information? 123 Michel Crouhy is a Partner at Black Diamond, Dan Galai is a Professor at the Hebrew University and Principal at

Sigma P.C.M., and Robert Mark is CEO of Black Diamond.



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Before proceeding further, let us define some fundamental concepts. Default, in theory, occurs when the asset value falls below the value of the firms liabilities (Merton, 1974). Default, however, is distinct from bankruptcy. Bankruptcy describes the situation in which the firm is liquidated, and the proceeds from the asset sale are distributed to the various claim holders according to pre-specified priority rules. Default, on the other hand, is usually defined as the event that a firm misses a payment on a coupon and/or the reimbursement of principal at debt maturity. Cross-default clauses on debt contracts are such that when the firm misses a single payment on a debt, it is declared in default on all its obligations. Since the early 1980s, Chapter 11 regulation in the United States has protected firms in default and helped to maintain them as going concerns during a period in which they attempt to restructure their activities and their financial structure. Figure III.B.5.1 compares the number of bankruptcies to the number of defaults during the period from 1973 to 2004. The data are for North American public companies, but note that legal procedures in enforcing the bankruptcy procedures in the case of a default event vary quite substantially across jurisdictions (see J.P. Morgan, 1997, Appendix G). Figure III.B.5.1: Bankruptcies and defaults in North American public companies 1973Q1 to 2004Q1

140 120 100 80 60 40 20 0

Bankruptcies

Defaults



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Over the last few years, a number of new approaches to credit risk modelling have been made public. The CreditMetrics approach (which was initiated by J.P.Morgan and was spun off to RiskMetrics Inc.) is based on the analysis of credit migration, i.e. the probability of moving from one credit grade to another, including default, within a given time horizon which is usually one year. CreditMetrics estimates the full, one-year forward distribution of the values of any bond or loan portfolio, where the changes in values are related to credit migration only. The past migration history of thousands of rated bonds is assumed to accurately describe the probability of migration in the next period. The credit migration framework is reviewed in Section III.B.5.3. Tom Wilson (1997a, 1997b) proposes an improvement to the credit migration approach, CreditPortfolioView, by allowing default probabilities to vary with the credit cycle. In this approach, default probabilities are a function of macro-variables such as unemployment, the level of interest rates, the growth rate in the economy, government expenses and foreign exchange rates. These macro-variables are the factors which, to a large extent, drive credit cycles. This methodology is reviewed in Section III.B.5.4. The structural approach to modelling portfolio credit risk offers an alternative to the credit migration approach. Here, the economic value of default is presented as a put option on the value of the firms assets. The contingent claim approach in introduced in Section III.B.5.5 KMV Corporation, a firm that specialises in credit risk analysis, has developed a credit risk methodology and extensive database to assess default probabilities and the loss distribution related to both default and migration risks. KMVs methodology differs from CreditMetrics in that it relies upon the expected default frequency for each issuer, rather than upon the average historical transition frequencies produced by the rating agencies for each credit class. The KMV approach is based on the asset value model originally proposed by Merton (1974). KMVs methodology, together with the contingent claim approach to measuring credit risk, is reviewed in Section III.B.5.6. At the end of 1997, Credit Suisse Financial Products released CreditRisk+, an approach that is based on actuarial science. CreditRisk+, which focuses on default alone rather than credit migration, is examined briefly in Section III.B.5.7. CreditRisk+ makes assumptions concerning the dynamics of default for individual bonds or loans, but ignores the causes of default, contrary to KMV and CreditMetrics.



275


III.B.5.2

What Actually Drives Credit Risk at the Portfolio Level?

Banks, as regulated institutions, are very focused on the quality of their credit portfolio. Banks must assign regulatory capital against credit risk. The current regulation requires banks to assign regulatory capital against each loan obligation, usually 8% of the principal amount. Future regulation, as described in Chapter III.B.6, will allow for better differentiation among obligors based on their ratings. However, the regulators will also look at the quality of the loan portfolio, and the level of concentration by industry and region (Pillar II in the New Basel Accord). But beyond the formal regulatory requirements, banks are judged and evaluated by their shareholders, as well as by their customers, especially the depositors. Therefore, banks have strong incentives to monitor the risk of their assets, and in particular the risk of their loan portfolio. The profitability of most banks largely depends on the performance of the loans they granted in the past. So what are the major factors that affect the performance of the loan portfolio? It should be emphasised that performance has (at least) two dimensions: return and risk. The risk of a loan portfolio can be tricky to assess since a bank can show a nice profitability over a few years due to high interest charges and low default rates, and then, once a default event (or events) occurs on a major exposure, the bank can incur a substantial loss, instantaneously wiping out those profits. The first factor affecting the portfolio is the credit standing of individual obligors. One bank may concentrate on prime, investment-grade obligors, granting loans only to the best credits, with very low probability of default for any obligor. Another may choose to concentrate on riskier, speculative-grade obligors who pay a much higher coupon rate on their debt. The critical issue for both types of institution is to charge the appropriate interest rate to each borrower that compensates the lender for the risk it undertakes. The second factor is concentration risk, or the extent to which the obligors are diversified across geography and industries. A bank with corporate clients mostly in commercial real estate is considered to be riskier than a bank with corporate loans distributed over many industries. Also, a bank serving only a narrow geographical area can be devastated by a slowdown in the economic activity of that particular region. This leads to a third important factor that affects the risk of the portfolio: the state of the economy. During good times of economic growth the frequency of defaults falls sharply compared to periods of recession. There is a propensity for things to go wrong at the same time, usually at the trough of the economic cycle. In addition, periods of high default rates such as 20012002 are characterised by low recoveries that lead to high loss rates.



276


The quality of the portfolio can also be affected by the maturity of the loans. Usually, longer loans are considered riskier than short-term loans. Time diversification can reduce the risk of the portfolio by spreading maturities over the economic cycle, as well as reducing liquidity risk. Liquidity risk is defined as the risk that the bank will run into difficulties when refinancing its assets, for instance by renewing deposits or by raising money through issuing debt instruments, because the market dried up or prices increased sharply. Risk assessment of the portfolio is needed to determine how much economic capital should be allocated against unexpected credit losses. Therefore, the future distribution of the values of the loan portfolio must be estimated. This task is not at all straightforward and is much more complicated than estimating the value of a portfolio of market traded instruments such as stocks and bonds. The major obstacle lies in the estimation of the correlations among potential default events. While we have a lot of data on market traded instruments, we do not have data on nontraded debt instruments. The data problem is also aggravated by statistical issues, for instance that default correlations are not directly observable. To overcome some of the estimation problems, most approaches imply default correlations from equity correlations as in Section III.B.5.3.2. Still, the estimation problem is huge since so many pairs of cross-correlations must be estimated for a portfolio of obligors. For example, a small portfolio of 1000 obligors requires the estimation of 1000? 99/2 = 499,500 correlations. This last problem is circumvented by using a multi-factor or a multi-index statistical model. The rate of return for each firm or stock is assumed to be generated by a linear combination of a few indices. For example, the indices can be related to a country or an industry. This approach reduces the calculation requirements to merely estimating the correlations among pairs of indices. All the simplifying assumptions are used in order to estimate the portfolios credit value-at-risk (credit VaR). The distribution of the rate of return of the portfolio of obligors is estimated, and the credit VaR is derived from a percentile of that distribution. The credit VaR of a loan portfolio is thus derived in a similar fashion to market risk, except that the risk horizon is usually much longer. It is simply the distance from the mean to the percentile of the forward distribution, at the desired confidence level. However, the future point in time is typically one year for both regulatory and economic credit risk capital, whereas for market VaR the risk horizon is 10 days for regulatory capital (but again, usually one year for economic capital). 124

124 The choice of a risk horizon is somewhat arbitrary. It is usually one year as it corresponds to the planning cycle and

the average time it would require to recapitalise the bank if it were to suffer a major unexpected loss.



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Economic capital is the financial cushion that a bank uses to absorb unexpected losses, including those related to credit events such as credit migration and/or default (see Chapter III.B.6). Figure III.B.5.2 illustrates how the capital charge related to credit risk can be derived from the portfolio value distribution, using the following notation: P(p) = value of the portfolio in the worst case scenario at the p% confidence level FV = forward value of the portfolio = V0 (1 + PR) V0 = current mark-to-market value of the portfolio PR = promised return on the portfolio EV = expected value of the portfolio = V0 (1 + ER) ER = expected return on the portfolio EL = expected loss = FV EV. Figure III.B.5.2: Credit VaR and economic capital attribution

P%

P(p)

EV

Economic Capital

FV EL

Because the expected loss is priced in the interest charged on loans, it is not part of required economic capital. The capital charge is instead a function of the unexpected losses: Economic Capital = EV P(p) When the risk horizon is one year, credit VaR and economic capital are equivalent. The bank should hold reserves against these unexpected losses at a given confidence level, say 0.01%, so that there is only a 1 in 10,000 chance that the bank will incur losses above the capital level over the period corresponding to the credit risk horizon, say, one year. The choice of a confidence level is generally associated with some target credit rating from a rating agency such as Moodys or Standard and Poors. Most banks today are targeting a AA debt rating, which implies a probability of default of 35 basis points, which then corresponds to a confidence level in the



278


range of 99.9599.97%. This confidence level is also the expression of the risk appetite of the bank.

III.B.5.3

Credit Migration Framework

Credit migration is a methodology based on the estimation of the forward distribution of changes in the value of a portfolio of loan and bond-type products at a given time horizon, usually one year. 125 The changes in value are related to the migration, upwards and downwards, of the credit quality of the obligor, as well as to default. This approach is based on historical data of ratings of many bonds by a rating agency, or on the internal database of a bank. Forward values and exposures are derived from deterministic forward curves of interest rates. The only uncertainty in CreditMetrics relates to credit migration, i.e. the process of moving up or down the credit spectrum. Market risk is ignored in this framework as forward values and exposures are derived from deterministic forward curves. A typical portfolio distribution is shown Figure III.B.5.3 it is far from being normal, contrary to market VaR. While it may be reasonable to assume that changes in portfolio values are normally distributed when due to market risk, credit returns are by their nature highly skewed and fattailed. An improvement in credit quality brings limited upside to an investor, while downgrades or defaults bring with them substantial downsides. Unlike market VaR, the percentile levels of the distribution cannot be estimated from the mean and variance only. The calculation of VaR for credit risk thus demands a simulation of the full distribution of the changes in the value of the portfolio. The CreditMetrics risk measurement framework consists of two main building blocks: VaR due to credit for a single financial instrument; and VaR at the portfolio level, which accounts for portfolio diversification effects. The first step is to specify a rating system, with rating categories, together with the probabilities of migrating from one credit quality to another over the credit risk horizon. This transition matrix is the key component of the credit migration approach. The matrix may take the form of the historical migration frequencies published by an external rating agency such as Moodys or Standard & Poors, or it may be based on the proprietary rating system internal to the bank. A strong assumption made by CreditMetrics is that all issuers within the same rating class are

125 CreditMetrics approach applies primarily to bonds and loans, which are both treated in the same manner. It can be easily extended to any type of financial claims such as receivables, financial letters of credit for which we can easily derive the forward value at the risk horizon for all credit ratings. For derivatives such as swaps or forwards the model needs to be somewhat adjusted or twisted, since there is no satisfactory way to derive the exposure, and the loss distribution, within the proposed framework (since it assumes deterministic interest rates).



279


homogeneous credit risks: they have the same transition probabilities and the same default probability. Figure III.B.5.3: Comparison of the probability distributions of credit returns and market returns

Typical credit returns

Typical market returns

Portfolio Value

Second, the risk horizon should be specified. This is usually taken to be one year. The third step consists of specifying the forward discount curve at the risk horizon for each credit category. In the case of default, the value of the instrument should be estimated in terms of the recovery rate, which is given as a percentage of face value or par. In the final step, this information is translated into the forward distribution of the changes in the portfolio value following credit migration. III.B.5.3.1

Credit VaR for a Single Bond/Loan

For a given bond in the portfolio, we estimate the distribution of changes in the bond value over a one-year period. Therefore, we have to estimate the assumed values of the bond a year from now for all possible migration events. The most probable event is that the bond will maintain its rating by the end of the year (e.g. a BBB bond has a probability of 86.93% of retaining its BBB rating after a year see table III.B.5.1) Another migration event is that the bond will be downgraded by one notch, e.g. a BBB bond has a probability of 5.3% of being downgraded to BB within the year. For each credit migration event we use the relevant forward zero-coupon curve, estimated for the new possible rating of the bond. These rates serve as discount factors, by which the value of the future cash-flows from the bond are discounted in order to find the value of the bond at the end of the year for the new possible rating. In our example the rating categories, as well as the transition matrix, are chosen from an external rating system (such as the S&P transition matrix in Table III.B.5.1) or an internal rating system.



280


Table III.B.5.1: Transition matrix: probabilities of credit rating migrating from one rating quality to another, within one year Initial

Rating at year-end (%)

Rating

AAA

AA

A

BBB

BB

B

CCC

Default

AAA

90.81

8.33

0.68

0.06

0.12

0

0

0

AA

0.70

90.65

7.79

0.64

0.06

0.14

0.02

0

A

0.09

2.27

91.05

5.52

0.74

0.26

0.01

0.06

BBB

0.02

0.33

5.95

86.93

5.30

1.17

1.12

0.18

BB

0.03

0.14

0.67

7.73

80.53

8.84

1.00

1.06

0

0.11

0.24

0.43

6.48

83.46

4.07

5.20

0.22

0

0.22

1.30

2.38

11.24

64.86

19.79

B CCC

Source: Standard & Poors CreditWeek (15 April 1996)

In the case of Standard & Poors, there are seven rating categories. (It should be noted that the rating agencies supply more granular statistics where each rating category is split into three subcategories, e.g. A+, A and A for Standard & Poors rating category A) The highest category is AAA, the lowest CCC. Default is defined as a situation in which the obligor cannot make a payment related to a bond or a loan obligation, whether the payment is a coupon payment or the redemption of the principal. The bond issuer in our example currently has a BBB rating. The shaded row in Table III.B.5.1 shows the probability, as estimated by Standard & Poors, that this BBB issuer will migrate over a period of one year to any one of the eight possible states, including default. Obviously, the most probable situation is that the obligor will remain in the same rating category, BBB; this has a probability of 86.93%. The probability of the issuer defaulting within one year is only 0.18%, while the probability of it being upgraded to AAA is also very small, 0.02%. Such a transition matrix is produced by the rating agencies for all initial ratings, based on the history of credit events that have occurred to the firms rated by those agencies. Moodys publishes similar information. Although ten or twenty years ago these two rating agencies concentrated on US companies, they now cover tens of thousands of companies around the world. Transition matrices for Japan, Europe and other regions are now becoming available. However there are still some regions where historical default data are insufficient to estimate transition matrices. In such regions the KMV methodology, which does not rely on transition matrices, is often the method of choice. In the US, the transition probabilities published by the agencies are based on more than 20 years of data across all industries. But even these data should be interpreted with care since they only represent average statistics across a heterogeneous sample of firms, and over several business Copyright ? 2004 The Authors and the Professional Risk Managers International Association


281


cycles. For this reason many banks prefer to rely on their own statistics, which relate more closely to the composition of their loan and bond portfolios. The realised transition and default probabilities also vary quite substantially over the years, depending upon whether the economy is in recession or is expanding (see Section III.B.5.4). When implementing a model that relies on transition probabilities, one may have to adjust the average historical values shown in Table III.B.5.1, to be consistent with ones assessment of the current economic environment. A study provided by Moodys (Carty and Lieberman, 1996) provides some idea of the variability of default rates over time. Historical default statistics (mean and standard deviation) by rating category for the population of obligors that they rated during the period 19701995 are shown in Table III.B.5.2. Clearly the default rates become more volatile as credit quality deteriorates. Thus one should expect the elements of the transition matrix corresponding to low grade issuers to change considerably over time, whilst transition probabilities for high grade issuers are unlikely to change much. Table III.B.5.2: One-year default rates by rating, 19701995 One year default rate Credit rating

Average (%)

Standard deviation (%)

Aaa

0.00

0.0

Aa

0.03

0.1

A

0.01

0.0

Baa

0.13

0.3

Ba

1.42

1.3

B

7.62

5.1

Source: Carty and Lieberman (1996)

Now consider the valuation of a bond. This is derived from the zero curve corresponding to the rating of the issuer. Since there are seven possible credit qualities, seven spread curves are required to price the bond in all possible states (Table III.B.5.3). All obligors within the same rating class are then marked to market using the same curve. The spot zero curve is used to determine the current spot value of the bond. The forward price of the bond one year from the present is derived from the forward zero curve, one year ahead, which is then applied to the residual cash-flows from year 1 to the maturity of the bond.



282


Table III.B.5.3: One-year forward zero curves for each credit rating (%) Category

Year 1

Year 2

Year 3

Year 4

AAA

3.60

4.17

4.73

5.12

AA

3.65

4.22

4.78

5.17

A

3.72

4.32

4.93

5.32

BBB

4.10

4.67

5.25

5.63

BB

5.55

6.02

6.78

7.27

B

6.05

7.02

8.03

8.52

CCC

15.05

15.02

14.03

13.52

Source: CreditMetrics, J.P. Morgan

From Chapter I.B.2 we know that the one-year forward price, VBBB, of the five-year 6% coupon bond, if the obligor remains rated BBB, is

VBBB

6

6 6 1.041 1.04672

6 1.05253

106 1.05634

107.53

where the discount rates are taken from Table III.B.5.3. The cash-flows are shown in Figure III.B.5.4. If we replicate the same calculations for each rating category we obtain the values shown in Table III.B.5.4.

126

Figure III.B.5.4: Cash flows for five-year 6% coupon bond 0

1

2

3

4

5

6

6

6

6

106

VBBB

Time

Cash flows

(Forward price = 107.53)

126 CreditMetrics calculates the forward value of the bonds, or loans, including compounded coupons paid out during

the year.



283


Table III.B.5.4: One-year forward values for a BBB bond Year-end rating

Value ($)

AAA

109.35

AA

109.17

A

108.64

BBB

107.53

BB

102.00

B

98.08

CCC

83.62

Default

51.11

Source: CreditMetrics, J.P. Morgan

We do not assume that everything is lost if the issuer defaults at the end of the year. Depending on the seniority of the instrument, a recovery rate of par value is recuperated by the investor. These recovery rates are estimated from historical data by the rating agencies. Table III.B.5.5 shows the expected recovery rates for bonds by different seniority classes as estimated by Moodys. 127 In simulations performed to assess the portfolio distribution, the recovery rates are not taken as fixed, but rather are drawn from a distribution of possible recovery rates. The distribution of the changes in the bond value, at the one-year horizon, due to an eventual change in credit quality is shown in Table III.B.5.6 and Figure III.B.5.5. Table III.B.5.5: Recovery rates by seniority class (% of face value, i.e. par) Seniority Class

Mean (%)

Standard Deviation (%)

Senior Secured

53.80

26.86

Senior Unsecured

51.13

25.45

Senior Subordinated

38.52

23.81

Subordinated

32.74

20.18

Junior Subordinated

17.09

10.90

Source: Carty and Lieberman (1996).

Table III.B.5.6: Distribution of the bond values, and changes in value of a BBB bond, in 1 year Year-end

Probability of

Forward

Change in

127 Cf. Carty and Lieberman (1996). See also Altman and Kishore (1996, 1998) for similar statistics.



284


rating

state:

price: V ($)

p(%)

value: V ($)

AAA

0.02

109.35

1.82

AA

0.33

109.17

1.64

A

5.95

108.64

1.11

BBB

86.93

107.53

0

BB

5.30

102.00

5.53

B

1.17

98.08

9.45

CCC

0.12

83.62

23.91

Default

0.18

51.11

56.42

Source: CreditMetrics, J.P. Morgan.

Figure III.B.5.5: Histogram of the one-year forward prices and changes in value of a BBB bond

Frequency 86.93

5.95 5.30 Probability of State (%)

1.17

.33 .18 .12 .02 Default

CCC

51.11

83.62

98.08

-56.42

-23.91

-9.45

B

BB

BBB A

102.0 107.53

-5.53

0

AA AAA

....

109.35

....

1.82

This distribution exhibits a long downside tail. The first percentile of the distribution of

Forward Price: V

Change in value: V

V,

which corresponds to credit VaR at the 99% confidence level, is 23.91. It is a much lower value than if we computed the first percentile assuming a normal distribution for credit VaR at the 99% confidence level would be only

128 The mean, , and the variance,

follows:

2,

V. In that case

7.43. 128

of the distribution for V can be calculated from the data in Table III.B.5.5 as



285


The above analysis is the basis for the evaluation of the portfolio of loans, along the lines described above. CreditMetrics proposes to use Monte Carlo simulations to assess the credit risk of a bond/loan portfolio due to the large amount of factors that must be taken into consideration. III.B.5.3.2

Estimation of Default and Rating Changes Correlations

So far we have shown how the future distribution of values for a given bond (or loan) is derived. In what follows we focus on how to estimate the potential changes in the value of a portfolio of creditors, when the changes are due to credit risk only, and credit risk is expressed as the potential ratings changes during the year. One important factor in the portfolio assessment is the correlation between changes in the credit ratings and the default correlation for any two obligors. In reality, the correlations between the changes in credit quality are not zero, and the overall credit VaR is quite sensitive to these correlations. Their accurate estimation is therefore one of the key determinants of portfolio optimisation. Default correlations might be expected to be higher for firms within the same industry, or in the same region, than for firms in unrelated sectors. In addition, correlations vary with the relative state of the economy in the business cycle. If there is a slowdown in the economy, or a recession, most of the assets of the obligors will decline in value and quality, and the likelihood of multiple defaults increases substantially. The opposite happens when the economy is performing well: default correlations go down. Thus, we cannot expect default and migration probabilities to remain stationary over time. There is clearly a need for a structural model that relates changes in default probabilities to fundamental variables. CreditMetrics derives the default and migration probabilities from a correlation model of the firms assets. CreditMetrics makes use of the stock price of a firm as a proxy for its asset value, as the true asset value is not directly observable. (This is another simplifying assumption in CreditMetrics that may affect the accuracy of the approach.) CreditMetrics estimates the correlations between the equity returns of various obligors, then it infers the correlations between changes in credit quality directly from the joint distribution of these equity returns.

mean( V )

pi Vi i

0.02% 1.82 0.46 2

0.33% 1.64

variance( V )

pi ( Vi

... 0.18% ( 56.42 ) )2

i

0.02%( 1.82

0.46 ) 2

0.33%( 1.64

0.46 ) 2

... 0.18%( 56.42 0.46 ) 2

i.e. = 2.99. The 0.01 percentile of a normal distribution N( ,

2)

is

8.95

2.33 , i.e. 7.43.



286


The theoretical framework underlying all this is the option pricing approach to the valuation of corporate securities first developed by Merton (1974). The model is described in detailed in Section III.B.5.5 as it forms the basis for the KMV approach. In Mertons model, the firm is assumed to have a very simple capital structure; it is financed by equity, St, and a single zerocoupon debt instrument maturing at time T, with face value F, and current market value Bt. The firms balance sheet is represented in Table III.B.5.7, where Vt is the value of all the assets and

Vt

Bt ( F ) S t .

Table III.B.5.7: Balance sheet of Mertons firm Assets

Liabilities / Equity

Risky Assets: Vt

Total:

Debt:

Bt(F)

Equity:

St

Vt

Vt

Figure III.B.5.6: Distribution of the firms assets value at maturity of the debt obligation

Probability of default

F

VT

Default point

In this framework, default occurs at the maturity of the debt obligation only when the value of assets is less than the payment, F, promised to the bondholders. Figure III.B.5.6 shows the distribution of the assets value at time T, the maturity of the zero-coupon debt, and the probability of default (i.e. the shaded area on the left-hand side of the default point, F). Mertons model is extended by CreditMetrics to include changes in credit quality as illustrated in Figure III.B.5.7. This generalisation consists of slicing the distribution of asset returns into bands in such a way that, if we draw randomly from this distribution, we reproduce exactly the migration frequencies as shown in the transition matrices that we discussed earlier. Copyright ? 2004 The Authors and the Professional Risk Managers International Association


287


Figure III.B.5.7 shows the distribution of the normalised assets rates of return, one year ahead. The distribution is normal with mean zero and unit variance. The credit rating thresholds are calculated using the transition probabilities in Table III.B.5.1 for a BB-rated obligor. The area in the right-hand tail of the distribution, down to ZAAA, corresponds to the probability that the obligor will be upgraded from BB to AAA, i.e. 0.03%. Then, the area between ZAA and ZAAA corresponds to the probability of being upgraded from BB to AA, etc. The area in the left-hand tail of the distribution, to the left of ZCCC, corresponds to the probability of default, i.e. 1.06%. Figure III.B.5.7: Generalisation of the Merton model to include rating changes

Standard normal distribution for a BB-rated firm

Rating: Prob (%): Z-threshold (s)

Firm remains BB

B

Default CCC

1.00 8.84

1.06

Z

C C C

Z

B

BBB

80.53

Z

B B

-2.30 -2.04 -1.23

Z

B B B

1.37

AA

A

7.73

AAA

0.67 0.14

Z

A

Z

A A

2.39 2.93

Z

0.03 A A A

3.43



288


Table III.B.5.8: Transition probabilities and credit quality thresholds for BB- and A-rated obligors A-rated obligor Rating in one year

BB-rated obligor

Probabilities

Thresholds:

Probabilities

Thresholds:

(%)

( )

(%)

( )

AAA

0.09

3.12

0.03

3.43

AA

2.27

1.98

0.14

2.93

A

91.05

1.51

0.67

2.39

BBB

5.52

2.30

7.73

1.37

BB

0.74

2.72

80.53

1.23

B

0.26

3.19

8.84

2.04

CCC

0.01

3.24

1.00

2.30

Default

0.06

1.06

Table III.B.5.8 shows the transition probabilities for two obligors rated BB and A respectively, and the corresponding credit quality thresholds. The thresholds are given in terms of normalised standard deviations. For example, for a BB-rated obligor the default threshold is 2.30 standard deviations from the mean rate of return. This generalisation of Mertons model is quite easy to implement. It assumes that the normalised log-returns over any period of time are normally distributed with a mean of 0 and a variance of 1, and the distribution is the same for all obligors within the same rating category. If pDef denotes the probability of the BB-rated obligor defaulting, then the critical asset value VDef is such that:

p Def

Pr(Vt

VDef )

which can be translated into a normalised threshold ZCCC, such that the area in the left-hand tail below ZCCC is pDef.. 129 ZCCC is simply the threshold point in the standard normal distribution, N(0,1), corresponding to a cumulative probability of pDef. Then, based on the option pricing model, the critical asset value V Def which triggers default is such that ZCCC = d2. This critical asset value VDef is also called the default point. 130

129 See the Appendix for the derivation of the proof. In the next section we define the distance to default as the

distance between the expected asset value and the default point. 130 Note that d2 is different from its equivalent in the BlackScholes formula since, here, we work with the actual instead of the risk-neutral return distributions, so that the drift term in d2 is the expected return on the firms assets, instead of the risk-free interest rate as in BlackScholes. See Chapter I.A.8 for the definition of d2 in BlackScholes and Appendix 1 for d2 in the above derivation.



289


Note that only the threshold levels are necessary to derive the joint migration probabilities, and these can be calculated without it being necessary to observe the asset value, and to estimate its mean and variance. To derive the critical asset value V Def we only need to estimate the expected asset return

and asset volatility . Accordingly ZB is the threshold point corresponding to a

cumulative probability of being either in default or in rating CCC, i.e. pDef +pCCC, etc. We mentioned above that, as asset returns are not directly observable, CreditMetrics makes use of equity returns as their proxy. Yet using equity returns in this way is equivalent to assuming that all the firms activities are financed by means of equity. This is a major drawback of the approach, especially when it is being applied to highly leveraged companies. For those companies, equity returns are substantially more volatile, and possibly less stationary, than the volatility of the firms assets. Now, assume that the correlation between the assets rates of return is known, and is denoted by , which is assumed to be equal to 0.2 in our example. The normalised log-returns on both assets follow a joint normal distribution:

f rBB , rA ;

1 2

1

2

exp

1 2(1

2

)

rBB2

2 rBBr A

r A2

We can therefore compute the probability of both obligors being in any particular combination of ratings. For example, we can compute the probability that they will remain in the same rating classes, i.e. BB and A, respectively:

Pr

1.23 rBB

1.37, 1.51 r A

1.98

0.7365

where rBB and rA are the rates of return on the assets of obligors BB and A, respectively, assumed normally distributed as in Figure III.B.5.7.131 For any two obligors the joint probability of both obligors defaulting is

p1,2 Pr[V1 VDef 1 ,V2 VDef 2 ] where V1 and V2 and denote the asset values for both obligors at time t, and VDef1 and VDef2 are the corresponding default points. This joint probability may be calculated in exactly the same way as the migration probabilities were calculated above, i.e. using the bivariate normal distribution.

131 See Chapter II.E for details on how to compute joint probabilities when the two random variables have a bivariate

normal distribution.



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Given this joint probability of default, and the individual probabilities of default for each obligor, p1 and p2, the default correlation can be calculated as:

corr( Def 1, Def 2 )

p1, 2

132

p1 p2

p1(1 p1 ) p2 (1 p2 )

(III.B.5.1)

We can illustrate the results with a numerical example. If the probabilities of default for obligors rated A and BB are PDef(A) = 0.0006 and PDef(BB) = 0.0106, respectively, and the correlation coefficient between the rates of return on the two assets is = 0.2, then the joint probability of default is only 0.000054.133 Now, using equation (III.B.5.1), we find that the correlation coefficient between the two default events is only 0.019. This example, with asset return correlation of 0.2 but default correlation of only 0.019, is not unusual. Asset returns correlations are approximately 10 times larger than default correlations for asset correlations in the range of 0.20.6%. This shows that the joint probability of default is in fact quite sensitive to pairwise asset return correlations, and it illustrates how important it is to estimate these data correctly if one is to assess the diversification effect within a portfolio. It can be shown that the impact of correlations on credit VaR is quite large. It is larger for portfolios with relatively low-grade credit quality than it is for high-grade portfolios. Indeed, as the credit quality of the portfolio deteriorates and the expected number of defaults increases, this number is magnified by an increase in default correlations. III.B.5.3.3

Credit VaR of a Bond/Loan Portfolio

The analytic approach that we sketched out above for a portfolio with bonds issued by two obligors is not practicable for large portfolios. Instead, CreditMetrics implements a Monte Carlo simulation to generate the full distribution of the portfolio values at the credit horizon of one year. The following steps are necessary: 1. Derive the asset return thresholds for each rating category. 2. Estimate the correlation between each pair of obligors asset returns. 3. Generate return scenarios according to their joint normal distribution. A standard technique that is often used to generate correlated normal variables is the Cholesky decomposition. 134 Each scenario is characterised by n standardised asset returns, one for each of the n obligors in the portfolio.

132 See Lucas (1995).

133 If the default events were independent the joint probability of default would simply be the product of the two default probabilities, i.e. 0.0006 ? 0.0106 = 0.000064. 134 A good reference on Monte Carlo simulations and the Cholesky decomposition is Fishman (1997, p. 223)



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4. For each scenario, and for each obligor, map the standardised asset return into the corresponding rating, according to the threshold levels derived in step 1. 5. Given the spread curves, which apply for each rating, revalue the portfolio. 6. Repeat the procedure a large number of times, say 100,000, and plot the distribution of the portfolio values to obtain a graph such as Figure III.B.5.1. 7. Finally, derive the percentiles of the distribution of the future values of the portfolio to obtain the credit VaR and/or credit economic capital as in Figure II.B.5.2. Estimating VaR for credit requires a very large number of simulations as the loss distribution is very skewed with very few observations in the tail. In order to reduce substantially the number of simulations, say by a factor of 510, while maintaining the same level of accuracy, it is recommended for practical applications to implement credit portfolio models with the use of variance reduction techniques. Importance sampling is a technique which produces remarkable results for credit risk (Glasserman et al., 2000).

III.B.5.4

Conditional Transition Probabilities CreditPortfolioView

CreditPortfolioView is a multi-factor model that is used to simulate the joint conditional distribution of default and migration probabilities for various rating groups in different industries, and for each country, conditional on the value of macro-economic factors. CreditPortfolioView is based on the observation that default probabilities and credit migration probabilities are linked to the economy. When the economy worsens both downgrades and defaults increase; when the economy becomes stronger, the contrary holds true.

In other words, credit cycles follow

business cycles closely. Since the shape of the economy is, to a large extent, driven by macro-economic factors, CreditPortfolioView proposes a methodology to link those macro-economic factors to default and migration probabilities. It employs the values of macro-economic factors such as the unemployment rate, the rate of growth in GDP, the level of long-term interest rates, foreign exchange rates, government expenditures and the aggregate savings rate. Provided that data are available, this methodology can be applied in each country to various sectors and various classes of obligors that react differently during the business cycle sectors such as construction, financial institutions, agriculture, and services.

It applies better to

speculative-grade obligors whose default probabilities vary substantially with the credit cycle, than to investment-grade obligors whose default probabilities are more stable.



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Conditional default probabilities are modelled as a logit function, whereby the independent variable is a country-specific index that depends upon current and lagged macro-economic variables. That is: 135

P j ,t

1 Y 1 e j ,t

where Pj,t is the conditional probability of default in period t, for speculative-grade obligors in country/industry j, and Yj,t is the country index value derived from a multi-factor model . 136 In order to derive the conditional transition matrix the (unconditional) transition matrix based on Moodys or Standard & Poors historical data will be used. These transition probabilities are unconditional in the sense that they are historical averages based on more than 20 years of data covering several business cycles, across many different countries and industries. As we discussed earlier, default probabilities for non-investment grade obligors are higher than average during a period of recession. Also credit downgrades increase, while upward migrations decrease. The opposite holds during a period of economic expansion. We can express this in the following way:

P j ,t P j ,t P j ,t P j ,t where

1 in economic recession (III.B.5.2)

1 in economic expansion

Pj,t is the unconditional (historical average) probability of default in period t, for

speculative-grade obligors in country/industry j. CreditPortfolioView proposes to use (III.B.5.2) to adjust the unconditional transition probabilities in order to produce a transition matrix Mt that is conditional on the state of the economy: Mt =M(Pj,t/ Pj,t) where the adjustment consists in shifting the probability mass toward downgraded and defaulted states when the ratio Pj,t/ Pj,t is greater than one, and in the opposite direction if the ratio is less than one. Since one can simulate Pj,t over any time horizon t = 1, , T, this approach can generate multi-period transition matrices:

MT

t 1,...,T

M P j ,t / P j ,t .

(III.B.5.3)

135 Note that the logit function ensures that the probability takes a value between 0 and 1. 136 J.P. Morgan (1997) provides an example of multi-factor model in the context of credit risk modelling. For a review

of multifactor models, see Elton and Gruber (1995) and Rudd and Clasing (1988).



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One can simulate the transition matrix (III.B.5.3) many times to generate a distribution of the cumulative conditional default probability such as that shown in Figure III.B.5.8 for any rating over any time period. The same Monte Carlo methodology can be used to produce the conditional cumulative distributions of migration probabilities over any time horizon. Figure III.B.5.8: Distribution of the cumulative conditional default probability, for a given rating, over a given time horizon, T expected (average) default probability

Frequency (%)

99th percentile 1

0

CreditPortfolioView and KMV (described in Section III.B.5.6) base their approach on the empirical observation that default and migration probabilities vary over time. KMV adopts a micro-economic approach that relates the probability of default of any obligor to the market value of its assets. CreditPortfolioView proposes a methodology that links macro-economic factors to default and migration probabilities.

The calibration of CreditPortfolioView thus

requires reliable default data for each country, and possibly for each industry sector within each country. Another limitation of the model is the ad hoc adjustment of the transition matrix. It is not clear that the proposed methodology performs better than a simple Bayesian model, where the revision of the transition probabilities would be based on the internal expertise accumulated by the credit department of the bank, and an internal appreciation of the current stage of the credit cycle (given the quality of the banks credit portfolio). These two approaches are somewhat related since the market value of the firms assets depends on the shape of the economy; it would be interesting to compare the transition matrices produced by both models.

III.B.5.5

The Contingent Claim Approach to Measuring Credit

Risk The CreditMetrics approach to measuring credit risk, as described previously, is rather appealing as a methodology. Unfortunately it has a major weakness: reliance on ratings transition



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probabilities that are based on average historical frequencies of defaults and credit migration. As a result, the accuracy of CreditMetrics calculations depends upon two critical assumptions: first, that all firms within the same rating class have the same default rate and the same spread curve, even when recovery rates differ among obligors; and second, that the actual default rate is equal to the historical average default rate. Credit rating changes and credit quality changes are taken to be identical. Credit rating and default rates are also synonymous, i.e. the rating changes when the default rate is adjusted, and vice versa. This view has been strongly challenged by researchers working for the consulting and software corporation KMV. 137 Indeed, the assumption cannot be true since we know that default rates evolve continuously, while ratings are adjusted in a discrete fashion. (This lag is because rating agencies necessarily take time to upgrade or downgrade companies whose default risk has changed.) What we call the structural approach offers an alternative to the credit migration approach. Here, the economic value of default is presented as a put option on the value of the firms assets. The merit of this approach is that each firm can be analysed individually based on its unique features. But this is also the principal drawback, since the information required for such an analysis is rarely available to the bank or the investor. One has to estimate the total value, and the risk (e.g. the volatility) of the firms assets. The option pricing approach, introduced by Merton (1974) in a seminal paper, builds on the limited liability rule which allows shareholders to default on their obligations while they surrender the firms assets to the various stakeholders, according to pre-specified priority rules. The firms liabilities are thus viewed as contingent claims issued against the firms assets, with the payoffs to the various debt-holders completely specified by seniority and safety covenants. Default occurs at debt maturity whenever the firms asset value falls short of debt value at that time. In this model, the loss rate is endogenously determined and depends on the firms asset value, volatility, and the default-free interest rate for the debt maturity. III.B.5.5.1

Structural Model of Default Risk: Mertons (1974) Model

To determine the value of the credit risk arising from a bank loan, we first make two assumptions: that the loan is the only debt instrument of the firm, and that the only other source of financing is equity. In this case, as we shall see below, the credit value is equal to the value of a put option on the value of assets of the firm, at a strike price equal to the face value of debt (including accrued interest), maturing at the maturity of the debt. By purchasing the put on the assets of the firm for the term of the debt, with a strike price equal to the face value of the loan, the bank can completely eliminate all the credit risk and convert the risky corporate loan into a 137 KMV is a trademark of KMV Corporation. The initials KMV stand for the surnames of Stephen Kealhofer, John McQuown and Oldrich Vasicek who founded KMV Corporation in 1989. Kealhofer and Vasicek are former academics from the University of California at Berkeley.



295


riskless loan. Thus, the cost of eliminating the credit risk associated with providing a loan to the firm is the value of this put option. Now, if we make the assumptions that are needed to apply the BlackScholes (BS) model (Black and Scholes, 1973) to equity and debt instruments, we can express the value of the credit risk in an option-like formula. Consider a firm with risky assets V, which is financed by equity, S, and by one debt obligation, maturing at time T with face value (including accrued interest) of F and market value B. If we assume that markets are frictionless, with no taxes, and there is no bankruptcy cost, then the value of the firms assets is simply the sum of the firms equity and debt. At time t = 0 then,

V0

S0

B0 .

(III.B.5.4)

The loan to the firm is subject to credit risk, namely the risk that at time T the value of the firms assets VT , will be below the obligation to the debt holders, F. Credit risk exists as long as the probability of default, Pr(VT < F), is greater than zero. From the viewpoint of a bank that makes a loan to the firm, this gives rise to a series of questions. Can the bank eliminate/reduce credit risk, and at what price? What is the economic cost of reducing credit risk? And, what are the factors affecting this cost? In this simple framework, credit risk is a function of the financial structure of the firm, i.e. its leverage ratio L

Fe-rT/V0, where V0 is the present value of the firms assets, and Fe

rT

is

the present value of the debt obligation at maturity, the volatility of the firms assets, and the time T to maturity of the debt. The model was initially suggested by Merton (1974) and further analysed by Galai and Masulis (1976). To understand why the credit value is equal to the value of a put option on the value of assets of the firm, at a strike price equal to the face value of debt (including accrued interest), and with maturity equal to the maturing of the debt, consider Table III.B.5.9. This shows that if the bank buys the put option with value P, the value at time T will be F whether VT

F or VT > F,

so credit risk is eliminated. In this way they can convert the risky corporate loan into a riskless loan with a face value of F.



296


Table III.B.5.9: Banks payoff at times 0 and T for making a loan and buying a put option Time Value of assets Banks position: (a) make a loan (b) buy a put Total

0 V0

VT

T

B0 P0 B0 P0

VT F VT F

VT > F

F

F 0 F

Thus the value of the put option is the cost of eliminating the credit risk associated with providing a loan to the firm. If we make the assumptions that are needed to apply the (BS) model to equity and debt instruments (see Galai and Masulis, 1976, for a detailed discussion of the assumptions), we can write the value of the put as:

P0

N ( d 1 )V0

Fe rt N ( d 2 ) ,

(III.B.5.5)

where P0 is the current value of the put, N(.) is the cumulative standard normal distribution, d1

ln(V0 / F ) ( r T

2

/ 2)T

ln(V0 / Fe rT ) T

2

T /2

, d2

d1

T ,

(III.B.5.6)

and is the standard deviation of the rate of return of the firms assets. The model illustrates that the credit risk, and its costs, is an increasing function of the volatility of the assets of the firm and the time interval T until debt is paid back, and a decreasing function of the risk-free interest rate r (the higher is r, the less costly it is to reduce credit risk). The cost of credit risk is also homogeneous function of the leverage ratio, which means that it stays constant for a scale expansion of Fe

rT

/V0 .

Note that, when the probability of default is greater than zero the yield to maturity on the debt yT must be greater than the risk-free rate r, so that the default spread

T = yT

r that compensates the

bond holders for the default risk that they bear is positive. The default spread can be regarded as a risk premium associated with holding risky bonds. It can be shown that, in the Merton (1974) framework, the default spread can be computed exactly as a function of the leverage ratio, the volatility of the underlying assets and the debt maturity. In fact: T = yT

r=

1 V0 ln N ( d 2 ) N ( d1 ) . T Fe rT

Note that the default spread decreases when the risk-free rate increases. The greater the risk-free rate, the less risky is the bond and the lower is the value of the put protection therefore, the lower is the risk premium. The numerical examples in Table III.B.5.10 show the default spread for various levels of volatility and different leverage ratios.



297


Table III.B.5.10: Default spread for corporate debt (for V0 = 100, T = 1, and r = 10% 138) Volatility of underlying asset: Leverage ratio: L

0.05

0.10

0.20

0.40

0.5

0

0

0

1.0%

0.6

0

0

0.1%

2.5%

0.7

0

0

0.4%

5.6%

0.8

0

0.1%

1.5%

8.4%

0.9

0.1%

0.8%

4.1%

12.5%

1.0

2.1%

3.1%

8.3%

17.3%

Example III.B.5.1 We show how the 5.6% default spread (marked in red) was obtained in Table III.B.5.10. Using equations (III.B.5.5) and (III.B.5.6) with V0 = 100, T = 1, r = 0.1 (i.e. 10%),

= 0.4 (i.e. 40%)

with the leverage ratio L = 70%, we obtain S0 = 33.37 for the value of equity and B0 = 66.63 for the value of the corporate risky debt. Since L = Fe

rT

/V0 , we have F = 77. Therefore the yield

on the loan is 77/66.63) 1 = 0.156 and there is a 5.6% risk premium to reflect the credit risk. The model also shows that the put value is P0 = 3.37. Hence the cost of eliminating the credit risk is $3.37 for $100 worth of the firms assets, where the face value (i.e. the principal amount plus the promised interest rate) of the one-year debt is 77. This cost drops to 25 cents when volatility decreases to 20% and to 0 for 10% volatility. The assets volatility is clearly a critical factor in determining credit risk. To demonstrate that the bank eliminates all its credit risk by buying the put, we can compute the yield on the banks position as

F /( B0 P ) 77 /( 66.63 3.37 ) 1.10 , which translates to a riskless yield of 10% per annum. III.B.5.5.2

Estimating Credit Risk as a Function of Equity Value

We have already shown that the cost of eliminating credit risk can be derived from the value of the firms assets. A practical problem arises over how easy it is to observe V. In some cases, if both equity and debt are traded, V can be reconstructed by adding the market values of both equity and debt. However, corporate loans are not often traded and so, to all intents and

138

10% is the annualised interest rate discreetly compounded, which is equivalent to 9.5% continuously compounded. Copyright ? 2004 The Authors and the Professional Risk Managers International Association


298


purposes, we can only observe equity. The question, then, is whether the risk of default can be hedged by trading shares and derivatives on the firms stock. In the Merton framework, equity itself is a contingent claim on the firms assets. Its value can be expressed as a function of the same parameters as the put option:

S VN ( d 1 ) Fe

rT

N (d 2 )

(III.B.5.7)

A put can be created synthetically by selling short N( d1) units of the firms assets, and buying F N( d2) units of government bonds maturing at T, with face value of F. If one sells short N( d1)/ N(d1) units of the stock S, one effectively creates a short position in the firms assets of N( d1) units, since:

N ( d1 ) S N (d1 )

VN ( d 1 ) Fe

rT

N (d 2 )

N ( d1 ) . N (d1 )

Therefore, if V is not directly traded or observed, one can create a put option dynamically by selling short the appropriate number of shares. The equivalence between the put and the synthetic put is valid over short time intervals, and must be readjusted frequently with changes in S and in time left to debt maturity. Example III.B.5.2 Using the data from Example III.B.5.1, N( d1)/ N(d1) = 0.137/0.863 = 0.159. This means that in order to insure against the default of a one-year loan with a maturity value of 77, for a firm with a current market value of assets of 100, the bank should sell short 0.159 of the outstanding equity. Note that the outstanding equity is equivalent to a short-term holding of N(d1) = 0.863 of the firms assets. Shorting 0.159 of equity is equivalent to shorting 0.863 of the firms assets. The question now is whether we can use a put option on equity in order to hedge the default risk. It should be remembered that equity itself reflects the default risk, and as a contingent claim its instantaneous volatility

S

can be expressed as: S

where and

S,V

S,V

=

(III.B.5.8)

S,V

= N(d1)V/S is the instantaneous elasticity of equity with respect to the firms value, 1. Since

S

is stochastic and changes with V, the conventional BS model cannot be

applied to the valuation of puts and call on S. The BS model requires

to be constant, or to

follow a deterministic path over the life of the option. However in practice, for long-term options, the estimated

S

from (III.B.5.8) is not expected to change widely from day to day.



299


Therefore, equation (III.B.5.8) can be used in the context of BS estimation of long-term options, even when the underlying instrument does not follow a stationary lognormal distribution.

III.B.5.6

The KMV Approach

KMV derives the expected default frequency (EDF), i.e. the default probability, for each obligor based on the Merton (1974) type of model. The probability of default is thus a function of the firms capital structure, the volatility of the asset returns and the current asset value. The EDF is firmspecific, and can be mapped onto any rating system to derive the equivalent rating of the obligor. EDFs can be viewed as a cardinal ranking of obligors relative to default risk, instead of the more conventional ordinal ranking proposed by rating agencies (which relies on letters such as AAA, AA, ). Contrary to CreditMetrics, KMVs model does not make any explicit reference to the transition probabilities which, in KMVs methodology, are already embedded in the EDFs. Indeed, each value of the EDF is associated with a spread curve and an implied credit rating. Credit risk in the KMV approach is essentially driven by the dynamics of the asset value of the issuer. Given the capital structure of the firm, 139 and once the stochastic process for the asset value has been specified, the actual probability of default for any time horizon, one year, two years, etc., can be derived. Figure III.B.5.9 depicts how the probability of default relates to the distribution of asset returns and the capital structure of the firm. We assume that the firm has a very simple capital structure. It is financed by means of equity, St, and a single zero-coupon debt instrument maturing at time T, with face value F, and current market value Bt. The firms balance sheet can be represented as follows: Vt

Bt ( F ) S t , where

Vt is the value of all the assets. The value of the firms assets, Vt, is assumed to follow a standard geometric Brownian motion. In this framework, default only occurs at maturity of the debt obligation, when the value of assets is less than the promised payment F to the bondholders. Figure III.B.5.9 shows the distribution of the assets value at time T, the maturity of the zerocoupon debt, and the probability of default which is the shaded area below F.

139 That

is, the composition of its liabilities: equity, short- and long-term debt, convertible bonds, etc. Copyright ? 2004 The Authors and the Professional Risk Managers International Association


300


Figure III.B.5.9: Distribution of the firms assets value at maturity of the debt obligation

Assets Value VT = V0 exp {(

T+

TZT}

E(VT) = V0 exp ( T)

VT

V0 F

Probability of default T

Time

The KMV approach is best applied to publicly traded companies, where the value of the equity is determined by the stock market. The information contained in the firms stock price and balance sheet can then be translated into an implied risk of default, as shown in the next section. The derivation of the actual probabilities of default proceeds in three stages: estimation of the market value and volatility of the firms assets; calculation of the distance to default, which is an index measure of default risk; and scaling of the distance to default to actual probabilities of default using a default database. III.B.5.6.1

Estimation of the Asset Value

and the Volatility of Asset Return

In the contingent claim approach to the pricing of corporate securities, the market value of the firms assets is assumed to be lognormally distributed, i.e. the log-asset return follows a normal distribution. 140 This assumption is quite robust and, according to KMVs own empirical studies, actual data conform quite well to this hypothesis. 141 In addition, the distribution of asset returns is stable over time, i.e. the volatility of asset returns remains relatively constant. As we discussed earlier, if all the liabilities of the firm were traded, and marked to market every day, then the task of assessing the market value of the firms assets and its volatility would be

140 Financial models consider essentially market values of assets, and not accounting values, or book values, which

only represent the historical cost of the physical assets, net of their depreciation. Only the market value is a good measure of the value of the firms ongoing business and it changes as market participants revise the firms future prospects. KMV models the market value of assets. In fact, there might be huge differences between both the market and the book values of total assets. For example, as of February 1998 KMV has estimated the market value of Microsoft assets at US$228.6 billion versus $16.8 billion for their book value, while for Trump Hotel and Casino the book value, which amounts to $,2.5 billion, is higher than the market value of $ 1.8 billion. 141 The exception is when the firms portfolio of businesses has changed substantially through mergers and acquisitions, or restructuring.



301


straightforward. The firms asset value would be simply the sum of the market values of the firms liabilities, and the volatility of the asset return could be simply derived from the historical time series of the reconstituted asset value. In practice, however, only the price of equity for most public firms is directly observable, and in some cases part of the debt is actively traded. The alternative approach to assets valuation consists of applying the option-pricing model to the valuation of corporate liabilities as suggested in Merton (1974). In order to make their model tractable KMV assume that the capital structure of a corporation is composed solely of equity, short-term debt (considered equivalent to cash), long-term debt (in perpetuity), and convertible preferred shares. 142 Given these simplifying assumptions, it is possible to derive analytical solutions for the value of equity, S, and its volatility,

S:

S = f (V, , L, c, r),

(III.B.5.9)

= g (V, , L, c, r),

(III.B.5.10)

S

where L denotes the leverage ratio in the capital structure, c is the average coupon paid on the long-term debt and r is the risk-free interest rate. If

S

were directly observable, like the stock price, we could simultaneously solve (III.B.5.9) and

(III.B.5.10) for V and . But the instantaneous equity volatility,

S,

is relatively unstable, and is in

fact quite sensitive to the change in asset value; there is no simple way to measure

S

precisely

from market data. Since only the value of equity S is directly observable, we can back out V from (III.B.5.9) so that it becomes a function of the observed equity value, or stock price, and the volatility of asset returns: V = h (S, , L, c, r )

(III.B.5.11)

Here volatility is an implicit function of V, S, L, c and r. So to calibrate the model for , KMV uses an iterative technique. III.B.5.6.2

Calculation of the Distance to Default

Using a sample of several hundred companies, KMV observed that firms default when the asset value reaches a level that is somewhere between the value of total liabilities and the value of short-term debt. Therefore, the tail of the distribution of asset values below total debt value may not be an accurate measure of the actual probability of default. Loss of accuracy may also result from factors such as the non-normality of the asset return distribution, and the simplifying assumptions made about the capital structure of the firm. This may be further aggravated if a 142 In the general case the resolution of this model may require the implementation of complex numerical techniques,

with no analytical solution, due to the complexity of the boundary conditions attached to the various liabilities. See, for example, Vasicek (1997).



302


company is able to draw on (otherwise unobservable) lines of credit. If the company is in distress, using these lines may (unexpectedly) increase its liabilities while providing the necessary cash to honour promised payments. For all these reasons, KMV implements an intermediate phase before computing the probabilities of default. As shown in Figure III.B.5.10, which is similar to Figure III.B.5.9, KMV computes an index called distance to default (DD). This is the number of standard deviations between the mean of the distribution of the asset value, and a critical threshold called the default point (DPT) which is set at the par value of current liabilities including short-term debt to be serviced over the time horizon (STD), plus half the long-term debt (LTD), i.e. STD + LTD/2. If the expected asset value in one year is E(V1) and is the standard deviation of future asset returns then DD

E(V1 ) DPT .

Figure III.B.5.10: Distance to default

Asset Value Asset Value Distribution E (VT ) VO e

T

VT

V0 F Probability of default Time

T

Given the lognormality assumption of asset values, the distance to default expressed in unit of asset return standard deviation at time horizon T, is

DD

ln V0 / DPTT

(

1

2

2

)T

T

(III.B.5.12)

where V0

= current market value of assets

DPTT

= default point at time horizon T = expected return on assets, net of cash outflows



303


= annualised asset volatility. It follows that the shaded area shown below the default point in Figure III.B.5.10 is equal to N( DD). III.B.5.6.3

Derivation of the Probabilities of Default from the Distance to Default

This last phase consists of mapping the distance to default to the actual probabilities of default, for a given time horizon. KMV calls these probabilities expected default frequencies. Using historical information about a large sample of firms, including firms that have defaulted, one can estimate, for each time horizon, the proportion of firms of a given ranking, say DD = 4, that actually defaulted after one year. This proportion, say 40 bp, or 0.4%, is the EDF as shown in Figure III.B.5.11. Figure III.B.5.11: Mapping of the distance to default into the EDFs, for a given time horizon EDF

40 bp 1

2

3

4

5

6

DD

Example III.B.5.3: Current market value of assets:

V0 = 1000

Net expected growth of assets per annum:

20%

Expected asset value in one year:

V0 ? 1.20 = 1200

Annualised asset volatility, :

100

Default point:

800

Then DD = (1200 800)/100 = 4. Assume that among the population of 5000 firms with a DD of 4 at one point in time, 20 defaulted one year later. Then EDF1 year = 20/5000 = 0.04 = 0.4% or 40 bp. The implied rating for this probability of default is BB+. Example III.B.5.4: Federal Express ($ figures are in billions of US$) This example is provided by KMV and relates to Federal Express on two different dates: November 1997 and February 1998. Copyright ? 2004 The Authors and the Professional Risk Managers International Association


304


November 1997 Market capitalisation

February 1998

$ 7.7

$ 7.3

Book liabilities

$ 4.7

$ 4.9

Market value of assets

$ 12.6

$ 12.2

Asset volatility

15%

17%

Default point

$ 3.4

$ 3.5

(price ? shares outstanding)

Distance to default (DD) EDF

12.6 3.4 0.15 12.6 0.06% (6 bp)

4.9

AA

12.2 3.5 0.17 12.2 0.11% (11 bp)

4.2

A

This example illustrates the main causes of changes for an EDF, i.e. variations in the stock price, the debt level (leverage ratio), and asset volatility (i.e. the perceived degree of uncertainty concerning the value of the business). III.B.5.6.4

EDF as a Predictor of Default

KMV has provided a Credit Monitor service for estimated EDFs since 1993. EDFs have proved to be a useful leading indicator of default, or at least of the degradation of the creditworthiness of issuers. When the financial situation of a company starts to deteriorate, EDFs tend to shoot up quickly until default occurs, as shown in Figure III.B.5.12. Figure III.B.5.13 shows the evolution of equity value and asset value, as well as the default point during the same period. On the vertical axis of both graphs the EDF is shown as a percentage, together with the corresponding Standard & Poors rating.



305


Figure III.B.5.12: EDF of a firm that defaulted versus EDFs of firms in various quartiles and the lower decile

Source: KMV Corporation

Note: The quartiles and decile represent a range of EDFs for a specific credit class. Figure III.B.5.13: Asset value, equity value, short- and long-term debt of a firm that defaulted




306


Figure III.B.5.14: EDF of a firm that defaulted, versus Standard & Poors rating

EDF

Default

S&P Credit Rating


KMV has analysed more than 2000 US companies that have defaulted or entered into bankruptcy over the last 20 years. These firms belonged to a large sample of more than 100,000 companyyears with data provided by Compustat. In all cases KMV has shown a sharp increase in the slope of the EDF a year or two before default. Changes in EDFs tend to anticipate by at least one year - the downgrading of the issuer by rating agencies such as Moodys and Standard & Poors (Figure III.B.5.14). Contrary to Moodys and Standard & Poors historical default statistics, EDFs are not biased by periods of high or low numbers of defaults. The distance to default can be observed to shorten during periods of recession, when default rates are high, and to increase during periods of prosperity characterised by low default rates. The loss distribution is generated in a way similar to CreditMetrics by simulating correlated defaults at the risk horizon, say one year.

III.B.5.7

The Actuarial Approach

In the structural models of default, default occurs when the asset value falls below a certain boundary such as a promised payment (e.g., the Merton, 1974, framework). By contrast, the actuarial model discussed in this section treat the firms bankruptcy process, including recovery, as exogenous. CreditRisk+, released in late 1997 by investment bank Credit Suisse Financial Products, is a purely actuarial model, based on mortality models of the insurance companies. This means that the probabilities of default that the model employs are based on historical statistical data of default experience by credit class.



307


Contrary to the structural approach to modelling default, the timing of default is assumed to take the bond-holders by surprise. Default is treated as an end of game (stopping time) which comes as a surprise, and the probability of such a surprise is known and follow a Poisson type of distribution. CreditRisk+ applies an actuarial science framework to the derivation of the loss distribution of a bond/loan portfolio. Only default risk is modelled; downgrade risk is ignored. Unlike the KMV approach to modelling default, there is no attempt to relate default risk to the capital structure of the firm. Also, no assumptions are made about the causes of default. It is assumed that: 1. for a loan, the probability of default in a given period, say one month, is the same as in any other month; 2. for a large number of obligors, the probability of default by any particular obligor is small, and the number of defaults that occur in any given period is independent of the number of defaults that occur in any other period. Under these assumptions, the probability distribution for the number of defaults during a given period of time (say, one year) is represented well by a Poisson distribution: 143 n

Pr(n defaults) =

where

e n!

for n = 0,1,2,,

(III.B.5.13)

is the average number of defaults per year.

The annual number of defaults n is a stochastic variable with mean

and standard deviation

.

The Poisson distribution has a useful property: it can be fully specified by means of a single parameter, . For example, if we assume

= 3 then the probability of no default in the next

year is: Pr(0 default) =

30 e 3 0!

0.05 5%

while the probability of exactly three defaults is: Pr (3 defaults) =

33 e 3 3!

0.224

22.4% .

143 In any portfolio there is, naturally, a finite number of obligors, say n; therefore, the Poisson distribution, which

specifies the probability of n defaults, for n = 1, , , is only an approximation. However, if n is large enough, then the sum of the probabilities of n + 1, n + 2, defaults become negligible.



308


However, we expect the mean default rate

to change over time, depending on the business

cycle. This suggests that the Poisson distribution can only be used to represent the default process if, as suggested by CreditRisk+, we make the additional assumption that the mean default rate is itself stochastic. In the event of default by an obligor, the counterparty incurs a loss that is equal to the amount owed by the obligor (its exposure, i.e. the marked-to-market value, if positive and zero, if negative at the time of default) less a recovery amount. In CreditRisk+, the exposure for each obligor is adjusted by the anticipated recovery rate in order to calculate the loss given default. These adjusted exposures are calculated outside the model (exogenous), rather than determined by it, and are independent of market risk and downgrade risk. In order to derive the loss distribution for a well-diversified portfolio, the losses (exposures, net of the recovery adjustments) are divided into bands. The level of exposure in each band is approximated by means of a single number. Example III.B.5.5: Suppose the bank holds a portfolio of loans and bonds from 500 different obligors, with exposures between $50,000 and $1 million. Notation Obligor

A

Exposure

LA

Probability of default

PA

Expected loss

A

= L A PA

In Table III.B.5.11 we show the exposures for the first six obligors. Table III.B.5.11: Exposure per obligor Exposure ($) Exposure Round-off exposure (in $100,000) Band Obligor (loss given default) (in $100,000) j A LA j j 1 2 3 4 5 6

150,000 460,000 435,000 370,000 190,000 480,000

1.5 4.6 4.35 3.7 1.9 4.8

2 5 4 4 2 5

2 5 4 4 2 5



309


The unit of exposure is assumed to be $100,000. Each band j, j = 1, , m, with m = 10, has an average common exposure:

= $100,000 ? j. In CreditRisk+, each band is viewed as an

j

independent portfolio of loans/bonds, for which we introduce the following notation: Notation Common exposure in band j in units of exposure

j

Expected loss in band j in units of exposure

j

Expected number of defaults in band j Denote by

A

j

the expected loss for obligor A in units of exposure, i.e.

A

=

A/L

where in this

case L = $100,000. Then j, the expected loss over a one-year period in band j, expressed in units of exposure, is simply the sum of the expected losses But since by definition be calculated, using

j

j

=

j j,

j/ j

A

of all the obligors that belong to band j.

the expected number of defaults per annum in band j may now

. Table III.B.5.12 provides an illustration of the results of these

calculations. Table III.B.5.12: Expected number of defaults per annum in each band Band: j

Number of obligors

j

j

1

30

1.5

1.5

2

40

8

4

3

50

6

2

4

70

25.2

6.3

5

100

35

7

6

60

14.4

2.4

7

50

38.5

5.5

8

40

19.2

2.4

9

40

25.2

2.8

10

20

4

0.4

To derive the distribution of losses for the entire portfolio, we follow the three steps outlined below. Step 1: Probability generating function for each band Each band is viewed as a separate portfolio of exposures. The probability generating function for any band, say band j, is by definition:



310


nL ) z n

Pr( loss

Gj(z)

Pr( n defaults ) z

n 0

n

j

,

n 0

where the losses are expressed in the unit of exposure. Since we have assumed that the number of defaults follows a Poisson distribution, we have:

e

G j (z )

j

n!

n 0

n j

z

nv j

e

jz

j

vj

.

Step 2: Probability generating function for the entire portfolio Since we have assumed that each band is a portfolio of exposures that is independent of the other bands, the probability generating function for the entire portfolio is simply the product of the probability generating function for each band: m

G( z )

where

m j 1

j

j 1

e

j

jz

vj

exp

m j 1

m j

j 1

jz

j

(III.B.5.14)

denotes the expected number of defaults for the entire portfolio.

Step 3: Loss distribution for the entire portfolio Given the probability generating function (III.B.5.14), it is straightforward to derive the loss distribution, since

Pr( loss of nL )

1 d n G(z ) |z n ! dz n

0

for n 1,2,... .

These probabilities can be expressed in closed form and depend only on

j

and j. 144

Credit VaR is then easily derived from the above loss distribution by first computing the percentile corresponding to the confidence level, and then subtracting the expected loss from this number. CreditRisk+ proposes several extensions of the basic one-period, one-factor model. First, the model can be easily extended to a multi-period framework. Second, the variability of default rates can be assumed to result from a number of background factors, each representing a sector of activity. Each factor k is represented by a random variable Xk which is the number of defaults in sector k, and which is assumed to be gamma distributed. The mean default rate for each obligor is then supposed to be a linear function of the background factors, Xk. These factors are 144 See Credit Suisse (1997), p.26.



311


further assumed to be independent. In all cases, CreditRisk+ derives a closed-form solution for the loss distribution of a bond/loan portfolio. CreditRisk+ has the advantage that it is relatively easy to implement. First, as we mentioned above, closed-form expressions can be derived for the probability of portfolio bond/loan losses, and this makes CreditRisk+ very attractive from a computational point of view. In addition, marginal risk contributions by obligor can be easily computed. Second, CreditRisk+ focuses on default, and therefore it requires relatively few estimates and inputs. For each instrument, only the probability of default and the exposure are required. Its principal limitation is the same as for the CreditMetrics and KMV approaches: the methodology assumes that credit risk has no relationship with the level of market risk. In addition, CreditRisk+ ignores what might be called migration risk; the exposure for each obligor is fixed and is not sensitive to possible future changes in the credit quality of the issuer, or to the variability of future interest rates. Even in its most general form, where the probability of default depends upon several stochastic background factors, the credit exposures are taken to be constant and are not related to changes in these factors. Finally, like the CreditMetrics and KMV approaches, CreditRisk+ is not able to cope satisfactorily with non-linear products such as options and foreign currency swaps.

III.B.5.8

Summary and Conclusion

In this chapter we have presented the key features of some of the more prominent new models of credit risk measurement in a portfolio context. At first sight, these approaches appear to be very different and likely to produce considerably different loan loss exposures and VaR figures. However, analytically and empirically, these models are not as different as they may first appear. Indeed, similar arguments stressing the structural similarities have been made by several authors such as Gordy (2000) and Koyluoglu and Hickman (1999). Comparative studies such as IIF/ISDA (2000) have stressed the sensitivity of the risk measures (expected and unexpected losses) to the key risk drivers, i.e. probabilities of default, loss given default and default correlations. This study also showed that when these models are run using consistent parameters, they produce results which fall in quite a narrow range.

References Altman, E I, and Kishore, V (1996) Almost everything you wanted to know about recoveries on defaulted bonds. Financial Analysts Journal, 52(6), pp. 5764.



312


Altman, E I, and Kishore, V (1998) Defaults and returns on high yield bonds: analysis through 1997. Working Paper S-98-1, New York University Salomon Center. Black, F, and Scholes, M (1973) The pricing of options and corporate liabilities. Journal of Political Economy, 81, pp. 637654. Carty, L, and Lieberman, D (1996) Defaulted Bank Loan Recoveries. Moodys Investors Service, Global Credit Research, Special Report. Credit Suisse (1997) CreditRisk+: A Credit Risk Management Framework. New York: Credit Suisse Financial Products. Crouhy, M, Galai, D, and Mark, R (2001) Risk Management. New York: McGraw-Hill. Elton, E J, and Gruber, M J (1995) Modern Portfolio Theory and Investment Analysis. New York: Wiley. Fishman G (1997) Monte Carlo: Concepts, Algorithms, and Applications. New York: Springer Seris in Operations Research, Springer. Galai, D, and Masulis, R W (1976) The option pricing model and the risk factor of stocks. Journal of Financial Economics, 3 (January/March), pp. 5382 Glasserman, P, Heidelberger, P, and Shahabuddin, P (2000) Variance reduction technique for estimating value-at-risk. Management Science, 46, pp. 13491364. Gordy, M (2000) A comparative anatomy of credit risk models. Journal of Banking and Finance, 1, pp. 119149. IIF/ISDA (2000) Modeling credit risk: Joint IIF/ISDA testing program. February. J.P. Morgan (1997) CreditMetrics. Technical Document. Koyluoglu, H U, and Hickman, A (1998) A generalized framework for credit risk portfolio models. Lucas, D (1995) Default correlation and credit analysis, Journal of Fixed Income, 4(4), pp. 76-87. Working Paper. New York: Oliver, Wymann and Co., July. Merton, R C (1974) On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance, 28, pp. 449470. Rudd, A, and Clasing Jr, H K (1988) Modern Portfolio Theory: The Principles of Investment Management. Orienda, CA: Andrew Rudd. Vasicek, O (1997) Credit valuation, Net Exposure, 1(1). Wilson, T (1997a) Portfolio credit risk I, Risk, 10(9), pp. 111-117. Wilson, T (1997b) Portfolio credit risk II, Risk, 10(10), pp. 56-61.



313




314


III.B.6 Credit Risk Capital Calculation Dan Rosen145

III.B.6.1

Introduction

As discussed in Chapter III.0, the primary role of capital in a bank, apart from the transfer of ownership, is to act as a buffer to absorb large unexpected losses, protect depositors and other claim holders and provide confidence to external investors and rating agencies on the financial health of the firm. In contrast, regulatory capital refers to the minimum capital requirements which banks are required to hold, based on regulations established by the banking supervisory authorities. From the perspective of the regulator, the objectives of capital adequacy requirements are to safeguard the security of the banking system and to ensure its ongoing viability, and to create a level playing field for internationally active banks. While the role of economic capital (EC), in general, is to act as a buffer against all the risks that may force a bank into insolvency, economic credit capital (ECC) can be viewed as a buffer against those risks specifically associated with obligor credit events such as default, credit downgrades and credit spread changes. In this chapter, we review the main concepts for estimating and allocating ECC as well as the current regulatory framework for credit capital. Chapter III.0 presents the chronology and describes the basic principles behind regulatory capital and gives an overview of the Basel Accord, the framework created by the Basel Committee on Banking Supervision (BCBS), which is now the basis for banking regulation around the globe. We focus in this chapter on the main principles for computing minimum regulatory credit capital requirements under the current Basel I Accord as well as under the current proposal for the new Basel accord, Basel II. In this chapter you will learn: how credit portfolio models must be defined and parameterised consistently to measure ECC from a bottom-up approach; the basic rules for computing minimum credit capital under the Basel I Accord; the rules for computing minimum capital requirement for credit risk in the Basel II Accord (Pillar I) we cover various types of exposure and comment on the special credit capital considerations under Pillar II;

145 Algorithmics Inc.



315


the Basel II treatment for internal ratings systems and probability of default estimation, as well as the minimum standards for credit monitoring processes and validation methods. Finally, Section III.B.6.7 covers some advanced topics: the applications of risk contribution methodologies ECC allocation, as well as the shortcomings of value-at-risk (VaR) for ECC and coherent risk measures.

III.B.6.2

Economic Credit Capital Calculation

ECC acts as a buffer that provides protection against the credit risk (potential credit losses) faced by an institution. Traditionally, capital is designed to absorb unexpected losses up to a certain confidence level, while credit reserves are set aside to absorb expected losses. As explained in Chapter III.0, the methodologies to estimate EC at the firm level can be generally classified into top-down or bottom-up approaches. In general, top-down approaches do not readily allow the decomposition of capital into its various constituents, such as market risk, credit risk and operational risk. Bottom-up approaches provide risk-sensitive measures of capital, which allow us to understand and manage these risks better. In this section, we discuss how credit portfolio models are applied to measure ECC from a bottom-up approach. The credit portfolio models that were described in Chapter III.B.5 provide the statistical distributions of potential credit losses of a portfolio over a given horizon. Hence, they offer the key tools to estimate the required ECC from a bottom-up approach. Chapter III.B.5 covers the fundamentals of credit portfolio models, as well as the main models used in industry. III.B.6.2.1

Economic Capital and the Credit Portfolio Model

A credit portfolio model must be defined and parameterised consistently with the definition of economic capital used by the firm. In particular, when devising an economic capital framework, we must explicitly consider the time horizon or holding period, the definition of credit losses, and the quantile (and definition of unexpected losses) covered by capital. The definitions of these three parameters are tightly linked to the actual definition of capital used by the firm. III.B.6.2.1.1 Time Horizon While trading activities tend to involve short time horizons (a few hours to a few days), credit activities generally involve longer horizons (a few months to a few years). Operational risk and insurance activities generally involve even longer horizons, spanning several years. It is common



316


practice for credit VaR measurement to assume a one-year horizon, although longer periods are sometimes employed. 146 Several reasons are cited for a one-year horizon, among them the following: It accords with the firms main accounting cycle. It is a reasonable period over which the firm will typically be able to renew any capital depleted through losses. It coincides with a reasonable period over which actions can be taken to mitigate losses for various credit assets. Credit reviews are usually performed annually. The borrower might be updating its financial information only on an annual basis. III.B.6.2.1.2 Credit Loss Definition Accounting rules, regulatory guidance and management policy combine to determine when a loss will be recognised. Loss recognition is straightforward in some instances, for example, for trading securities that are marked-to-market regularly or operational losses that are recognised when they occur. In these instances, there is usually little question as to the function of economic capital. Management has more leeway in other instances. With regards to credit exposures, management may choose to measure economic capital against default events only (default mode) or against both default and credit migration events (mark-to-market mode): (i) Default-only credit losses. In this case we are only concerned whether a loan (or other instrument) would be repaid or not. This type of measurement is currently the most prevalent, typically used for traditional (accrual accounted, hold-to-maturity) banking book activities. Credit portfolio models require several building blocks, or components, at the enterprise level. These include the following: Estimates of the probabilities of default (PD), loss given default (LGD), and exposures at default (EAD) for individual corporate, banking and sovereign exposures; similarly, such estimates are required for homogeneous buckets of retail or small- and mediumsized enterprise (SME) exposures. A portfolio model with specific assumptions about the co-dependence of o

credit events (e.g. asset values or PD correlations)

o

exposures, LGDs and credit events.

146 Note that the optimal time horizons for given portfolios might vary. However, when applied on an enterprise

basis, it is important to set a common methodology to estimate the desired capital. Also, while it is most common to apply credit portfolio models in a single-step setting, the multi-step applications are becoming increasingly popular. Finally, the choices of time horizon and credit loss definition can be tightly linked.



317


(ii) Mark-to-market credit losses. In this case, the models used are closer to market risk models, capturing the losses due to defaults and the changes in mark-to-market due to deterioration in credit quality (or credit migration). Therefore, in addition to information needed for default-only models (PDs, EADs, LGDs and correlations), they require market pricing information (e.g. credit spreads) and more general information on the instruments structures for marking to market. Mark-to-market models are more commonly used for portfolios which are not held to maturity, or for which reliable pricing information might be available, such as bond and credit derivatives portfolios. The choice of loss definition and time horizon is tightly linked. For example, consider credit risk in lending activities. If capital is held against defaults only, it makes sense that the horizon should be the entire life of the loan. If capital is held against deterioration in value, then some time period would be selected, independent of the life of the underlying loans (e.g. one year). Both choices allow for longer-maturity instruments to carry higher capital charges. III.B.6.2.1.3

Quantile of the Loss Distribution

Credit VaR should capture the ECC that shareholders should invest, in order to limit the probability of default of the firm over a given horizon. This involves defining a high, predetermined quantile of the credit loss distribution so that shareholders will be highly confident that credit losses will not exceed this amount. The quantile, which defines the confidence interval, is chosen to provide the right level of protection to debt holders and hence achieve a desired rating, as well as providing the necessary confidence to other claim holders, such as depositors. 147 The horizon and quantile are thus key policy parameters set by senior management and the board of directors. For example, a bank that wishes to be consistent with an AA rating from an external credit agency might chose a 99.97% confidence level over a oneyear horizon, since the one-year default probability of an AA-rated firm is 0.03%. III.B.6.2.2

Expected and Unexpected Losses

In its most common definition, economic capital is designed to absorb only unexpected losses (UL) up to a certain confidence level. Credit reserves are traditionally set aside to absorb expected losses (EL) during the life of a transaction. Thus, it is common practice to estimate economic capital as the chosen confidence interval minus the estimated expected EL.

The key rationale for

subtracting EL is that credit products are already priced such that net interest margins less non-

147 But note there can be a trade-off between achieving this objective and providing high returns on capital for

shareholders.



318


interest expenses are at least enough to cover estimated EL (and must also cover a desired return to capital). Therefore, in this case, the credit VaR measure that is relevant to estimating EC covers only unexpected losses: Credit VaR ( L )

Q ( L ) EL ,

(III.B.6.1)

where Q ( L ) denotes the % quantile (e.g. 0.03%) of the portfolio loss distribution at the horizon. In Chapter III.0 we show that subtracting EL as given by the expected end-of-period value from the worse-case losses represents a simplifying approximation to estimate EC. This is the approach commonly taken by practitioners and generally leads to conservative estimates. 148 More precisely, the credit VaR measure appropriate for EC should in fact measure loss relative to the portfolios initial mark-to-market (MtM) value and not relative to the EL in its end-of-period distribution. 149 This highlights the importance for banks of moving towards full MtM portfolio credit risk models and of establishing accurate estimates of MtM values of credit portfolios. Also, the credit VaR measure normally ignores the interest payments that must be made on the funding debt. These payments must be added explicitly to the EC measure. The estimation of the interest compensation calculation and the credit MtM value of the portfolio generally require the use of an asset pricing model. III.B.6.2.3

Enterprise Credit Capital and Risk Aggregation

A large firm, such as a bank, acquires credit risk through various businesses and activities, including retail banking, commercial lending, bonds, derivatives and credit derivatives trading. Such an institution is likely to have one methodology for its larger commercial loans and another for its retail credits. In general, a bank may have any number of methodologies for various credit segments. If each area is modelled separately, then the amounts of ECC estimated for each area need to be combined. In making the combination, the firm needs to incorporate, either implicitly or explicitly, various correlation assumptions. The choices of aggregation can be divided into three main categories: Sum of stand-alone capital for each business unit or portfolio. This methodology essentially assumes perfect correlation across business lines and does not allow for diversification from them. It is consistent, in general, with one-factor portfolio models (such as the Basel II underlying portfolio model).

148 For a detailed discussion, see Kupiec (2002). 149 Capital allocation credit VaR measures in this context can be negative.



319


Ad hoc cross-business correlation. In order to allow for some cross-business diversification, a firm might aggregate the individual stand-alone capital estimates using analytical models and simple cross-business (asset) correlation estimates. Full enterprise credit portfolio modelling. Current multi-factor credit models and technology allow the computation of credit loss distributions of large enterprise portfolios. This can be accomplished through semi-analytical methods (e.g. fast Fourier transforms, saddlepoint methods), or full-blown Monte Carlo simulation. Various financial institutions today are starting to apply either in-house or commercial credit portfolio models to compute ECC at the enterprise level.

III.B.6.3

Regulatory Credit Capital: Basel I

In this section you will learn the key principles of the Basel I Accord for computing minimum capital requirements for credit risk. We further discuss some of its shortcomings and the motivation for regulatory arbitrage. III.B.6.3.1

Minimum Credit Capital Requirements under Basel I

The 1988 Basel I Accord focused mainly on credit risk, establishing minimum capital standards that linked capital requirements to the credit exposures of banks (Basel Committee of Banking Supervision 1988). Prior to its implementation in 1992, bank capital was regulated through simple, ad hoc capital standards. While generally prescriptive, Basel I left various choices to be made by local regulators, thus resulting in several variations of the implementation across jurisdictions. The calculation of regulatory credit capital requirements has three steps: converting exposures to credit-equivalent assets, computing loan equivalents, and applying the capital adequacy ratio. Step 1: Credit-Equivalent Assets The objective in this step is to express all on-balance sheet and off-balance sheet credit exposures in comparable numbers. This is achieved by converting off-balance sheet exposures into equivalent credit assets, which better reflect the amounts that could be lost if a transaction were to default. The general rules for this are as follows: For contingent banking book assets, such as letters of credit, etc., asset equivalents are obtained by multiplying the exposure by a percentage, which broadly reflects the likelihood of its conversion into an actual exposure. For example, the conversion factor of an undrawn standby letter of credit is 50%.



320


For derivatives in the trading book, 150 asset equivalents are given by the instruments total exposure, obtained through the so-called method of add-ons: Total Exposure = Actual Exposure + Potential Exposure

(III.B.6.2)

Actual Exposure = max(0, Mark-to-Market) Potential Exposure = Notional

Counterparty Add-on

The potential exposure attempts to capture the change in value of derivatives, resulting from market fluctuations, which lead to higher credit losses should the counterparty default. Counterparty add-ons to measure potential exposures are given prescriptively, for example as in Table III.B.6.1. Table III.B.6.1: Add-ons for derivatives exposures in Basel I Exchange rate

Precious metals

Other

except gold

commodities

6.00%

7.00%

10.00%

5.00%

8.00%

7.00%

12.00%

7.50%

10.00%

8.00%

15.00%

Residual maturity

Interest rate

One year or less

0.00%

1.00%

0.50% 1.50%

Over one year to five years Over five years

and gold

Equity

Thus, for example, a three-year foreign exchange forward contract carries a total exposure given by its current mark-to-market plus 5% of its notional; an interest-rate swap with remaining nine-month maturity is deemed to carry a 0% addon, and hence its total exposure is given by its current mark-to-market. Basel I allows also for partial recognition of mitigation techniques such as netting when the proper agreements are in place. In this case, actual exposure is computed as the netted mark-tomarket values of all transactions, and the total add-on is adjusted as follows: Netted Add-on = Gross Add-on ? (0.4 + 0.6 ? NGR Ratio),

(III.B.6.3)

where NGR denotes the net-to-gross ratio, that is, the netted mark-to-market of the transactions with a counterparty divided by the gross mark-to-market (the sum of all the positive mark-tomarket transaction values).

150 This was originally defined in BCBS (1995).



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Example III.B.6.1 Consider a counterparty with three derivatives transactions as shown in Table III.B.6.2. With a gross MtM of $500m and a netted MtM of $200m, the NGR is 0.4. Netting in this case reduces the credit equivalent for this portfolio by (535 222.4)/535) = 58%. Table III.B.6.2: Example credit equivalents in the trading book Transaction Instrument

Residual maturity

Notional

MtM

Add-on (%)

Add-on

Credit equivalent

1

IR swap

3 years

1000

500

0.50

5

505

2

FX forward

1.5 years

500

-100

5

25

25

3

FX option

5 months

500

-200

1

5

5

Gross

500

35

535

Netted

200

22.4

222.4

NGR ratio

0.4

Table III.B.6.3: Basel I risk weights (examples) 0%

Cash Claims on central governments, central banks denominated in national currency

0, 10, 20, or 50% 20%

Claims on domestic public-sector entities, excluding central government, and loans guaranteed by securities issued by such entities Claims on multilateral development banks Claims on banks incorporated in the OECD and loans guaranteed by OECD

50%

Loans fully secured by mortgage on residential property

100%

Claims on private sector Claims on banks incorporated outside the OECD with a residual maturity of over one year



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Step 2: Risk-Weighted Assets Risk-weighted assets (RWAs) are obtained by multiplying the exposures (or credit equivalents) by a risk weight. Risk weights broadly attempt to reflect the credit riskiness of the asset. Example risk weights are given in Table III.B.6.3. Thus, a loan to a corporate, regardless of its credit rating, carries a 100% risk weight, while a credit exposure to an OECD government has a 0% weight. Step 3: Capital Adequacy Ratio Minimum Capital Requirement The minimum capital requirements are obtained by multiplying the sum of all the RWAs by the capital adequacy ratio of 8% (also referred to as the Cook ratio): Capital

RWAk

8%.

(III.B.6.4)

k

Example III.B.6.2 Following on from Example III.B.6.1, assume that the counterparty is a UK bank. Then the risk weight is 20%, which leads to minimum capital requirements Min capital requirements = $222.4m ? 0.2 ? 0.08 = $3.56m. The reduction of regulatory capital from the application of netting for this portfolio is also 58% (as with the credit equivalent). III.B.6.3.2

Weaknesses of the Basel I Accord for Credit Risk

As mentioned in Chapter III.0, a great strength of Basel I is the simplicity of the framework, which allowed it to be implemented in countries with different banking and accounting practices. Its simplicity also has been its major weakness, as the accord does not effectively align regulatory capital requirements closely with an institutions risk. Some criticisms with regard to credit risk include the following: Lack of credit quality differentiation. All corporate credits elicit a risk weight of 100% regardless of their credit quality. Furthermore, high-rated corporate exposures may carry much higher capital than low-rated sovereign exposures. For example, a loan to an AAArated corporate such as GE carries five times more capital than a loan to a bank in Korea or Turkey (20% risk weight). This indirectly provides incentives for banks to take bad credits and avoid high-rated credits (with lower returns). Lack of proper maturity differentiation. For example, revolving credit exposures with a term of less than one year do not get a regulatory capital charge. Similarly, a facility of 366 days bears the same capital charge as a long-term facility. This has led to very simple regulatory arbitrage through the systematic creation of rolling 364-day facilities.



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Insufficient incentives for credit mitigations techniques. The accord does not fully recognise the risk reduction achieved through credit mitigation techniques such as netting, collateral, guarantees and credit derivatives. Lack of recognition of portfolio effects in credit risk. The accord does not provide any capital benefits for diversification across assets and businesses. III.B.6.3.3

Regulatory Arbitrage

The lack of differentiation in the accord, together with the financial engineering advances in credit risk over the last decade, have led to the widespread development of regulatory capital arbitrage. This refers to the process by which regulatory capital is reduced through instruments such as credit derivatives or securitisation, without an equivalent reduction of the actual risk being taken. Through regulatory arbitrage instruments, for example, banks typically transfer lowrisk exposures from their banking book to their trading book, or simply place them outside the regulated banking system.

III.B.6.4

Regulatory Credit Capital: Basel II

We introduce the latest proposals of the current Basel II Accord for credit risk capital. In this section you will learn about the Pillar I rules for computing minimum capital requirement for credit capital in the Basel II Accord. We cover the standardised and internal ratings based approaches for different types of exposures and conclude with a brief comment on the special credit capital considerations under Pillar II of the new Accord. III.B.6.4.1

Latest Proposal for Minimum Credit Capital requirements

In 1999, the BCBS issued a proposal for a new capital adequacy framework (Basel II or BIS II Accord). The third consultative paper (CP3) on the new accord was released in April 2003 (BCBS, 2003). The Final version of the accord was published in June 2004 (BCBS, 2004). 151 The implementation of the accord will take effect between the end of 2006 and the end of 2007. Basel II attempts to improve capital adequacy framework along two important dimensions: First, the development of a capital regulation that encompasses not only minimum capital requirements but also supervisory review and market discipline. Second, a substantial increase the risk sensitivity of the minimum capital requirements.

151 A small number of open issues are still to be resolved during 2004.



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The reader is referred to Chapter III.0 for a general discussion on the Basel II Accord. In this section we summarise the key principles and formulae for the computation of minimum capital requirements for credit risk under Pillar I of the new accord. For greater detail, the reader is referred to the BCBS papers which can be found at www.bis.org . As with Basel I, minimum capital requirements consist of three components: 4. definition of capital (no major changes from Basel I); 5. definition of RWA; 6. minimum ratio of capital/RWA (remains 8%). Basel II proposes substantive changes to the treatment of RWAs for credit risk relative to Basel I. It moves away from a one-size-fits-all approach through the introduction of three distinct options for the calculation of credit risk. These approaches present increasing complexity and risk-sensitivity. Banks and supervisors can thus select the approaches that are most appropriate to the stage of development of banks operations and of the financial market infrastructure. Similar to Basel I, total minimum capital requirements are obtained by multiplying the riskweighted assets by the capital adequacy ration of 8%, as in equation (III.B.6.4) 8: Capital

RWAk

8% .

k

The calculation of RWAs can be done through two types of approach: the standardised approach and the internal ratings based (IRB) approach. The IRB approach has two variants, called foundation and advanced IRB. III.B.6.4.2

The Standardised Approach in Basel II

This approach is similar to Basel I in that Basel II requires banks to slot their credit exposures into supervisory categories based on observable characteristics of the exposures (e.g. whether it is a corporate loan or a residential mortgage loan), and then establishes fixed risk weights corresponding to each supervisory category. Important differences from Basel I include the following: Use of external ratings. The standardised approach allows the use of external credit assessments to enhance risk sensitivity. The risk weights for sovereign, interbank, and

8 In addition, in (BCBS, 2004) the committee introduced a scaling factor. Where aggregate capital is lower than under

Basel I, the new requirements must be scaled by a factor, currently estimated at 1.06, such that overall capital levels do not fall.



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corporate exposures are differentiated based on external credit assessments. For sovereign exposures, these credit assessments may include those developed by OECD export credit agencies or private rating agencies. The use of external ratings for corporate exposures is optional. Where no external rating is applied to an exposure, the approach mandates that in most cases a risk weighting of 100% be used (as in Basel I). In such instances, supervisors are to ensure that the capital requirement is adequate given the default experience of the exposure type. For example, for claims on corporates the risk weights are given in Table III.B.6.4. Table III.B.6.4: Standardised risk weights for corporate exposures Credit assessment Risk weight

AAA to AA 20%

A+ to A BBB+ to BB 50%

100%

Below B

Unrated

150%

100%

Loans past-due. A loan considered past-due requires a risk weight of 150%, unless a threshold amount of specific provisions has already been set aside against the loan. Credit mitigants. The approach recognises an expanded range of credit risk mitigants: collateral, guarantees, and credit derivatives. The approach expands the range of eligible collateral beyond OECD sovereign issues to include most types of financial instruments. It also sets several approaches for assessing the degree of capital reduction based on the market risk of the collateral instrument. Finally, it expands the range of recognised guarantors to include all firms that meet a threshold external credit rating. Retail exposures. The risk weights for residential mortgage exposures are reduced relative to Basel I, as are those for other retail exposures, which receive a lower risk weight than that for unrated corporate exposures. In addition, some loans to SMEs may be included within the retail treatment, subject to meeting various criteria. Through several options, the standardised approach attempts to improve the risk sensitivity of the RWAs. Basel II also provides a simplified standardised approach, where circumstances may not warrant a broad range of options. Banks under the simplified methods must also comply with the supervisory review and market discipline requirements of Basel II. Example III.B.6.3 Based on Table III.B.6.4, a loan to a corporate obligor, with a AA rating from an external agency, would have a capital requirement of one-fifth under the Basel II standardised approach



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compared to Basel I (a risk weight of 20% versus 100%). Similarly, a loan to an A-rated obligor would see its capital requirement going to one-half (50% weight). III.B.6.4.3

Internal Ratings Based Approaches: Introduction

In the IRB approaches, banks internal assessments of key risk drivers serve as primary inputs to the capital calculation, leading to more risk-sensitive capital requirements. This is a substantial difference from both Basel I and the standardised approach. However, the IRB approach does not fully allow for internal portfolio models to calculate capital requirements. Instead, the risk weights (and thus the capital charges) are determined through the combination of quantitative inputs provided by banks and formulae specified by the accord. These formulae, or risk-weight functions, are based on a simple credit portfolio model, and thus align more closely with modern risk management techniques. The IRB approach includes two variants: a foundation approach and an advanced approach. The IRB approaches cover a wide range of portfolios, with the mechanics of the calculation varying somewhat across exposure types. In the remainder of this section we present the key inputs and principles behind the regulatory risk-weight formulae, and highlight the differences between the foundation and advanced IRB approaches by portfolio, where applicable. We present these concepts first for wholesale exposures (corporate, bank and sovereigns); then we briefly cover retail exposures and SMEs, as well as specialised lending and equity exposures. 9 III.B.6.4.4

IRB for Corporate, Bank and Sovereign Exposures

The IRB calculation of RWAs relies on four quantitative inputs, referred to as the risk components: Probability of default (PD): the likelihood that the borrower will default over one year. Exposure at default (EAD): the loan amount that could be lost upon default; for loan commitments this is the amount of the facility likely to be drawn if a default occurs. Loss given default (LGD): the proportion of the exposure that will be lost if a default occurs. Maturity (M): the remaining economic maturity of the exposure. Given these four inputs, the corporate IRB risk-weight function produces a capital requirement for each exposure. The RWA for a given exposure is given by RWA 12.5 EAD K .

(III.B.6.5)

9 Risk weight functions represent the latest version of the Accord (BCBS, 2004)



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The capital requirement, K, is the minimum capital per unit exposure, and is given by 10 K

LGD

N

N 1 ( PD )

RN

1

0.999

1 R

PD

MF ( M , PD ) ,

(III.B.6.6)

where N(.) denotes the cumulative normal distribution and N1(.) is its inverse. The formula for K is based on a simple credit portfolio model, with some adjustments as follows: The term N(.) represents the 99.9% default losses of an infinitely granular homogeneous portfolio of unit exposure and 100% LGD, under a one-factor Merton-type credit model. 11 The term LGD

N .

PD

denotes the unexpected default losses 12 of the infinitely

granular portfolio (already adjusted for loss given default); that is, the 99.9% losses minus the expected losses (EL = PD ? LGD). The parameter R denotes the one-factor asset correlation for the homogeneous portfolio in the credit portfolio model. It is obtained from a calibration exercise by the BCBS: R

0.12

1 e 50 PD 1 e 50

0.24 1

1 e 50 PD 1 e 50

.

(III.B.6.7)

R is a decreasing function of the default probability ranging from 24% to 12%, with higher-quality obligors showing higher systemic risk than lower-quality obligors. Figure III.B.6.1 gives the correlation parameter R for corporate exposures (as well as for retail). The final component of the capital requirement in (III.B.6.6) is the maturity function, MF, which is given by MF ( M , PD )

1 ( M 2.5) b ( PD ) 1 1.5b( PD )

(III.B.6.8)

with b ( PD ) [0.11852 0.05478 log( PD )]2 (the function b is referred to as the maturity adjustment). The maturity function MF is equal to one for loans of one-year maturity. Obtained from a calibration exercise by the BCBS, MF empirically adjusts further the default losses (given by N(.) ) to the MtM losses of loans of higher maturity than one year. 10 Note that the 12.5 multiplier cancels the 8% term in the capital. This simply allows the RWA to be expressed

consistently with the Basel I formulae. 11 See Chapter III.B.5 for credit portfolio models and Gordy (2003) for a detailed treatment of this formula. 12 The original formulae in CP3 (BCBS, 2003) did not subtract expected losses. After a period of consultation on the role of EL, the final version bases the risk weights exclusively on unexpected losses.



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For corporate, bank and sovereign exposures the foundation IRB and advanced IRB approaches differ primarily in terms of the inputs that are provided by a bank based on its own estimates and those that have been specified by the supervisor. These differences are summarised in Table III.B.6.5. Table III.B.6.5: Foundation and advanced IRB for corporate, bank and sovereign exposures Data Input

Foundation IRB

Probability of

Provided by bank based on own

default (PD)

estimates

Loss given default (LGD) Exposure at default (EAD)

Maturity (M)

Supervisory values set by the

Advanced IRB Provided by bank based on own estimates

Provided by bank based on own estimates

Committee Supervisory values set by the


Committee Supervisory values set by the


Committee,

(with an allowance to exclude certain

or, at national discretion, provided by

exposures)

bank based on own estimates (with an allowance to exclude certain exposures)

Thus, all IRB banks must provide internal estimates of PD. In addition, advanced IRB banks must provide internal estimates of LGD and EAD, while foundation IRB banks will make use of supervisory values that depend on the nature of the exposure. For example, under the foundation approach: LGD=45% for senior claims, LGD=75% for subordinated claims. These initial LGDs are then adjusted to reflect eligible collateral and guarantees provided for each transaction. In BCBS (2004), the committee decided to adopt a more stringent definition of LGD, where LGDs must be determined based on economic downturn values, and not averages over the cycle or current values. A major element of the IRB framework pertains to the treatment of credit risk mitigants: collateral, guarantees and credit derivatives. The LGD parameter provides a great deal of flexibility to assess the potential value of credit risk mitigation techniques. For foundation IRB



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banks, therefore, the different supervisory LGD values reflect the presence of different types of collateral. Advanced IRB banks have even greater flexibility to assess the value of different types of collateral. With respect to transactions involving financial collateral, the IRB approach seeks to ensure that banks are using a recognised approach to assess the risk that such collateral could change in value, and thus a specific set of methods is provided, as in the standardised approach. In the case of trading book exposures, Basel II outlines the same treatment of EADs as in Basel I, calculating loan equivalents through the use of add-ons and allowing partial recognition of netting and mitigation (e.g. as in equations (III.B.6.2) and (III.B.6.3)). However, practitioners and industry associations like ISDA have pointed out the limitation of such a technique in terms of its accuracy and risk sensitivity, its recognition of mitigation and natural offsets, and the overconservative capital it demands for trading exposures, compared to loans. 13 As industry pressure is building to allow for internal models for trading book EADs, the BCBS is currently revising this topic. 14 Advanced IRB banks will generally provide their own estimates of effective maturity for these exposures, although there are some exceptions where supervisors can allow fixed maturity assumptions. For foundation IRB banks, supervisors can choose on a national basis whether to apply fixed maturity assumptions or to provide their own estimates of remaining maturity. III.B.6.4.5

IRB for Retail Exposures

For retail exposures, there is only a single, advanced IRB approach and no foundation IRB alternative. Retail exposures are classified into three primary product categories, with a separate risk-weight formula for each: residential mortgages exposures (RMEs); qualifying revolving retail exposures (QRREs); other retail exposures (OREs). The QRRE category refers to unsecured revolving credits, which include many credit card relationships. The other retail category refers to all other non-mortgage consumer lending, including exposures to small businesses.

13 For example, short of allowing for full portfolio models, Canabarro et al. (2003) propose to use as a loan equivalent

exposure for a given counterparty the expected positive exposure (EPE) derived from an internal model (e.g. from a Monte Carlo simulation), perhaps increased by a small percentage. 14 The SEC in the USA has issued a proposed capital rule for broker-dealers, aligned with Basel II requirements, which allows for internal models; see Securities and Exchange Commission (2004).



330


The key inputs to the IRB retail formulae are PD, LGD and EAD, all of which are to be provided by the bank based on its internal estimates (no maturity component). In contrast to corporate exposures, these values are not estimated for individual exposures, but instead for pools of similar exposures. Given these three inputs, the retail IRB risk-weight function produces a specific capital requirement for each homogeneous pool of exposures. The risk-weighted assets for a given exposure are also given by expression (III.B.6.5). The formula for the capital requirement K for all three retail product categories is K

LGD

N

N 1( PD )

RN

1

0.999

1 R

PD

(III.B.6.9)

where again N denotes the cumulative normal distribution and N1 is its inverse. The term N(.) PD is the same as for corporate exposures: the 99.9% unexpected default losses of an infinitely granular homogeneous portfolio of unit exposure and 100% LGD. As there is no maturity adjustment for retail exposures, capital only covers default risk. The parameter R denotes the one-factor asset correlation for the homogeneous portfolio and is different for each product category, according to a calibration exercise by the BCBS: residential mortgages QRREs

R 15%

15

other retail

R

4%

R

0.03

1 e 35PD 1 e 35

0.16 1

1 e 35PD 1 e 35

Figure III.B.6.1 gives the correlation parameter R for retail as well as corporate exposures. Correlations are in practice smaller for retail than wholesale. Corporate exposures have declining correlations ranging from 24% to 12% . Retail correlations are generally smaller, flat at 4% and 15% for QRREs and mortgages, and from 16% to 3% for other retail.

15 In CP3, the correlation for QRREs was originally given by

1 e 50 PD 1 e 50 PD 0.11 1 50 1 e 1 e 50 Industry groups recommended revisions for the retail correlation curves based on best practices (e.g. RMA, 2003), suggesting that correlations for retail exposures were lower and not dependent on PD, as with wholesale exposures. In response, the final version of the accord uses a flat 4% correlation. R

0.02



331


Figure III.B.6.1: Correlation parameter,

0.24

Corporate

0.2

Mortgages

0.16

QRRE -CP3

0.12

Other Retail QRRE - Final

0.08 0.04 0 0

0.05

0.1

In (III.B.5.9), as with wholesale, EL = LGD

0.15

0.2

PD

PD is excluded from minimum capital, but is to

be monitored through rules that determine the value of a banks total regulatory capital. 16 III.B.6.4.6

IRB for SME Exposures

Exposures not classified as purely retail, but with turnover of less than 50m, are classified as SMEs. They are further divided into: retail SMEs, where exposures are less than 1m (they may be treated as retail exposures if they are managed by the bank in the same way as retail exposures); corporate SMEs, where the exposure is more than 1m. SMEs that fall into the corporate approach will apply the corporate IRB risk-weight formula, with an optional firm-size adjustment, which leads to a discount in minimum capital. The value of the adjustment is a function of turnover of the borrower (up to 50 million). The average discount will be about 10%, and can range between 1% and 20%. Within this range, a mid-sized firm with a 2% PD would be weighted at 100%. Retail SMEs can be treated using the retail IRB formulae. This requires showing that their exposures are below the 1m threshold and that the bank indeed manages them as aggregated retail (lower-risk, small, diversified loans managed on a pooled basis).

16 In this sense, banks will be required to compare loan loss provisions (general and specific) to EL, and deduct any

shortfall from its capital; excess provisions are credited through Tier 2 capital.



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III.B.6.4.7

IRB for Specialised Lending and Equity Exposures

Specialised lending is associated with the financing of individual projects where the repayment is highly dependent on the performance of the underlying pool or collateral. Two options are given to treat these exposures: For all but one of the specialised lending subcategories, if banks can meet the minimum criteria for the estimation of the relevant inputs, they can use the corporate IRB framework to calculate risk weights. For high-volatility commercial real estate (HVCRE), IRB banks that can estimate the required inputs will use a separate risk weight that is more conservative than the general corporate risk weight. Since the hurdles for meeting these criteria for this set of exposures may be more difficult in practice, CP3 also includes an additional option that only requires that a bank classify such exposures into five distinct quality grades, and provide a specific risk weight for each of these grades. IRB banks must separately treat their equity exposures. Two distinct approaches are given: The first approach builds on the PD/LGD approach for corporate exposures and requires banks to provide their own PD estimates for the associated equity exposures. This approach, however, mandates the use of a 90% LGD value and also imposes various other limitations, including a minimum risk weight of 100% in many circumstances. The second approach provides the opportunity to model the potential decrease in the market value of equity holdings over a quarterly holding period. A simplified version of this approach with fixed risk weights for public and private equities is also included. III.B.6.4.8

Comments on Pillar II

Pillar II of the Basel Accord (supervisory review) is based on a series of guiding principles which point to the need for banks to assess their capital adequacy positions relative to their overall risks, and for supervisors to review and take appropriate actions in response to those assessments. Important new components of Pillar II for credit risk include the treatment of the following: Stress testing. Banks adopting the IRB approach to credit risk will be required to perform a meaningfully conservative stress tests of their own design to estimate the potential increases in capital requirements during a stress scenario. Stress-test results are to be used as a means of ensuring that banks hold a sufficient capital buffer to protect against adverse or uncertain economic conditions. To the extent that there is a capital shortfall, supervisors may, for example, require a bank to reduce its risks or increase its existing capital resources to cover its minimum capital requirements plus the results of a recalculated stress test.



333


Concentration risk. Minimum capital risk weights assume that the portfolio is large and well diversified. Within Pillar II, banks will need to assess the degree of concentration risk they incur. Residual risks arising from the use of collateral, guarantees and credit derivatives, as well as specific securitisation exposures (such as significant risk transfer and considerations related to the use of call provisions and early amortisation features).

III.B.6.5

Basel II: Credit Model Estimation and Validation

In this section we broadly introduce the Basel II methodology for probability of default estimation, the distinction between point-in-time and through-the-cycle ratings, the minimum standards for credit monitoring processes and the validation methods. III.B.6.5.1

Methodology for PD Estimation

Basel II requires that PDs be derived from a two-stage process: 1. Each obligor must be classified into a risk bucket, corresponding to its internal rating grade. Obligors within a bucket share the same credit quality as assessed by the banks internal credit rating system, which must be based on clear rating criteria. 2. A pooled PD is calculated for each bucket. For minimum capital calculations, this PD is assigned to each obligor in a given bucket. Pooled PDs must be long-run averages of one-year realised default rates for borrowers in the bucket. III.B.6.5.2

Point-in-Time and Through-the-Cycle Ratings

Since pooled PDs are assigned to rating buckets (rather than to individual obligors directly), they can differ meaningfully from individual obligors PDs that result from a forecasting model. Thus, the credit ratings approach can have a substantial effect on the on the minimum capital requirements. In this sense, credit rating approaches can be classified into two categories (see, for example, BCBS, 2000): In a point-in-time (PIT) rating approach, obligors are classified into rating grades based on the best available current credit quality information. The internal rating reflects an assessment of the borrowers current condition and/or most likely future condition over the time horizon. Thus, a rating changes as the borrowers conditions changes over the credit/business cycle. In general, PIT ratings tend to rise during business expansions as obligors creditworthiness improve and tend to fall during recessions. In a through-the-cycle (TTC) rating approach, obligors are assigned to rating grades based on their ability to remain solvent over a full business cycle or during stress events. A borrowers riskiness assessment is thus based on a worst-case, bottom of the cycle Copyright ? 2004 The Authors and the Professional Risk Managers International Association


334


scenario. Since they emphasise stress conditions, borrowers ratings would tend to be more stable over the credit/business cycle. In practice, there is large variation between banks rating approaches. Furthermore, the terms PIT and TTC are often defined poorly and used differently across institutions. While not explicitly requiring that rating systems be PIT or TTC, Basel II does hint at a preference for TTC approaches. 17 III.B.6.5.3

Minimum Standards for Quantification and Credit Monitoring Processes

Basel II gives banks great flexibility in determining how obligors are assigned to buckets, but it establishes minimum standards for their credit monitoring processes. Banks can rely on their own internal data, or data derived from external sources as long as they can demonstrate the relevance of such data to their own exposures. In practical terms, banks will be expected to have in place a process that enables them to collect, store and utilise loss statistics over time in a reliable manner. Other minimum standards include the following: Internal rating systems should accurately and consistently differentiate degrees of risk. Banks must define clearly and objectively the criteria for their rating categories. A strong control environment must be in place to ensure that banks rating systems perform as intended and that the resulting ratings are accurate. Banks must have independent and transparent ratings processes and internal reviews. III.B.6.5.4

Validation of Estimates

Basel II requires that banks must have a robust system in place to validate the accuracy and consistency of rating systems, processes, and the estimation of all relevant risk components. While the accord does not go into methodological details, regulators broadly describe two empirical approaches for validating PDs: Benchmarking involves comparing reported PDs for similar obligors across banks and other external systems. Thus, banks that do not estimate PDs effectively or systematically misrepresent them will report substantial differences with respect to their peers.

17 From BCBS (2004, paragraph 415): A borrower rating must represent the banks assessment of the borrowers ability and willingness to contractually perform despite adverse economic conditions or the occurrence of unexpected events. For example, a bank may base rating assignments on specific, appropriate stress scenarios. Alternatively, a bank may take into account borrower characteristics that are reflective of the borrowers vulnerability to adverse economic conditions or unexpected events, without explicitly specifying a stress scenario. The range of economic conditions that are considered when making assessments must be consistent with current conditions and those that are likely to occur over a business cycle within the respective industry/geographic region.



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Backtesting compares the pooled PD for a grade with the actual observed (out of sample) default frequencies for that grade. If a grades pooled PD is truly an estimate of the longrun average of the grades observed yearly default frequencies, over time the two should converge. However, in practice, this convergence could take many years, thus providing important technical challenges.

III.B.6.6

Basel II: Securitisation

A securitisation is a financial structure where cash flows from an underlying pool of exposures are used to service one or more stratified positions, or tranches, reflecting different degrees of credit risk. Payments of the structure depend on the performance of the underlying exposures. In a traditional securitisation, the underlying pool commonly contains standard credit instruments such as bonds or loans. In contrast, a synthetic securitisation uses credit derivatives or guarantees in a funded (e.g. credit-linked notes) or an unfunded (e.g. credit default swaps) way. Common examples of securitisation structures are collateralised bond obligations, collateralised loan obligations and asset-backed securities. Securitisation by its very nature relates to the transfer of risks associated with the credit exposures of a bank to other parties. In this respect, it provides better risk diversification and contributes to enhancing financial stability. See Chapter I.B.6, and Section I.B.6.6 in particular, for more details on these securitisation structures. Some securitisations have enabled banks under the current Basel I Accord to avoid maintaining capital commensurate with the risks to which they are exposed. This is commonly referred to as regulatory arbitrage: the avoidance of minimum regulatory capital charges through the sale or securitisation of a banks assets for which the true risk (and hence economic capital) is much lower than regulatory capital. In contrast, Basel II provides a specific treatment for securitisation, which requires banks to look to the economic substance of a securitisation transaction when determining the appropriate capital requirement in both the standardised and IRB treatments. Under the standardised approach, banks must assign risk weights prescribed by the accord according to various criteria such as a facility type and its external rating (if available). Banks that apply the standardised approach to the type of exposures securitised must also use it under the securitisation framework. Some examples of capital charges under the standardised approach are as follows: A structure with a long-term rating of BBB is assigned a capital of 8% times its notional while an A-rated is assigned 4% (100% and 50% risk weights, respectively). Most unrated structures have a capital factor of 100% (exceptions might be some most senior tranches). Copyright ? 2004 The Authors and the Professional Risk Managers International Association


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Eligible liquidity facilities satisfying some basic criteria might be assigned a lower capital by means of a credit conversion factor (CCF) which is less than 100% (total capital is the product of the notional, the capital factor and the CCF). Banks applying the IRB approach for the type of exposures securitised must also apply it to securitisations. The IRB approach proposes three methods for calculation: Ratings-based approach (RBA). In this case capital factors are tabulated based on a tranches credit rating and thickness, as well as the granularity of the underlying pool. Thus, for example, a AAA thick tranche with granular pool is assigned 0.56% capital (a 7% risk weight), while a BB+ tranche gets 20% capital (250% risk weight). Internal assessment approach (IAA). Subject to meeting various operational requirements, a bank may use its internal assessments of the credit quality of the securitisation exposures that it extends to asset-backed commercial paper programmes (e.g. liquidity facilities and credit enhancements). The internal assessment is then mapped to an equivalent external rating, which is used to determine the appropriate risk weights under the RBA. Supervisory formula approach (SFA). This approach is an attempt to provide a closed-form capital charge, based on a bottom-up risk assessment of the structure. The methodology used to determine the formula is based on similar mathematical modelling of the problem to that used to derive the IRB capital charge of individual exposures (see equation (III.B.6.6)). In essence, the SFA is based on five bank-supplied parameters: o

Kirb the capital charge of the underlying pool of exposures, had the assets not been securitised;

o

L the credit enhancement supporting a given tranche (i.e. how big is the buffer absorbing credit losses, before they hit the tranche);

o

T the thickness of the tranche;

o

N the effective number of exposures in the pool;

o

LGD the exposure-weighted average LGD of the pool.

The reader is further referred to various BCBS documents on securitisation; those interested in the mathematics of the SFA are referred to Gordy and Jones (2003) and Pykhtin and Dev (2002 and 2003). Basel II prescribes the use of these three approaches in a hierarchical manner. The RBA must be applied to securitisation exposures that are rated, or where a rating can be. Where an external or an inferred rating is not available, either the SFA or the IAA must be applied. Securitisation exposures to which none of these approaches can be applied must be deducted.



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III.B.6.7

Advanced Topics on Economic Credit Capital 18

In this section we review the application of credit risk contribution methodologies for ECC allocation, and the shortcomings of VaR for ECC and coherent risk measures. III.B.6.7.1

Credit Capital Allocation and Marginal Credit Risk Contributions

In this section, we briefly highlight key issues on the application of the methodologies introduced in Section III.0.4.2 for computing marginal credit risk contributions for ECC allocation. In addition to computing the total ECC for portfolio, it is important to develop methodologies to attribute this capital a posteriori to various sub-portfolios such as the firms activities, business units and even individual transactions and allocate it a priori in an optimal fashion, to maximise risk-adjusted returns. EC allocation down the portfolio is required for management decision support and business planning, performance measurement and risk-based compensation, pricing, profitability assessment and limits, building optimal riskreturn portfolios and strategies. There is no unique method to allocate ECC. Section III.0.4.2 classifies EC contributions into stand-alone contributions, incremental contributions, and marginal contributions. 19 Every methodology has its advantages and disadvantages, and might be more appropriate for a particular managerial application. The most common approach used today to attribute ECC on a diversified basis is based on the marginal contribution to the volatility (or standard deviation) of the portfolio losses. Such allocations are generally ineffective for credit risk, since loss distributions are far from normally distributed, producing inconsistent capital charges, and in some cases a loans capital charge can even exceed its exposure (see Praschnik et al., 2001; Kalkbrener et al., 2004). Given the definition of ECC, the natural choice for allocating capital is the risk contributions to a VaR-based measure. However, VaR has several shortcomings since it is not a coherent risk measure (see Section III.B.6.7.2). Specifically, while VaR is sub-additive for normal distributions, this is not true in general. This limitation is particularly relevant for credit losses, which may be far from normal and not even smooth. 20 Furthermore, the pointwise nature of VaR has generally lead to difficulties in computing accurate and stable risk contributions with simulation. Recently,

18 This section is not mandatory for the exam, but is added for completeness. 19 The reader is reminded that there is currently no universal terminology for these methodologies in the literature. 20 The discreteness of individual credit losses leads to non-smooth profiles and marginal contributions.



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several authors have proposed the use of expected shortfall (ES) for attributing ECC (see, for example, Kalkbrener et al., 2004). As a coherent risk measure, ES represents a good alternative both for measuring and allocating capital. In particular, ES yields additive and diversifying capital allocations. This requires, however, a modification of the standard interpretation of ECC to act as a buffer for an expected loss conditional on exceeding a certain quantile. As explained in Section III.0.4, marginal contributions require the computation of a derivative of the risk measure (see equation (III.0.16)). The general theory behind the definition and computation of these derivatives in terms of quantile measures (e.g. VaR, ES) has recently been developed (see Gouri閞oux et al., 2000; Tasche, 2000, 2002). Several semi-analytical approaches have also recently been proposed for VaR or ES contributions (e.g. Martin et al.., 2001; Kurth and Tasche, 2003). While computing the conditional expectations can be challenging when credit losses are estimated from a Monte Carlo simulation, various methodologies have been devised in recent years for VaR and ES contributions in simulation models (see Kalkbrener et al., 2004; Hallerbach, 2003; Mausser and Rosen, 2004). Simulation provides the flexibility to support more realistic credit models, which include diversification through multiple factors, more flexible codependence structures, multiple asset classes and default models, stochastic (correlated) modelling of exposures, and loss given default. III.B.6.7.2

Shortcomings of VaR for ECC and Coherent Risk Measures

In its common interpretation, ECC is a buffer that provides protection against potential credit losses, at a confidence level that is less than 100% (e.g. 99.9%). The confidence level is consistent with the desired credit rating of the firm. This leads to measures of capital that reflect a given quantile of a credit loss distribution. Given this definition, VaR is an intuitive measure for ECC. However, VaR has several shortcomings since it is not a coherent risk measure (in the sense of Artzner et al., 1999). In particular, VaR is not sub-additive in general. A risk measure

is said to be sub-additive if (X

Y)

(X )

(Y )

(III.B.6.10)

for any two portfolios X and Y. Thus, sub-additivity is a property of risk measures required to account for portfolio diversification.



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While VaR is always sub-additive for normal loss distributions, in more general cases the total portfolio VaR might be higher than the sum of stand-alone VaRs. This is particularly relevant to credit loss distributions, which are far from normal and not smooth, given the discreteness of individual credit losses Example III.B.6.4 Consider a simple one-year BBB loan with a notional of $100. The obligor has a PD of 0.5% and a 50% LGD. Expected losses are $100 ? 0.5 ? 0.005 = $0.25. However, the 99% VaR is 0 (we are more than 99% certain that we will not incur a loss). This leads to a 'negative' unexpected loss and, thus, the stand-alone capital of this position is negative. Now consider a portfolio that invests $100 equally in 10 one-year loans to different BBB obligors. Assume obligor defaults are independent and a 50% LGD for all of them. In this case, EL is still $0.25. There is now also a 95.11% probability of no defaults and a 99.89% chance that there is at least one defaulted loan. Thus, the 99% VaR is equal to $5 ($10 notional ? 50% LGD). Based on VaR, the total credit capital to support this portfolio is $4.75. However the stand-alone VaR of each loan is zero (thus the stand-alone capital of each loan remains, in principle, $0.25). The theory of coherent risk measures (Artzner et al., 1999) is well developed and has become popular among academics and practitioners. Coherent risk measures such as expected shortfall present a good alternative both for measuring and allocating capital. This requires, of course, the modification of the standard definition and interpretation of capital as a buffer to cover x% losses, to one where the buffer would cover the expected losses conditional on reaching an x% loss.

III.B.6.8


This chapter reviews the main concepts for estimating and allocating ECC as well as the current regulatory framework for credit capital. Credit portfolio methodologies are the key tools to compute ECC from a bottom-up approach. Today, they are used broadly by practitioners for estimating ECC and managing credit risk at the portfolio level. Furthermore, various institutions are starting to apply credit portfolio frameworks to compute and manage ECC at the enterprise level. Credit portfolio models must be defined and parameterised consistently with the ECC definition of the firm. This definition includes the time horizon, the type of credit loss (default only or mark-to-market) and the confidence level (or quantile) of the loss distribution. Since ECC is



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designed to absorb unexpected losses up to a certain confidence level, it is commonly estimated by a VaR-type measure (at the defined confidence level) which subtracts expected losses. We further address some potential shortcomings of VaR for measuring risk as well as for allocating capital, and discuss other measures such as expected shortfall. In the past, regulatory credit capital has differed significantly from ECC. However, the new Basel II Accord for banking regulation has introduced a closer alignment of regulatory credit capital with current best-practice credit risk management and ECC measurement. While today falling short of allowing the use of credit portfolio models to estimate regulatory credit capital, Basel II has introduced various approaches for minimum capital requirements of increased complexity and alignment with the credit riskiness of an institution. In particular, for the first time, it allows banks to use internal models for estimating key credit risk components (PDs, exposures and LGDs). Furthermore, the IRB risk-weight formulae are based on solid credit portfolio modelling principles. Finally, with its three-pillar foundation, Basel II focuses not only on the computation of regulatory capital, but also on a holistic approach to managing risk at the enterprise level.

References Artzner, P, Delbaen, F, Eber, J-M, and Heath, D (1999) Coherent measures of risk, Mathematical Finance, 9(3), pp. 203228. Basel Committee on Banking Supervision (1988) International convergence of capital measurement and capital standards. Available at http://www.bis.org Basel Committee on Banking Supervision (1995) Basel capital accord: treatment of potential exposure for off-balance-sheet items. Available at http://www.bis.org Basel Committee on Banking Supervision (2000) Range of practice in banks internal ratings systems. Discussion paper, available at http://www.bis.org Basel Committee on Banking Supervision (2003) The new Basel capital accord: Consultative document. Available at http://www.bis.org Basel Committee on Banking Supervision (2004) International convergence of capital measurement and capital standards: A revised framework. Available at http://www.bis.org Canabarro, E, Picoult, E, and Wilde, T (2003) Analysing counterparty risk. Risk, September, pp. 117122. Gordy, M (2003) A risk-factor model foundation for ratings-based bank capital rules, Journal of Financial Intermediation, 12(3), pp. 199232. Gordy, M, and Jones, D (2003) Random tranches, Risk, March, pp. 7883. Gouri閞oux, C, Laurent, J-P, and Scaillet, O (2000) Sensitivity analysis of values at risk, Journal of Empirical Finance, 7(34), pp. 225245.



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Hallerbach, W G (2003) Decomposing portfolio value-at-risk: a general analysis, Journal of Risk, 5(2), pp. 118. Kalkbrener, M, Lotter, H, and Overbeck, L (2004) Sensible and efficient capital allocation for credit portfolios, Risk, January, pp. S19S24. Kurth, A, and Tasche, D (2003) Contributions to credit risk. Risk, March, pp. 8488. Kupiec, P (2002) Calibrating your intuition: Capital allocation for market and credit risk. IMF Working Paper WP/02/99, available at http://www.imf.org Martin, R, Thompson, K, and Browne, C (2001) VAR: who contributes and how much? Risk, August, pp 99102. Mausser, H, and Rosen, D (2004) Scenario-based risk management tools. In S W Wallace and W T Ziemba (eds), Applications of Stochastic Programming. Philadelphia: SIAM. Praschnik, J, Hayt, G, and Principato, A (2001) Calculating the contribution, Risk, 14(10), pp. S25S27. Pykthtin, M, and Dev, A (2002) Credit risk in asset securitisations: an analytical model, Risk, May, pp. S1620. Pykthtin, M, and Dev, A (2003) Coarse-grained CDOs. Risk, January, pp. 113-116. Risk Management Association (2003) Retail credit economic capital estimation best practices. Working Paper, Risk Management Association, available at http://www.rmahq.org Securities and Exchange Commission (2004) Proposed rule: Alternative net capital requirements for broker-dealers that are part of consolidated supervised entities. 17 CFR Part 240. http://www.sec.gov/rules/proposed/34-48690.htm Tasche, D (2000) Conditional expectation as quantile derivative. Working paper, Technische Universit鋞M黱chen. Tasche, D (2002) Expected shortfall and beyond. Working paper, Technische Universit鋞 M黱chen.



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III.C.1 The Operational Risk Management Framework Michael K. Ong 152 In this chapter I provide a brief outline of how to establish an operational risk management framework within an institution. Operational risk has received a lot of attention recently although it is not an entirely new field of risk management. Many of the biggest losses in the financial industry and the corporate arena can be attributed, in one way or another, to operational risk failures. The chapter begins by highlighting some of the better-known losses in the recent past and argues why it is important for individual institutions to define what operational risk means to them. The Basel II proposals, scheduled for promulgation in 2006, have also provided some guidance (primarily to banking institutions) on the types of operational risk failure and their associated loss event types. The chapter then discusses the goals and scope of an operational risk management framework. It outlines the key components of operational risk and presents some useful tools, e.g., the risk catalogue and risk scorecard, for identifying specific operational risk failures. Finally, it explains how to make the risk assessment process work through the involvement of senior management and every business unit within the institution.

III.C.1.1

Introduction

Operational risk management has become increasingly important for financial institutions over the past several years. The need for a better understanding of operational risk is driven primarily by two factors, namely, the growing sophistication of financial technology and the rapid deregulation and globalisation of the financial industry. These factors contribute to the increasing complexity of banking activities and, therefore, heighten the operational risk profile of the financial services industry. Over the past few years, a significant number of high-impact and highprofile losses, some leading to the demise of once revered, well-respected institutions, have pointed consistently to failure in operational risk management. This seemingly sudden awareness of operational risk management is quite ironic considering that operational risk has always been an integral risk associated with doing business. Operational risk is as old as the banking industry itself, the rating agency Fitch reports, and yet, the industry has only recently arrived at a definition of what it is. The report goes on to say that in its rating analysis of banks, Fitch will be looking for evidence of a clearly articulated definition of operational risk, examining the quality of an organizations structure and operational risk culture, 152 Professor of Finance and Executive Director of the Center for Financial Markets, Stuart Graduate School of Business, Illinois Institute of Technology. The author wishes to extend his sincerest thanks to the editors, Carol Alexander and Elizabeth Sheedy, for their careful editing of this chapter.



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the development of its approach to the identification and assessment of key risks, data collection efforts, and overall approach to operational risk quantification and management (Ramadurai et al., 2004). In addition, Moodys believes that operational risk management improves the quality and stability of earnings, thereby enhancing the competitive position of the bank and facilitating its long-term survival (Moodys Investors Service, 2003). Moodys goes on to comment that: The control of operational risk is fundamentally concerned with good management, which involves a tenacious process of vigilance and continuous improvement. This is a value-adding activity that impacts, either directly or indirectly, on bottom-line performance. It must, therefore, be a key consideration for any business. Since operational risk will affect credit ratings, share prices, and organisational reputation, analysts will increasingly include it in their assessment of the management, their strategy and the expected long-term performance of the business. Thus rating agencies are now clearly interested in how financial institutions manage their operational risk. In fact, how institutions manage their operational risks is likely to influence how they will be rated by the rating agencies. Against the background of greater complexity and opaqueness in the banking industry due to technological advancement, the Basel Committee for Banking Supervision (2003) cites the emergence of new forms of risk that require attention immediately: Developing banking practices suggest that risks other than credit, interest rate and market risk can be substantial. Examples of these new and growing risks faced by banks include: If not properly controlled, the greater use of more highly automated technology has the potential to transform risks from manual processing errors to system failure risks, as greater reliance is placed on globally integrated systems; Growth of e-commerce brings with it potential risks (e.g., internal and external fraud and system security issues) that are not yet fully understood; Large-scale acquisitions, mergers, de-mergers and consolidations test the viability of new or newly integrated systems; The emergence of banks acting as large-volume service providers creates the need for continual maintenance of high-grade internal controls and back-up systems; Banks may engage in risk mitigation techniques (e.g., collateral, credit derivatives, netting arrangements and asset securitisations) to optimise their exposure to market risk and credit risk, but which in turn may produce other forms of risk (e.g. legal risk); and Growing use of outsourcing arrangements and the participation in clearing and settlement systems can mitigate some risks but can also present significant other risks to banks.



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The diverse set of risks listed above can be grouped under the heading of operations risk. The emergence of the types of risks listed above by the Basel Committee forms the basis for regulatory pressure currently felt by many major financial institutions. The Financial Services Authority (2003) reported that, even as late as in mid-2003, the financial industry was still in the early stages of developing operational risk frameworks. In its survey, the FSA reported that a majority of firms stated that their primary motivation for developing the operational framework was increased regulatory focus, with regulation a more significant driver in smaller firms than in major financial groups. Should the impetus for developing a sound operational risk management framework be driven primarily by emerging concerns raised by rating agencies or the threat of greater scrutiny by regulatory authorities? In fact, should operational risk attain the limelight it is currently basking under because of the impending capital charge being deliberated in Basel II? I think not. These motivations in isolation are unlikely to lead to successful implementation of an operational risk management function. A coherent, sound and successful operational risk management framework can only come from an internal realisation and desire amongst senior management that this is a value-adding activity that ultimately impacts the bottom line of the institution.

III.C.1.2

Evidence of Operational Failures

Table III.C.1.1 lists some of the largest derivatives losses on the Street during the 1990s. In all cases the losses are attributable, at least in part, to operational risk. Losses resulted from flaws in the risk management framework of the institutions concerned. One of the most dramatic and well-documented derivatives losses was the collapse in 1995 of Barings, Britains oldest merchant bank (200 years!). One person (Nick Leeson), based in Singapore (several thousand miles away from corporate headquarters), managed to circumvent internal systems over an extended period of time to hatch and hide his trading schemes. It ultimately resulted in over $1.2bn of losses. This all pointed to senior managements abject failure to institute proper managerial, financial and operational control over the institution. Since the banks risk management and control functions were very weak, the system of checks and balances failed at several operational and managerial junctures and in more than one location where the bank operated. The Barings debacle is not the story of just one single solitary rogue trader, but rather the breakdown of an entire organisation that had failed to exercise sufficient oversight and control of its people at all levels, its lack of clear directions and accountability for the processes



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within the bank, and the failure of technology to detect trading and booking anomalies for an extended period of time. Yet all of these derivatives losses combined pale in comparison to the S&L crisis ($150bn) of the 1980s, the non-performing real estate loans of Japanese banks ($500bn) in the early 1990s, the Credit Lyonnais bankruptcy ($24bn) due to bad debt in 1996, and the more recent asset management frauds at Deutsche Morgan Grenfell, Jardine Fleming, etc. 153 The early 2000s witnessed the multi-billion dollar collapse of Enron, WorldCom, Tyco, Parmalat, and many other fallen angels which, in more ways than one, brought about a sense of urgency for better corporate governance. Corporate governance in essence calls for greater accountability of senior management in an effort to combat corporate fraud. And corporate fraud is one common instance of operational risk failure. Table III.C.1.1: Publicly disclosed derivatives losses in the 1990s Company/Entity Air Products Askin Securities Baring Brothers Cargill (Minnetonka Fund) Codelco Chile

Loss Amount ($m) 113 600 1240.5 100 200

Glaxo Holdings PLC Long Term Capital Management Metallgesellschaft Orange County

150 4000 1340 2000

Proctor & Gamble

157

Area of loss Leverage & currency swaps Mortgage-backed securities Options Mortgage derivatives Copper & precious metals futures and forwards Mortgage derivatives Currency & interest rate derivatives Energy derivatives Reverse repurchase agreements & leveraged structured notes Leveraged German marks spread

US dollars

Source: Exhibit 1 in McCarthy (2000), taken from Brian Kettel, Derivatives: Valuable Tool or Wild Beast? Copyright ? 1999 by Global Treasury News (www.gtnews.com).

153Details of other financial scandals can be found at http://www.ex.ac.uk/~rdavies/arian/scandals/.



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Much more recently, improper trading in mutual funds has cost banks and some funds management companies millions of dollars in fines. For instance, in mid-March of 2004 Bank of America Corp. and its merger partner, FleetBoston Financial Corporation, agreed to pay a collective sum of $675m to settle charges with securities regulators that they had defrauded shareholders by allowing select investors to trade improperly in their mutual funds. The Boston Globe reported on 16 March 2004: With this agreement, mutual fund firms have now reached settlements totalling $1.65 billion, eclipsing the $1.4 billion Wall Street firms agreed to pay [in 2003] to settle charges their analysts issued biased research to win investment banking business, New York Attorney General Eliot Spitzer said.

III.C.1.3

Defining Operational Risk

What is operational risk? This depends on what an institution wishes to gain from its operational risk management function. No two institutions will have exactly the same definition of what operational risk means since there are unique facets such as composition of the business portfolio, internal culture, risk appetite, etc., that differentiate the types of operational risks the institutions are exposed to. Nevertheless, there are some very clear commonalities that are shared by different financial institutions. There are many highfalutin and facetious ways to define operational risk. The most important element to take into account, however, is to choose a definition that is in line with the institutions philosophy and sound management culture of taking proactive stances in managing the risks of the enterprise. One of the earliest definitions in the financial industry broadly defines operational risk in financial institutions as the risk that external events, or deficiencies in internal controls or information systems, will result in an economic loss whether the loss is anticipated to some extent or entirely unexpected. There are two obvious observations here. This early industry definition identifies both the expected and unexpected losses attributable to operational mishaps. In simple terms, expected losses are those losses incurred during the natural course of doing business, and unexpected losses are usually associated with big surprises resulting from lapses in management and breakdown in controls. The scope of operational risk in this early definition is quite broad and extends to all facets and aspects of risk associated with both internal and external events, tangible resources such as information technology and systems, and intangibles such as people and process. This early industry definition eventually became the cornerstone of the official definition from Basel II. The first Basel definition of operational risk was simply the risk of direct or indirect loss



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resulting from inadequate or failed internal processes, people and systems or external events. This definition includes legal risk, but Basel II explicitly excluded strategic and reputational risk. These exclusions are very important aspects of the daily operation of any financial institution, but are admittedly much more difficult to assess and manage. Concerns were expressed about the exact meaning of direct and indirect loss. Consequently the current Basel II definition drops this distinction but provides clear guidance on which losses are relevant for regulatory capital purposes. This is achieved by defining the types of loss events that should be recorded in internal loss data. In its September 2001 press release for the Working Paper on the Regulatory Treatment of Operational Risk, the Risk Management Group (RMG) of the Basel Committee on Banking Supervision defined operational risk as the risk of loss resulting from inadequate or failed internal processes, people and systems or from external events.

According to the RMG press release this is a causal-based definition: It is important to note that this definition is based on the underlying causes of operational risk. It seeks to identify why a loss happened and at the broadest level includes the breakdown by four causes: people, processes, systems and external factors. This causal-based definition, and more detailed specifications of it, is particularly useful for the discipline of managing operational risk within institutions. However, for the purpose of operational risk loss quantification and the pooling of loss data across banks, it is necessary to rely on definitions that are readily measurable and comparable. Given the current state of industry practice, this has led banks and supervisors to move towards the distinction between operational risk causes, actual measurable events (which may be due to a number of causes, many of which may not be fully understood), and the P&L effects (costs) of those events. Operational risk can be analysed at each of these levels. The Basel II definition is primarily for capital adequacy purposes. That is, a key output of the regulatory framework is a measure of the amount of capital required by a financial institution as a buffer against unexpected operational risks.

III.C.1.4

Types of Operational Risk

The types of operational risk encountered daily within an institution are quite diverse and plentiful. What are the main types of operational risk financial institutions need to be wary of? The RMG struggled with this very same issue when it embarked on its event-by-event loss data collection exercise in June 2002. 154 In its press release, the Basel Committee on Banking

154 Having had two previous quantitative impact studies (QIS) in the previous years, the RMG decided that this more

recent loss data collection exercise would specifically concentrate on very granular loss data.



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Supervision (2002) described the goals of this exercise: The primary purpose of this survey is to collect granular (event-by-event) operational risk loss data to help the Committee determine the appropriate form and structure of the AMA (Advanced Measurement Approach). To facilitate the collection of comparable loss data at both the granular and aggregate levels across banks, the Committee is again using its detailed framework for classifying losses. In the framework, losses are classified in terms of a matrix comprising eight standard business lines and seven loss event categories. These seven event categories are then further divided into 20 sub-categories and the Committee would like to receive data on individual loss events classified at this second level of detail if available. The eight standard business lines are: corporate finance; trading and sales; retail banking; commercial banking; payment and settlement; agency services; asset management; and retail brokerage. Table III.C.1.2: Basel II definition of business lines



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In Table III.C.1.2 we see that investment banking as a primary business unit is further split up into two level 1 sub-units: corporate finance; and trading and sales. Furthermore, within the trading and sales sub-unit, there are further level 2 sub-delineations: sales; market making; proprietary positions; and treasury activities. Each of these sub-units is classified based on its respective business functions, such as fixed income, foreign exchange, equity, commodities, credit, funding, brokerage, and so on. The Committee proposes looking at seven loss event categories associated with each business unit. These are depicted in Table III.C.1.3. The level 1 event types are the practical and obvious events: internal fraud; external fraud; employment practices and workplace safety; clients, products and business practices; damage to physical assets; business disruption and system failures; and execution, delivery and process management. Furthermore, within the level 1 event type category, there are at least two level 2 sub-categories, 20 sub-categories in total. Each of these sub-categories is again classified based on the associated business activities. Table III.C.1.3: Basel II definition of loss event types



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Table III.C.1.3 (Continued): Basel II definition of loss event types

For internal purposes it is very important to establish suitable definitions of the different types of operational risk that are relevant for each individual institution. Once suitable definitions of event types are decided upon, the institution can proceed to establish a structure for the operational risk management framework. It is interesting to note that almost all of the so-called internationally active banks that are the primary focus of Basel II have their own unique definitions of operational risk event types that are tailored to their particular businesses, corporate culture and risk appetite. For example, one internationally active bank defines operational risk as: The risk of inadequate identification of and/or response to shortcomings in organizational structure, systems, transaction processing, external threats, internal controls, security measures and/or human error, negatively affecting the banks ability to realize its objectives. A smaller domestic bank defines operational risk simply as: The risk associated with the potential for systems failure in a given market.

III.C.1.5

Aims and Scope of Operational Risk Management

The fundamental goal of operational risk management should be risk prevention. The assessment (meaning the quantitative measurement) of operational risk is of secondary importance. Because complete elimination of operational risk failures is not feasible, our operational risk management framework must aim to minimise the potential for loss through whatever means possible. Indeed, the risk management of the entire institution as an enterprise should focus more on the operational aspects of the different business activities with the important provision that



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the other key risk management functions within the enterprise (e.g., market risk, credit risk, audit, and compliance) are already firmly grounded. Regardless of how an institution chooses to define operational risk, it is vitally important at the outset to explicitly articulate what its target objectives and key concerns are. I suggest the following important objectives when establishing an operational risk management function: 1.

To formally and explicitly define and explain what the words operational risk mean to the institution.

2.

To avoid potential catastrophic losses.

3.

To enable the institution to anticipate all kinds of risks more effectively, thereby preventing failures from happening.

4.

To generate a broader understanding of enterprise-wide operational risk issues at all levels and business units of the institution in addition to the more commonly monitored credit risk and market risk.

5.

To make the institution less vulnerable to such breakdowns in internal controls and corporate governance as fraud, error, or failure to perform in a timely manner which could cause the interests of the institution to be unduly compromised.

6.

To identify problem areas in the institution before they become critical.

7.

To prevent operational mishaps from occurring.

8.

To establish clarity of peoples roles, responsibilities and accountability.

9.

To strengthen management oversight at all levels.

10. To identify business units in the institution with high volumes, high turnover (i.e., transactions per unit time), high degree of structural change, and highly complex support systems. Such business units are especially susceptible to operational risk. 11. To empower business units with the responsibility and accountability of the business risks they assume on a daily basis. 12. To provide objective measurements of performance for operational risk management. 13. To monitor the danger signs of both income and expense volatilities. 14. To effect a change of behaviour within the institution and to enhance the culture of control and compliance within the enterprise. 15. To ensure that there is compliance to all risk policies of the institution and to modify risk policies where appropriate. 16. To provide objective information so that all services offered by the institution take account of operational risks. 17. To ensure that there is a clear, orderly and concise measure of due diligence on all risktaking and non-risk-taking activities of the institution.



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18. To provide the executive committee 155 regularly with a concise state of the enterprise report for strategic and planning purposes. The stated objectives tacitly assume that the institution already has in place robust credit risk and market risk management functions, supported by audit, compliance and risk control oversight. Note that the operational risk management objectives delineated above should apply to all business units, including those responsible for market risk and credit risk management. In practice, the scope of an operational risk management function within an institution should aim to encompass virtually any aspect of the business process undertaken by the enterprise. The scope must transcend those business activities that are traditionally most susceptible to operations risk that is, those activities with high volume, high turnover, and highly complex support systems, e.g. trading units, back office, and payment systems. It is true that the business activities sharing these characteristics have the greatest exposure to operational risk failures. Nevertheless, other business activities could potentially sustain economic losses of similar magnitude. Table III.C.1.4: Two broad categories of operational risk Operational strategic risk (external)

Operational failure risk (internal)

Defined as the risk of choosing an

Defined as the risk encountered in the pursuit of a

inappropriate strategy in response to

particular chosen strategy due to:

external factors such as: political taxation regulation societal competition

people process technology others Source: adapted from Crouhy et al. (1998)

Should operational risk management encompass external events? The goal of operational risk management must be to focus on internal processes (as opposed to external events) since only internal processes are within the control of the firm. The firms response to external events is, however, a valid concern for operational risk management. Hence we examine operational risk from two interrelated perspectives. Table III.C.1.4 distinguishes between operational strategic

155 In this context, the Executive Committee, composed only of very senior members of management, is assumed to

be the highest governing body of the institution.



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risk (i.e., a flawed internal response to external stimuli) and operational failure risk. Failure to comply with externally dictated strategic risk factors such as changes in tax laws 156 and new derivatives accounting treatment (e.g., FASB 133) ultimately translates to an internal operational risk failure. Once senior management issues the call to action in response to an external stimulus, there must be no internal breakdown in the people, processes and technologies supporting the strategic call to action.

III.C.1.6

Key Components of Operational Risk

In view of the very wide scope of an institutions operational risk management function, we might want to concentrate on some key components of operational risk, such as the following. (i) Core operational capability Risks to the institutions core operational capability include the risk of premises, people or systems becoming unavailable due to: natural disasters, fire, bombs or technical glitches; loss of utilities such as power, water or transportation; employee disputes such as strikes; loss of key operational personnel; and the loss or inadequacy of systems capabilities due to computer viruses or Y2K issues. All of the aforementioned events seriously disrupt the institutions core competency in supporting its long-term and stable operations, thereby representing a considerable exposure in terms of their possible impact on the institutions future earnings and credibility. The good news is, for the most part, that many of the risks mentioned above are largely insurable at some cost to the institution. Insurance is, therefore, a useful risk mitigant for these kinds of failure risk. 157 (ii) People 158 An institutions most important assets are its good people. Unfortunately, people also contribute a myriad of problems through: human error; fraud, 159 lack of honesty 160 and integrity; lack of

156 For example, an institution operating in another country might encounter some unexpected tax liability due to an

unforeseen change in local taxation rules that was not anticipated by the accounting department. This is definitely an operational risk item that could lead to large unanticipated fines and tax liabilities. 157 Basel II currently does not recognise insurance as a risk mitigant, except in some restricted cases within the Advanced Measurement Approach. This is somewhat odd considering the fact that banks have routinely used insurance as a risk management tool. 158 Unfortunately, the category of people is by far the largest cause of operational risk failures. 159 An FDIC study found that fraud was the main contributing factor to 25% of 92 bank failures in the period 1960 77. The proportion rises to 83.9% if one includes insider fraud i.e., improper lending to individuals or groups connected with the bank. A review by the Bank of England suggested that, in the UK in the period 198496, fraudulent concealment was a major contributory factor in 7 out of 22 cases of bank problems. In many of these cases, these bank frauds were perpetrated by senior managers of the banks themselves. 160 On the subject of integrity and honesty, rogue trading generally surfaces as the most obvious case of dishonesty and fraud; however, we need to recognise that the problem of fraud also extends to teller theft, mailroom theft, illegal funds transfers, and insider fraud mentioned in footnote 8, etc. Some well-known cases of securities fraud and rogue trading are: Kidder-Peabody (Jett, $340m, 1994), Orange County (Citron, $1.6bn, 1994), Barings (Leeson, $1.2bn,



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cooperation and teamwork; in-fighting, jealousies, and rumour-mongering; personal sabotage; office politics; 161 lack of segregation and risk of collaboration; lack of professionalism and customer focus; over-reliance on key individuals, insufficient skills, training and education; insubordination; employment disputes; poor management and supervision; and a lack of culture of control, discipline and compliance. 162 (iii) Client relationships An institution derives much of its value from its reputation and the services it provides to its client base. Any damage to an institutions reputation has the potential to disrupt revenue flow. From an operational risk perspective, the institution needs to assess how disreputable activities might harm its client relationships. Examples include: money laundering; Nazi gold; 163 improper client suitability and lack of disclosure; 164 false valuations of client assets to mislead or conceal losses; collusional relationships with broker-dealers; cosy association with highly-leveraged institutions; and dishonest practices amidst competition that can harm the institutions reputation. (iv) Transactional and booking systems Operational risk failures are no longer limited to settlement risk in the trading accounts or back office. More recently, with advances in automation, transactional issues also include: data capture and processing; deal confirmation 165 and contractual documentation, e.g., ISDA master agreements; collateral management; and general processing and payment/settlement errors which not only disrupt the flow of business but also put an institution at risk of litigation. In addition, corporate banking activities contributing to operational failures may include: correspondent banking; payment services; treasury services; private trust and executor services; structured finance; custody; and leasing. From a retail banking perspective, there is even more room for potential operational failures associated with such retail banking activities as: mortgage servicing; funds management; deposit taking; sending; foreign exchange; custody; credit cards; ATMs; private banking; and insurance. The transactional processes associated with handling retail customers are many: payments; cheque clearing; cash handling and teller errors; credit analysis; account opening;

1995), Daiwa (Iguchi, $1.1bn, 1995), Sumitomo (Hamanaka, $1.8bn, 1995) and many other cases involving varying amounts of losses. 161 Ask yourself this question: how many institution-wide problems and inefficiencies were caused by internal fights and office politics? 162 Rogue trading is not the only contributor to well-publicised derivatives losses. Table III.C.1.1 is a list of the largest derivatives losses attributable largely to failures in risk management where operational risk played a major role. 163 This has become an important reputation risk issue among a few big European banks in the recent past. 164 Among the most highly publicised client suitability and disclosure cases are the Gibson Greetings and Procter & Gamble lawsuits against Bankers Trust (1994) and the Orange County collective legal debacle with Merrill Lynch, Morgan Stanley Dean Witter, and Nomura Securities (1994). 165 The best-known documented case of booking errors occurred at Salomon Brothers in the mid-1990s where a backoffice confirmation for a trade was erroneously booked several orders of magnitude larger than the intended trade.



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documentation; mortgage applications; interest charges; processing credit/debit card transactions; processing insurance claims; and payroll processes. In addition, associated with each of these business processes is heavy reliance on a sound and stable systems infrastructure within the institution. (v) Reconciliation and accounting Of course the reconciliation of transactions at different levels of institutional activities is important from a bookkeeping perspective. Another important aspect of reconciliation and accounting is that it enables the institution to identify areas of inefficient capital allocation. More broadly, a finance or accounting department, in a quagmire of bureaucracy and mere paperpushing, can do the institution a lot of harm by failing to provide senior management with a precise picture of the state of the institutions finances. The inability to reconcile properly the revenue-generating activities with the general ledger per se inhibits the institution from strategically assessing the performances of its business units and their growth potential. 166 In addition, it also undermines the institutions ability properly to allocate its scarce resources, such as capital. Finally, the inability of the finance, legal or accounting departments fully to assess the implications of regulatory changes puts the institution at a serious disadvantage with regard to tax shelters and favourable legal treatment of the institutions assets and liabilities. (vi) Change and new activities To stagnate is to fall behind. But, in responding to rapid industry developments, the institution should not stumble and fall when initiating new business activities. For example, the introduction of the euro or the change in regulatory accounting rules, e.g., FAS 133, 167 requires the institution to be more vigilant in implementing new technology, re-engineering its processes, expanding its staff, accepting new clients, launching new products or entering into new markets. More recently, we have new regulatory directives for anti-money laundering and terrorist funding, the SarbanesOxley Act of 2002 which requires all listed institutions to enforce more effective corporate governance and more effective financial statements reporting, and many other new directives concerning securities fraud promoted by the Securities and Exchange Commission. Inability to adapt to change may damage an institutions reputation or disrupt the continuity of its old businesses. 168

166 Ask yourself this question: through our current finance and accounting systems, do we know for sure where we are

generating the greatest revenue for the least amount of risks that we take? 167 Ask yourself this question: how quickly can the institution conform to the FAS 133 directives without unduly taxing its resources and disrupting its day-to-day business operations? 168 Ask yourself this question: can the institution leverage its current resources and technology and enter into a new market without unduly incurring additional expenses? If the answer is negative, there is operational risk in the headline risk categories involving people, processes and technology.



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(vii) Expense and revenue volatility A sure sign of operational failures associated with management control is a rapid increase in expenses and significant deviations from budget. Expense increases (including bonuses, salaries, and systems infrastructure spending) without adequate return signals a potential breakdown and inefficiencies in people, process and technology. Rapid increase in expenses is not necessarily a desirable symbol of growth but, in my years of observation, it is also a sure sign of lax accounting and a complacent management on the verge of going out of control. 169 It is normally associated with a banks undesirable corporate culture of wanton waste and lack of accountability. On a related matter, excessive revenue volatility is the result of at least two related factors: the inability to respond properly to external market conditions; and the failure to control the operational cost base. Each of these key factors has its obvious attendant operational failures in people, process and technology in responding to external risk factors. I need not elaborate further on this.

III.C.1.7

Supervisory Guidance on Operational Risk

The Basel Committee on Banking Supervision has provided preliminary supervisory guidelines for the management and supervision of operational risk. The guidelines are intended to serve as best or sound practices within the financial industry. The Committee recognises that the exact approach for operational risk management chosen by an individual bank will depend on a range of factors, including its size and sophistication and the nature and complexity of its activities. However, despite these differences, clear strategies and oversight by the board of directors and senior management, a strong operational risk culture and internal control culture (including, among other things, clear lines of responsibility and segregation of duties), effective internal reporting, and contingency planning are all crucial elements of an effective operational risk management framework for banks of any size and scope. The Committee therefore believes that the principles outlined in this paper establish sound practices relevant to all banks (Basel Committee on Banking Supervision, 2003). The Committee also recognizes that internal operational risk culture is taken to mean the combined set of individual and corporate values, attitudes, competencies and behaviours that determine a firms commitment to and style of operational risk management. Recognising the different nature of operational risk in different institutions, the Committee defines the management of operational risk to mean the identification, assessment, monitoring and control/mitigation of risk. To this end, the sound practice paper of February 2003 is structured around ten basic principles grouped into four main themes:

169 Ask yourself this question: in the past three years, did the increase in expenditure result in a greater market share or

revenue to the institution? If the answer is negative, there is operational risk involving inefficiency and misallocation of precious resources.



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Developing an appropriate risk management environment. Risk management: identification, assessment, monitoring, and mitigation/control. Role of supervisors. Role of disclosure. By imposing the ultimate responsibility on senior management via the board of directors, the first theme emphasizes the importance of cultivating a risk-awareness culture within the institution as dictated directly from the highest level of the organization the board of directors. The second theme maintains that the management of risk is comprised of four important complementary activities. The first concerns itself with surveillance and identification of risk within the institution, followed by a thorough assessment and monitoring of events as they unfold, and finally, devising mechanisms to control and mitigate these risks, even before they occur. As stated earlier, prevention is key. Recognising that the safety and soundness of the financial system is a collaborative effort, the third theme emphasises the important role regulatory supervisors play in the risk management process of financial institutions. Finally, risk management can only be facilitated properly if the process is clear and transparent at the outset. Public disclosure to the market, therefore, serves as an important binding constraint on the behaviour of financial institutions as they provide the public with their intermediary functions.

III.C.1.8

Identifying Operational Risk the Risk Catalogue

How can an institution identify potential operational risks lurking within the organization? In the recent past, many institutions have been surprised to discover that even the most obvious types of operational risk are widely prevalent within the organisation, unnecessarily squandering precious resources and hindering productivity. The first step in the identification process is to require that each business unit (or operational unit) shall be assessed using a so-called risk catalogue which adequately identifies all the risk categories relevant to the specific unit being assessed. It clearly identifies possible operational failures under three categories: people, process and technology.



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Table III.C.1.5: Risk catalogue for Business Unit A 170

Pe ople Risk Incompetenc y Inadequate Head C ounts Key Personnel Management Communication

Internal Polit ics, Conf lict of Interest , Lack of Cooperation Collusion and Connivance Fraud

Pr ocess Risk A. Model Risk Model or Methodology Error Pricing or Mark-to-Model Error Availabil ity of Loss Reserves Model Complexit y

B. Transaction Risk Execution Error Booking Error Collateral, Confirmation, Matching, and Ne tting Err or Product Complexi ty

Capacity Risk Valuati on Risk Erroneous Discl osure Risk Fraud

C. Operations Control Risk Li mi t Exceedances Volume Ri sk Se curi ty Risk Posit ion Report ing Risk Profi t and Loss Reporting Ri sk

Technology Risk Systems Fai lure Netw ork Fail ure Systems Inadequacy Compati bility Risk Supplier/Vendor Risk

Programming Error Data Corr uption Disaster Recovery Risk Sy stems Age Sy stems Support

Consider a simple example: suppose we have determined that Business Unit A, due to its intrinsic business activities, is subject to some sources of operational risk. We can then check them off against our generic risk catalogue as shown in Table III.C.1.5. This checklist can then form the basis for assessing the loss frequency and loss severity of the different event types in the business unit using the risk scorecard (see next section). We shall also see, in Section III.C.1.10, that the control process requires the identification of pertinent operational risk failures at two different levels: independent management oversight and self-assessments by individual business units.

III.C.1.9

The Operational Risk Assessment Process

It is important to keep in mind that an operational risk assessment process without the aim of proactive management is an exercise in futility. What drives our desire to measure must be our belief that a sound and active risk management structure is in place or will be in place in the future. As a precursor to building a sound risk management structure, measurement can be the tool by which senior managers are convinced that such a structure is needed. It also helps an institution to identify the key problem areas of operational risk and this helps target the resources that will be allocated.

170 Risk types shown are for illustration only. In practice, different business units may have different types of risks

they are most concerned with. For instance, there is presumably no model risk in retail banking or in leasing.



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For each business unit, the risk assessment process follows four fundamental steps: (1) inputs to risk catalogue; (2) risk assessment scorecard; (3) review and validation; and (4) outputs of risk assessment process. Step 1: Inputs to Risk Catalogue Operational risk should be evaluated net of risk mitigants. For example, if the institution has insurance to cover a potential breakdown, then the degree of risk must be properly adjusted by the insurance premium paid. To obtain a measure of net operational risk, the required inputs to the risk catalogue must be able to adequately assess both the frequency of failure occurrences and the severity of loss given that a failure occurs: The assessment for frequency of occurrences may come from both internal and external reports, such as: audit reports; external audit reports; regulatory reports; management reports; expense reports; deviation from business plans, operational plans, and budgets, etc.; and expert opinion and industry best practices. An assessment for severity of loss may come from: management interviews, both pre and post mortem; variances on budgets; insurance claims; and loss history, whenever possible. 171 Step 2: Risk Assessment Scorecard Using the risk catalogue and the inputs from step 1, each business or operational unit will be assessed using a risk scorecard. The risk scorecard will appropriately identify and assess the nature of operational risk based on the following broad points: Risk categories people, process, technology, and external dependencies. Connectivity and interdependencies. Because the headline risk categories of people, process and technology cannot be looked at in isolation, their cumulative effects and interdependencies must be carefully identified and accounted for. Change, complexity, and complacency. The sources that drive the headline risk categories may be due to: a change in the work environment, e.g., the introduction of new technology to the business unit; the complexity of products, process or technology; or the complacency factor due to ineffective management of the unit. 171 A few major banks are beginning to gather their own internal loss experiences the outcome will not be known until many years from now. Loss history should also cover credit losses as a result of operational mishaps, loss due to theft and fraud, and losses strictly due to errors. Admittedly, this is an extremely difficult task and the financial industry is still struggling with how to collect these loss data. In addition, the RMG has also analysed data collected from the numerous participating banks through the 2002 Operational Risk Loss Data Collection Exercise (LDCE) in June of 2002. The 2002 LDCE was an extension and refinement of two previous data collection exercises sponsored by the RMG, which primarily focused on banks internal capital allocations for operational risk and their overall operation risk loss experience during the period from 1998 to 2000.



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Frequency and severity assessments. Quantifying the likelihood of breakdown in operational processes is very difficult. It may be simply rated as very likely, not likely, very unlikely and so forth, or a question relating to the expected number of loss events may be posed. Severity of loss describes the potential monetary loss to the institution, given the occurrence of an operational failure. Since actual loss history may be difficult to come by, some institutions subjectively attach a range of loss (e.g., between $5 million to $10 million for certain failures). More details on the recommendations for frequency and severity of self-assessments are given in Chapter III.C.3. Net operational risk. Operational risks should be evaluated net of risk mitigants. For instance, the potential monetary amount lost due to certain insurable operational failures can be reduced through the use of risk mitigants, e.g., insurance and underwriting. We need to find out which bank activities are currently covered by insurance policies and by how much. In addition, a catalogue of insurable bank activities needs to be prepared. Net risk assessment. The combination of all the ingredients in the risk scorecard enumerated above gives the overall net risk assessment. Step 3: Review and Validation After the risk assessment process is completed (via the risk catalogues) and risk scorecards for each business unit are produced, it is the responsibility of the operational risk management committee 172 to review the assessment results with the management of the respective business unit and other key officers of the institution. The responsibilities of the committee may include: Formulating a set of operational risk policies and guidelines clearly delineating the actions needed to correct and prevent the operational problems and issues identified. Determining the important differences between the unit's own self-assessment and the independent assessment. Opining on the ratings in the risk scorecards before publication. In conjunction with audit and compliance departments, issuing a mandatory report and list of recommendations to the affected business units. Issuing summary risk reporting about the enterprise to the executive committee. Step 4:

Outputs of Risk Assessment Process

There are several possible outputs from the operational risk assessment process. We concentrate on only three broad items. (i)

Improved Risk Reporting and Analysis:

172 In many financial institutions, the operational risk management committee is a committee within either the risk

management function or the audit function.



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As an ongoing goal of the operational risk assessment framework, the institution should endeavour to streamline its risk reporting processes among the different risk-monitoring units of the institution (i.e., audit, compliance, and risk control). These reports should be viewed as a concise summary of specific audit and compliance reports which are already instituted within the financial institution. The most useful of these reporting tools are the risk catalogue, risk scorecards and heat maps which are used to highlight the relative information on operational risk exposures across the institution. An example of a heat map is given in Figure III.C.1.6. This shows that Business Unit D, relative to all the other business units in the institution, has a moderately high likelihood of incurring a large amount of loss due to operational risk failures. This means that it requires a relatively large amount of economic capital to sustain its business activities. Figure III.C.1.6: Example of a heat map This quadrant requires urgent attention

VH Business Unit E

Business Unit D

H M

Business Unit B

Business Unit C

L VL

Business Unit A

5

10

15

20

25

30

35

40

Severity of Loss ($MM)



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(ii)

OCC Exam Chart

The Office of the Comptroller of the Currency (OCC), in its September 1995 press release, highlighted its examination procedure of banks to cover nine principal categories of risk. They are: credit risk; interest-rate risk; liquidity risk; price risk; foreign exchange risk; transaction risk; compliance risk; strategic risk; and reputation risk. (The OCC mandate is directed at banks with financial derivatives activities, therefore the nine categories of risk need not apply to every business units of the financial institution.) Using the risk scorecard and other reports, we can graphically represent the evaluation of a business unit in a concise manner in line with the OCC examination procedure. This is illustrated by the OCC exam chart in Figure III.C.1.7. Figure III.C.1.7: OCC exam chart Inherent Business Risk for Business Unit C High Extensive

Average Average

Low Minimal

Inherent Business Risk Adequacy of Management Controls

(iii) Capital Attribution By attributing economic capital to operational risks we can ensure that business units which are more prone to operational failures are assigned a greater allocation of capital commensurate to the risks that they take. Admittedly, this is a difficult and subjective task and a whole chapter of this handbook is devoted to it (see Chapter III.0). It is also important to note that the current Basel II proposals call for a regulatory capital charge for operational risk, in addition to the capital charge already required for both market risk and credit risk (see Chapter III.C.3). Whether there is wisdom behind a capital charge for operational risk remains to be seen (Ong, 1998). After all, major financial catastrophes resulting from breakdowns in people, process and technology cannot be prevented entirely. Thus, operational risk events cannot be eliminated altogether, regardless of the amount of capital charge levied on operational risk.



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III.C.1.10

The Operational Risk Control Process

With hindsight the well-publicised derivatives losses listed in Section III.C.1.2 were all preventable. Outside of derivatives activities, losses attributable to fraud in other lines of business are likewise preventable. Subsequently, countless studies have continued to point to the following failures in risk control as the ultimate culprits. (i)

Lax management structure: lack of adequate management oversight and accountability; no segregation of duties; too many delays in systems development; disrespect for audit reports; and ignorance.

(ii)

Inadequate assessment of risk: on and off-balance sheet activities: lack of stress-testing for unexpected market moves; inappropriate setting of limits; too much risk relative to capital; and lack of risk-adjusted return measurement.

(iii)

Lack of transparency: inaccurate information on capital, solvency and liquidity; inadequate accounting policies; lack of benchmarks and comparability; lack of approvals, verifications and reconciliations; and lack of review of operating performance.

(iv)

Inadequate communication of information between levels of management: lack of escalation process in times of crises; paying too much lip service in the various risk committees; lack of procedures for monitoring and correcting deficiencies; and poor communication between different risk-monitoring groups.

(v)

Inadequate or ineffective audit and compliance programmes.

To help facilitate the identification of operational risk failures within the institution by business unit, it is important that the operational risk management function is streamlined alongside the risk control, compliance, and audit functions of the institution. Lessons that we have learned from many highly publicised financial fiascos all point to the need for the following: (i)

Independent management oversight. This includes audit oversight, risk control and compliance functions, and most definitely senior management involvement. Each overseer plays the role of independent risk monitor, considering such operational performance measures as volume, turnover, settlement failures, delays, errors, compliance to market and credit limits, income and expense volatility, effectiveness of line management, accounting anomalies, and other higher-level controls. For many business activities of the holding company, this information is already available within the institution.

(ii)

Self-assessments by individual business units. Line management has the best knowledge of its own people, the day-to-day processes it has to go through, the integrity of the



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systems supporting the business unit, and the external circumstances that could cause its people, process and technology to fail. A self-assessment by the individual business units is, therefore, a key first step in the operational risk assessment process.

III.C.1.11

Some Final Thoughts

Operational risk failures can wreak havoc within an organization if not properly identified, assessed, monitored, controlled and mitigated. Furthermore, if not sufficiently contained, operational risk also has a tendency to spill over and cause systemic risk to the broader markets. In spite of its importance for containing operational risk, it is still more art than science. Operational risk management continues to be one of the least developed areas of enterprise risk management in spite of the heightened attention it has received. Perhaps since operational risk is fundamentally qualitative in nature, it might never be as developed as other risk areas. While much progress has been made over the past several years, at least primarily within the financial industry, the fundamental operational risk management framework continues to be confused by many people. Many people with quantitative background have used the Basel II proposals as their impetus for furthering the argument for more operational risk modelling, People with compliance, audit and risk control backgrounds tend to interpret the Basel II guidance as an opportunity to codify additional policies, thereby reducing operational risk management to a mere set of rules and regulations. In 2006 there will be a new regulatory capital charge for operational risk, but no regulators in their right mind would think that operational risk management is about levying capital charges. While capital charges are important, they are only one small component of prudent risk management, and they are not a good substitute for sound judgement. An increase in capital will not itself reduce risk; only management action can achieve that, a Moodys Special Comment reported (Moodys Investors Service, 2003). My personal experience as head of enterprise risk management and chief risk officer for two of the ten largest banks in the world has taught me that most operational risk failures are preventable. The processes outlined in this chapter are based on my experience of how to prevent operational risk mishaps. Experience tells me that the most important aspect of the operational risk management framework is still sound corporate governance and proactive senior management involvement. After all, the control of operational risk is concerned fundamentally with good management. In practice, good management means vigilance, patience, and persistence in improving the risk management process. And this is what operational risk management is all about.



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References Basel Committee on Banking Supervision (2002) Operational Risk Loss Data Collection Exercise 2002, 4 June. Basel Committee on Banking Supervision (2003) Sound Practices for the Management and Supervision of Operational Risk, February. Crouhy, M, Galai, D and Mark, R (1998) Key steps in building consistent operational risk measurement and management. In Operational Risk and Financial Institutions. London: Risk Books. Financial Services Authority (2003) Building a Framework for Operational Risk Management: The FSAs Observations, July. McCarthy, E. (2000) Derivatives revisited. Journal of Accountancy, 189(5). See http://www.aicpa.org/pubs/jofa/may2000/mccarthy.htm Moodys Investors Service (2003) Moodys Analytical Framework for Operational Risk Management of Banks. Special Comment, January. Ong, M (1998) On the quantification of operational risk a short polemic. In Operational Risk and Financial Institutions. London: Risk Books. Ramadurai, K., Olseon, K, Andrews, D, Scott, G., and Beck, T. (2004) The Oldest Tale but the Newest Story: Operational Risk. Special Report, FitchRatings, January.



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III.C.2 Operational Risk Process Models James Lam 173

III.C.2.1

Introduction

Management and board attention to operational risk management (ORM) has never been greater. While businesses have always faced operational risks, the discipline of ORM is still in the early stages of development. The focus on operational risk has been driven by a number of important factors: Corporate disasters.

The need for ORM first gained the attention of risk management

professionals in the 1990s when they realized that the root causes underlying the major financial disasters Barings, Kidder, Daiwa, etc. were operational risks and not financial risks. More recent corporate failures such as Enron and WorldCom, as well as the markettiming and late-trading problems plaguing the mutual fund industry, have reinforced the importance of ORM. In the aftermath of these disasters, the standards for corporate governance and risk management have increased. These new standards impact not only corporate executives and boards, but also key stakeholders such as stock analysts, rating agencies, and regulators. Regulatory actions.

In response to the corporate disasters, regulators have dramatically

increased their examination and enforcement standards. New regulations with significant operational risk requirements include Sarbanes-Oxley (in particular, Section 302 on certification of chief executives and chief financial officers and Section 404 on internal controls), the Patriot Act, anti-money laundering and bank secrecy acts, and other corporate governance rules adopted by the stock exchanges. Additionally, the new Basel initiative (Basel II) has established a direct linkage between minimum regulatory capital and a banks underlying risks, including explicit treatment for operational risk. Industry initiatives. A number of industry initiatives have been organized around the world to establish frameworks and standards for corporate governance and risk management. These initiatives include the Treadway Report (United States, 1993) that produced the Committee of Sponsoring Organizations (COSO) framework of internal controls, while the Turnbull Report (United Kingdom, 1999) and the Dey Report (Canada, 1994) developed similar

173 President, James Lam & Associates; founding member, Blue Ribbon Panel, PRMIA; and Senior Research Fellow,

Beijing University.



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guidelines. It is noteworthy that the Turnbull and Dey reports were supported by the stock exchanges in London and Toronto, respectively. In 2004, COSO is scheduled to release a major study on enterprise-wide risk management (ERM), which will include key ERM principles and advocate its application within a sound corporate governance framework. Corporate programs.

Corporations have achieved significant benefits from their risk

management programs; among these are stock price improvement, debt rating upgrades, early warning of risks, loss reduction, and regulatory capital relief. While ORM programs are relatively new, early adapters have reported sustained reduction in operational losses and error rates (one company reported a sustained 80% reduction in operational risk losses). Other reported benefits include improved customer service and operational efficiency. These results demonstrate that investments in operational risk controls can produce direct benefits that are multiples of the costs, as well as indirect benefits such as prevention of crises that divert management attention and cause reputational damage. Technology developments. Over the past decade, technology developments have transformed how businesses operate.

Examples include using the Internet to communicate with

customers and facilitate commerce; developing customer relationship management applications to better serve customer segments; and outsourcing IT operations and business processes to improve efficiency. In risk-intensive industries, such as financial and energy services, corporations have also developed sophisticated models and databases to measure all types of risk. While these technology developments provide business benefits, they also present new and complex risks, such as information security, data integrity, cyber-crime, cyber-terrorism, systems availability, and model risk. These risks require operational risk controls for day-to-day operations, as well as disaster recovery planning for unlikely but potentially disastrous events. Going forward, the key trends and developments highlighted above should continue to assert significant pressure on corporate boards and executives to improve their risk management capabilities, especially in the area of operational risk. The focus of this chapter is on the development and application of operational risk process models. We will discuss the following questions: How to develop and apply operational risk process models? What are the specific quantitative and qualitative tools used by companies today? How to link these tools with economic capital allocation? What are the actions management can take to mitigate operational risk?



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At the end of this chapter, we will use IT outsourcing as an example to illustrate how an operational risk process can be established.

III.C.2.2

The Overall Process

In developing and applying operational risk process models, risk managers should first take advantage of other related programmes that can provide valuable information or tools. While the discipline of ORM is relatively new, businesses have always had to ensure that their operations are effective and efficient. As such, many companies have implemented programmes to identify, monitor, and improve their business processes. These programmes often fall under the monikers of re-engineering or total quality management, and these corporate-wide efforts produce detailed process maps and performance metrics. In addition to process improvement, companies have also implemented risk assessment processes to identify key operational risks.

These risk

assessments are either performed by the business and operating units themselves (known as control self-assessments), or by independent internal or external audit groups. As a starting point, the methodologies and results from these initiatives process maps, performance metrics, audit ratings can be used to gain a deeper understanding of the general scope and specific issues that the ORM program must address. With this knowledge, the development of operational risk process models should include the following four steps: Step 1: Establish the objectives and requirements of key stakeholders. The design of operational risk process models should always start with the end goal(s) in mind: what are the key business and operational objectives for the company? These objectives can generally be grouped into three categories, business performance, financial performance, and compliance. Business performance objectives include product innovation, customer acquisition and retention, and market share. Financial objectives include earnings growth, risk-adjusted profitability, and shareholder value. Compliance objectives should encompass internal risk policies and limits, as well as external regulatory and legal requirements. For example, one of the key objectives of a capital markets trading business is to maintain their market risk exposures within boardapproved risk policy limits. Step 2: Identify the core processes that support these objectives. Most companies view their businesses vertically in terms of operating units, support functions, products, or customer segments. However, companies must manage their business processes horizontally to fully address the operational risks that may prevent them from achieving their key objectives. This is because the core processes of any company customer acquisition, product delivery, cash



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management, etc. involve the participation of various entities within and outside of the organization. To better understand these linkages, process maps should be developed for the core processes of the company. These process maps should be driven by the objectives of the company, and highlight the specific interdependencies, such as work flows, data flows, and/or cash flows. It is also important to note that risk management is a process. As an example, Figure III.C.2.1 shows a process map for daily market risk measurement. This process map shows the work flows between the front office (traders), the middle office (risk management), and the back office (accounting and IT). It also shows two time-critical objectives: an account reconciliation between the open positions report from the front office and the accounting report by 10 a.m., and a daily market risk report showing risk exposures against limits by 4.30 p.m. Figure III.C.2.1: Daily market risk measurement process map Open positions report

10:00 am

Market risk datab ase

Account Recon ciliation Financial market datab ase

Account ing dat a

Formulate a nd Execut e H edgin g Strategies

Deal capture

Daily Accountin g Accruals

4:30 pm Market Risk System

Daily Ris k Report v s. Policy Limit s

Exception Manag eme nt and Reporting

Y es Limits Exceed ed ?

No Daily Reporting and Revie w

Back Testing



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Step 3: Define performance and risk metrics, including goals and MAPs. For each core process of the company, performance metrics and risk metrics should be clearly defined. For example, the systems availability of a core application is essential for day-to-day operations. A company might set 100% systems availability as a goal and 99.99% as minimum acceptable performance (MAP). Similarly, goals and MAPs should be established for all key performance and risk metrics. As such, all of the companys operations can be monitored against specific benchmarks. Over time, management can respond proactively to specific processes that perform below MAP.

For

processes that perform consistently above goal, then the goal and MAP for those processes can be raised to encourage continuous improvement. To follow on with our example, an operational risk metric for the daily market risk measurement process may be the percentage of time that the daily market risk report is produced by 4.30 p.m. Management can establish 99% as the goal and 95% as the MAP. As such, the goal is to produce the daily market risk report by 4.30 p.m. on 250 out of the 253 trading days in a year, with a MAP of 240 days. Step 4: Implement organizational and risk mitigation strategies. With a clear understanding of stakeholder objectives and supporting core processes, and performance of those processes against performance standards, the company is well positioned to execute the appropriate ORM strategies. These strategies may include: new training programs; new IT applications; process redesigns; management restructuring; integration of audit, compliance, security and ORM activities; specific investigations and corrective actions; and risk transfer through insurance programmes. In our example, suppose management noticed that the daily market risk report was late 4 times in a month, a below-MAP performance given that the frequency is greater than 13 times per year. An investigation revealed that the main reason, or root cause, is that the traders are late in updating their daily trades. Risk mitigation strategies may include discussion forums to resolve any misunderstandings or conflicts, or hiring new trading assistants to support the traders. General Electric is well known for its workouts in which cross-functional teams are organized to discuss and resolve any operational issues in an open forum. To highlight the importance of this process, senior executives usually attend the last session of these workouts, to obtain an inperson report from the team leaders on how they plan to address any outstanding issues. More sophisticated companies go beyond these four steps in their ORM programmes, and allocate economic capital to each business unit based on its operational risks. The direct linkage between capital requirements and operational risks is one of the key developments in Basel II. The allocation of capital to operational risks provides a number of benefits:



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Management can measure risk-adjusted profitability consistently across different business units and products. In fact, performance models that do not fully adjust for risks (such as economic value added models) would overstate the profitability of high-risk businesses and understate the profitability of low-risk businesses. Organizational incentives, in the form of lower capital charges, are provided to business units that effectively manage their operational risks. One of the key objectives of any risk model is to motivate appropriate behaviour. In the evaluation of risk transfer strategies, such as insurance, management can compare the cost of risk retention (i.e., economic capital times the cost of capital) and the cost of risk transfer (i.e., net cost of the insurance strategy). As we will discuss later, the allocation of economic capital to operational risk will enhance the evaluation of the costs and benefits of these strategic alternatives.

III.C.2.3

Specific Tools

Given the wide scope of operational risk, a company should employ a range of qualitative and quantitative tools to assess, measure, and manage operational risks. Below is a summary of the basic ORM tools that companies use today: (i) Loss-incident database. A company should record operational losses and also keep a record of operational incidents for two main reasons. First, losses are measurable and can be used to indicate trends (e.g., trend in the loss/revenue ratio). Incidents record other events that should be noted, even if they did not result in an operational loss. Second, every loss and incident within a company represents a learning opportunity, without which past mistakes are more likely to be repeated. As such, the loss-incident database should be used to support the identification of operational risk exposures, the development of risk-based audits (in which high-risk business units are audited more frequently), as well as to facilitate the sharing of lessons learned within the company. Additionally, there are several industry initiatives to develop more robust loss-event databases, but it is too early to tell which one(s) will become the industry standard. It is unlikely, however, that the management of operational risk will ever become a wholly data-driven process; given the nature of operational risk, it will always be more of a management issue than a measurement issue. (ii) Control self-assessment. A control self-assessment (as distinct from a risk self-assessment see Section II.C.2) is an internal, subjective analysis of the key risks, the controls available to mitigate these risks, and the management implications. It is important for all of the business units to assess their current situation in terms of a control self-assessment to develop a clear picture of



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how to proceed in the ORM process. Given that they fully participated in the assessment process, they would also have a greater sense of ownership to address outstanding opportunities or issues. Tools that support self-assessments include questionnaires, issue-specific interviews, team meetings, and facilitated workshops. The output is an inventory of key risk exposures, key control initiatives, and sometimes even a Letterman-style top 10 risks. The following are questions that might be included in a control self-assessment: 1. What are the key business and financial objectives for the business unit in the next 12 months? 2. What are the key risks that may prevent you from attaining these objectives? 3. What policies, procedures and controls do you have in place to ensure that risks are within acceptable levels? 4. What new risk management initiatives do you have planned for the next 12 months? 5. What metrics, tests and reviews provide you with assurance that the policies, procedures and controls specified in 3 and 4 above are indeed effective? (iii) Risk mapping. Building on the work from control self-assessments, the companys key risk exposures can be ranked with respect to their probability and severity so that management can have a comparative view in the form of a two-dimensional risk map. Figure III.C.2.2 shows an example of a risk map.

Some ORM professionals argue that companies should be most

concerned about events of low probability and high severity because management does not have sufficient experience in dealing with these events, whereas they have more experience in dealing with events of high probability and high severity. For operations that are more complex (e.g., outsourcing arrangements, special-purpose vehicles), risk-based process maps can be produced to show how various risk exposures can arise. These maps will aid in the identification of the risks encountered in each business unit, indicating problem spots, such as single points of failures or where errors often occur. These maps will also enable each business unit to develop and prioritize their risk management initiatives to address the most important risks. (iv) Key risk indicators. Risk indicators are quantitative measures that are linked to operational risks for a specific process. Examples include customer complaints for a sales or service unit, trading errors for a trading function, unreconciled items for an accounting function, or system downtime for an IT function. These risk indicators are usually developed by the individual business units and closely tied to their business objectives. Early-warning indicators should also be developed to provide management with leading signals (e.g., employee absenteeism and turnover as an early warning indicator of future operational errors). As discussed earlier, the establishment of goals and MAPs will provide useful performance benchmarks against which the key risk indicators can be measured.



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Figure III.C.2.2: Risk Map Risk

Probability Severity

1. Investment/Credit

High

High

2. Human Capital

Low

High

3. Valuation

Medium

High

4. Liquidity朏 unding

Medium

Medium

5. Operational

Low

High

6. Reputational

Medium

High

7. Economic

Medium

High

Low

High

Medium

Medium

High

Low

8. Compliance 9. Leverage 10. Interest Rate

High

1

10

4 9

3 7 6

Probability

2 5 8 Low Low

High Severity

Other sources of valuable information for risk identification and assessment include internal audit reports, external assessments (external auditors, regulators), employee exit interviews, and customer and employee surveys. Operational risk professionals also find it useful to distinguish between key risk indicators (KRIs) and key risk drivers (KRDs). KRIs are ex-post indicators of operational risk performance, in that management has no direct control over their outcomes. On the other hand, KRDs are levers that management has direct control over. Examples of KRDs include number of training hours, number of automated versus manual processes, time to fill open positions, and time to resolve outstanding audit findings. As such, KRDs can be best thought of as controllable factors that will influence future KRIs.

III.C.2.4

Advanced Models

When ORM first came on the scene a few years ago, there were basically two distinct schools of thought.

One school subscribed to the notion that you cannot manage what you cannot

measure, and they focused on quantitative tools such as loss distributions, risk indicators, and economic capital models. The other school believed that operational risk cannot be quantified effectively, and they focused on more humanistic, qualitative approaches such as selfassessments, risk maps, and audit findings. 174 Today, operational risk professionals realize that best practices must integrate both quantitative and qualitative tools.

174 See Lam (2003a).



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In addition to the basic risk identification and assessment tools discussed above, leading companies employ advanced operational risk models. Unlike market risk and credit risk, where risk measurement methodologies have been developed and tested for many years, there are no widely accepted models for operational risk measurement. In selecting a methodology (or combination of methodologies), each company should first establish its objectives and resources and choose accordingly. Different methodologies imply different interpretations of operational risk, and require various inputs to be useful. Given that there is likely to be no single solution, a combination of methodologies will allow the disadvantages of one model to be balanced by the strengths of another, allowing a more robust overall measurement to be developed. Some of the most common methodologies, including their strengths and weaknesses, are discussed in this section (see Hernandez et al., 2000, for a detailed discussion of key strengths and weaknesses of various ORM models). III.C.2.4.1

Top-down models

The top-down approach to operational risk assessment calculates the implied operational risk of a business by using data that are usually readily available, such as the overall financial performance of the company or that of the industry in which it operates. Top-down models use relatively simple calculations and analyses to arrive at a general picture of the operational risks encountered by a company.

These top-down models benefit from the sophisticated

methodologies already developed for credit and market risk. Examples of top-down models on operational risk include the implied capital model, the income volatility model, the economic pricing model, and the analogue model: (i) Implied capital model. This methodology assumes that the domain of operational risk is that which lies outside of credit and market risk. Thus, the capital allocated to operational risk must be the result of subtracting the capital attributable to credit and market risk from the total allocation of capital. Although this model provides an easily calculated number for operational risk, its simplicity presents several disadvantages. First, total risk capital must be estimated given the companys actual capital and the relationship between its actual debt rating and target debt rating. Second, it ignores the interrelationships between operational risk capital and market risk and credit risk capital. Finally, this model does not explicitly capture the causes and effects for operational risk. (ii) Income volatility model. This model is similar to the capital allocation model, but it goes one step further, by looking at the primary determinant of capital allocation income volatility. The volatility attributable to operational risk is calculated in the same way as in the capital allocation model by subtracting the credit and market risk components from the total income volatility.



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One of the advantages of this model is that of data availability: historical credit and market risk data are usually easily obtained, and total income volatility can be observed. However, this model also has several shortcomings, the most dramatic of which is that it ignores the rapid evolution of firms and industries. Structural changes, such as new technologies or new regulations, are not captured in this model. The income volatility model also fails to capture softer measures such as opportunity costs or reputation damage. In addition, it fails to capture the low-frequency, highseverity risks, as is true in all of the top-down approaches. (iii) Economic pricing model. The capital asset pricing model (CAPM) is probably the most widely used of economic models, and can be used to determine a distribution of the pricing of operational risk relative to the other determinants for capital (see Chapter I.A.4). The CAPM assumes that all market information is captured in the share price; thus the effect of publicized operational losses can be determined by evaluating the market capitalization of a company. The advantage of this approach is that it incorporates both discrete risks and softer issues such as reputational damage and effects of forgone opportunities. With this approach, a companys stock price volatility due to operational risk is derived by taking the companys total stock price volatility and subtracting from it the stock price volatility due to credit risk and market risk. However, the CAPM approach presents an incomplete and simplistic view of operational risk. It provides only an aggregate view of capital adequacy, not information about specific operational risks. Furthermore, the level of operational risk exposure is not affected by particular controls and business risk characteristics, so there is no motivation to improve operations, and while tailend risks are incorporated in the model, they are not thoroughly accounted for. This is a significant omission. Such incidents can do more than just diminish the value of a business: they can lead to the end of the business completely. Finally, this model does not help in anticipating, and therefore avoiding, incidents of operational risk. (iv) Analogue model. The analogue model is based on the assumption that one can look at external institutions with similar business structures and operations to derive operational risk measures for ones own organization. This model can be extended to look for the causes and effects of operational losses at such institutions. This method offers one way to proceed when a company does not have a robust database of operational risk losses. However, it takes some credulity to assume that the high-level numbers of another institution can accurately measure ones own operational risk, and many are suspicious of this approach. In the words of one analyst: [The] intangibles within an institution its risk-taking appetite, the character of its senior executives, the bonus structure of its traders put so many wild cards into the operational risk equation that similarities in business volume, transaction volume, documented risk policies and other qualities that can be scored are swamped.



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III.C.2.4.2

Bottom-up models

The bottom-up methodology applies loss amounts and/or causal factors to predict operational losses in the future. It requires a company to clearly define the different categories of operational risk that it faces, gather detailed data on each of these risk categories and then quantify the risk. A company often needs to augment its internal data with an external loss-event database. The final output of this bottom-up approach is a loss distribution that enables operational risk capital to be estimated for a given confidence level (see Chapter III.C.3) A number of surveys have indicated an increasing preference for risk-based bottom-up methodologies over the top-down approaches. The Basel II requirements should further encourage banks to develop bottom-up models. The data needed for this methodology can also be used to derive a business risk profile. For example, turnover or error rates can be tracked over time and combined with changes in business activities to construct a more robust picture of the business operational risk profile. By tracking these KRIs over time, the company can assess its operational risk exposure on an ongoing basis and can upgrade specific controls as needed. Furthermore, continuous tracking provides a companys management with better information about its operations and increases awareness of the causes of operational risk. However, bottom-up models present several difficulties. Mapping loss data from the company with loss data from other companies is complex, given the differences in business mix, size, scope and operating environment. Even mapping internal losses to specific risk types is difficult because losses are frequently reported as aggregates from multiple risk sources that are difficult to isolate. For example, an operational loss on a trading floor might result from personnel risk, lack of trading controls, expanding overseas business, lack of back- and front-office segregation, volatile markets, senior management confusion, and incompetence. In addition, robust internal historical loss data may not be available, particularly for low-frequency, high-severity events. Bottom-up models are usually based on statistical analysis and scenario analysis. Classical statistical models require an ample supply of operational loss data that are relevant to the business unit. The lack of appropriate internal data is therefore the greatest obstacle to the widespread application of this methodology; the use of external data as a proxy poses several problems, as mentioned earlier. However, the analytical power of this tool will hopefully become more widely applicable in the near future as increased awareness of operational risk leads to improvements in data collection and extensions of the classical statistical methodology. Scenario analysis offers several benefits that are not addressed by the classical statistical models. A scenario analysis is used to capture diverse opinions, concerns and experience/expertise of key managers and represents them in a business model. Scenario analysis is a useful tool in capturing



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the qualitative and quantitative dimensions of operational risk.

Risk maps allow the

representation of a wide variety of loss situations, and capture the details of the loss scenarios envisioned by the managers surveyed. Risk maps of each business unit identify where operational risk exposures exist, the severity of the associated risks, whether any controls are in place, and the type of control: damage, preventive, or detective. Cause and effect relationships can be captured with this methodology. The shortcoming of such a model, however, is in its subjectivity, which creates a potential for recording data inconsistently and/or for biasing conclusions if one is not careful. At the beginning of this section we discussed the need to balance the qualitative and quantitative tools. For example, control self-assessments require that business units are honest and forthright about their major operational risks, which can often be embarrassing problems that they would rather not discuss (let alone highlight for senior management!).

To counterbalance this

shortcoming, business units should be required to not only tell me but also to show me. This can be accomplished through validation processes, such as: pre-established operational risk indictors that are monitored against goals and MAPs; periodic tests to ensure that actual losses and incidents (ex post) result from operational risks that were being monitored through KRIs or at least discussed in the control self-assessments (ex ante); comparisons between control self-assessments and independent assessments such as internal audits, external audits, regulatory reviews, and customer surveys. In fact, Section 404 of the Sarbanes-Oxley Act requiring management assessment of internal controls for financial reporting, as well as auditor attestation, was designed to ensure such validation.

III.C.2.5

Key Attributes of the ORM Framework

Today, ORM practitioners recognize the pitfalls of using only one approach to modelling operational risk either top-down or bottom-up and that best practice ORM incorporates elements of both approaches. We will now discuss the attributes of a unified ORM framework, and then how these attributes can underpin a seven-factor economic capital model. A unified ORM framework should satisfy two basic requirements. First, it should support both the measurement and management of operational risks. Second, the ORM framework should incorporate the interdependencies across credit, market and operational risks as part of an overall ERM program.

Based on these two requirements, the key attributes of a unified ORM

framework include the following:



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(i) Integrating qualitative and quantitative tools. The nature of operational risk (i.e., the risk of loss due to people, processes, systems and external events 175 is complex and dynamic. As such, the advantage of qualitative tools is that they can incorporate human experience and judgement in order to capture risks that are subjective. For example, what are the operational risks associated with a new product? On the other hand, the advantage of quantitative tools is that they provide objective indicators that can be used to show aggregate losses, exposures, and trends against established targets. A unified ORM framework should incorporate both advantages, as well as integrate the institutions various risk management and oversight activities (e.g., ORM, audit, compliance, quality, insurance). (ii) Providing early warnings and escalations. Operational risk cannot be managed effectively based only on backward-looking indicators such as losses, error rates, and incidents.

The ORM

framework should provide early warning indicators of emerging risk issues. A quantitative example is that an increase in employee absenteeism may be an early warning for increasing turnover and human errors. A qualitative example is competitive intelligence that indicates significant investments in a new technology by a key competitor that, if successful, would render the firms existing technology obsolete. An ORM framework should establish early warning indicators, as well as effective escalation processes so that management can take the appropriate actions. For example, a money management company established specific escalation processes such that the higher the number of customers impacted by an incident, the higher the level of management is notified. This ensures that bad news travels up the organization and that the appropriate level of management responds in a timely manner. (iii) Influencing business activities. One of the most important attributes of an ORM framework is that it influences business actions and decisions. Such influence can be asserted through: (1) corporate policies with respect to guidelines for, and restrictions on, business activities, such as acceptable versus unacceptable sales practices; (2) teamwork between the line units and ORM in new business and product development processes; (3) risk response plans based on ORM indicators and escalations; (4) adjustments in economic capital given operational risk performance and risk mitigation strategies; and (5) positive and negative incentives to motivate appropriate business behaviour. This attribute ensures that operational risks are managed on an ongoing basis, and that specific consequences are in place to provide organizational reinforcements. As excellent example of using positive incentives is when GE tied one-third of senior management compensation to the achievement of quality management objectives as part of the companys six sigma programme.

175 The definition of operational risk in this chapter includes business risk, which is notably absent in Pillar I of the

Basel II proposals.



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(iv) Reflecting environmental changes. Just as credit risk and market risk frameworks reflect changes in underlying default rates and market prices, an ORM framework should reflect changes in the operational risk environment. For example, increases in industry-wide operational risk losses and incidents may indicate an increase in systemic risk. A number of industry loss-event databases are being developed that can provide this type of information. Other environmental changes include new legal and regulatory requirements, such as those established by the Sarbanes-Oxley Act, the Patriot Act and the Basle II proposals. A company that lacks the processes and systems to comply with these new requirements is likely to face greater operational risk with respect to regulatory scrutiny and legal penalties. (v) Incorporating risk interdependencies. There are important interdependencies within and across risk types. For example, credit risk is the primary concern for most banks, but inadequate loan documentation (an operational risk) is likely to increase loss severity in the event of a borrower default. An ERM programme should address such interdependencies in the design of early warning indicators, the development of scenario analysis, and the implementation of risk response plans. For example, financial institutions must simultaneously manage market risk and operational risk during stressed market conditions (e.g., the Russian crisis during the autumn of 1998). Given that there is a high correlation between volatile prices (a key driver for market risk) and transactional volumes (a key driver for operational risk) during stressed periods, financial institutions should establish early warning indicators and risk response plans. Examples of early warning indicators are shown in Figure III.C.2.3. If these indicators exceed a critical level, management should implement pre-established contingency plans, such as reduction of trading limits to reduce market risk exposures and activation of back-up sites to increase processing capacity.



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Figure III.C.2.3: Early warning indicators

Risk Category

Credit Risk

Market risk

Early warning indicators Borrower/counter party stock price declines Widening of credit spreads in the debt and credit derivatives markets Increases in actual and implied price volatilities Breakdowns in historical price relationships and patterns

Business/ Operational Risk

Spikes in business growth, profitability, and complexity/change

Enterprisewide Risk

Increases in any risk concentrations and/or organizational powers

High and undesirable turnover rates

Changes in intra- and inter-risk correlations

As we will discuss in the next section, these interdependencies should also affect the determination of economic capital, and in ways that might not be obvious.

III.C.2.6

Integrated Economic Capital Model

Given the five attributes of a unified ORM framework discussed above, what factors should determine economic capital for operational risk? Figure III.C.2.4 shows a seven-factor approach to calculating operational risk capital. While this is conceptual, it can be adapted to a firms specific business mix, size, and risk profile.



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Figure III.C.2.4: Operational risk capital calculation 2 1 Revenue Multiplier

3

O perating Margin

Revenue volatility Fixed vs. variable expenses

Internal Indicators

External Indicators

Customers Regulators Event risk exposures

4

Operational Risk Capital

5 Model Risk 6 Systemic Risk

Credit Risk Capital

7 Financial Risk Capital Multiplier

Market Risk Capital

Losses/incidents Risk indicators Risk assessments Audit ratings

Model reliance Back test results

Industry loss experience Banking and settlement failures

Captures linkages between operational risk and financial risk

Let us discuss each one of these factors in turn. (i) Revenue multiplier. This is a top-down estimate of the amount of operational risk capital required by a business or operating unit. Such an estimate can be derived from observing analogues of publicly traded companies in same or similar businesses, while adjusting for market risk and credit risk. For example, Capital One may be a credit card company analogue, while First Nationwide may be one for mortgage companies. Outsourcing firms such as IBM or EDS may be analogues for internal IT functions. The central question is if the business or operating unit were a standalone business, how much capital would it need for operational risk capital? The revenue multiplier 176 assumes an average operational risk profile, which can then be adjusted upwards or downwards by the other factors below. (ii) Operating margin. This factor incorporates the degree to which the firms operating margin is more or less volatile than average, and is often referred to as business risk. A firms inability to 176 For certain businesses, a top-down proxy based on activity or volume might be more appropriate.



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generate sufficient revenue to cover expenses (net of unexpected credit and market risk losses) is a major reason why it needs to hold operational risk capital. For example, business variables that can increase the required operational risk capital include greater volatility in business volume, weak power to set prices, and higher fixed versus variable expenses. (iii) Internal indicators. This adjustment reflects the effectiveness of internal controls. A scorecard should be developed for the internal quantitative and qualitative indicators, with individual weightings, that would provide an overall adjustment to operational risk capital.

Internal

indicators would include losses, incidents, risk metrics (e.g., error rates, unreconciled items), early warnings, internal audit ratings, risk maps, etc. The economic impact of contingency plans and insurance programmes should also be factored in. Each key indicator should also be associated with specific goals and MAPs. (iv) External indicators. As with internal indicators, a scorecard of external indicators should be developed.

External indicators would include customer satisfaction scores and complaints,

external audit comments, and regulatory exam findings.

This scorecard would also track

exposures to external events, such as fires, earthquakes and acts of terrorism. Firms that rely on external vendors should also incorporate vendor performance relative to service level agreements. Goals and MAPs for external indicators should also be established. (v) Model risk. This factor reflects the degree to which a firm relies on models, and the quality of such models. The primary input is back-testing results against predetermined criteria. A firm should include all models that drive management decisions and actions, such as pricing and valuation models, scenario and simulation models, and risk management models. For firms that do not rely on models, this may simply be one of the internal indicators. (vi) Systemic risk. This factor adjusts for dramatic shocks in the business environment, such as industry-wide losses and incidents, and banking and settlement failures.

Systemic risk is

especially important for highly interconnected industries such as financial services and energy services, where trading activities and counterparty exposures within the industry are significant. Past examples include the Long-Term Capital Management collapse, Y2K readiness, and the Enron bankruptcy. In each of these situations, companies were concerned not only about their direct exposures, but also the exposures of their business partners and counterparties. (vii) Financial risk multiplier. This factor is meant to capture the compounding effects between operational, credit, and market risks. It is not portfolio diversification, which may lead to a reduction in aggregate economic capital at the enterprise-wide level. In fact, it is a compounding



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factor that many risk managers ignore. Regulators refer to this compounding factor as spillover effects. Cumming and Hirtle (2001) argued that the confluence of variables including market liquidity problems, lack of corporate limberness, and reputational and contagion effects, could result in the aggregate risk of a firm exceeding the sum of its individual risks. The financial risk multiplier is meant to capture such spillover effects. An argument can also be made that a variety of operational risk exposures (e.g., rogue trader, inadequate loan documentation, unsavoury sales practices) are compounded in a firm with significant market risk and credit risk exposures. After all, a rogue trader can do much more damage at a bank than at a retail store. The practice of ORM has come a long way in the past several years. It still has a long way to go. At the annual 2003 operational risk conference organized by the Risk Management Association, Eric Rosengren of the Federal Reserve Bank of Boston said that only three of the 20 largest US banks qualify for the advanced management approach for operational risk under Basle II, which is supposed to lead to reduced capital charges. However, the development of ORM is more than a regulatory compliance issue.

Early adopters of more sophisticated ORM have reported

significant business benefits, including improved customer service, greater operating efficiency and reduced losses. To fully realize these benefits, it is clear that the further development of ORM practices must integrate quantitative and qualitative tools.

III.C.2.7

Management Actions

Assessing and measuring operational risk is important, but pointless unless directed towards the improved management of operational risk by enhancing internal controls and controlling key risk factors. Simply stated, the goal of ORM is to help management to achieve their business objectives. Once a measurement framework is in place, the next step is to implement a process that identifies actions that will reduce operational losses. These actions include adding human resources, increasing training and development, improving and/or automating processes, changing organizational structure and incentives, adding internal controls (e.g., more frequent or more extensive monitoring), and upgrading systems capabilities. The key to effective operational risk mitigation is to establish a cross-functional rapid response team that will address and resolve any emerging operational risk issues. At one business unit at Fidelity Investments these teams were called turbo teams, and they responded immediately when operational risk indicators fall below MAP and reported back to management on their assessments and actions within a few days or weeks. Finally, a mechanism for evaluating and prioritizing potential improvements must be created. Costbenefit analysis and readiness assessments are useful tools that should be included in the



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evaluation process. For example, business executives often turn to IT, or more specifically automation, as the answer to process improvements. However, the potential benefits must be weighted against the total costs of the project (e.g., development, testing, training, implementation and ongoing maintenance). More importantly, management must ensure that the organization is ready to take advantage of the technology solutions. Automation of poorly designed processes can result in significant operational risks in the future. Some of the operational risk measurement approaches discussed above should naturally lead to improved operational risk management at the business unit level. A business unit can monitor and improve its operational risk levels by setting operational goals, exposure limits and MAPs on key operational processes. For example, suppose a brokerage group on average processes a million trades per day. This group may specify that its operational goal is that failed trades be less than less than 50 per day, while its MAP is no more than 100 failed trades per day. Additionally, the group may specify that no more than 40% of daily trades can be processed by one operational centre in order to spread its reliance across multiple operational centres. The allocation of economic capital for operational risk, if it captures both performance and behaviour effects, should motivate business units to improve their ORM in order to reduce their capital charges. For example, a business may set up procedures through which employees may respond immediately to operational problems and implement the controls necessary to monitor and improve performance. A key requirement for risk mitigation is to understand the root causes of operational risks, such as lack of training or inadequate systems, and then focus corrective actions on these root causes. Business units that take the appropriate actions should receive a credit in their economic capital charges. One of the key objectives of any ORM programme is to ensure that bad news travels up an organization. In other words, the quantification and modelling of operational risk should lead to more timely communication and escalation of operational risk issues. This can be accomplished through the various process models and quantitative tools discussed above.

Additionally,

management should clearly communicate when they should be informed through a cascading set of escalation triggers, which would lead to the appropriate decisions and actions on the part of management. Escalation triggers can be defined in terms of the KRIs, such as the level of operational losses, the number of errors, significant policy violations, and number of customers impacted by an incident. These escalation triggers and procedures should be incorporated into the companys policies and procedures, as well as training programmes and on-line support tools, so that all employees understand what is expected of them.



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Besides risk mitigation through operational processes and controls, and economic capital incentives for ORM, there are other financial solutions that management may consider. Companies can establish reserves to cover their expected operational losses. These reserves are considered a form of self-insurance. Expected losses should be embedded in the pricing of a product, indeed market and credit risks are already incorporated into some transaction prices as a matter of practice. Including an additional adjustment for operational risk makes for a more comprehensive picture and allows for more accurate risk-adjusted pricing. For example, if a business unit performs 10,000 transactions annually, with an expected loss of $80,000 a year due to operational factors, then an adjustment of $8 per transaction could cover such losses. Additionally, the cost of capital for operational risk (and other risks) should be incorporated into the pricing of a transaction. Pricing can also be driven by the target levels of returns that the company expects a product to achieve given competitive pricing and market share objectives.

III.C.2.8

Risk Transfer

For critical operational risk exposure, a company must decide if the best strategy is to implement internal controls and/or execute risk transfer strategies. The two are not mutually exclusive and are often complementary. For example, most companies implement workplace safety procedures (an internal control) and purchase workers compensation insurance (a risk transfer strategy). In fact, the former can reduce the cost of the latter. Another example is product liability. A company can strengthen product development controls as well as purchase product liability insurance. Some risk transfer strategies are intended as backstops to internal controls. For example, directors and officers liability insurance provides protection against wrongful acts. In the past, insurance managers would purchase such backstop insurance policies based on the structure, cost, and provider rating and service level. In the context of ERM and ORM, a company should:

identify its operational risk exposures and quantify its probabilities, severities and economic capital requirements; integrate its operational risk with its credit risk and market risk in order to assess its enterprise-wide riskreturn profile; establish operational risk limits (e.g., MAPs, economic capital concentration); implement internal controls and develop risk transfer and financing strategies;



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evaluate alternative providers and structures based on costbenefit economics (i.e., comparing the cost of risk retention and risk transfer). The economic capital framework discussed above is also a useful tool for evaluating the impact of different risk transfer strategies. For example, in executing any risk transfer strategy the economic benefits include lower expected losses and reduced loss volatility, while the economic costs include insurance premiums, as well as higher counterparty credit exposures. In a sense, the company is both ceding risk and ceding return, resulting in a ceded RAROC. By comparing the ceded RAROCs of various risk transfer strategies, a company can compare different structures, prices and counterparties on an apples-to-apples basis and select the most optimal transaction(s). Moreover, a risk transfer strategy with a ceded RAROC below the firms cost of equity would add to shareholder value, and vice versa. Figure III.C.2.5 provides the framework for a ceded RAROC analysis. Figure III.C.2.5: Ceded RAROC analysis

Different Structures

Common Cost/Benefit Framework Ceded RAROC =

Derivatives

Return Economic Capital

Return Pay cashflows or insurance premium Include transaction and ongoing management costs Reduce Economic Capital benefit

Structured Finance

Economic Capital Reduce Economic Capital held for risk Increase Economic Capital counterparty exposure Increase operating risk Economic Capital

Insurance

Let us apply the framework using the following example of purchasing a product liability insurance policy at an annual premium of $100,000: 1. Annual insurance premium 2. Annual management cost

$100,000 20,000



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3. Economic capital benefit 177 4. Net reduction in economic capital 178

60,000 2,000,000

In this example, the ceded RAROC is 9% [(100,000 + 20,000 + 60,000) / 2,000,000]. This represents the effective cost of risk transfer, which can be compared to the effective costs of alternative risk transfer strategies as well as the cost of risk retention. For instance, if the companys cost of economic capital were 10%, then this transaction would add to shareholder value because the ceded RAROC (cost of risk transfer) is below the cost of risk retention.

III.C.2.9

IT Outsourcing

IT outsourcing is widely considered one of the major business imperatives in todays business world, often described as a mega-trend. The META Group estimates that IT outsourcing is a US$150 billion market, compared to $200220 billion for software and hardware.

More

importantly, the growth of IT outsourcing is expected to outpace that of software and hardware for the next few years at least. Let us discuss how one might establish an operational risk process for IT outsourcing based on the processes and tools discussed earlier. III.C.2.9.1

Stakeholder Objectives

Buyers of outsourcing should first establish an overall outsourcing strategy. This strategy would identify the IT systems and/or applications that are outsource candidates, the approach to evaluating alternative outsource providers and solutions, and the decision processes and cost benefit analyses that will result in specific outsourcing transactions. This strategy should also discuss how the outsourcing strategy will support the overall business strategy of the company, including the specific business and financial objectives that outsourcing is expected to achieve. Buyers of outsourcing services often cite the following expected benefits: cost savings; access to IT skills and advanced technologies; quality of services; resource allocation to core activities; scalability and flexibility in IT resources; shorter time to market; enhance e-business applications.

177 The economic capital benefit represents a funding credit. It is used in matched-maturity funds transfer systems to

recognize the interest income from the investment of capital funds. In our example, we assume a funding rate of 3%. 178 The net reduction in economic capital includes the gross reduction of economic capital for the risk exposure that is

being insured, minus the increase in counterparty and operational risk capital.



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As a sign of more realistic expectations on the part of buyers, expected cost savings cited by buyers have come down from 4070% a few years ago to 1020% today. However, even with reduced expectations, outsourcing arrangements often fail to meet the buyers requirements. A 2004 IT outsourcing study by DiamondCluster 179 noted that 21% of buyers said that they had prematurely terminated an outsourcing agreement in the last 12 months. The most common reasons cited for cancelling outsourcing arrangements include: provider having financial difficulties (credit risk); provider failure to deliver on commitments (operational risk); buyer consolidation of outsourcing vendors (business risk). In addition to these risks, buyers must address a complex set of significant operational risks such as geopolitical risk, regulatory compliance, data integrity and security, and reputational risk. While project delays and cost overruns are the most common reported outcomes from these risks, it might be useful to review a couple of more specific examples. In the area of information security, a woman from Pakistan recently obtained sensitive patient information from the University of California, San Francisco, Medical Center through a medical transcription subcontractor that she worked for, and threatened to post the files on the Internet unless she was paid more money. On the political side, there is a significant backlash against exporting jobs through outsourcing. In response to political pressure, the state of New Jersey has passed legislation to ban IT and business process outsourcing by state government. Perhaps one of the most critical risks in IT outsourcing arrangements is the appropriate alignment of objectives between the parties.

Given the long-term nature of outsourcing

contracts, buyers must seek out a provider that not only offers the optimal technologies and services, but also possesses compatible business culture, professional standards, and business objectives. III.C.2.9.2

Key Processes

The key processes associated with IT outsourcing include: (i) Evaluation and selection of outsourcing provider.

The process includes documentation of

requirements in a request for proposal (RFP), sending out RFPs and evaluating the responses, and negotiating the contract and service level agreement (SLA). This step can take six months to a year, and cost 25% of the annual cost of the contract.

179 DiamondCluster 2004 Global IT Outsourcing Study, available at www.diamondcluster.com.



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(ii) Transitioning to the outsourcing environment. The transition period is perhaps the most challenging stage of an outsourcing initiative. Steps include bringing the provider company professionals onsite for training and knowledge transfer, transferring or terminating existing employees, and establishing the required infrastructure (hardware, software, communication protocols) and operational processes. This step can take an additional three to twelve months, and cost 515% of the annual contract. (iii) Ongoing management of the outsourcing contract.

On a day-to-day basis, the buyer must manage

work processes and communications, monitor provider performance, audit operations, perform quality tests, and integrate the output with other in-house or outsourced IT operations. This requires significant project management resources, and can cost 510% of the contract. Additionally, the ongoing costs of risk should be considered, such as incremental insurance expense and economic capital costs. Each of the above processes, especially ongoing management, should be fully documented in policies and procedures and illustrated in process maps. As such, roles and responsibilities of both parties are clearly established. III.C.2.9.3

Performance Monitoring

As discussed earlier, performance and risk metrics should be developed as part of an ongoing outsourcing review process. Performance goals and MAPs should be incorporated into SLAs and a scorecard should be developed to track actual performance against goals and MAPs. The DiamondCluster study noted that 70% of buyers and 69% of providers monitor performance at least monthly, with the following quantitative metrics as being the most common: on-time delivery; cost effectiveness; end-user satisfaction; timely, quality staffing; service availability; time to process requests; defect rates; standards compliance; size of request backlog. In addition to the above performance metrics, the buyer should monitor the providers key business and risk metrics, such as market share, earnings, stock price performance, debt rating



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and financial ratios. Performance and risk monitoring should not be viewed as a reporting exercise, but as a proactive process to ensure overall project success, which may include some of the risk mitigation strategies discussed below.

III.C.2.9.4

Risk Mitigation

One of the first and most important steps in mitigating outsourcing risk is to fully evaluate the economic costs and benefits at the onset, including consideration of all critical risk factors. As noted in a survey of 500 human resources executives by Hewitt Associates, 92% of the firms had moved jobs overseas to cut costs. However, less than half of those companies studied the tax environments of the offshore country and only 34% considered the expense of shutting down US facilities.

Let us take a simple example to see how an outsourcing contract should be evaluated. Suppose a US company is considering moving one of its application development projects offshore to China, which provides a tax-adjusted 50% labour arbitrage on a $10 million contract. The following table shows the costbenefit analysis for both a best case and worst case:

($ Thousands)

Best Case

Worst Case

1. Projected labour cost savings

$5000 (50%)

$5000 (50%)

2. Vendor selection

200 (2%)

500 (5%)

3. Transition costs

500 (5%)

1500 (15%)

4. Ongoing management

500 (5%)

1000 (10%)

1000 (10%)

1500 (15%)

$2800 (28%)

$500 (5%)

5. Risk costs

180

Net savings:

The above example shows that the adjusted net savings range from 28% in the best case to only 5% in the worst case. Such an analysis should be performed for different outsourcing strategies and various providers. Finally, the projected range of net savings should then be considered against less tangible risks such as reputational risks and geopolitical risks.

180 Risk costs include incremental insurance expense for the outsourcing arrangement and the cost of incremental

operational risk capital. The latter should include exit cost, which is a function of the probability of project failure and the cost of switching operations in-house or to another provider.



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In addition to performing a full costbenefit analysis of outsourcing opportunities, other risk mitigation strategies include: Developing a hybrid outsourcing strategy. Outsourcing experts suggest that the optimal blend of in-house and outsourced IT resources is generally 2030% in-house and 7080% outsourced. At a significant scale, even the outsourced component should be diversified across different providers, countries and locations. Other considerations include the mix between nearshore operations (e.g., Canada, Mexico, in the case of a US firm) and offshore operations (e.g., India, China), as well as the possibility of setting up captive outsourcing companies. Taking an incremental outsourcing approach.

Early outsourcing contracts were large and very

ambitious arrangements that often failed to live up to the buyer and/or provider expectations. For buyers new to outsourcing or experienced buyers working with a new provider, a more practical approach is to outsource incrementally to ensure a compatible relationship in terms of expectations, corporate cultures, and technology and service requirements.

Once the initial work is performed at a satisfactory level, then the

outsourcing relationship can be expanded over time.

Negotiating a flexible winwin contract. It is critical that the outsourcing contract is attractive to both parties. For example, an arrangement that is overly favourable to the buyer might not get the appropriate level of service and attention of the provider. However, the provider should have skin in the game to provide the right incentives for ongoing performance. For example, the META Group estimates that by 2006, 35% of outsourcing contracts will adopt output-based pricing instead of time and materials (input-based pricing). Also, given the rapid changes in technologies, customer preferences and business requirements, contract terms should incorporate sufficient flexibility so that they do not become obsolete prematurely. Establishing exit strategies and contingency plans. Companies should establish exit strategies and contingency plans in the event that the outsourcing contract expires, or the provider does not deliver as expected, or business conditions require the termination of the contract. These exit strategies and contingency plans should be developed in the early stages and with the participation of the provider(s), because their cooperation will be needed to execute such



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plans. Moreover, in the post September 11 world, contingency plans for disaster recovery should be fully developed and tested by the provider and/or buyer.

Developing a compelling stakeholder communication strategy. Outsourcing should continue to be a sensitive political issue for the foreseeable future. Forrester Research estimates that over the next 12 years, 3.3 million US jobs, accounting for $100 billion in wages, will move offshore. Such numbers will likely fuel the current backlash.

To minimize reputational risks,

companies should develop a well thought-out communication strategy with respect to their outsourcing initiatives.

Besides communicating to external groups such as customers,

unions, and governmental entities, internal communication is important given the potential for disgruntled employees to undermine outsourcing initiatives. To develop, coordinate and implement project management controls and the above risk mitigation strategies, companies are putting in place centralized programme management offices (PMOs) as governance structures. These PMOs take a portfolio approach in allocating resources and monitoring vendor performance to ensure optimal performance. The PMOs also represent a centre of excellence for project management skills and resources, including sourcing and managing external resources such as consultants, tax experts and lawyers. Regardless of whether a PMO is established, companies involved in, or planning to initiate, outsourcing arrangements should establish the appropriate operational risk controls discussed in the chapter. Otherwise, given the strategic importance of IT and outsourcing, the next best thing might very well become the companys worst nightmare.

References Cumming, C M, and Hirtle, B J (2001) The challenges of risk management in diversified financial companies. Federal Reserve Bank of New York Economic Policy Review, March . Hernandez, J V, Sanchez, L M, and Ceske, R (2000) Quantifying event risk: the next convergence. Journal of Risk Finance, 1(3), pp. 923. Lam, J (2003a) A unified management and capital framework for operational risk. RMA Journal, Feb., pp. 2629. Lam, J (2003b) Enterprise Risk Management from Incentives to Controls. Hoboken, NJ: Wiley.



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III.C.3 Operational Value-at-Risk Carol Alexander 181 Many firms may wish to apply an advanced measurement approach (AMA) to assess their capital to cover operational risk. Under the new Basel Accord that comes into force at the end of 2006, banks will at first apply a top-down method (either the basic indicator or standardised approach) to assess their operational risk regulatory capital. However, by the end of 2007 they will be able to apply an AMA and hopefully reduce their regulatory capital charge provided they meet certain qualitative and quantitative criteria. Rating agencies are another driving force behind the implementation of AMA. Banks and corporates that aim for a high credit rating require an accurate assessment of their operational risks to convince rating agencies that capitalization is adequate. The top-down methods provide only a crude estimate of operational risk, based on the unrealistic assumption that operational risks increase proportionally with gross income (see Section III.C.2.4.1). Quantitative risk management requires an understanding of the value-at-risk (VaR) models that are used to assess market, credit and operational risk capital. Operational VaR modelling is the subject of this chapter: Section III.C.3.1 outlines the loss model approach to computing operational risk capital (ORC). Sections III.C.3.2 and III.C.3.3 examine how to apply some standard functional forms for the frequency distribution and severity distribution. Sections III.C.3.4 and III.C.3.5 describe how each component of ORC is estimated using (a) analytic and (b) simulation methods, and then Section III.C.3.6 explains how the component ORC estimates are aggregated over all business lines and event types to obtain the total ORC estimate for the firm. Section III.C.3.7 concludes

III.C.3.1

The Loss Model Approach

The actuarial loss model approach has recently become accepted by the industry as the generic AMA for the determination of operational risk regulatory capital for the new Basel 2 Accord (see Sections III.0.3 and III.C.1.7 for further details). Consequently, the loss model approach may also be favoured by rating agencies for firms of high credit quality. But even without this external pressure, many firms will want to adopt operational loss models as a key element of good risk management practice.

181 Chair of Risk Management and Director of Research, ISMA Centre, Business School, University of Reading, UK.



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Prior to implementing an AMA the firm must identify events that are linked to operational risks, and map these events to an operational risk matrix such as that based on the Basel 2 consultative documents (see Basel Committee on Banking Supervision, 2001) and which is shown in Table III.C.3.1. Each element in the matrix defines an operational risk type by its business line and operational event category. In the AMA, operational risk capital is first assessed separately for each risk type for which the AMA is the designated approach. 182 Then the component estimates are aggregated to obtain the total AMA operational risk capital for the firm. The AMA could be chosen for only the most important risk types and this depends on the nature of the business: that is, the definition of the important event types and business lines will be specific to the firms operations. For example, operational losses for clearing and settlements firms may be concentrated in processing risks and systems risks. Table III.C.3.1: The operational risk matrix Internal External Fraud Fraud

Employment Practices & Workplace Safety

Clients, Products & Business Practices

Damage to Physical Assets

Business Disruption & System Failures

Execution, Delivery & Process Management

Corporate Finance Trading & Sales Retail Banking Commercial Banking Payment & Settlement Agency & Custody Asset Management Retail Brokerage The definition of business units and event types for the operational risk matrix can be specific to the firm. It will be natural to follow pre-existing internal definitions of business units and to define event types that capture the important operational risks. It should also take account of the granularity that is required for the AMA calculations.

Increasing levels of granularity are

necessary to include the impact of insurance cover, which may only be available for some event types in certain lines of business. 182 Corporates are, of course, free to pick and choose which risk types they assess using AMA. Indeed, banks are also

afforded some flexibility: under the new Basel Accord they can choose the AMA for some risk types and apply the standardized approach to others. However, once a risk type has been chosen for AMA modelling, the bank will not be allowed in future to apply a more basic risk capital assessment method.



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It is desirable to isolate those elements of the matrix that are likely to be dominant in the final aggregation, as these risk types should be the main priority for risk control. Unfortunately, these can be precisely those risks for which the data are very subjective, consisting of expert opinions or risk self-assessments that have a large element of uncertainty. Somehow, this uncertainty in the data must be included in the risk model. Qualitative judgements must be translated into quantitative assessments of risk capital using appropriate statistical methodologies. Many firms now aim to do this through a risk self-assessment process. A risk self-assessment gives a forward-looking, subjective estimate of the loss model parameters. Risk self-assessment programs can be facilitated in the same way as control self-assessments (see Section III.C.1.9). Operational risks may be categorised in terms of: frequency, the number of loss events during a certain time period; and severity, the impact of the event in terms of financial loss. Risks with very low frequency and high severity, such as a massive fraud or a terrorist attack, could jeopardise the whole future of the firm. These are the risks associated with losses that will lie in the very upper tail of the total loss distribution. Risk capital is not really designed to cover these risks. However, they might be insurable. Risks with high frequency and low severity, which include credit card fraud and processing risks, can have a high expected loss but will have relatively low unexpected loss. That is, the range of loss outcomes is relatively narrow. If expected losses for the high-frequency, low-severity risks are covered by the general provisions of the business, the implication is that ORC requirements for these risk types will be relatively low. If this is not the case, then expected losses should be included in the risk capital. Unless expected losses are very high, the risk capital will still be lower than that for medium-frequency, medium-severity risks. These latter risks are the legal risks, the minor frauds, the fines from improper practices, the large system failures and so forth. In general, these should be the main focus of the AMA. An example of loss data is shown in Table III.C.3.2. For simplicity and only for the purposes of this illustration, the exact date of the loss is not actually recorded, only the quarter into which it falls. This allows one to classify data by quarterly frequency or by semi-annual or annual frequency but not by monthly or lower frequencies. Suppose we choose the quarterly period, so the frequency distribution will be of the number of loss events per quarter. First we ignore the severity data, and consider only the dates of the loss events. There were no quarters in which no loss events occurred, no quarters in which 1 loss event occurs, but there was one quarter in which 2 loss events occurred (2001:Q3) and two quarters in which 3 loss events occurred (2000:Q1 and 2000:Q3). Continuing counting in this way, we can draw an empirical frequency



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density. Secondly, returning to the example loss data, we now ignore the date of loss, consider only the loss amounts and hence construct the empirical severity density. Table III.C.3.2: Example of historical loss experience data Date 2000:Q1 2000:Q1 2000:Q1 2000:Q2 2000:Q2 2000:Q2 2000:Q2 2000:Q3 2000:Q3 2000:Q3 2000:Q4 2000:Q4 2000:Q4 2000:Q4

Loss (? 00) 4.45 13.08 29.38 25.92 39.10 12.92 1.24 8.01 12.17 13.88 53.37 5.89 1.32 7.11

Date 2001:Q1 2001:Q1 2001:Q1 2001:Q1 2001:Q1 2001:Q1 2001:Q2 2001:Q2 2001:Q2 2001:Q2 2001:Q2 2001:Q3 2001:Q3 2001:Q4 2001:Q4 2001:Q4 2001:Q4 2001:Q4 2001:Q4 2001:Q4 2001:Q4 2001:Q4

Loss (? 00) 7.51 1.17 1.35 105.45 37.24 16.55 7.34 1.35 1.50 1.19 2.80 3.00 6.82 1.73 231.65 5.00 3.10 26.45 12.62 2.32 71.12 1.73

Date 2002:Q1 2002:Q1 2002:Q1 2002:Q1 2002:Q1 2002:Q1 2002:Q1 2002:Q1 2002:Q1 2002:Q1 2002:Q2 2002:Q2 2002:Q2 2002:Q2 2002:Q2 2002:Q3 2002:Q3 2002:Q3 2002:Q3 2002:Q3 2002:Q3 2002:Q4 2002:Q4 2002:Q4 2002:Q4

Loss (? 00) 1.12 4.06 34.55 10.24 24.17 11.01 3.89 187.50 13.21 4.49 2.10 2.20 2.31 25.00 3.81 1.48 1.57 20.33 43.78 3.62 45.72 142.59 20.73 31.96 55.60

For some real loss experience data based on a record going back over 24 months of operational loss events, Figure III.C.3.1 illustrates the empirical frequency density (the number of loss events per month). It shows that, out of the 24 months, there was only one month in which less than 10 loss events occurred; there were five months for which between 10 and 19 loss events occurred; three months for which between 20 and 29 loss events occurred, and so forth. Figure III.C.3.1: Example of (monthly) frequency distribution Frequency 6 5 4 3 2 1 0 1

10

20

30

40

50

60

70

80

90

100

110

120

130

140

Num be r of Loss Events



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Figure III.C.3.2 illustrates the empirical severity density obtained from the same data. Only losses in excess of 1000 are recorded, so the severity data are truncated at the lower end. Special methods need to be applied to detruncate the severity data so that the full severity density can be recovered. Figure III.C.3.2: Example of loss severity distribution Severity 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0

20

40

60

80

100

120

Loss Give n Eve nt

In summary, for a given operational risk type we construct a discrete frequency density h(n) of the number of loss events n per period, and a continuous density g(l ) representing the loss severity, L. The density function for the total loss distribution f(x) is then given by compounding these two densities, usually under the assumption that loss frequency and loss severity are independent. More details of the method for compounding these densities will be given in Section III.C.3.5 below. Figure III.C.3.3 gives a diagrammatic representation of the relationship between the total loss distribution and its underlying frequency and severity distributions. The basic time period for the frequency density here is one year, and in this case the total loss distribution is also called the annual loss distribution. This is a convenient terminology because later we need to aggregate several total loss distributions over different risk types into a total total loss distribution. However, by referring to each component as an annual loss distribution (or a quarterly loss distribution if the frequency is measured at quarterly intervals, or monthly and so on) there is no confusion about what is meant by the total loss distribution.



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Figure III.C.3.3: Representation of the loss model

Unexpected Loss Expected Loss

99.9th percentile

Frequency Distribution

No. Loss Events Per Period

Loss Severity Distribution

Loss Given Event

Marked on the density function for the total loss distribution are the expected loss, that is, the mean of this distribution; and the unexpected loss at the 99.9th percentile, that is, the unexpected loss at the th percentile is the difference between the upper th percentile and the mean of the annual loss distribution ORC is held to cover all losses, other than highly exceptional losses, that are not already covered by the normal cost of the business. The definition of what one means by highly exceptional translates into the definition of a percentile of the loss distribution, such that only losses exceeding this amount are highly exceptional. One should attempt to control these, using scenario analysis. As already mentioned, often expected losses are already included in the normal cost of business. For example, the expected loss from credit card fraud may be included in the balance sheet under operating costs. In that case ORC should cover only the unexpected loss at some predefined percentile of the loss distribution. The ORC is thus defined using the VaR metric, for some risk horizon (defined below) and at some percentile. The Basel Committee recommend 99.9% and a one-year horizon for the calculation of ORC in banks; internally, for companies wishing to maintain a high credit rating, it is common to use an even higher percentile that is consistent with their desired credit rating. For instance, a firm that targets a AA rating will typically be measuring economic capital at the 99.97th percentile.



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III.C.3.2

The Frequency Distribution

The total operational loss refers to a fixed time period over which these events are to be observed. This time period is usually called the risk horizon of the loss model to emphasise that it is a forward-looking time interval starting from today. For regulatory purposes, both operational and credit risk horizons are set at one year, but for internal purposes it is also common to use risk horizons of less than or more than one year. For ease of exposition we shall henceforth only refer to the one-year horizon hence the AMA aims to model the annual loss distribution. Having defined a risk horizon, the probability of a loss event, which has no time dimension, can be translated into the loss frequency, that is, the number of loss events occurring during the risk horizon. In particular, the expected loss frequency, denoted lambda ( ), is the product of the expected total number of events, N, during the risk horizon, including events for which no operational loss was made, and the expected loss probability, p: = Np. Sometimes it is convenient to forecast

(III.C.3.1)

directly this is the case when we cannot quantify the

total number of events N and we only observe loss events and in other cases it is best to forecast N and p separately for example, N could be the target number of transactions over the next year, and in that case it is p, not , that we should attempt to forecast using loss experience and/or risk self-assessment data. Loss frequency (the number of loss events occurring during the risk horizon) is a discrete random variable: it can only take the values 0, 1, 2, , N. A fundamental density for such a discrete random variable (which nevertheless is only appropriate under certain assumptions) 183 is the well-known binomial density (see Section II.E.4.1),

h( n )

N n p (1 n

p )N

n

n = 0, 1, , N.

(III.C.3.2)

The binomial frequency is only used when a value for N can be specified. For example, for Trading & Sales/Client, Products and Business Practice Risk, N could be the target number of deals over the next year; for Retail Banking/External Fraud, N could be the total number of credit cards in issuance during the risk horizon. 184 183 We must assume that the probability of a loss event is the same for all the events in this risk type, and therefore

equal to p; and that operational events are independent of each other. 184 Note that here N would be so large that the Poisson distribution can be used instead of the binomial distribution. However, firms may still wish to employ the binomial distribution when, for instance, their target for N over the next year is quite different from the historical value of N.



401


It is not always possible to specify N. However, since p is normally small the binomial distribution can often be well approximated by the Poisson distribution, which has the single parameter

the expected frequency, as in (III.C.3.1) above. Incidentally,

is also equal to the

variance of the Poisson distribution. The Poisson distribution has the density function (see Section II.E.4.2) n

h( n )

exp( n!

)

n = 0, 1, 2, .

(III.C.3.3)

If the empirical frequency density is not well modelled by a Poisson distribution for example, one could equate the mean frequency observed empirically with , but then find that the sample variance is significantly different from

an alternative, more flexible functional form is the

negative binomial distribution, with density function h(n )

which has mean

n 1 n

and variance

n

1 1

1

n = 0, 1, 2, ,

(III.C.3.4)

2.

There is absolutely no point in applying a statistical test to decide which of the frequency distributions provides the closest fit to loss data: the binomial is only applicable when a number of events can be quantified (and is small) and the negative binomial has two parameters so it will always fit better than the Poisson. However, this does not imply that one should always choose the negative binomial as the frequency functional form. In contrast to market and credit risk, for operational risk the precise fitting of data by choosing the best functional form is not a main source of model risk. In fact the model risk arising from inappropriate and/or ad hoc methods when handling the data is a much more important source of operational VaR model risk. 185 The choice of functional form for the frequency distribution should depend on both the type of data and the source(s) of the data internal and/or external loss data and/or risk selfassessments. It is very difficult to design psychologically meaningful questions in a risk selfassessment that are compatible with a negative binomial frequency. On the other hand, the question what is the expected number of loss events next year? will directly invoke the parameter

for the Poisson distribution. Also, it is difficult to apply the binomial distribution to

external consortium data because the consortium will not normally be recording a value of N for each bank. These restrictions imply that it is often the Poisson distribution that is used in practice. 185 But of course, in common with market and credit risk, the main model risk is aggregation risk. This stems from

making inappropriate assumptions regarding dependencies when aggregating risks. This has by far the most influence on the total VaR estimate. See, for instance, Chapter III.A.3.



402


Example III.C.3.1: Estimating a Poisson frequency from historical data (i) High-frequency risk type. Suppose historical loss events give the following data on just the number of loss events recorded each month, over the last 2 years: Month

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of Loss Events 20 13 24 26 25 21 17 13 21 30 16 24 31 20 19 21 14 14 15 18 16 21 22 19

The total number of loss events recorded was 480, so the average number of loss events per month is 20. The monthly frequency distribution is therefore estimated as a Poisson distribution with = 20. The density function is shown in red in Figure III.C.3.4. (ii) Low(er)-frequency risk type. Suppose historical loss events give the following data on just the number of loss events recorded each month, over the last 2 years: Month

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of Loss Events 0 12 3 0 8 4 10 1 0 9 2 10 3 5 3 5 1 7 4 7 10 5 7 4

The total number of loss events recorded is now only 120, so the average number of loss events per month is 5. The monthly frequency distribution is therefore estimated as a Poisson distribution with = 5. The density function is shown in blue in Figure III.C.3.4. Figure III.C.3.4: Poisson frequency densities for high-frequency and low-frequency risks 0.2 0.18 Lam bda = 5

0.16

Lam bda = 20

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 1

Number of loss events per month



403


Notes: 1. The Poisson annual frequency density is obtained by multiplying the estimated lambda from monthly data (20 and 5 in the above example) by 12. 2. Lower-frequency risks have more skewed and leptokurtic frequency densities than highfrequency risks. This property influences the compound distribution, so that the annual loss distribution will also be highly skewed and leptokurtic for the low-frequency, highseverity risks.

III.C.3.3

The Severity Distribution

Now consider how to fit a severity distribution, such as that shown in Figure III.C.3.2. Various functional forms are available for continuous random variables like severity, as described in Chapter II.E. The lognormal distribution for loss severity, L, has the density function: g (l )

2

1 ln l 2

1 exp 2 l

(l > 0) .

(III.C.3.5)

Here the log of the loss severity (or log severity for short, denoted ln L) is normally distributed with mean

and variance

. This is a very common distribution in financial mathematics, and more

details are given in Section II.E.4.5. High-frequency risks can have severity distributions that are relatively lognormal, but lowfrequency risks can have severity distributions that are too skewed and leptokurtic to be well captured by the lognormal density function. Common choices, therefore, are the gamma density,

g(l )

l

1

exp

l/

(l > 0),

( )

(III.C.3.6)

where (.) denotes the gamma function, and the two-parameter hyperbolic density,

g(l )

2

exp 2 B(

l2 )

(l > 0),

(III.C.3.7)

where B(.) denotes the Bessel function of the first kind. On first sight these may look daunting, but Excel does have in-built functions for these common distributions (with the obvious names). Other functional forms for the severity that have been considered by some banks include the generalised hyperbolic, lognormal mixtures, and general mixture distributions.



404


Again, there is absolutely no point in applying a statistical test to decide which of these frequency distributions provides the closest fit to loss data: the generalised four-parameter hyperbolic distribution will always fit best. However, this does not imply that one should always choose the generalised hyperbolic distribution as the severity functional form. Again, this choice again depends on both the type of data and the source(s) of the data internal and/or external loss data and/or risk self-assessments. For example, it is very difficult to design a risk self-assessment that is compatible with any severity density having more than two parameters. In some databases, for instance those constructed from public, newsworthy events, only very extreme losses are recorded. There is an implicit high threshold for the losses included in the database and thus some analysts have attempted to model the severity of these losses using the generalised Pareto and other distributions from the class of distributions in extreme-value theory (EVT). Whilst EVT may have found useful applications to high-frequency, tic-by-tic financial market data, one should not forget that it was introduced (almost 50 years ago) to model the distributions of extreme values in repetitive, independent and identically distributed processes, such as those observed in the physical sciences; see, for instance, Gumbel (1958) and Embrechts et al. (1991). To attempt to fit a generalised Pareto distribution, or any other extremevalue distribution, to the sparse and fragmented data that are available for very large operational losses is, in my view, a triumph of hope over reason. Nevertheless, some AMAs are currently attempting to incorporate extreme-value densities, and readers should therefore have some understanding of them. The peaks-over-threshold (POT) model applies when losses over a high and predefined threshold u are recorded. The distribution function Gu of the excess losses, X u, has a simple relation to the distribution F(x) of the loss severity X. In fact Gu(y) = prob(X u < y X > u) = [ F(y + u) F(u) ] / [ 1 F(u) ].

(III.C.3.8)

For many choices of underlying distribution F(x) the distribution Gu(y) will belong to the class of generalised Pareto distributions (GPDs) given by: 1 exp(y/ )

if = 0,

Gu(y) =

(III.C.3.9) 1 (1 + y/ )1/

The parameters

and

if

0.

will depend on the type of underlying loss distribution F(x) and on the

choice of threshold u. Some generalised Pareto densities for different values of

and are shown

in Figures III.C.3.5 and III.C.3.6. Note that the primary effect of increasing

is to increase the



405


range of the density (Figure III.C.3.5), and that

is called the tail index precisely because as

increases so does the weight in the tails of the GPD (see Figure III.C.3.6). Figure III.C.3.5: Generalised Pareto densities ( = 1)

0.1 0.09 0.08 0.07

=1

0.06 0.05 0.04 0.03 0.02 0.01 0

=5

=2

0

20

Figure III.C.3.6: Generalised Pareto densities ( = 1)

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

=0 = 0.9

0



20

406


III.C.3.4

The Internal Measurement Approach

Under certain assumptions (which are rather strong, but nevertheless appear to be admissible by regulators), there are simple analytic formulae for the expected loss and the unexpected loss in the annual loss distribution. These formulae are based on what the Basel Committee has called the internal measurement approach (IMA).

The basic formula for the IMA risk capital

calculation given in the proposed Basel 2 Accord is: ORC = gamma

expected annual loss =

NpL,

(III.C.3.10)

where N is a volume indicator (a proxy for the number of operational events), p is the expected probability of a loss event, L is the loss given event, and , gamma, is a multiplier that depends on the operational risk type. Note that NpL only corresponds to the expected annual loss when the loss frequency is binomially distributed and the loss severity is not regarded as a random variable. More generally, with a Poisson frequency distribution having expected frequency ORC =

Np):

L.

(III.C.3.11)

Note that a very strong assumption of the IMA is that each time a loss is incurred, exactly the same amount is lost (within a given risk type). Introduction of severity uncertainty, as in the loss distribution approach (LDA) described below, will always increase the ORC, often by a factor of 5 or more. Thus the IMA provides only a useful benchmark, a lower bound for the operational risk capital calculated using the full simulation method that we shall describe presently. I have shown elsewhere (Alexander, 2003, p. 151) how the value of

in (III.C.3.11) can be

calculated and provide statistical tables for : it only depends on the percentile and the expected loss frequency. Instead of following Consultative Paper 2.5 (Basel Committee, 2001) and writing unexpected loss as a multiple ( ) of expected loss, I write unexpected loss as a multiple phi ( ) of the loss standard deviation. That is, ORC =

standard deviation of annual loss.

(III.C.3.12)

Recall that ORC is measured by a VaR metric, and is either unexpected loss at the 99.9th percentile or the 99.9th percentile itself, the latter being the case when expected losses are not already provisioned. Thus (III.C.3.12) can be rewritten as:



407


= [(99.9%-ile

mean)/standard deviation]

if expected losses are provisioned, (III.C.3.13)

= [(99.9%-ile /standard deviation)]

otherwise.

This formulation shows that, once we have estimated

(either from a risk self-assessment or

from loss data, as explained in Section III.C.3.2) it is easy to calibrate . Because loss severity is assumed to be the same amount, L, every time a loss is made, all the randomness in the annual loss distribution comes only from the frequency distribution. Hence all quantities on the righthand side of (III.C.3.13) the 99.9th percentile, the mean and the standard deviation are just L times the respective quantities of the frequency density. Thus L cancels, and the right-hand side of (III.C.3.13) really just refers to the 99.9th percentile, the mean and the standard deviation of the frequency density. Having obtained

we can obtain

as follows: compare (III.C.3.10) and (III.C.3.12). Since ORC

is the same in both, equating them gives: expected annual loss =

standard deviation of annual loss

so is a multiple of : in fact =

(standard deviation/mean).

The same argument as above (i.e. that L cancels) implies that and

are related through a factor

(standard deviation/mean) of the frequency density. For instance, when the frequency is Poisson, which has mean and variance , =

.

(III.C.3.14)

In this way, we construct statistical tables for the gamma factor in the IMA, a different table for each choice of frequency distribution. Table III.C.3.3 gives the Poisson gamma tables ( is also shown).



408


Table III.C.3.3: The gamma in the IMA (Poisson frequency)

99.9%-ile

99.9%-ile

99.9%-ile

99.9%-ile

100

50

40

30

20

10

131.805

72.751

60.452

47.812

34.714

20.662

3.180

3.218

3.234

3.252

3.290

3.372

0.318

0.455

0.511

0.594

0.736

1.066

8

6

5

4

3

2

17.630

14.449

12.771

10.956

9.127

7.113

3.405

3.449

3.475

3.478

3.537

3.615

1.204

1.408

1.554

1.739

2.042

2.556

1

0.9

0.8

0.7

0.6

0.5

4.868

4.551

4.234

3.914

3.584

3.255

3.868

3.848

3.839

3.841

3.853

3.896

3.868

4.056

4.292

4.591

4.974

5.510

0.4

0.3

0.2

0.1

0.05

0.01

2.908

2.490

2.072

1.421

1.065

0.904

3.965

3.998

4.187

4.176

4.541

8.940

6.269

7.300

9.362

13.205

20.306

89.401

Source: Alexander (2003). Table reproduced by kind permission of Pearson Education (Financial Times-Prentice Hall).

For instance, when

= 100 (i.e. a very high-frequency risk) the 99.9th percentile of the Poisson

distribution is 131.805. Thus, for a capital charge based only on unexpected loss, not on expected loss, we have = (131.805 l00)/ 100 = 31.805/10 = 3.1805, 3.1805/ 100

Note that

0.31805 .

does not change much with , but that does. For low-frequency risks is very large

indeed, but for high-frequency risks it is very low. As

increases above 100, for very high-

frequency risks, the frequency distribution approaches the normal distribution so

tends to a

lower limit of 3.09 (this is the 99.9th percentile in the standard normal distribution); however, tends to a lower limit of zero! Copyright ? 2004 The Authors and the Professional Risk Managers International Association


409


Also note that the ORC should increase as the square root of the expected frequency. For example, in the Poisson frequency, combining (III.C.3.11) with (III.C.3.14) gives ORC = we know that

L, and

is around 3 or 4, unless the expected number of loss events is less than 1 every

50 years. So in the AMA the ORC will not be linearly related to the size of the banks operations, as it is under the basic indicator or standardised approach. Hence doubling the size of ones operations should only lead to an increase of 2 in the capital charge; this may be an incentive for banks to use the AMA when they are permitted to do so at the end of 2007. The ORC also is linearly related to loss severity yet another reason why high-severity risks (which are by definition also low-frequency) will attract higher capital charges than low-severity risks; the following two examples illustrate this point. Example III.C.3.2: Credit card operational risk Question: 50,000 credit cards will be in issuance during the forthcoming year and the expected probability of a credit card fraud on any card is 0.05. If every fraud gives rise to an operational loss of 1000, calculate (a) the expected loss, (b) the ORC at the 99.9th percentile, with and without the assumption that the expected loss is already covered by the normal cost of the business. Answer: (a)

= 50,000

0.05 = 2500 so expected loss = 2500

1000 = 2.5 million.

(b) For such a high we have = 3.1 and ORC is estimated using (III.C.2.10). (c) Standard deviation of loss is ORC = 3.1

L = 2500

1000 = 50,000 and so

50,000 = 155,000 (assuming expected loss is provisioned for),

ORC = 155,000 + 2.5 million = 2.655m (if expected loss must be included). This example shows that for high-frequency, low-severity operational risks the expected loss is much greater than the unexpected loss. The expected losses are normally already provisioned in the normal cost of being in the credit card business. In that case the ORC is there to cover the unexpected losses. The ORC for this type of risk will therefore be very small, and this risk type will have little impact on the total ORC for a retail bank. Indeed, the next example shows that the total ORC will be dominated by the low-frequency, high-severity operational risks: Example III.C.3.3: Transactions errors versus internal fraud Show that the expected loss is the same in the following two cases, but that the ORC is far greater for the internal fraud: Case A: 30,000 transactions are processed in the back office and the expected probability of a human error giving rise to an operational loss is 0.01. Each time a loss occurs the operational loss is 1000. Copyright ? 2004 The Authors and the Professional Risk Managers International Association


410


Case B: 60 deals are made in Corporate Finance and the probability of internal fraud resulting in any one of these making an operational loss is 0.0005. However, if such a loss is made, it will amount to 10m. The ORC calculations based on (III.C.2.9) in Table III.C.3.4 below assume that expected loss is provisioned for; clearly, the same basic observation holds whether or not the additional sum of 300,000 is added to the ORC figures. That is, even if the ORC estimates were 0.35369m and 7.22474m respectively, the low-frequency, high-severity risk type is the one that totally dominates the total ORC. Table III.C.3.4: Results for Example III.C.3.3 Case A

Case B

N

30,000

60

p

0.01

0.0005

L

1,000

10,000,000

300,000

300,000

Expected Loss (

)

Standard Deviation (= (Np)L)

17,320.51 1,732,051

(= Np)

(= / ) ORC (m)

III.C.3.5

300

0.03

3.1

3.998

0.178979 23.08246 0.05369

6.92474

The Loss Distribution Approach

The standard assumption that frequency and severity are independent may not be very realistic, but it makes the construction of the compound distribution very simple. Monte Carlo simulation (see Section II.E.4.3) is used to generate the total loss distribution as the compound of the frequency and severity distribution. The simulation algorithm is as follows: 1. Take a random draw from the frequency distribution: suppose this simulates n loss events per period. 2. Take n random draws from the severity distribution: denote these simulated losses by L1, L2, Ln. 3. Sum the n simulated losses to obtain a total loss X = L1 + L2 + + Ln. 4. Return to step 1, and repeat several thousand times: thus obtain X1, , XM where the number of simulations M is very large. 5. Form the histogram of X1, , XM: this represents the simulated total loss distribution.



411


6. The ORC for this risk type is then the difference between the 99.9th percentile and the mean of the simulated annual loss distribution, or just the 99.9th percentile, if expected losses are not provisioned for elsewhere. Figure III.C.3.7 illustrates the first two steps in the simulation algorithm. 186 Figure III.C.3.7: Simulating the annual loss distribution 1

1 Severity CDF Frequency CDF

0

0 L 2 L n L 1

n Total annual loss from one simulation =

Li = TL

Repeat to obtain TL 1 ,,TL 10000 ORC Estimate = Percentile (TL 1 ,.,TL 10000 , 0.999) Average (TL 1 ,.,TL 10000 )

Elsewhere (Alexander, 2003, Ch. 7) I have compared the LDA estimate of ORC with the result of applying the IMA formula of Section III.C.3.4. I considered two cases: Without an assumption of loss severity uncertainty in the LDA, the result will be identical to the IMA estimate, provided enough simulations are used in the LDA calculation. The IMA estimate can be modified to include the assumption of loss severity uncertainty, in which case it is multiplied by the factor (1 + r 2), where r = (severity standard deviation/severity mean).

(III.C.3.15)

The resulting modified IMA estimate is similar to the LDA estimate. This shows that operational risk capital will increase more or less in proportion with the severity standard deviation. So when loss severity becomes more uncertain, this has a direct, almost linear impact on the capital charge. A very strong conclusion can be drawn. That is, for any given risk type: the

186 The use of empirical frequency and severity distributions is not advised, even if sufficient data are available to

generate these distributions. There are two reasons for this. Firstly, the simulated annual loss distribution will not be an accurate representation if the same frequencies and severities are repeatedly sampled. Secondly, there will be no ability to carry out scenario analysis in the model unless one specifies and fits the parameters of a functional form for the severity and frequency distributions.



412


LDA ORC is always far greater than the ORC calculated from an analytic formula based on the assumption that severity is non-random (and this includes the IMA formula). This difference will be most pronounced for operational risks where the loss severity is highly uncertain (so that r in (III.C.3.15) is large): for these risk types, the risk capital calculation based on the LDA can easily give an ORC estimate that is 10 or 20 times larger than the IMA estimate.

III.C.3.6

Aggregating ORC

Having estimated the ORC for each risk type for which the AMA is the designated approach, we must now add up these ORC estimates to obtain the total ORC for the firm. The aggregation of risks not just operational risks is currently a hot topic of research. It is very difficult to aggregate risks (a) when they are assessed using a VaR metric at an early stage; 187 (b) because the total we get for the risks is very much influenced by the assumptions we make about the dependencies between the risks; and (c) because dependencies between risks are very difficult to assess. Thus risk aggregation is very difficult even within market and credit risks it is a very thorny issue in operational risk! Regarding dependency assumptions, the Basel Committee (2001) has stated: The bank will be permitted to recognize empirical correlations in operational risk losses across business lines and event types, provided that it can demonstrate that its systems for measuring correlations are sound and implemented with integrity. Dependencies between operational risks are common. 188 Indeed, they occur whenever two operational risk types share a common key risk driver (see Section III.C.2.3). For instance, risk drivers associated with human risks such as pay, training, management, workload affect many types of operational risks, including employment practices, transactions processing, legal risks, and fraud. Often, when aggregating risks, particularly when the VaR metric is applied, banks will make just two simple assumptions: Full dependency: This implies risks should simply be added to obtain the total risk; this is an approximate upper bound for the total risk and is often used for regulatory risk capital calculations (to err on the conservative side). 187 Unlike the variance risk metric, percentiles obey no simple standard rules. It is easy to aggregate variances (and to allow for correlation in the process) but it is not so simple to aggregate VaR. 188 I prefer not to use the term correlation instead I use the term dependency. Operational risks are likely to be dependent, in the sense that if one changes then so will the other, but they are not correlated. Correlation is a metric that refers to jointly stationary random variables with elliptical distributions like the multivariate normal (see Section II.E.4.7) but there is no evidence that operational losses behave in this way.



413


No dependency: This is the assumption of independence. It implies that the total risk is the square root of the sum of the squares of the component risks. For aggregating different types of operational risks, this will give an approximate lower bound for the total risk capital since it is unlikely that there will be many large negative dependencies between different operational risks. The next example shows that small changes in correlation between operational risk types will have a huge effect on the total risk estimate: for example, the total risk capital based on the aggregation of only two operational risk types can easily be doubled or halved depending on the assumption made about their dependency. Example III.C.3.4: Effect of correlation on risk aggregation Consider the two annual loss distributions with density functions shown in Figure III.C.3.8. Figure III.C.3.9 shows the total loss under correlations of

= 0.5, 0, 0.5, respectively. Then

Table III.C.3.5 shows that the expected loss is hardly affected by the assumptions made about co-dependencies of these two risks: it is approximately 22.4 in each case. However the unexpected loss at the 99.9th percentile (and at the 99th percentile) is very much affected by the assumption one makes about dependency. Table III.C.3.5: Risk capital estimates under different correlation assumptions = 0.5

=0

= 0.5

Expected Loss

22.3909

22.3951

22.3977

41.7658

48.7665

54.1660

19.3749

26.3714

31.7683

99.9th Percentile Unexpected Loss

Of course, the values of the correlation parameter were chosen arbitrarily in Example III.C.3.4. But it has shown that small changes in correlation can produce estimates of total operational risk capital that is doubled or halved even when aggregating only two annual loss distributions. Obviously the effect of correlation assumptions on the aggregation of many annual loss distributions to the total annual loss for the firm will be quite enormous.



414


Figure III.C.3.8: Two annual loss densities 189 0.15

0.12

0.09

0.06

0.03

0 0

5

10

15

20

25

Figure III.C.3.9: The total loss distribution under different assumptions for correlation 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0

20 rho = 0

III.C.3.7

40 rho = 0.5

60

80

rho = - 0.5

Concluding Remarks

Operational risks often show a positive dependency because they experience the same directional influence from common key risk drivers. These drivers can be linked to human risks, systems risks and so forth. Thus the total risk estimate will be somewhere between a lower bound given by the square root of the sum of the squared risk estimates and an upper bound given by the sum of the risk estimates. These two bounds are usually far apart and it is very difficult to say exactly where, within these bounds, the total risk will be. Hence aggregation risk is by far the most important source of model risk in an operational VaR model. 189

The bimodal density has been fitted by a mixture of two normal densities: with probability 0.3 the normal has mean 14 and standard deviation 2.5 and with probability 0.7 the normal has mean 6 and standard deviation 2. The other annual loss is gamma distributed with = 7 and = 2.



415


The next most important source of model risk is the way that data are obtained and, subsequently, how these data are handled. The most important risk types for a risk estimate (as opposed to an expected loss estimate) are the low-frequency, high-severity risk types, such as internal fraud, acts of God (such as earthquakes), massive legal suits and so forth. Internal historical loss data on these risk types are simply not available as the frequency is just too low. The only way to include these risk types in the risk estimate is to use subjective data: such as a risk self-assessment, or external data from public sources, or from a data consortium. But risk self-assessment questionnaires require careful design and even more careful control of the responses, and external data need processing using proper statistical methods. When data are subjective the proper statistical methods to use are Bayesian methods (see Chapter II.E). However, at the moment we are witnessing attempts to apply some of the traditional classical methods that have been developed in market risk to operational risk analysis, as if the sparse and unreliable data on operational losses should be treated like the plentiful and reliable data on market prices. Operational risks are not at all like market or credit risks operational risk analysis is about human behaviour, not market or firm behaviour! Thus we really should not be talking about correlations, or tail distributions with data such as these. And software consultants who focus on fitting the best functional form to operational risk data are missing the big picture (and making a fast buck in the process!). Getting the right assumptions about functional forms may be an important source of model risk in market and credit risk analysis but this is really not an important issue in operational risk assessment. The important issues are how to model dependencies and how to obtain reliable data. We may indeed apply a VaR metric to an operational loss distribution, but that does not mean that operational risk analysis is a statistical science with proper economic foundations, like market and credit risk. It is not. Operational risk is a behavioural science.

References Alexander, C (ed.) (2003) Operational Risk: Regulation, Analysis and Management. London: FTPrentice Hall (Pearson Education). Basle Committee on Banking Supervision (2001) Working Paper on the Regulatory Treatment of Operational Risk, Consultative Paper 2.5, September., available from www.bis.org Gumbel, E J (1958) Statistics of Extremes. New York: Columbia University Press. Embrechts, P, Kl黳pelberg, C, and Mikosch, T (1991) Modeling Extremal Events. Berlin: SpringerVerlag.



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