Barraone Performance Analytics Guide

Performance

Analytics Guide

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This document and all of the information contained in it, including without limitation all text, data, graphs, charts (collectively, the “Information”) is the property of MSCl Inc., its subsidiaries (including without limitation Barra, Inc. and the RiskMetrics Group, Inc.) and/or their subsidiaries (including without limitation the FEA, ISS, and CFRA companies) (alone or with one or more of them, “MSCI”), or their direct or indirect suppliers or any third party involved in the making or compiling of the Information (collectively (including MSCI), the “MSCI Parties” or individually, an “MSCI Party”), as applicable, and is provided for informational purposes only. The Information may not be reproduced or redisseminated in whole or in part without prior written permission from the applicable MSCI Party. The Information may not be used to verify or correct other data, to create indices, risk models or analytics, or in connection with issuing, offering, sponsoring, managing or marketing any securities, portfolios, financial products or other investment vehicles based on, linked to, tracking or otherwise derived from any MSCI products or data. Historical data and analysis should not be taken as an indication or guarantee of any future performance, analysis, forecast or prediction. None of the Information constitutes an offer to sell (or a solicitation of an offer to buy), or a promotion or recommendation of, any security, financial product or other investment vehicle or any trading strategy, and none of the MSCI Parties endorses, approves or otherwise expresses any opinion regarding any issuer, securities, financial products or instruments or trading strategies. None of the Information, MSCI indices, models or other products or services is intended to constitute investment advice or a recommendation to make (or refrain from making) any kind of investment decision and may not be relied on as such. The user of the Information assumes the entire risk of any use it may make or permit to be made of the Information. NONE OF THE MSCI PARTIES MAKES ANY EXPRESS OR IMPLIED WARRANTIES OR REPRESENTATIONS WITH RESPECT TO THE INFORMATION (OR THE RESULTS TO BE OBTAINED BY THE USE THEREOF), AND TO THE MAXIMUM EXTENT PERMITTED BY LAW, MSCI, ON ITS BEHALF AND ON THE BEHALF OF EACH MSCI PARTY, HEREBY EXPRESSLY DISCLAIMS ALL IMPLIED WARRANTIES (INCLUDING, WITHOUT LIMITATION, ANY IMPLIED WARRANTIES OF ORIGINALITY, ACCURACY, TIMELINESS, NON-INFRINGEMENT, COMPLETENESS, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE) WITH RESPECT TO ANY OF THE INFORMATION. Without limiting any of the foregoing and to the maximum extent permitted by law, in no event shall any of the MSCI Parties have any liability regarding any of the Information for any direct, indirect, special, punitive, consequential (including lost profits) or any other damages even if notified of the possibility of such damages. The foregoing shall not exclude or limit any liability that may not by applicable law be excluded or limited, including without limitation (as applicable), any liability for death or personal injury to the extent that such injury results from the negligence or willful default of itself, its servants, agents or sub-contractors. Any use of or access to products, services or information of MSCI requires a license from MSCI. MSCI, Barra, RiskMetrics, ISS, CFRA, FEA, EAFE, Aegis, Cosmos, BarraOne, and all other MSCI product names are the trademarks, registered trademarks, or service marks of MSCI in the United States and other jurisdictions. The Global Industry Classification Standard (GICS) was developed by and is the exclusive property of MSCI and Standard & Poor’s. “Global Industry Classification Standard (GICS)” is a service mark of MSCI and Standard & Poor’s.

About MSCI Inc. MSCI Inc. is a leading provider of investment decision support tools to investors globally, including asset managers, banks, hedge funds and pension funds. MSCI products and services include indices, portfolio risk and performance analytics, and governance tools. The company’s flagship product offerings are: the MSCI indices which include over 120,000 daily indices covering more than 70 countries; Barra portfolio risk and performance analytics covering global equity and fixed income markets; RiskMetrics market and credit risk analytics; ISS out-sourced proxy research, voting and vote reporting services; CFRA forensic accounting risk research, legal/regulatory risk assessment, and due-diligence; and FEA valuation models and risk management software for the energy and commodities markets. MSCI is headquartered in New York, with research and commercial offices around the world.

© 2010 MSCI. All rights reserved.

RV0610

Rev. 18 Aug 2010

Table of Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 1.1 Intended Audience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 1.3 Feature Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 1.4 Holdings-Based Performance Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 2 Business Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 2.1 Allocation-Selection Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 2.1.1 Attributes Calculated by the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 2.1.2 Asset-Level Attributes and Subportfolio-Level Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 2.1.3 Asset Selection Attribute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 2.1.4 Asset Allocation Attribute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 2.1.5 Interaction Attribute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 2.1.6 Subportfolios with no Benchmark Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 2.1.6.1 Benchmark Subportfolio Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 2.1.7 Overlay Attribute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 2.1.8 Special Rule for Subportfolios with Zero Portfolio Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 2.1.9 Compounding the Attributes over Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 2.1.9.1 Notional Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 2.1.9.2 Multiple-Period Allocation-Selection Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 2.1.9.3 Formal Statement of the Pro Rata Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 2.1.10 Summary of Allocation-Selection Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 2.2 Asset Selection Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 2.2.1 Asset Selection Attribute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 2.2.2 Unlisted Attribute. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18 2.2.3 Single-Day Asset Selection Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 2.2.4 Compounding the Attributes over Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 2.2.4.1 Multiple-Day Asset Selection Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 2.2.4.2 Formal Statement of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22

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2.3 Fixed Income Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 2.3.1 Introduction: The Key Requirement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 2.3.2 Term Structure Attribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 2.3.2.1 Two Possible Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24 2.3.3 Single Term Structure Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26 2.3.3.1 Return from each of the key rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26 2.3.3.2 Return from parallel and non-parallel movements . . . . . . . . . . . . . . . . . . . . . . . . . . . .27 2.3.4 Multiple Term Structure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27 2.3.5 Attribution of Spread Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 2.3.5.1 Single-Spread Market Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28 2.3.5.2 Multiple-Spread Market Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30 2.4 Arithmetic and Geometric Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30 2.4.1 Adjustment of Single-Period Geometric Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 2.5 Currency Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 2.6 Contribution Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 2.6.1 Single Day . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36 2.6.2 Multiple Days . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37 2.6.3 Compounding of Arithmetic Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37 2.6.4 Compounding of Geometric Contributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 2.6.5 Contribution Data that BarraOne Performance Produces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 2.7 Ex Post Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 2.7.1 Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 2.7.1.1 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 2.7.1.2 Sample Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 2.7.1.3 Covariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40 2.7.1.4 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41 2.7.2 Beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41 2.7.3 Portfolio Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42 2.7.4 Tracking Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42 2.7.5 Information Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 2.8 Report Column Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 2.8.1 Allocation-Selection Attribution and Asset Selection Attribution Reports . . . . . . . . . . . . . . .43 2.8.2 Fixed Income Attribution Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

BARRAONE PERFORMANCE II

Table of Contents

3 Factor-Based Performance Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 3.1.1 Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 3.2.1 Asset Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 3.2.2 Asset Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46 3.3 Factor-Based Performance Attribution Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49 3.3.1 Hybrid Brinson-Factor Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49 3.3.1.1 Benchmark Group Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54 3.3.1.2 Portfolio Group Return. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55 3.3.1.3 Active Group Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56 3.3.1.4 Selection Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57 3.3.2 Factor-Based Drilldown into Selection Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59 3.3.2.1 Factor Return Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59 3.3.2.2 Trading Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61 3.3.3 Currency Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61 3.3.3.1 Aggregation by Currency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61 3.4 Multiple-Period Linking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 3.4.1 Logarithmic Linking Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 3.4.2 Multiple-Period Cumulative Views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64 3.4.3 Multiple-Period Annualized Views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66 3.4.4 Linking Mixed-Period Factor Returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67 3.5 Realized Risk and Risk-Adjusted Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68 3.5.1 Information Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68 3.5.2 t Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69 3.5.3 Portfolio Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70 3.5.4 Beta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71 3.5.5 Tracking Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71 3.6 Fixed Income Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71 3.7 Report Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72 4 Annotated Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79

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1 Introduction 1.1 Intended Audience This document is intended primarily for performance analysts who will be responsible for using BarraOne Performance to calculate and report on the sources of the performance of investment portfolios. However, this document will also provide useful information to portfolio managers, back office personnel, marketing staff, and other people who use performance information or portfolio data. Additionally, the material in this document will provide useful background material for software developers and other IT personnel who will be dealing with BarraOne Performance.

1.2 Overview BarraOne Performance provides a means for BarraOne clients to analyze the sources of return for their portfolios in the following report types: 1

Allocation-Selection Attribution (explains active return)—along with Contribution of subportfolios and assets within these subportfolios (explains total portfolio return)

2

Asset Selection Attribution (explains active return)—along with Contribution (explains total portfolio return) of assets in the portfolio

3

Fixed Income (Single Market and Multiple Market)

4

Factor Attribution (equity only)

1.3 Feature Summary BarraOne Performance offers the following features: •

Enables BarraOne users to produce customized performance analysis reports in batch mode (scheduled or unscheduled)

•

Supports the following types of analysis:

•

•

Attribution, explaining portfolio active return vs. its benchmark, and

•

Contribution: explaining how different parts (assets, subportfolios) of the portfolio contributed to total portfolio return.

Supports attribution for all asset classes supported in BarraOne and covered in the BIM factor models. However, the ability to cover any particular asset class depends on the availability of daily returns for those assets. User-supplied returns are supported for any assets for which Barra does not provide daily returns. Mutual funds, hedge funds, and ETFs are treated as single positions.

ANALYTICS GUIDE Intended Audience

Introduction

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, Notes: •

Any returns attributable to dividends or coupon payments must be accounted for in the returns supplied in the attribution process. Barra-supplied equity returns will be corporate-action adjusted. The user must, however, adjust holdings accordingly (e.g., after 21 split). Most index vendor-reported asset level total returns include corporate actions, such as dividends or coupon payments. If the client is providing returns for fixed income instruments, the return contribution of a coupon payment should be included in the return supplied.

•

Returns on cash securities are due only to exchange rate returns. Cash assets are assumed to have zero local return.

BarraOne Performance offers the following attribution models: •

An allocation-selection model (equity Brinson model) that explains active portfolio return in terms of allocation to subportfolios, and selection of assets within each subportfolio based on daily asset-level returns.

•

An asset-level model that measures asset selection from active bottom-up asset positions.

•

A hybrid fixed income attribution model that explains active return due to interest rate changes using yield curve key rates, and active return due to spreads, using a Brinson model that is modified for fixed income.

1.4 Holdings-Based Performance Attribution BarraOne Performance uses a holdings-based methodology. In holdings-based performance attribution, users provide portfolio snapshots with daily or any lesser frequency. Monthly or weekly snapshots are most common. Users load in the number of units held. User-supplied returns are supported for any assets for which Barra does not provide daily returns. The calculations in BarraOne Performance use daily data, though clients do not need to load daily portfolios (the portfolio holdings are assumed to remain constant until a new list of holdings is loaded). On days where portfolio snapshots are provided, the market values for the holdings are computed from the number of shares multiplied by the price. If holdings are not loaded on a given day, then they are assumed constant, and new prices on that day are used to compute market value. In holdings-based performance attribution, the portfolio-level and subportfolio-level returns are computed by summing contributions from holdings, where each contribution is computed from the weight and the return for the holding.

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BARRAONE PERFORMANCE Introduction

Holdings-Based Performance Attribution

where: R(t1, t) = the total portfolio-level or subportfolio-level return for day t MV(t1) = the total portfolio-level or subportfolio-level closing market value on day t1 mvi(t1) = the closing market value of the ith holding on day t1 ri(t1, t) = the return of the ith holding for day t

2 Business Logic This section explains the business logic that BarraOne Performance implements. It includes all of the important definitions and formulas that determine how BarraOne Performance calculations work.

2.1 Allocation-Selection Attribution Allocation-selection attribution decomposes the active return of a portfolio into subportfolios and assets within these subportfolios. The subportfolios can be based on any asset-level attribute so that it reflects the allocation decisions in your investment process. Some examples of subportfolio grouping are Region, Country, GICS Sector, Region-Country, Country-GICS Sector, GICS Sector-GICS Industry, Country-Asset Class, etc. Below each of the lowest elements in this subportfolio tree, you would find each of the assets that comprise that subportfolio. By convention, individual assets are always classified into the lowest elements of the subportfolio tree, and are never classified into an “intermediate-level” subportfolio. Other conceivable subportfolio trees include: •

The MSCI World Index decomposed first into industries and then into countries;

•

The MSCI World Index decomposed into regions, then industries; or

•

A domestic equity index broken into “analyst groupings” using a custom attribute, where each group contains the assets that a particular analyst has been assigned to monitor.

The possible ways to break down a portfolio are limited only by your imagination. You can use either the System-level asset attributes or user-supplied asset attributes for grouping purposes.

ANALYTICS GUIDE Allocation-Selection Attribution

Business Logic

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BarraOne Performance’s allocation-selection attribution model is—at its core—the well-known Brinson model1. There are a couple of minor elaborations to the model: The first is an alternative calculation, known as the overlay attribute, that is used for subportfolios in which a portfolio invests even though the subportfolios are not part of its benchmark. The second is the way BarraOne Performance adjusts multiple-period attribution results so the attributes will sum to the active return. However, these elaborations are extensions to the model, and they do not change its fundamental character. 2.1.1 Attributes Calculated by the Model BarraOne Performance explains the active return contributed from each subportfolio using the following attributes: 1

Subportfolio Allocation: the value added by overweighting subportfolios that outperform the benchmark, or underweighting subportfolios that underperform the benchmark;

2

Asset Selection: the value added due to a subportfolio in the portfolio outperforming the subportfolio in its benchmark (this is essentially a bottom-up source of added value, since it arises from selecting individual securities within each subportfolio); and

3

Interaction: this measures the inevitable interaction between asset selection and subportfolio allocation.

The interaction attribute is sometimes considered problematic. Its meaning is not as intuitive as the meaning of the other attributes. To avoid the difficulty of explaining what interaction means, some firms have traditionally hidden the interaction term by adding it to asset selection or asset allocation. If the investment process involves a top-down decision process (subportfolio allocation) followed by a bottomup decision process (asset selection within each subportfolio), it is inevitable that there will be some interaction between the decisions. The interaction term measures an effect that occurs when a portfolio is managed through both top-down and bottom-up decisions. 2.1.2 Asset-Level Attributes and Subportfolio-Level Attributes The allocation-selection attribution model focuses on explaining the active return due to subportfolio bets, but it also explains the active return due to asset bets within each subportfolio. This intra-subportfolio return is explained by the asset selection attribute. This attribute sums from the asset level to the subportfolio level, and then up to the portfolio level. Hence, it is considered a “bottom-up” attribute. On the other hand, BarraOne Performance calculates the subportfolio allocation attribute and the interaction attribute only at the subportfolio level. Hence, they are considered “top-down” attributes. More precisely, subportfolio allocation is a “top-down” attribute, and the interaction attribute measures the active return generated by the interaction between top-down subportfolio allocation and the bottom-up attribute.

1. See Brinson and Fachler 1985.

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BARRAONE PERFORMANCE Business Logic

Allocation-Selection Attribution

2.1.3 Asset Selection Attribute The asset selection attribute for a subportfolio is essentially: •

the benchmark weight for the subportfolio; multiplied by

•

the extent to which the portfolio subportfolio outperforms the benchmark subportfolio.

Intuitively, this attribute takes the outperformance of a portfolio subportfolio and weights it according to the benchmark. This therefore shows the value added by the portfolio manager, on the assumption that the asset allocation decisions are neutral. At the subportfolio level, BarraOne Performance defines the daily asset selection attribute for the ith subportfolio in portfolio p between two consecutive days t1 and t as:

where:

At the asset level, BarraOne Performance defines the daily asset selection attribute for the jth asset in the ith subportfolio of portfolio p between two consecutive days t1 and t as:

where:

2.1.4 Asset Allocation Attribute The asset allocation attribute for a subportfolio is essentially: •

the overweighting of the subportfolio; multiplied by

•

the extent to which the subportfolio’s benchmark return outperforms the total portfolio benchmark return.


Business Logic

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Intuitively, it is easy to see that this attribute depends only on benchmark returns rather than on actual portfolio returns. Therefore, this attribute is measuring the asset allocator’s skill at predicting benchmark subportfolio returns, not their skill at predicting portfolio subportfolio returns. This distinction will also be important in understanding what the interaction attribute means. BarraOne Performance defines the daily asset allocation attribute for the ith subportfolio in portfolio p between two consecutive days t1 and t as:

where:

2.1.5 Interaction Attribute The interaction attribute for a subportfolio is essentially: •

the overweighting of the subportfolio; multiplied by

•

the extent to which the subportfolio outperforms its benchmark return.

This term is called “interaction,” because: •

on one hand, the asset allocator determines the overweighting of each subportfolio;

•

on the other hand, the subportfolio manager determines each subportfolio’s performance relative to its benchmark.

BarraOne Performance defines the daily interaction attribute for the ith subportfolio in portfolio p between two consecutive days t1 and t as:

where:

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The interaction attribute is sometimes controversial, because people do not always find it intuitive. Indeed, many firms have chosen over the years to avoid the problem of explaining the interaction term by somehow “getting rid of it.” This is usually achieved by adding it to asset selection or to subportfolio allocation, or perhaps by somehow prorating it across these two attributes. However, a number of experts recommend that you separately calculate and disclose the interaction attribute. The chief reasons for this are: •

the interaction term can be very informative per se. See Laker (2000) for a discussion of how to interpret the interaction term.

•

a user of performance reports can treat the interaction however the user wishes, if it is disclosed separately. However, once the interaction term has been added to one of the other attributes, there is no way for anyone reading a performance report to obtain a separate number for interaction.

There are some possible objective justifications for adding interaction to one of the other terms in certain specific cases. For example: •

if you expect the asset selectors in each subportfolio to be cognizant of the active weight for their subportfolio, it would be reasonable to add the interaction term to asset selection; and

•

if you expect the asset allocator to bet not just on the benchmark performance of different subportfolios, but also on their active performance, it would be reasonable to add the interaction to the asset allocation term.

While these justifications might arguably be applicable in certain circumstances, they will not always be applicable. This is why BarraOne Performance provides the interaction attribute separately. If you wish to add interaction to asset selection term, BarraOne Performance enables you to do this in your report (in which case it is combined daily before the multiple-period adjustment is applied). Also, if you want to avoid the interaction term, bear in mind that BarraOne Performance’s asset selection attribution model does not have an interaction term, since it is a purely bottom-up attribution model. 2.1.6 Subportfolios with no Benchmark Return When a portfolio invests in a subportfolio that does not form part of its benchmark, questions arise about how to treat that subportfolio in an attribution analysis: 1

What benchmark return should you assume for a subportfolio that is not in the portfolio’s benchmark? This is one particular question for which answer is not obvious.

2

When a subportfolio is not in the portfolio’s benchmark, does it make sense just to “plug in” the usual equations for performance attribution?

3

Will the usual attributes (asset allocation, asset selection, and interaction) make sense in the case of a subportfolio that is not in the portfolio’s benchmark? This is the most important question from the perspective of the people who will be using the attribution analysis.

The answers to these questions are not independent of one another. For example, if the choice of benchmark return is arbitrary, then the values of the attributes will vary accordingly, which might possibly affect the overall interpretation of the attribution analysis.


Business Logic

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2.1.6.1 Benchmark Subportfolio Return There are two approaches to attributing the active performance from a portfolio investing in a non-benchmark subportfolio: 1

using a known benchmark return for the subportfolio; or

2

treating the non-benchmark investment as a tactical departure from the benchmark asset classes into a potentially wide range of instruments for which there is no clear subportfolio benchmark return (often this will be the case for tactical asset allocation overlay investments).

To implement the first approach using BarraOne Performance, you can specify a benchmark for the subportfolio with a weight of zero. This permits BarraOne Performance to determine the benchmark return for that subportfolio, and since none of the other inputs to the attribution equations is absent, it is possible to calculate sensible attributes for the subportfolio with a zero benchmark weight. The second approach is where BarraOne Performance’s concept of calculating an overlay attribute arises. This happens when the portfolio invests in a subportfolio that is not mentioned in the benchmark specification. In such a case, BarraOne Performance knows that the benchmark weight for that subportfolio must be zero, but it does not know the benchmark subportfolio return1. Note that BarraOne Performance has to do something in the case where a portfolio invests in a subportfolio for which you have specified no benchmark, and some of the possibilities (for example, refusing to do attribution at all, or arbitrarily assigning a benchmark return of zero) are undesirable. Assume that a non-benchmark investment has been in a subportfolio with a known benchmark. The calculation for the subportfolio not in the benchmark uses the same method as the other subportfolios that are in the benchmark. In accordance with the standard equations, the asset selection for a subportfolio with zero benchmark weight is inevitably zero. Apart from that, there is nothing special about the results for the subportfolio not in the benchmark. However, what happens to the analysis when BarraOne Performance does not know the benchmark return for the subportfolio2? First, the sum of the attributes for the subportfolio will be the same, no matter what benchmark return you choose to select. You can prove this using algebra, or more easily, by plugging in arbitrary benchmark returns for the subportfolio in the portfolio. Second, the sum of the attributes for the subportfolio will be distributed differently between asset allocation and interaction depending on the subportfolio benchmark return that you choose (again, you can prove this by plugging in different subportfolio benchmark returns). This demonstrates that choosing an arbitrary subportfolio benchmark return is not desirable. 1. It follows from elementary arithmetic that the subportfolio benchmark return is irrelevant to the portfolio's benchmark return when the subportfolio benchmark weight is zero. Zero multiplied by anything is zero. 2. This might happen either because you have not gone to the trouble of spelling out which benchmark will apply to this subportfolio (with a zero weight), or because the subportfolio is so broad that it does not have any single meaningful benchmark, such as when the subportfolio is used for a tactical asset allocation overlay with very wide discretion.

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BarraOne Performance resolves this dilemma by separating the active performance for the subportfolio and expressing it as a completely different attribute, known as “overlay.” BarraOne then reclassifies the overlay attribute as asset allocation and adds it to the allocation attribute after the multiple-period adjustment is made. Assume that you have not included the non-benchmark subportfolio as part of the benchmark specification, and therefore BarraOne Performance has no way of knowing what the benchmark return for that subportfolio is. The actual calculations (internal to BarraOne Performance) work the same. The sole difference to the core calculation is that when BarraOne Performance is treating a subportfolio as an overlay subportfolio, it assumes that the benchmark subportfolio return is by definition the total benchmark return. Therefore, the core calculation is only slightly different. The only difference is that the “trick” of defining the benchmark subportfolio return as being equal to the benchmark portfolio return has guaranteed that the active performance for the subportfolio flows through only to the interaction attribute, which BarraOne Performance then “reclassifies” as the overlay attribute. , Note: BarraOne Performance displays “N/A” for the benchmark subportfolio return, despite the fact that it has assumed a value for its own internal calculation purposes. Why bother with the “trick” of assuming that the benchmark subportfolio return is equal to the benchmark portfolio return, when the total active performance for this subportfolio will be the same no matter what benchmark subportfolio return is assumed? The answer concerns multiple-period attribution. When BarraOne Performance deals with the error arising from the cross-product terms in a multiple-period attribution, it uses two notional portfolios. One notional portfolio return is calculated as the sum product of portfolio returns and benchmark weights, while another notional portfolio return is calculated as the sum product of benchmark returns and portfolio weights. Therefore, BarraOne Performance has to decide what benchmark return to plug in for an overlay subportfolio. By consistently assuming that the benchmark subportfolio return is equal to the benchmark portfolio return for overlay subportfolios, BarraOne Performance simultaneously obtains a single attribute for each overlay subportfolio and computes a notional portfolio return that is consistent with that attribute. Therefore, when BarraOne Performance is dealing with the multiple-period problem, it can combine the overlay attributes with the interaction attributes and obtain a result that is completely consistent and preserves the integrity of each different attribute. As discussed above, BarraOne Performance’s allocation-selection attribution model automatically separates into the overlay attribute the active performance due to investment in a subportfolio that is not mentioned in the benchmark specification. 2.1.7 Overlay Attribute The definitions above for asset allocation, asset selection, and interaction all apply to non-overlay subportfolios. However, for an overlay subportfolio, those attributes will be by definition zero, and the active return due to the subportfolio will appear only in the overlay attribute.


Business Logic

9

While the overlay attribute is calculated separately from the other attributes, BarraOne Performance adds the overlay attribute into asset allocation for reporting purposes. For overlay subportfolios, BarraOne Performance makes the following assumptions: •

the subportfolio benchmark weight is zero; and

•

the subportfolio benchmark return is equal to the total portfolio benchmark return.

Therefore, from an intuitive standpoint, you can see that a long position in an overlay subportfolio that outperforms the portfolio benchmark (or a short position in an overlay subportfolio that underperforms the portfolio benchmark) will result in a positive attribute. This seems to be a natural way of analyzing the performance of tactical asset allocation decisions. BarraOne Performance defines the daily overlay attribute for the ith subportfolio in portfolio p between two consecutive days t1 and t as:

where:

You can see how the overlay attribute fits with the broader mathematical framework that BarraOne Performance uses for allocation-selection attribution. If it is assumed (as BarraOne Performance assumes for and , the mathematical consequences for the “nonoverlay subportfolios) that overlay” attributes are as follows:

and

and

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Thus, asset allocation and asset selection become zero, and the active return due to this subportfolio is explained solely by the interaction term. Since it is unintuitive (but mathematically correct) to call this the interaction term, BarraOne Performance instead reclassifies it as the overlay attribute. As mentioned above, BarraOne Performance presents the subportfolio overlay attribute as an asset allocation effect in performance reports. The overlay term is added to allocation after the multiple-period adjustment. , Note: When the portfolio invests in a benchmark subportfolio, the overlay attribute that is computed for assets within the subportfolio that have zero benchmark weight is presented as an asset selection effect in performance reports. The overlay term in this case is added to asset selection after the multiple-period adjustment. 2.1.8 Special Rule for Subportfolios with Zero Portfolio Weight One special case is how to treat a subportfolio where the portfolio weight is zero (i.e., there are no holdings in the subportfolio on that particular day). This special case poses no problem for overlay subportfolios, since a glance at the algebra will reveal that the overlay attribute will always be zero when the portfolio weight is zero. However, for non-overlay subportfolios, BarraOne Performance has to find a sensible way to deal with the fact that there is no portfolio subportfolio return when the subportfolio weight is zero1. A quick review of the definition for each attribute shows that the asset selection and interaction terms both depend on the portfolio subportfolio return. As it turns out, a very sensible rule in this case is to assume that the portfolio subportfolio return is equal to the ). When that assumption is benchmark subportfolio return (in other words, assume that applied to the equations for each attribute, the following results are obtained for asset selection and interaction:

and

This makes sense intuitively, since how could any value other than zero be assigned to the asset selection attribute when no asset selection had taken place? Moreover, if asset selection is zero, it makes sense for the interaction term to be zero. As a result, the active return for a subportfolio whose weight is zero drops into the asset allocation attribute, where it belongs. After all, the decision to have no holding in a subportfolio is entirely an asset allocation decision.

1. If this sounds a bit theoretical and mathematical, think of a practical example. If Portfolio XYZ has no holding in the equity subportfolio, what is its return for that subportfolio? Moreover, what will be the consequences of using that return in attribution calculations? The answer is not immediately obvious.


Business Logic

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The last few pages have described the work BarraOne Performance does to calculate the attributes for a single day. For multiple-period attributions (i.e., attributions over periods exceeding one day), BarraOne Performance has to do more work. 2.1.9 Compounding the Attributes over Time BarraOne Performance uses a method sometimes called the “notional portfolio” method for linking returns over multiple periods. The idea is that, at the portfolio level, there is an exact value for the total allocation, selection, and interaction effects. The subportfolio attribution effects are adjusted so that they sum to the exact portfolio-level values. On any single day, the attributes that BarraOne Performance calculates will be equal to the active return of the portfolio or subportfolio that you are analyzing. However, one of the classic problems in performance attribution is how to calculate the results over multiple periods (in BarraOne Performance, a period is one day). The essence of the problem is that, while the attributes may sum to the active return over any single period, they generally do not sum to the active return once they are compounded over multiple periods. This discrepancy is sometimes called the “cross-product term.” There is a precise method for determining the total asset allocation, asset selection, and interaction over any period, no matter how long. The next section describes how BarraOne Performance uses that method. 2.1.9.1 Notional Portfolios There is an elegant solution to the cross-products problem, in which the solution focuses on the following dimensions: 1

whether benchmark weights or portfolio weights are used; and

2

whether benchmark returns or portfolio returns are used.

By combining these two separate dimensions, four different portfolios can be described, as shown below:

Portfolio Subportfolio Returns Benchmark Subportfolio Returns Portfolio Subportfolio Weights Benchmark Subportfolio Weights

A

B

Portfolio

Active AA Portfolio

C

D

Active SS Portfolio

Benchmark

1

Returns for portfolio A are calculated as the sum product of portfolio subportfolio weights and portfolio subportfolio returns. This is the portfolio return.

2

Portfolio B is a notional portfolio in which the weights are dictated by the portfolio weights, but in each subportfolio it obtains the benchmark return. This notional portfolio reveals the performance of the asset allocator.

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3

Portfolio C is a notional portfolio that holds every subportfolio at benchmark weight, but which obtains the portfolio subportfolio return in each subportfolio. This notional portfolio reveals the performance of the subportfolio managers.

4

Finally, the returns for portfolio D are the sum product of benchmark subportfolio returns and benchmark subportfolio weights. In other words, this is the benchmark performance.

BarraOne Performance already has daily returns for portfolios A and D, since these are the portfolio and benchmark total returns. Therefore, it needs only to compute daily returns for the notional portfolios B and C. Mathematically, the returns for portfolios A, B, C, and D over any single-day period can be defined as:

Each of these returns is for the total portfolio, not for an individual subportfolio. Of the four sets of returns, two are “real” (the portfolio return and the benchmark return), while the other two are notional portfolios. These notional portfolios are useful, since they enable the calculation of an answer for each attribute, even when doing the attribution over a long period, where the cross-product term could be expected to harm the results. This is done as follows: First, obtain returns for these notional portfolios A, B, C, and D over any period by compounding the daily returns. Algebraically, the returns for each portfolio over the period between day t and day tk (where k 1) are:


Business Logic

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Second, use these compounded notional portfolio returns over any period to calculate a total for each attribute (at the portfolio level) over the period, in accordance with these formulas:

It is easy to see intuitively how each of these equations provides an answer over any period: •

The asset selection is the extent to which the “Active Asset Selection” portfolio outperformed the benchmark. Recall that the “Active Asset Selection” portfolio holds every subportfolio at the benchmark weight (and therefore by definition has a neutral asset allocation), but achieves the portfolio subportfolio returns.

•

The asset allocation is the extent to which the “Active Asset Allocation” portfolio outperformed the benchmark.

•

The interaction term is not as amenable to an intuitive interpretation. Perhaps the easiest way to see that this term makes sense is by noting that, in the following equation, the sum of the attributes adds up to the active return. This tends to confirm that the interaction term measures the portion of active performance that cannot reasonably be attributed either to asset allocation or to asset selection.

Although this process looks complicated, it is simple to show that these attributes sum to the active return over any period. For the sake of brevity, the period (tk, t) is omitted from each variable, and the total active return is expressed as the sum of the portfolio-level attributes:

This calculation has therefore provided a measurement of the total asset selection, asset allocation, and interaction for a period of any length. The beauty of this method is that it provides values for each attribute (at the total portfolio level), and that these attributes reconcile perfectly with the active return, without the need for any adjustment to achieve reconciliation. However, to obtain the attributes over the period for each subportfolio, a limited amount of adjustment is required. Given the values for the total portfolio-level asset selection, asset allocation, and interaction, the problem is how reasonably to obtain attributes for each subportfolio that will sum to these values. The method that BarraOne Performance uses is to prorate the discrepancy between the precise values and the compounded “raw” attributes.

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2.1.9.2 Multiple-Period Allocation-Selection Attribution Armed with the answers for the total value-added attributable to asset selection, asset allocation, and the interaction between them, all that remains is to prorate for each attribute the discrepancy between the total of the compounded subportfolio attributes and the precise totals that were calculated. The following section provides a formal statement of the algorithm that BarraOne Performance uses to allocate these adjustments across the subportfolios. The method is to prorate the discrepancy, based on the absolute value of each subportfolio’s “raw” compounded attribute. The method of prorating the discrepancy based on the absolute value of each subportfolio’s attribute—while admittedly arbitrary—seems to do little damage to the integrity of the subportfolio attributes. It is hard to find any case where this method leads to pathological results. 2.1.9.3 Formal Statement of the Pro Rata Calculation The following formally states the method that BarraOne Performance uses to arrive at each subportfolio’s attributes in a multiple-period attribution. First, compound the calculated daily subportfolio attributes to obtain the “raw” compounded attributes for each subportfolio:

Second, calculate for each attribute the discrepancy between the precise result and the sum of the “raw” compounded subportfolio results. These discrepancies are known as the “cross-product terms.” The crossproduct term for each attribute over the period from day tk to day t (where t>0) is:


Business Logic

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The third and final step is to compute for each subportfolio an adjusted value for asset selection, asset allocation, and interaction by prorating the cross-product term:

, Note: When a subportfolio in an allocation-selection report contains only non-benchmark assets, the contribution to active return in the attribution is assumed to be an allocation effect. For these assets, the only security-level attribution effect is currency. 2.1.10 Summary of Allocation-Selection Attribution In summary, the allocation-selection attribution algorithm that BarraOne Performance uses is complicated, and it would not be practical to implement manually using a spreadsheet. Fortunately, however, the algorithm is already implemented for you by BarraOne Performance. The principal advantage of this algorithm is that it produces answers for the total value added by asset selection, asset allocation, and interaction over periods of any length. While the interaction term is not always intuitive, it provides an accurate measurement of a phenomenon that is unavoidable in a top-down investment process. If you want to attribute the portfolio’s performance in accordance with a bottom-up process, the best solution will be to use an asset selection attribution.

2.2 Asset Selection Attribution Asset selection attribution decomposes the performance of a portfolio according to each individual asset that the portfolio holds. The active return for each asset is allocated across the following attributes: 1

Asset Selection: the active return due to overweighting assets that beat the total benchmark return, or underweighting assets that underperform the total benchmark.

2

Unlisted: the active return due to investing in assets that are not in the benchmark. The definition for “assets that are not in the benchmark” extends to anything the portfolio holds that is not in the benchmark. This can include holdings of cash in an equity portfolio, derivatives, assets that are too small to be included

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Asset Selection Attribution

in the benchmark, or assets that are in another benchmark. Therefore, the term “unlisted” can be a little confusing. It is worth remembering that the precise definition of this attribute includes any holding that is not in the benchmark (i.e., that does not have a non-zero weight in the benchmark) on any given day. , Notes: •

BarraOne Performance combines the asset selection attribute with the unlisted attribute into a single Selection attribute for purposes of reporting. Unlisted is added after the multiple-period adjustment is made.

•

BarraOne Performance does not calculate a timing attribute, because it uses a holdings-based rather than a transaction-based approach to performance; thus, effects due to intra-day trading are not captured.

•

There is no interaction term for asset selection attribution, since the attributes are mutually exclusive and do not interact.

On any given day, either an asset will have a non-zero weight in the benchmark or it will not. Therefore, on each day, an asset will either produce an attribute for asset selection (when it is in the benchmark), or an attribute for unlisted (when it is not). What assets will appear in an asset-level attribution for a given portfolio? On any particular day, the set of assets in an asset-level attribution analysis for the portfolio will comprise the union of two sets: 1

the set of assets that are held in the portfolio; and

2

the set of assets that are in the in the portfolio benchmark.

In other words, an asset will be included in the analysis when either it is held in the portfolio or it is in the portfolio benchmark. When an asset is not in the benchmark, then that means that the benchmark weight is zero. Similarly, when an asset is not held in the portfolio, the portfolio weight will be zero. As you can imagine, for portfolios measured against the more popular market benchmarks, the number of lines in an asset selection attribution quickly becomes hundreds and hundreds. This can initially seem a bit unintuitive, especially for concentrated portfolios that hold only one or two dozen assets. However, it does reveal a fundamental aspect of the asset selection attribution paradigm: the decisions about which assets not to hold are just as much active decisions as the decisions about which assets to hold. This is really the key concept behind attribution analysis in general. Every deviation from the benchmark is an active decision, since the benchmark is by definition the neutral position. It is fascinating to observe the sources of excess return in actively managed equity portfolios: frequently, the biggest bets are the assets that the portfolio does not hold at all. 2.2.1 Asset Selection Attribute The asset selection attribute measures, for assets that are in the benchmark, the active performance due to overweighting assets that outperform the total benchmark return, or underweighting assets that underperform the total benchmark return.

ANALYTICS GUIDE Asset Selection Attribution

Business Logic

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This attribute does not depend at all on the portfolio asset return. BarraOne Performance defines the daily asset selection attribute for the jth asset in portfolio p (only for assets that are in the benchmark on day t) between two consecutive days t1 and t as:

where:

The interpretation of this attribute is intuitive. If the portfolio is 1% overweight asset XYZ on a day when the total benchmark return is 2% and the benchmark asset return for asset XYZ is 3%, then the asset selection attribute for asset XYZ on that day will be 1%(3%2%)=1 basis point. As always with asset selection attribution, this seems like a small performance effect to measure. However, each basis point cumulates over every asset and every day to explain the total active return. When an asset is not in the benchmark, BarraOne Performance assigns a value of zero to this attribute and instead computes the “unlisted” attribute. The reason for this is that asset-level returns are unlikely to be available for an asset that is not in the benchmark. 2.2.2 Unlisted Attribute This attribute measures the total contribution to active performance resulting from a holding in an asset that is not in the benchmark. Since the asset is not in the benchmark, it follows by definition that its weight in the benchmark is zero (i.e., ). Furthermore, there will be no benchmark asset return for an asset that is not in the benchmark. This can be dealt with by making the reasonable assumption that the benchmark return for ). This assumption has the practical this asset will be the total benchmark return (i.e., effect of assigning positive value-added to a holding in a non-benchmark asset if and only if that asset outperforms the benchmark. This makes sense intuitively. Mathematically, these two assumptions have an interesting effect when they are applied to the equation for asset selection. The effect on the asset selection equation is:

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This seems to be a sensible formula for assessing the contribution to active return from investing in a nonbenchmark asset. The value added by this holding is the portfolio weight for the holding multiplied by the extent to which the holding has outperformed the overall benchmark. The formula for the unlisted attribute is shown below. BarraOne Performance defines the daily unlisted attribute for the jth asset in portfolio p (only for assets that are not in the benchmark on day t) between two consecutive days t1 and t as:

where:

Therefore, there is no need for any special mathematics to calculate the unlisted attribute. If the benchmark asset return is assumed equal to the overall benchmark return, the equations inevitably produce a zero asset selection effect that provides the desired result. BarraOne computes the unlisted attribute for all assets in the portfolio that are not part of the benchmark index. This unlisted attribute is then added to the asset selection attribute for the asset after the multiple-period adjustment. Thus, BarraOne does not display the unlisted attribute as a separate item in the reports; rather, it rolls this attribute into the asset selection attribute in the reports. 2.2.3 Single-Day Asset Selection Attribution Note that the total benchmark return is the sum product of the asset-level benchmark returns and benchmark weights, and that the total portfolio return is the sum product of the asset-level returns and portfolio weights. Those relationships are essential for making the attribution algorithm work. The attributes sum to the active return (i.e., the difference between the portfolio subportfolio return and the benchmark subportfolio return), as they should if there are no errors in the data. For an asset not held in the benchmark, its asset selection attribute is by definition zero. Its contribution to performance comes out solely in the unlisted attribute. For all assets in the benchmark, their unlisted attributes are by definition zero, and their contribution to performance comes out in the asset selection attribute.


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2.2.4 Compounding the Attributes over Time The previous sections have explained how BarraOne Performance does an asset selection attribution on any single day. However, it is rare for people to be interested in looking at an attribution analysis over a single day. Usually, people are interested in looking at an attribution analysis for a period of several months, or even several years. A notorious problem makes it difficult to link a number of single-period attribution analyses together to obtain an attribution report over a longer period. This is known as the “cross-products” problem. The easiest way to explain this problem is by compounding some daily attribution results over a few days. On each single day, the attributes sum to explain the active return. However, the active return over the threeday period (calculated by compounding the benchmark and portfolio returns, then taking the difference of these compounded returns) will differ from the sum of the compounded asset selection attribute. The reason is a matter of arithmetic. On any single day, the asset selection attribute equals the active return. However, when compounded over time, they are no longer equal. For example, 2 + 4 = 6, but if each of the terms is squared, the result is 4 + 16 = 36, which obviously is not correct. Fortunately, for typical investment returns, the error is not as large as this. However, it is an error that cannot be ignored. If the error is 4.86 basis points over three days, it is easy to imagine how it could grow to hundreds of basis points over a period of years. The following section explains the method that BarraOne Performance uses to ensure that the overall value for asset selection is correct, no matter how long the analysis period. 2.2.4.1 Multiple-Day Asset Selection Attribution While there is no method for determining values for each of the asset-level attributes, there is an method for calculating total asset selection (on one hand) and total unlisted (on the other). This method depends on the following conceptualization of portfolio and benchmark returns:

Portfolio Asset Weights Benchmark Asset Weights

Portfolio Asset Returns

Benchmark Asset Returns

A

B

Portfolio

Active Asset Selection Portfolio D Benchmark

The portfolios A, B, and D are notional combinations of portfolio or benchmark weights with portfolio or benchmark returns. However, if the calculations have been done correctly, A will happen to correspond with the actual portfolio return (the sum product of portfolio asset weights and portfolio asset returns), and D will happen to correspond with the actual benchmark return (the sum product of benchmark asset weights and benchmark asset returns).

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Notional portfolio B is the key to resolving the cross-products problem. Since BarraOne Performance already has returns for portfolio A (the portfolio return) and portfolio D (the benchmark return) the only extra data that BarraOne Performance needs to compute for resolving the cross-products problem is the return for notional portfolio B. This can be considered to be an “Active Asset Selection Portfolio.” It holds each asset at portfolio weight, but gains the benchmark return on it. Therefore, it is a pure measure of the active performance due to asset selection. Note that the “special rule” for unlisted asset applies equally to calculating the return for this notional portfolio. You will recall (as explained in “Unlisted Attribute” on page 18) that for assets that are not in the benchmark, BarraOne Performance’s asset selection attribution assumes that the benchmark asset return is equal to the total benchmark return. This works out nicely in calculating the performance of this “Active Asset Selection Portfolio,” since any non-benchmark asset will therefore be assumed to have a portfolio asset return equal to the total benchmark return, which means it has no impact on the asset selection attribute (recall that a nonbenchmark asset by definition has an asset attribute of zero; its contribution goes into the unlisted attribute). If this is not intuitive, consider the extreme case where a portfolio held 100% non-benchmark assets. For each asset, it would be assumed that the portfolio asset return was equal to the total benchmark return. Therefore, the sum product of the portfolio asset weights and the (assumed) benchmark asset returns would therefore inevitably be equal to the total benchmark return. This would suggest in turn that the active performance of the “Active Asset Selection Portfolio” had been zero, which meshes neatly with the definition that asset selection is always zero for non-benchmark assets. As the above discussion has already suggested, a precise value can be determined for the total asset selection over any period as the performance difference between portfolio B and the benchmark over that period. A value for unlisted can also be determined. This amount is the difference between the compound portfolio return and the compound return for portfolio B. Summarizing the progress so far, it has been determined that the total value added by asset selection over the period and the total value added by unlisted over the period. If these answers are expressed to full precision, they reconcile with the actual active return over the period. From this point onward, the calculation ceases to be exact, and instead adopts a pragmatic approximation. BarraOne Performance prorates away the discrepancy over the attributes. In this approximation, BarraOne Performance weights each attribute based on its absolute value. The difference between these values must be prorated across the calculated asset selection attribution for each asset. Similarly, in the case of unlisted, the difference1 between these values must be prorated across the calculated unlisted for each asset. To summarize, the total value for asset selection and the total for unlisted are exact answers. The method for prorating the cross-product error is arbitrary, but it is hard to think of any case in which it would significantly distort the results. 1. Note: The numbers do not reconcile at this degree of precision due to rounding errors.


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2.2.4.2 Formal Statement of the Algorithm All that remains is to state the pro rata calculation more formally. This section does not introduce any new concepts, so the reader who lacks an appetite for algebra can safely skip ahead. The previous section defined the notional portfolios that are required for calculating totals for the asset selection attribution attributes over any arbitrary period between days tk and day t. Portfolio A is the total portfolio return (for either the subportfolio or the portfolio, depending on whether the subportfolio or the portfolio is attributed), and portfolio D is the total benchmark return (for either the subportfolio or the portfolio, depending on whether the subportfolio or the portfolio is attributed). Portfolio B is the sum product of the portfolio asset weights and the benchmark asset returns. Mathematically:

The previous section defined the total asset selection over any period as: •

the return for notional portfolio B; minus

•

the return for notional portfolio D (the benchmark).

It also defined the total unlisted as: •

the return for notional portfolio A (the portfolio); minus

•

the return for notional portfolio B.

In algebraic notation, this definition becomes:

It is easy to show that these attributes sum to the active return:

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To get the raw compounded attributes over the period from day tk to day t, the daily attributes over that period are compounded:

Then the following are determined: •

the total cross-product term for asset selection; and

•

the total cross-product term for unlisted

by taking the difference between the total attributes and these raw compounded attributes. The total cross-product term for asset selection is:

The total cross-product term for unlisted is:

To bring the raw compounded attributes into line with the known totals for asset selection and unlisted, the cross-product terms are prorated over the raw attributes, using the absolute value of each attribute as the weight. Therefore the adjusted asset selection attribute for asset j between day tk and day t is:

Finally, the adjusted unlisted attribute for asset j between day tk and day t is:


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2.3 Fixed Income Attribution 2.3.1 Introduction: The Key Requirement BarraOne Performance can attribute the active return of portfolios containing any type of asset, as long as it has asset holdings and returns. BarraOne Performance can perform this attribution using several variants of the Brinson methodology, explaining active return as the result of subportfolio allocation and asset selection decisions. While this methodology is very popular for attributing the return of equity portfolios, it suffers from a serious shortcoming when it comes to the attribution of fixed income portfolios. Because so much of the return for bonds comes from term structure changes, most Brinson attributions of bond portfolios are really about differences in term structure exposures between subportfolios or securities, rather than about subportfolio allocation or asset selection decisions. The key requirement for fixed income attribution is the ability to isolate the return due to changes in the term structure. One potential way around this problem within the constraints of the allocation/selection framework is to use duration buckets as our subportfolios. The resulting allocation effect can be interpreted as the return from term structure decisions. A recursive attribution structure can even be created with industry or rating buckets within each duration bucket. If this is done, allocation effects can be calculated for each of these within-durationbucket subportfolio buckets, as well as asset selection effects within them. If these second-level effects are rescaled appropriately, they explain the selection effects associated with the top-level duration buckets. This approach may prove satisfactory for some fixed income attribution needs, as it represents one way of isolating the return impact of term structure changes. However, one of the main reasons for using performance attribution is to provide feedback on the investment process. In order to do that, the attribution must explain active return in a manner consistent with the investment process. If the point is to assess which decisions worked out and which ones did not, then the attribution needs to be in the same language as the decisions themselves. While many investment managers do find it useful to analyze the distribution of portfolio and benchmark value across duration buckets, it is much more common to manage term structure risk at the aggregate portfolio level, by managing exposure to term structure “factors.” In this case, duration bucket allocation effects are not a good proxy for evaluating term structure management decisions. In the next section, the approach implemented in BarraOne Performance is explored, an approach more consistent with most fixed income investment management processes. 2.3.2 Term Structure Attribution 2.3.2.1 Two Possible Approaches Factor-based term structure attribution requires answers to two questions: •

Which factors should be used?

•

Which method should be used to quantify asset returns from term structure changes?

As explained below, the first is an issue that is left to the BarraOne Performance client. For the second, it is necessary to choose between valuation-based and exposure-based methods.

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Fixed Income Attribution

Valuation Method On one hand, a valuation model could be used to quantify the impact on asset value of the change in each term structure factor. Typically, this is done by changing the initial term structure in a series of steps, each of which captures the impact on the term structure of a particular factor change. For example, if a shift/twist/butterfly term structure model were used, the period’s shift would be applied to the initial term structure, the twist to the shifted term structure, the butterfly to the shifted/twisted term structure, and finally, the shifted/twisted/ butterflied term structure would be changed to the actual, ending term structure. The securities would be revalued using each of the successive term structures, and by comparing that value with the value at the prior step, the return due to each term structure factor would be quantified. In addition, the difference between the valuations with the ending and beginning term structures gives the total return due to term structure changes. Exposure Method On the other hand, asset exposures to some set of term structure factors, along with period returns to those factors, could be used to calculate asset returns from the factors. If the set of factor returns completely explains the term structure changes, then this method also enables each asset’s total term structure return to be quantified. An example of this approach is to view each asset’s total term structure return as equal to:

where: KRD = the key rate durations of the asset at each maturity t, representing its price sensitivity to a change in the key rate for maturity t

rt = the changes for each key rate of maturity t Each of the methods has advantages and disadvantages, some of which are described in the table below:

Valuation Method

Exposure Method

Processing Speed

Requires multiple valuations for each asset; computationally demanding and time consuming

Leverages exposure calculations already required for portfolio management purposes, in order to estimate returns quickly

Accuracy

Does not assume linear response to factor returns

Assumes linear response to factor returns: potential for inaccuracy in cases of extreme rate movements and/or assets with large degree of embedded optionality

Flexibility

Providing factor flexibility is difficult

Starting from KRDs facilitates flexibility in how results are reported

ANALYTICS GUIDE Fixed Income Attribution

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Given the large number of assets in most bond indices and the complexity of valuation for many bonds (e.g., structured products, bonds with embedded optionality), processing speed is an important consideration. This is especially true given the time pressures under which performance departments already operate. While the accuracy issue may come into play in some extreme cases, our testing shows that, for typical situations, the term structure returns calculated with the exposure method are virtually identical to those calculated with the valuation method. This should not be surprising, since key rate durations capture the first order impact of non-linearity of asset response. Note also that, since BarraOne Performance’s attribution links daily results, the potential for extreme rate movements is minimized. Finally, it is important to keep in mind the purpose of the whole exercise. It is not bond rich-cheap analysis; it is merely an attempt to provide a reasonable estimate of how much of each bond’s total return came from term structure changes. After all, even the results from the valuation method are estimates dependent on parameters of the valuation model. In the next section, different ways are shown in which term structure attribution results can be presented when starting from returns to key rate durations. Experience has shown that offering this flexibility of presentation is critical, because different firms, and even different managers within the same firm, look at term structure dynamics in different ways. To cite just one example, a set of factors suitable for analyzing a portfolio whose benchmark includes bonds of all maturities may not be the right set of factors for a portfolio whose benchmark includes bonds of 1-year to 5-years maturity. Because of these considerations, the fixed income attribution model in BarraOne Performance adopts the exposure method for term structure attribution. The processing time it saves and the flexibility it facilitates more than compensate for the possibility of asset-level inaccuracies in extreme cases. 2.3.3 Single Term Structure Analysis So far, the discussion has been about how to isolate the portion of each asset’s return resulting from term structure changes. The point of doing this is to combine these returns for all the assets in the portfolio and the benchmark, in order to quantify how much active portfolio return came from term structure management decisions. BarraOne Performance offers the following ways of presenting these aggregate results: •

Show the return from each of the key rates

•

Decompose term structure return into portions due to parallel and non-parallel term structure movements

2.3.3.1 Return from each of the key rates The portfolio or benchmark returns from a specific key rate, rt, is rtKRDt . The portfolio or benchmark total term structure return is the sum of the returns from all of the individual key rates, and the active return is the difference between the portfolio and benchmark returns. Since rt (the change in the key rate) is portfolio independent (it can be viewed as the return to the factor rt ), differences in returns between the portfolio and the benchmark will be due solely to differences in key rate durations.

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2.3.3.2 Return from parallel and non-parallel movements Rather than managing exposures to individual key rates, many managers prefer to manage exposures to aggregate term structure changes. Typically, this translates into managing exposure to changes in the overall level of rates, on one hand, and to other term structure movements, on the other. In performance attribution terms, this translates into a desire to separate the return due to parallel rate movements from that due to nonparallel movements. Starting from the definition of each asset’s total term structure return:

a parallel term structure movement can be defined as one where all rt have the same value, rpar. BarraOne Performance defines parallel movement using the following equation to calculate rpar, given any set of rt, where the weights, wt, are equal:

Given rpar, the return from parallel term structure changes can be calculated in the same way that total return from term structure changes is calculated. Each asset’s return from non-parallel movements is the difference between its total term structure return and its return from the parallel movement. 2.3.4 Multiple Term Structure Analysis The section above specifies term structure attribution for those cases where the analysis involves only a single term structure. However, in many cases, the analysis will involve multiple term structures. In particular, there are two types of situations when this will be the case: •

multiple-currency analysis, where each currency market has its own term structure, and

•

single-currency analysis, where the user views different portions of the portfolio as being priced off of different curves.

In these cases, most users will still want to see results for each term structure market on a standalone basis, as described in “Single Term Structure Analysis” on page 26. At the same time, BarraOne Performance provides a Summary Term Structure Return view, which pulls together the effects of the various term structure markets. BarraOne Performance rolls up the term structure returns calculated for each subportfolio on a standalone basis to the total portfolio level. This is done in each term structure market by multiplying every individual component of portfolio, benchmark, and active return by the weight of that term structure market in the portfolio. The re-weighted within-market active term structure return is equal to the combined selection and interaction effects for a Brinson attribution performed on the total portfolio’s term structure return.


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2.3.5 Attribution of Spread Return 2.3.5.1 Single-Spread Market Analysis Having isolated each asset’s term structure return, its spread return has also been isolated: it is that which is left over after subtracting term structure return from total return. The sources of active spread then need to be explained. In general, the portfolio’s active spread is explained by quantifying the impact of allocation and selection decisions and the interactions between the two. Allocation decisions are decisions to distribute the portfolio’s exposure to spread effects between the categories that represent the choices within a given spread market; selection decisions are decisions made within a framework of allocation decisions. It might seem natural to continue with a factor-based approach here, similar to the approach used to explain term structure return. For example, a set of factors, such as those associated with different subportfolios and/or ratings, could be predefined. Then, asset exposures to the factors (often as dummy variables) would be defined, and returns to the factors estimated (perhaps by estimating the spreads associated with the different factors at different dates, through a regression on asset prices, using a valuation model). Given the asset exposures to the factors and the factor returns, the asset returns could be quantified from the factors. Alternatively, this could be done by successively revaluing the portfolio as the spreads are changed from their beginning to their ending values. After taking into account the returns from all of the factors in the model, there will undoubtedly be a remainder of unexplained return. This remainder term could be called “return due to asset selection,” or “trading return,” or “specific return.” There are a couple of problems with this approach. The first one is its inflexibility. The factors are predefined and may not fit all investment processes. For example, it is unlikely that a set of factors developed for an aggregate portfolio will include factors specialized enough to explain the decisions of a mortgage specialist. The second problem has to do with its dependence on a valuation model. The return left over after exhausting all the model factors can be called anything, but the truth of the matter is that it is a residual that indicates the relationship between the valuation model and the market, in addition to indicating manager skill. It is not possible to separate these two effects. In addition, in the world of fixed income, where active returns are relatively small, having a significant portion attributed to what is in effect a residual is not very satisfying. To address these shortcomings, active spread return is explained with a modified Brinson model. While all bond managers need to contend with the dominant impact of interest rates, they can take many different paths to add value over and above what they add through term structure management. The beauty of the Brinson model is its flexibility: it enables you to explain active return based on allocation of portfolio value to whatever categories are relevant to the decision process. In addition, the asset selection attribute in the Brinson method does not depend on a valuation model; it is the result of differing returns within a category between the portfolio and its benchmark. At the same time, the Brinson model was devised for equities. It relies on value (i.e., weights) as the measure of exposure to a particular class of assets. In many fixed income contexts, however, value alone is an inadequate measure of exposure: the duration of the assets in the category also needs to be taken into account. A manager who is overweight a subportfolio relative to the manager’s benchmark may actually have much less active exposure than appears at first sight, if the assets the manager holds are shorter than the benchmark’s; the

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manager could even be underexposed. This insight leads to the modification of the Brinson method as specified in the formulas in the table below. The basic idea behind the modified method is that wherever the Brinson method uses weight, the modified method uses contribution to spread duration (weight times spread duration); wherever the Brinson method uses return, the modified method uses return per unit of spread duration. Remember, of course, that the return in question here is not total return, but only spread return. Because of the modification of the formulas, an additional attribution effect is required so that the sum of effects across categories will equal the active spread return. This is the Market Exposure Effect, shown in the first row of the table. It measures active spread return due to differences in overall exposure to a market.

Effect

Classic Brinson

Modified Brinson

Market Exposure N/A

Allocation

Asset Selection

Interaction

Asset Selection + Interaction

In the table, w’s are weights, r’s are spread returns, and d’s are spread durations. Lowercase letters are category values; uppercase values are total portfolio/benchmark values. The p subscript designates values for the portfolio; the b subscript designates values for the benchmark.) , Note: BarraOne Performance calculates the selection and interaction attributes separately. However, BarraOne Performance spread attribution reports combine interaction with selection after the multipleperiod adjustment is applied.


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2.3.5.2 Multiple-Spread Market Analysis In many cases, clients want to view the spread return as coming from multiple spread markets and want to understand the impact of decisions within each of those spread markets. Here are some examples: •

A multiple-currency/term structure manager wants to analyze the sources of spread return within each currency/term structure market by analyzing the spread returns in terms of each market’s set of broad asset classes. In this case, the term structure market for each asset is identical with its spread market, but that is not necessarily always true.

•

A single-currency/term structure manager wants to analyze the sources of spread return within each of several broad asset classes. In this investigation, the manager may want to use one set of categories for corporate bonds, another set for MBS, and may not want to explore the spread return from government bonds in detail. In this case, a single term structure market includes multiple spread markets.

•

A single-currency manager chooses to view different asset classes as priced off of different term structures. (An asset’s term structure cannot be defined by means of its currency.) The manager may view treasuries and agencies as priced off a treasury curve, corporates and MBS off a swap curve, munis off a muni curve, and inflation-protected bonds off a real curve. For purposes of spread attribution, the manager may not investigate the spread return due to treasuries, agencies, or IPBs any further, while using different category schemes for corporates and MBS (even though both asset classes are associated with the same term structure for purposes of term structure attribution), as well as munis. (A one-to-one relationship between an asset’s term structure market and its spread market is not absolute.)

In many cases, users will want to see results for each spread market on a standalone basis, as described in “Single-Spread Market Analysis” on page 28. BarraOne Performance also provides a Summary Spread Return view, which pulls together the effects of the various spread markets. BarraOne Performance rolls up the spread returns calculated for each subportfolio on a standalone basis to the total portfolio level in the same way that it addresses the multiple term structure problem, i.e., by multiplying every individual component of portfolio, benchmark, and active spread return by the weight of the relevant spread market in the portfolio. The re-weighted, within-market active spread return is equal to the combined selection and interaction effects for a Brinson attribution performed on the total portfolio’s spread return.

2.4 Arithmetic and Geometric Attribution BarraOne Performance supports both arithmetic and geometric definitions of active return. The discussions above of Asset and Allocation-Selection Attribution are based on an arithmetic definition of portfolio active return. Each of those attribution models attempts to explain the sources of Ra as defined in the formula below:

where: Ra = portfolio active return Rp = portfolio total return Rb = benchmark total return

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Arithmetic and Geometric Attribution

Some firms, however, prefer to define active return geometrically, as follows:

The difference between the active return formulas is a good indicator of how the formulas discussed above for single-period arithmetic attribution effects need to change in a geometric framework. The general idea is that, wherever an arithmetic attribution effect is calculated using differences, the corresponding geometric attribution effect is calculated using ratios. To see this more clearly, look at the table below, which puts the formulas for calculating arithmetic effects next to the corresponding geometric formula. In some cases, the arithmetic formulas are expanded, to make it easier to see the parallels with the geometric formulas.

Arithmetic

Geometric (Note: Attribution Effects Require Adjustment)

Active Return

Attribution Allocation-Selection Attribution Allocation (subportfolio level)

or alternatively

Selection (subportfolio level)

or alternatively

Interaction (subportfolio level)

or alternatively

ANALYTICS GUIDE Arithmetic and Geometric Attribution

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Arithmetic

Geometric (Note: Attribution Effects Require Adjustment)

Selection (asset level, rolled up to subportfolio) or alternatively

Interaction (subportfolio-level calculation that uses rolled up asset-level calculations)

Asset Selection Attribution Selection (asset level)

or alternatively

Unlisted (asset level)

or alternatively

where: Ra = Portfolio active return Rp = Portfolio total return Rb = Benchmark total return aai = Allocation effect from subportfolio i

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ssi = Selection effect from subportfolio i (allocation-selection attribution) iai = Interaction effect for subportfolio i ssj,i = Selection effect for asset j in subportfolio i (allocation-selection attribution) ssj = Selection effect for asset j (asset selection attribution) unj = Unlisted effect for asset j (asset selection attribution) w ip = Weight of subportfolio i in portfolio p w ib = Weight of subportfolio i in benchmark b r ib = Return of subportfolio i in benchmark b r ip = Return of subportfolio i in portfolio p w jb,i = Weight of asset j in subportfolio i of benchmark b w jp,i = Weight of asset j in subportfolio i of portfolio p r jb = Return of asset j in benchmark b r jp = Return of asset j in portfolio p w jb = Weight of asset j in benchmark b w jp = Weight of asset j in portfolio p 2.4.1 Adjustment of Single-Period Geometric Effects One of the virtues of arithmetic attribution effects is that the attribution effects for a particular single-period attribution sum to the active return for the single period. Over multiple, linked single periods, however, the compounded single-period attribution effects do not sum to the total portfolio active return, due to crossproduct effects. As noted in the discussions of arithmetic attribution (Asset and Subportfolio), BarraOne Performance has a method for adjusting the raw compounded attribution effects so that they do sum to the multiple-period portfolio active return.


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One of the virtues of geometric attribution effects is that, over multiple periods, the linked single-period effects do roll up (in a geometric way) to the multiple-period portfolio active return. While this is true, less often noted is the need to deal with the cross products that result from geometrically combining attribution effects within a single period. For any attribution effect, the asset allocation effects of the various subportfolios are

because of cross-products between the various attribution effects. (A value for Raa, the portfolio-level allocation effect, can be calculated separately, as described below.) BarraOne Performance’s solution to this problem is analogous to its solution to the cross-product problem encountered in the context of arithmetic attribution. The approach has four steps: •

Calculate values for each of the single-period portfolio-level attribution effects (e.g., allocation, selection, interaction).

•

Combine the subportfolio-level or asset-level effects to get each of the unadjusted, “rolled-up,” singleperiod, portfolio-level attribution effects.

•

Calculate the difference between the exact and the unadjusted portfolio-level attribution effects.

•

Allocate the difference between the two to the asset-level or subportfolio-level effects, as appropriate, in a manner proportional to their absolute value.

BarraOne Performance calculates the values of the single-period, portfolio-level attribution effects by using the returns of the four notional portfolios discussed in the arithmetic context (i.e., the portfolio, the active asset allocation portfolio, the active asset selection portfolio, and the benchmark). As a reminder, the return formulas for the notional portfolios are shown below for the Allocation-Selection Attribution case:

Name

Return Calculation

Portfolio

Active Asset Allocation

Active Asset Selection

Benchmark

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The difference in the geometric context is that the geometric difference between notional portfolio returns is used to calculate portfolio-level attributes. The table below shows the differences between the arithmetic and geometric methods for the Allocation-Selection Attribution case:

Portfolio-Level Effect

Arithmetic

Geometric

Allocation

Selection

Interaction

The geometric difference between these portfolio-level attributes and the values obtained by “rolling up” the subportfolio-level or asset-level attributes are calculated to derive the overall rescaling that needs to be applied across all the components of an attribution effect:

This is the ratio of the total portfolio-level attribution effect to the compounded, raw component attribution effects. The numerator is the attribution effect calculated at the portfolio level, based on notional portfolios. A set of nonnegative exponents is used to divide the rescaling across the component returns:

The sum of these exponents is 1, so that

The intuition is similar to that guiding our adjustment to multiple-period arithmetic attributions: components whose attribution effects have larger absolute values receive a larger portion of the adjustment.


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This approach avoids the arbitrariness of the methods outlined by Bacon and McLaren. It also avoids the problem of attribution effects with a raw value of zero taking on an adjusted non-zero value, which is a problem for the adjustment method proposed by Menchero (which makes ).

2.5 Currency Attribution Currency attribution in BarraOne Performance follows the traditional approach, i.e., the currency effect for a given sector k is calculated as follows:

where: rk = Portfolio return (base or local, as indicated) in sector k Rk = Benchmark return (base or local, as indicated) in sector k wk = Portfolio weight in sector k Wk= Benchmark weight in sector k Since the term in the first parentheses is the portfolio’s active return in terms of base currency, and since the term in the second parentheses is the portfolio’s active return in terms of local currency, the expression also can be expressed as follows:

2.6 Contribution Calculations Contribution data explains the total return (as opposed to the active return) of a portfolio or subportfolio. It does this by decomposing the total return into the contributions made by each subportfolio or asset. It would be possible to conceptualize contribution data as being like attribution data where the benchmark return was zero. However, contribution data is even simpler than that, since nothing about the benchmark—neither the benchmark returns, nor the benchmark weights—is in any way relevant to contribution data. 2.6.1 Single Day The contribution of each asset to the portfolio is its opening weight multiplied by its return. The sum of each asset’s contribution is equal to the total return.

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Currency Attribution

In a single-period case, the contribution calculation is nothing more than the fundamental rule that the total return equals the sum product of the asset weights and asset returns. Each asset’s contribution is the product of the asset weight and the asset return. The total return is then the sum of the contributions. 2.6.2 Multiple Days Even when a contribution calculation is being performed over multiple days, the formula of opening weight multiplied by return will provide an answer for the contribution of each asset to the total return. If a day is divided into two parts, five parts, or ten parts, the answer would still depend purely on the opening weights and the return over the whole day. The total return over any period is the sum of the contributions for each asset. Each asset’s contribution is its opening weight multiplied by its return over the period. Note that if a contribution is calculated on each day, and if the results are then compounded, the daily contributions will generally not compound to agree with the results. This is similar to the “cross-products” problem that exists in performance attribution. The next section contains a discussion of how BarraOne Performance deals with this problem. 2.6.3 Compounding of Arithmetic Contributions This approach uses a method very similar to that used to adjust “raw” attribution effects in the case of multipleperiod arithmetic attributions. The basic idea is as follows: •

BarraOne Performance calculates the contribution of the relevant portfolio components for each day by multiplying weights and returns.

•

BarraOne Performance sums these single-day contributions to get the total portfolio return for the day.

•

For each portfolio component, BarraOne Performance compounds its contributions across the days that make up the analysis period. These compounded values are the “raw” contributions from each portfolio component.

•

BarraOne Performance compounds the single-day total portfolio returns to get the true time-weighted portfolio total return for the analysis period.

•

BarraOne Performance sums the “raw” contributions of all the portfolio components. This yields a number that will differ from the true time-weighted portfolio total return.

•

BarraOne Performance calculates the difference between the summed “raw” contributions and the true time-weighted portfolio total return. A portion of this difference needs to be allocated to each of the “raw” contributions to derive adjusted contributions that will sum to the true time-weighted portfolio total return.

•

Each “raw” contribution’s share of the difference to be allocated is equal to the ratio of its absolute value to the sum of the absolute values of all the “raw” contributions.

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2.6.4 Compounding of Geometric Contributions This approach solves the “cross-product” problem in much the same way that it is addressed by BarraOne Performance in the context of geometric attribution. Unlike the arithmetic approach, which adjusts “raw” contributions derived by compounding multiple single-period contributions, this approach adjusts singleperiod contributions so that they compound (rather than add) to produce the single-period total portfolio return. The resultant adjusted single-period contributions not only compound to produce the total portfolio return, they also compound over time to produce linked multiple-period contributions that themselves compound to produce the portfolio total return for the entire analysis period. The basic idea is as follows: •

BarraOne Performance calculates the unadjusted contribution of the relevant portfolio components for each day, by multiplying weights and returns.

•

BarraOne Performance sums these single-day contributions to get the total portfolio return for the day.

•

BarraOne Performance compounds the unadjusted contributions from the portfolio components to get an “unadjusted” total portfolio return. This number will differ from the true single-day portfolio return.

•

For each day, BarraOne Performance calculates the geometric difference between the true single-day portfolio return and the “unadjusted” total portfolio return. This ratio represents the total amount to be allocated between each of the “unadjusted” contributions. Once this allocation is done, the adjusted contributions will compound to produce the true single-day portfolio return. Formally, the difference to be allocated is shown below:

•

A set of nonnegative exponents is used to divide the rescaling between the unadjusted component contributions:

•

The sum of these exponents is 1, so that

2.6.5 Contribution Data that BarraOne Performance Produces BarraOne Performance can produce the following types of contribution data: 1

Contribution of subportfolios to a portfolio;

2

Contribution of assets to a portfolio; and

3

Contribution of assets to a subportfolio.

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Contribution Calculations

2.7 Ex Post Risk Measures BarraOne Performance calculates the ex post risk measures described in this section. , Note: BarraOne requires a minimum number of return periods to calculate risk measures. “N/A” will appear in the summary table when there are insufficient data for the calculation, such as: •

If monthly return frequency is specified when creating the performance case, risk measures will not be calculated for periods of fewer than six months.

•

For weekly return frequency, risk measures will not be calculated for periods of fewer than six weeks.

•

For daily return frequency, risk measures will not be calculated for periods of fewer than 20 calendar days (i.e., including non-business days).

2.7.1 Building Blocks Variance, standard deviation, and correlation are the building blocks of ex post risk measures. 2.7.1.1 Variance Variance is an essential building block in many statistical calculations. The formula for calculating variance is:

where:

2.7.1.2 Sample Standard Deviation The standard deviation is a risk measure used very commonly in financial markets. The formula for calculating the standard deviation is:

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39

where:

Annualization of Standard Deviation: If t is daily:

If t is weekly:

If t is monthly:

2.7.1.3 Covariance Covariance is a key measure of the relationship between two different series of numbers. The formula for calculating covariance is:

where:

In Microsoft Excel, covariance can be calculated using the worksheet function COVAR. , Note: To calculate the covariance of two time series, they should be of equal length.

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Ex Post Risk Measures

2.7.1.4 Correlation Most people are familiar with the concept of correlation. It is especially important in Modern Portfolio Theory, since the most effective way of reducing risk via diversification is by selecting assets whose returns are as uncorrelated as possible. When two sets of returns are perfectly correlated, their correlation is 1. When they fluctuate independently of one another, their correlation is zero. The formula that Barra Enterprise Performance uses for calculating correlation is:

where:

In Microsoft Excel, correlation can be calculated using the worksheet function CORREL. 2.7.2 Beta Beta has several applications in performance analysis. It is a concept that comes from the Capital Asset Pricing Model (CAPM). One application for Beta is measuring the sensitivity of equities to an index. Under the CAPM, the risk of a equity can be divided into two components: 1

the risk that is correlated with the index (this risk cannot be eliminated by diversification to other equities in the index); and

2

risks that are uncorrelated with the index (these risks can be diversified away).

A typical equity has a Beta of exactly 1. An equity with a Beta of 1 would be expected (other things being equal) to rise and fall in line with the index. An equity with a Beta of 1.5 would be expected to rise and fall by more than the index. For example, if the index rose by 10%, a equity with Beta = 1.5 would be expected to rise (other things being equal) by 15%. Another common application of Beta is measuring the sensitivity of a fund to its benchmark. For example, a fund with Beta < 1 would be expected (other things being equal) to underperform its benchmark in rising markets, and to outperform its benchmark in falling markets. There are different ways of calculating Beta. BarraOne Performance supports Beta calculations based on daily, weekly, and monthly returns. The formula for calculating the Beta of one return series with another is:

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41

where:

Beta does not need to be annualized or rescaled for daily, weekly, monthly return series. 2.7.3 Portfolio Volatility BarraOne Performance calculates Portfolio volatility using the following formula:

For the sake of comparability, BarraOne Performance annualizes all Portfolio Volatility that it calculates. As with the other risk measures, BarraOne Performance can calculate portfolio volatility using daily returns, weekly returns, or monthly returns. 2.7.4 Tracking Error Tracking error is just the standard deviation of a fund’s active returns. To avoid ambiguity, by “active return” we mean a fund’s gross return minus its benchmark return. Intuitively, it is possible to see that perhaps the term “tracking error” arose from indexed funds management, since presumably it might be regarded as an error if an indexed fund’s gross return did not closely match its benchmark return. Practically speaking, in active funds management, tracking error tends to indicate how actively a fund is being managed. It is easy to see that the returns for a fund with high tracking error fluctuate around its benchmark returns more than returns for a fund with low tracking error. BarraOne Performance calculates tracking error in the following way:

where:

For the sake of comparability, BarraOne Performance annualizes all tracking errors that it calculates. As with the other risk measures, BarraOne Performance can calculate tracking error using daily returns, weekly returns, or monthly returns.

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Ex Post Risk Measures

2.7.5 Information Ratio The information ratio measures the active return achieved per unit of active risk (as standard deviation). As the name suggests, the information ratio measures the information content of an active investment process. Being a standardized measure, it allows for an “apples-to-apples” comparison between different periods and between different active managers (or active strategies). Information ratio is frequently interpreted as a measurement of skill in an active fund manager (or indeed even in the case of an index manager). BarraOne Performance calculates the information ratio using this formula:

For information ratios to be comparable, they should be measured over the same period. It is conventional to calculate information ratio on an annualized basis:

2.8 Report Column Definitions This section describes selected columns available in BarraOne Performance Reports. 2.8.1 Allocation-Selection Attribution and Asset Selection Attribution Reports Portfolio Return Base: Portfolio return in terms of the reporting currency specified in the export set. Portfolio Return Local: Portfolio return from the perspective of the asset’s local market. Portfolio Return Is Full Period: At the asset level, indicates if the asset was held in the portfolio for the entire analysis period. At the portfolio level, indicates if the portfolio had assets in the given allocation group for the entire analysis period. Benchmark Return Base: Benchmark return in terms of the reporting currency specified in the export set. Benchmark Return Local: Benchmark return from the perspective of the asset’s local market. Benchmark Return Is Full Period: At the asset level, indicates if the asset was held in the benchmark for the entire analysis period. At the portfolio level, indicates if the benchmark had assets in the given allocation group for the entire analysis period.

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Active Return Base: Active return in terms of the reporting currency specified in the export set. Active Return Local: Active return from the perspective of the asset’s local market. Active Return Local Relative: At the asset level, the difference between an asset’s portfolio return local and total benchmark return local; if the asset is not in the benchmark, the performance report shows N/A. At the allocation level, the difference between the allocation group's benchmark return local and total benchmark return local; for allocations with zero benchmark weight, the difference between the group’s portfolio return local and total benchmark return local. At the asset level within allocation groups, the difference between an asset’s portfolio return local and the allocation group’s benchmark return local; if the asset is in a group with zero benchmark weight, the performance report shows N/A. Contribution Base: Contribution to portfolio return in terms of the reporting currency specified in the export set. Contribution Local: Contribution to portfolio return from the perspective of the asset’s local market. Contribution Currency: Currency contribution to portfolio total return. 2.8.2 Fixed Income Attribution Reports Relative Term Structure Return: The difference between benchmark allocation term structure return and benchmark total term structure return. Parallel: The portion of selection due to parallel curve movements within the market, assuming equal weighting of spot rate changes. Non-Parallel: The portion of selection due to non-parallel curve movements within the market, assuming equal weighting of spot rate changes. Relative Spread Return: The difference between benchmark allocation spread return and benchmark total spread return. For allocation groups with zero benchmark weight, the difference between portfolio allocation spread return and benchmark total spread return.

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Report Column Definitions

3 Factor-Based Performance Attribution 3.1 Introduction Factor-based performance attribution looks at the source of portfolio active returns that are attributed to the the factors to which the portfolio is exposed. This enables users to have a consistent view of their portfolios by looking at risk and returns that are attributed to the same set of sources. Factor attribution in BarraOne s limited only to equity and equity derivative assets; that is, it is limited to assets that are exposed to equity factors. 3.1.1 Reports BarraOne factor-based performance attribution first performs a top-level country allocation breakdown and then explains the factor attribution within each single country. This fits the single-country attribution model to the BIM framework. The factor attribution reports have been designed and placed in three broad categories: Summary Reports: These reports provide a top-level view of the portfolio and benchmark and are designed to provide a full period overview and a trailing period snapshot of the portfolio (MTD, QTD, and YTD information, and monthly and cumulative snapshots). Multiple Market Reports: Multi-market reports provide additional detail about the sources of the portfolio total and active return by showing the contribution from each of the markets to which the portfolio is exposed. Single Market Reports: These reports provide the most granular view of the portfolio by providing factor-level and asset-level detail.

3.2 Background 3.2.1 Asset Positions Consider a portfolio valued with respect to a fixed base currency (the numeraire) holding given positions in assets such as equities, ETF shares, currency forwards, equity index futures, equity options, equity futures, etc. The position can be measured in three different ways: 1

Size: The number of units (shares or contracts) held. Size always measures asset position and is independent of numeraire or any other currency considerations.

2

Value: The value of the asset held denominated in a specified currency, where value is the size of the position is multiplied by the share price. Value measures the position when the unit price is non-zero. In the case of forwards, futures, or swaps, the contract price is often zero; in this case, the value does not measure the contract position, because the value is zero regardless of the number of contracts held.

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Factor-Based Performance Attribution

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3

Weight: The ratio of the value of the asset to the value of the portfolio (also called the base value). Weight measures the position when the portfolio has a non-zero value and the unit price is non-zero. Note that the weights of all assets held in the portfolio sum to one, and that short positions are represented by negative weights.

For the purposes of performance attribution, we require that all asset positions be measurable by weight. 3.2.2 Asset Returns Consider a nonzero-value asset (equity, bond, ETF share, money market account, etc.) that is denominated in some currency. There are several useful notions of the “return” of the asset in a particular period: 1

The local return is the percent change in the value of the asset in the specified currency, which is computed by:

Note that the initial value of the asset is assumed to be nonzero. In the case where the asset is shares of equity, the local return is independent of the number of shares and is given by the percent change in the spot price per share, which is computed by:

2

46

The local excess return is:

BARRAONE PERFORMANCE Factor-Based Performance Attribution

Background

3

The return of the asset denominated in the base currency depends on the exchange rate fluctuations over the investment period. The exchange rate return is:

4

The base currency return is:

Solving the local excess return equation for the local return of asset n and substituting it for the first term above yields:

5

The base currency excess return is given by:

Background

ANALYTICS GUIDE Factor-Based Performance Attribution

47

and substituting the equation for base currency return for the first term yields:

This equation attributes the base currency excess return of the asset to contributions from three terms: the local excess return, the currency excess return, and the crossproduct. 6

The portfolio (base currency buy-and-hold) return is given by:

7

The portfolio (base currency buy-and-hold) excess return (subtracting the base currency risk-free rate from both sides of the equation for portfolio (base currency buy-and-hold) return) is given by:

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Background

Substituting the base currency asset excess returns attribution into this equation, we find the base currency portfolio excess return is attributed to contributions due to local excess returns, currency excess returns, and crossproduct:

, Note: When all assets are denominated in the base currency, then the currency excess returns and the crossproduct are both zero. In this case, this equation reduces to the familiar form:

3.3 Factor-Based Performance Attribution Details 3.3.1 Hybrid Brinson-Factor Attribution In BarraOne, the asset market is partitioned into separate groups such that each asset belongs to one group. For example, we could partition equities by economic sector, by country, or by market capitalization. The only attribute currently used for allocation grouping is Country. Once the groups are declared, a portfolio can be constructed in two ways: 1

Direct investment into assets as denoted by asset weights

2

Allocation into groups as denoted by group weights followed by selection of assets as denoted by asset weights such that:

The Brinson-Fachler attribution is based on the second investment process listed above, and it expresses active return as a sum of contributions due to group allocation and asset selection.

ANALYTICS GUIDE Factor-Based Performance Attribution Details


49

For each group in which the portfolio holds assets, and where the group weight is non-zero, the group local excess return is:

If the portfolio does not hold assets in the group, then the group return is not defined by the previous equation. We adopt the empty-group convention such that:

These conventions will attribute the entire returns contribution of the group to allocation effect. The portfolio local excess return is attributed to group returns by:

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Factor-Based Performance Attribution Details

The active local excess return is attributed to allocation effect and selection effect by:

A drilldown into the active group return:



51

is given, performing a bottom-up, security-level attribution with respect to relative security weights:

To maintain the empty-group convention, if the portfolio has zero weight in a group, then the portfolio relative security weights are declared to be equal to those of the benchmark, and vice versa if the benchmark has weight zero. If both portfolio and benchmark have zero weight in a group, then all relative weights for the group are declared to be zero. The active group return is attributed by:

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The relation between portfolio factor exposures and asset weights is given by:

In a hybrid allocation-factor drilldown attribution scheme, given pure benchmark and portfolio weights, there are a number of different weights, returns, and factor exposures to provide, as described below. Portfolio Return

Benchmark Return

Active Return

Currency Effect

Allocation Effect

Common Factor Return Contribution

Industry Return Contribution

Selection Effect

Specific Return Contribution

Style Return Contribution



53

3.3.1.1 Benchmark Group Return

The benchmark group factor exposures are provided by passing the benchmark relative weights to the equation above, i.e.,

54



3.3.1.2 Portfolio Group Return

The portfolio group factor exposures are provided by passing the benchmark relative weights to the equation above, i.e.,



55

3.3.1.3 Active Group Return

The active group factor exposures are provided by passing the active relative weights to the equation above, i.e.,

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3.3.1.4 Selection Effect There are two alternative methods to selection effect: 1

The group selection effect can be attributed to a rescaling of the attribution of active group return:

This can be restated as follows:

The selection factor exposures are obtained by rescaling the active group factor exposures:



57

2

Alternatively, one can compute portfolio and benchmark asset selection weights:

In terms of selection weights, the group selection effect becomes:

The selection factor exposures are provided by passing the active selection weights to the formula that describes the relation between portfolio factor exposures and asset weights:

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The portfolio and benchmark contributions are computed via the asset selection weights method. In a multiple-period view, all contributions are linked using active linking coefficients. 3.3.2 Factor-Based Drilldown into Selection Effects 3.3.2.1 Factor Return Attribution In the factor model framework, the local excess asset returns for non-cash assets are expressed as the weighted sum of common factor returns and returns that are specific (or “idiosyncratic”) to the assets under consideration:

As shown earlier, the local excess return of a portfolio given by asset weights is:

Cash assets are assumed to have zero local excess returns, so we regard this equation as a sum over non-cash assets. Substituting the equation for local excess asset returns yields:



59

The common factors may be grouped into broad categories. Here we consider a particular grouping scheme: Style factors and Industry factors. In terms of these factor groups, the previous equation can be rewritten as:

This represents the attribution of portfolio local excess returns in terms of style, industry, and specific returns. As shown earlier, the active local excess return of a portfolio is measured relative to a benchmark portfolio:

By applying the equation for factor groups to compute the portfolio and the benchmark returns, we may attribute the active local excess return:

This equation represents the attribution of active local excess returns in terms of style, industry, and specific returns.

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3.3.2.2 Trading Effect While most countries have daily factor returns, some have only monthly factor returns. For countries with monthly factor returns, for the following reports there is a special row called “Trading Effect”: 1) Attribution Summary, 2) Trailing Period, 3) Monthly, and 4) Cumulative. This “monthly trading effect” ensures that the results in each country’s factor reports sum to the daily selection in the multiple market reports and in the “country selection” displayed in each country’s “Attribution View” report. Since most countries have daily factor returns, the monthly trading effect is not required, and the row will not be present. The calculation is provided in “Linking Mixed-Period Factor Returns” on page 67. 3.3.3 Currency Attribution Suppose a portfolio valued in a fixed base currency (the numeraire) is globally invested in securities denominated in various currencies. As described earlier, the base currency portfolio excess return is attributed to contributions due to local market excess returns, currency excess returns, and crossproduct:

The currency attribution is performed by aggregating the currency excess returns and the crossproducts, currency-by-currency, as follows. 3.3.3.1 Aggregation by Currency Define the portfolio exposure to a currency to be:



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It is customary to express the currency exposure in terms of cash weights:

(where the sum is over all cash instruments1 held in the currency) and ex cash weights:

(where the sum is over all assets ex cash denominated in the currency) as:

The attribution of currency excess returns by contributions due to currency groups is given by:

Aggregating the crossproduct by currency, the attribution of the portfolio crossproduct is obtained:

where the local market excess returns contribution in the currency is:

Since cash has local excess return zero, it is customary to write, alternatively:

where the ex ex cash currency group local excess returns are:

1. Here we are regarding a “cash instrument” as any asset with a return completely determined by the risk-free rates and the exchange rates of return. This includes: cash, forward currency contracts (long weight in one currency, short weight in another currency), and probably currency swaps, since these are modeled as a portfolio of forward currency contracts.

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The attribution of active local excess returns by active local excess returns, active currency excess returns, and active crossproduct is:

3.4 Multiple-Period Linking Given a specified range of dates and a fixed single-period attribution methodology applied to each single period, we need to cumulate the single-period attribution effects to give a performance attribution for the entire range of dates. Suppose that the attribution at period given by the single-period methodology is represented by:

, Note: The equation above is not sufficient to determine the linking coefficients. Substituting the first equation above, substituting it into the second, and exchanging the order of summation, we obtain:

where each multiple-period cumulative contribution is given by linking the single period contributions:

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3.4.1 Logarithmic Linking Coefficients Logarithmic linking coefficients are an alternative to optimized coefficients given by:

The log linking coefficients solve the linking equation. However, optimized coefficients are preferred, because log linking coefficients tend to have a greater dispersion than optimized coefficients, and in particular they tend to overweight periods with lower-than-average active return and to underweight those with higher-thanaverage active return. 3.4.2 Multiple-Period Cumulative Views Every single-period returns attribution view has a corresponding multiple-period cumulative view obtained using the following rules: 1

Multiple-period weights and exposures are the average of the corresponding single-period weights and exposures.

2

Multiple-period cumulative returns (i.e., factor returns, specific returns, portfolio group returns) are computed by compounding the single-period returns.

3

Multiple-period cumulative relative returns (i.e., active return, group local excess returns, relative group returns) are computed by applying the formula defining the relative return. For example, multiple-period cumulative active return is equal to multiple-period cumulative portfolio return minus multiple-period cumulative benchmark return. There is an exception1 for portfolio and benchmark local excess returns, detailed below.

4

Multiple-period cumulative effects and contributions are the linking-coefficient-weighted sums of the corresponding single-period effects and contributions. Warning: For active effects and contributions, use active linking coefficients. For portfolio effects and contributions, use portfolio linking coefficients.

1. This is an exception in appearance only. Strictly speaking, local excess return is a returns contribution to base-currency return.

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Multiple-Period Linking

For Step 3, in the case of groups comprising assets denominated in multiple currencies, the compound group return is:

the compound group risk-free return is:

Then the cumulative group excess return is:

The portfolio and benchmark cumulative local excess return are reported as:

, Notes: •

Note the use of benchmark and portfolio linking coefficients, rather than active linking coefficients. They ensure that the cumulative values of portfolio local returns and benchmark local returns, quoted in the Brinson view, agree with the values quoted in the non-active attributions of portfolio return and benchmark return. A side effect is that the difference of cumulative local returns will not equal the cumulative active return. This is acceptable, since portfolio local returns and benchmark local returns are standalone quantities and not active return attribution effects: Linking of active return attributions is applied only to active return attribution effects. See the comment below about the failure of singleperiod algebraic relations in multiple-period cumulative views.

•

The algebraic relations that hold in a single-period view will not hold in a multiple-period cumulative view. In particular, the multiple-period cumulative effects cannot be determined from the average weights and cumulative returns using the formulas computing the singe-period effects from the weights and returns. The reason is that averaging and compounding lose sequencing information: a high weight and a high return followed by a low weight and a low return result in a different effect than a high weight and a low return followed by a low weight and a high return. However, each of these sequences results in the same average weight and compound return.



65

3.4.3 Multiple-Period Annualized Views The goal of annualization is to arrive at a standard measure of return that can be used to compare performance across analysis periods of differing lengths. Every single-period return attribution view has a corresponding multiple-period annualized view obtained using the following rules: 1

Multiple-period weights and exposures are equal to those in the multiple-period cumulative view.

2

Multiple-period annualized returns (i.e., portfolio local returns, benchmark local returns, factor returns, specific returns, portfolio group returns) are computed from those in the multiple-period cumulative view by:

3

Multiple-period annualized relative returns (i.e., local excess returns, active return, relative group returns) are computed by applying the formula defining the relative return. For example, multiple-period annualized active return is equal to multiple-period annualized portfolio return minus multiple-period annualized benchmark return.

4

Multiple-period annualized effects and contributions are computed from those in the multiple-period cumulative view by rescaling by the annualization ratio:

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Multiple-Period Linking

3.4.4 Linking Mixed-Period Factor Returns Suppose group 1 and group 2 both have daily asset returns, but group 1 has only daily factor returns and group 2 has only monthly factor returns. For the analysis period, we compute the linked Brinson attribution of daily buy-and-hold returns:

Compute a drill-down into the selection effect for market 2 as follows. First determine the nearest monthly analysis period: For example, if the analysis period is Oct. 3 through Nov. 23, then the nearest monthly analysis period is Oct. 1 through Nov. 301. For the nearest monthly analysis period, we compute the linked Brinson attribution of monthly buy-and-hold returns. Hence, we may write the linked daily selection effect in market 2 as a sum of the linked monthly buyand-hold selection effect and the linked intra-month trading effect:

1. If the analysis period includes days past the 15th day of the current month, then the nearest monthly analysis period must be taken to end on the last day of the previous month. This is due to the fact that the current month has no published monthly factor returns.



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3.5 Realized Risk and Risk-Adjusted Return 3.5.1 Information Ratio Definition The standalone information ratio of a random variable is defined to be the mean divided by the volatility:

The annualized standalone information ratio of attribution effect is estimated by:

Define the annualized standalone volatility of the multiple-period annualized contribution for an attribute computed via multiple-period linking by:1

1. Model risk forecasts are quoted on an annualized basis.

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Realized Risk and Risk-Adjusted Return

3.5.2 t Statistic The t-stat is a measure of the statistical significance of a sample-based estimate of the mean of a normally distributed random variable:

In particular, it measures the distance from the estimate to 0 as a number of estimated standard deviations of the estimate. Thus, if the t-stat of the estimate is 2 or more, then the mean is 0 with probability less than 5%1. When the variable is observable, then the standard error of the estimate is itself estimated by the realized standard deviation of the variable over the square root of the number of samples:

We use the t-stat formula above to estimate the standard error of the estimate, even when the random variable is not observable. Combing the two equations above, we have that the t-stat is estimated by:

We want to quote a t-stat for each return contribution as a means of gauging the statistical significance of a non-zero attribution effect. In this case, the random variable in question is the single-period attribution effect, and so the equation above becomes:

1. This is only an approximation to the true probability, because the standard error is only a sample-based estimate of the standard deviation of the sample mean. See the refinement (7-11) on p. 372 of Johnson and Wichern that accounts for this use of the standard error. This shows that the t-stat test is correct for practical purposes.



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where the information ratio is quoted on a single-period basis. The relation between a single-period information ratio and an annualized information ratio is:

Substituting the previous equation into the one before, and applying the annualized information ratio estimate, produces the estimated t-stat for an attribute effect:

3.5.3 Portfolio Volatility The goal of ex post volatility attribution is to attribute to the return sources the realized volatility of a portfolio over a period:

The realized portfolio volatility is given by the standard deviation of the time series of realized returns over the entire period:

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The portfolio volatility in the equation above can be attributed to its sources using the following equation:

The equation above is quite general. For instance, it could be used to attribute realized tracking error:

3.5.4 Beta For information on Beta, refer to “Beta” on page 41. 3.5.5 Tracking Error For information on tracking error, refer to “Tracking Error” on page 42.

3.6 Fixed Income Assets Fixed income asset and derivatives are supported in factor-based attribution, but with certain assumptions: •

The contribution from fixed income assets (in total portfolio return space) is computed and properly supported.

•

The attribution of fixed income assets to factors is not supported, and the only source of total/active return is “Asset Selection.” In other words, the contribution from a fixed income asset is attributed to “Asset Selection,” with no contribution from common factors. This active return is the excess (over the risk-free rate) active return.

•

For assets that have both equity and fixed income exposures (convertible bonds, mutual funds, composites, ETFs), the fixed income factor exposures will be ignored in the attribution calculation.

Fixed Income Assets


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3.7 Report Definitions •

Active compound group return = compound portfolio local excess return - compound benchmark local excess return

•

Active contribution (factor drilldown) = active exposure  factor return

•

Active contribution (equity drilldown) = active weight  specific return

•

Active contribution (top-level decomposition summary view) = portfolio contribution - benchmark contribution

•

Active exposure = portfolio exposure - benchmark exposure

•

Active relative weight = portfolio relative weight - benchmark relative weight

•

Active group factor exposure = portfolio group factor exposure - benchmark group factor exposure

•

Active group return = portfolio local excess return - benchmark local excess return

•

Active selection weight = portfolio group weight  active relative weight

•

Active weight = portfolio weight - benchmark weight

•

Allocation effect = active weight  relative group return

•

Asset factor exposure = exposure of asset n to factor l (from the asset factor exposure matrix)

•

Asset group factor exposure contribution = active relative weight  asset factor exposure =

•

Average active weight = average portfolio weight - average benchmark weight

•

Average benchmark weight =

•

Average portfolio weight =

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Report Definitions

•

Benchmark local excess return =

•

Benchmark relative weight =

•

Benchmark group factor exposure =

•

Benchmark weight =

ANALYTICS GUIDE Report Definitions


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•

Compound benchmark local return =

•

Compound benchmark local excess return = compound benchmark local return - compound benchmark risk-free return

•

Compound benchmark risk-free return =

•

Compound portfolio local return =

•

Compound portfolio local excess return = compound portfolio local return - compound portfolio risk-free return

•

Compound portfolio risk-free return =

•

Currency excess return =

•

Currency relative return = currency excess return - benchmark currency excess returns

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Report Definitions

•

Exchange rate return =

•

Information Ratio contribution = returns contribution  standalone volatility =

•

Linked allocation effect =

•

Linked selection effect =

•

Linked total effect = linked allocation effect + linked selection effect

•

Portfolio contribution (factor drilldown) = portfolio exposure  factor return

•

Portfolio contribution (equity drilldown) = portfolio weight  specific return



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•

Portfolio local excess return =

•

Portfolio relative weight =

x Pjn  relative security weight of asset n in portfolio P with respect to group weight of group j in portfolio P x Pjn 

wnP when W jP  0 P Wj

x Pjn 

wnB when W jP  0 B Wj

wnP  asset weight of asset n in portfolio P W jP  group weight of group j in portfolio P wnB  asset weight of asset n in benchmark B W jB  group weight of group j in benchmark B •

Portfolio group factor exposure =

•

Portfolio weight =

•

Relative compound group return = compound benchmark local excess return - benchmark cumulative local excess return

76


Report Definitions

•

Relative group return = benchmark group local excess return - benchmark local excess return

•

Relative group return (cash) = - (benchmark local excess return)

•

Returns contribution =

•

Risk contribution =

•

Risk free return = risk-free currency k return of asset n

•

Selection contribution (factor contribution) = selection factor exposure  factor return =

•

Selection contribution (specific return contribution) = active selection weight  specific return

•

Selection effect = portfolio weight  active group return

•

Selection factor exposure =



77

•

Selection factor exposure contribution =

•

Standalone Information Ratio = returns contribution  standalone volatility =

•

Standalone volatility =

•

T-stat =

•

Total (allocation and selection effect) = allocation + selection

78


Report Definitions

4 Annotated Bibliography In recent years, the volume of publications on topics concerning portfolio performance attribution has started to grow rapidly. This bibliography is by no means comprehensive. However, it will guide you to some of the most important ideas in the field. Brinson, Gary P. and Nimrod Fachler, “Measuring Non-U.S. Equity Portfolio Performance,” Journal of Portfolio Management, Spring 1985 pp. 73-76. This is the classic work on allocation-selection portfolio attribution. Davies, O. and Damien Laker, “Multiple-Period Performance Attribution Using the Brinson Model,” Journal of Performance Measurement, Fall 2001. Davis, Ben and Jose Menchero. “Beyond Brinson: Establishing the Link Between Sector and Factor Models,” MSCI Barra Research Insight, April 2010. Laker, Damien, “Implementing Daily Asset-level Attribution: A Case Study,” Journal of Performance Measurement, Winter 2000/2001. Laker, Damien, “Incorporating Transaction Cost Measurement into Performance Attribution,” Journal of Performance Measurement, Summer 2001. Laker, Damien, “What is this Thing Called Interaction?” Journal of Performance Measurement, Fall 2000. Menchero, Jose and Ben Davis. “Multi-Currency Perfromance Attribution,” Journal of Performance Measurement, Fall 2009.


Annotated Bibliography

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Barraone Performance Analytics Guide

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