SG Handbook Inflation 2008

Quantitative Strategy

Inflation Market Handbook

January 2008

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(44) 20 7676 7707

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Table of Contents

History.....................................................................................................................................................9 Volumes ............................. ............................... ............................... ................................ ..................... 13 Market participants ........................... ............................... ............................... .............................. ...... 14

Introduction..........................................................................................................................................18 How to measure inflation? .................................................. ................................................................ ................................... 18 Introducing real interest rates ................................................................ .................................................................... ............21

Calculation of indices..........................................................................................................................22 US CPI .......................................................... ................................................................. ........................................................ 22 Euro HICP ........................................................ .............................................................. ........................................................ 24 French CPI ( Indice Indice des prix à la consommation, IPC ) ....................................................... ..................................................... 27 UK RPI (Retail Price Index).....................................................................................................................................................27 Further information ...................................................... ............................................................ .............................................. 28

Definition ............................... .............................. ............................... .................................. ................ 31 Measurement ........................... .............................. ............................... ............................... ................ 32 Case study............................................................................................................................................36 Seasonality in the euro zone..................................................................................................................................................36 US seasonality .......................................................... ................................................................ ............................................. 38

Overview...............................................................................................................................................41 From inflation bonds to inflation swaps .............................................................. ................................................................. ..41 From inflation inflation swaps to inflation volatility ..................................................................... ......................................................... 43

Inflation-linked bonds ............................... ................................ ................................ .......................... 45 Product Mechanism...............................................................................................................................................................45 Description and conventions...........................................................................................................................................45 Lag and indexation ....................................................... ................................................................ ................................... 47 Key pricing and valuation concepts.......................................................................................................................................48

Invoice price and quotation .......................................................... ...................................................................... .............48 Linkers yield, inflation breakeven.....................................................................................................................................49 Risk premium...................................................................................................................................................................50 Duration and beta............................................................................................................................................................51 Carry and forward price...................................................................................................................................................54

Inflation Swaps .................................................................................................................................... 58 Real, inflation and standard swap markets............................................................................................................................58 Inflation and real swaps: characteristics and mechanisms....................................................................................................59 Zero coupon swaps.........................................................................................................................................................59 Year-on-Year inflation swaps .................................................. ............................................................... .........................62 Real swaps ............................................................. ........................................................... .............................................. 63 Building a CPI forward curve ........................................................ ................................................................. ........................65

Inflation-linked asset swaps...............................................................................................................70 Asset swaps definitions ........................................................ .............................................................. ................................... 70 Par/par and proceeds asset swaps.................................................................................................................................70 Accreting asset swaps ........................................................... ................................................................. ........................74 Early redemption asset swaps.........................................................................................................................................75 Another asset swap measure for bonds: Z-spread .......................................................... ............................................... 75

Inflation-linked options ....................................................................................................................... 78 Standard options ............................................................. .................................................................... .................................. 78 Inflation zero coupon caps and floors ................................................................... .......................................................... 78 Inflation year-on-year caps and floors.............................................................................................................................79 Real rate swaptions ........................................................... ...................................................................... ........................80 Strategies with caps and floors ............................................................. ................................................................... .............81

Inflation-linked futures ........................................................................................................................ 83 CME future.............................................................................................................................................................................83 Eurex future............................................................................................................................................................................85

Background to Pricing Models...........................................................................................................87 Foreign Currency Analogy .................................................................................................................. 89 Market Models ..................................................................................................................................... 92 Short-Rate Models .............................................................................................................................. 95

Why another model?..............................................................................................................................................................95 Model definition ....................................................... ................................................................. ............................................. 95 A possible improvement: inflation ratio as a state variable........................................................................... .........................97

Which model for which purpose? ...................................................................................................... 99

20Y EUR revenue swap ..................................................................................................................... 102 10Y EUR Livret A swap......................................................................................................................103 10Y EUR TFR swap............................................................................................................................104 10Y EUR swap spread France/Europe ............................................................................................ 105 10Y EUR swap switch (spread France/Europe) .............................................................................. 106 5Y EUR range accrual ....................................................................................................................... 107 10Y EUR swap corridor.....................................................................................................................108 20Y EUR Kheops................................................................................................................................109 10Y EUR HICP index-linked leverage slope .................................................................................... 110 Hybrid inflation/rate performance swap (HIRPS) ........................................................................... 111 20Y EUR Hybrid performance swap ................................................................................................ 112

The combined effects of international prices and demography have made inflation a growing concern in modern economies. Oil and commodities prices are being pushed up by global growth and the development of emerging countries, as demand for energy and agricultural resources increases. The symbolic $100 threshold for a barrel of Brent was breached in January 2008; at the same time, gold sky-rocketed to $900 per ounce while the prices of wheat, corn, soy beans and other agricultural commodities continued to rise. In this context, inflation numbers in Europe and in the United States were close to the highest for a decade. In the light of the subprime and financial crisis, which is ongoing at the time of writing, the stagflation theme is increasingly present in the newspapers, reflecting the combined effect of economic downturn and inflation pressures. This puts regulators in the tricky situation of having to choose between keeping inflation under control by increasing interest rates or sustaining economic growth by cutting them. And although we have been used to an inflation-controlled environment since the 1990s, we should not forget that inflation can reach substantial levels, as it did during the two oil crises in the 1970s when US inflation was well over 10%. At the same time, the population in western countries is ageing and more and more people are concerned about their pension schemes. Regulators are developing frameworks to guarantee pensions in real terms, requiring pension funds to hedge their assets against inflation. In this context, the inflation market is growing larger every year, with more sovereigns issuing more inflation-linked bonds and more investors interested in derivative products such as swaps and options. As with any developing market, every year brings innovations both in terms of products and theoretical research. This handbook reviews the mechanisms and past and future developments of the inflation market together with the markets impact. It can be read on two levels: the main text presents the major aspects of inflation while the technical boxes focus on some advanced aspects of the subjects developed. The handbook is split into six sections: The first section is a market review: when and how did the inflation market appear and what were the main steps in its development? How big is it? Who is interested in buying or selling inflation? 

In the second section we show how inflation is measured: what is an index price, who is responsible for measuring inflation and how do they do it? 

The third section concentrates on a very important technical aspect of inflation measurement, seasonality . We give a detailed definition of seasonality, look at ways of measuring it and analyse its evolution in Europe and the US. 

In the fourth section we present the products available to potential investors in the inflation market. This section offers an overview of all cash and vanilla products including inflation-linked bonds, inflation swaps, inflation options and inflation futures. 

In the fifth section we look at the different models available for pricing inflation derivatives. As this is a very recent market, quantitative research in this area is still in its infancy and most models are still in development. 



The final section provides examples of the Société Générale’s structured product offer.

History Inflation-linked derivatives appeared fairly recently. Indeed, the concept of inflation itself and its integration into a general economic theory only emerged in the work of 20th century economists such as J.M. Keynes and I. Fisher. The first inflation products to appear in the market were bonds and futures. Pre-1998: Birth of the inflation cash market

Inflation-Linked bonds (ILB) were first launched in the UK in 1981, closely followed by Australia in 1983. The first issue from Canada in 1991 was particularly important for the ILB market, as the bond format was particularly attractive. It described the bond in real terms so that the bond yield could be calculated without any assumptions about future inflation rates. After the US chose this format in 1997 for the first TIPS (Treasury Inflation Protected Security) issuance, followed by France in 1998, the Canadian model rapidly became the market standard. Sweden issued its first linker in 1994 and moved quickly to the Canadian model after the US and French issues. The UK refused to switch to this format on several occasions but finally changed its mind in 2005. Bonds are the main instruments providing liquidity and breadth in the inflation derivatives market. But inflation futures - the first inflation derivatives which have generated some interest - could also be a source of liquidity. In 1986, the Coffee, Sugar and Cocoa exchange launched a future based on the American CPI index. It met with relative success, with more than 10,000 contracts traded over 2 years. Unfortunately, the underlying market of inflation-linked bonds was still in its infancy and the future was eventually delisted. In 1997 the Chicago Board of Trade tried to launch an inflation-indexed Treasury note future based on the newly-introduced US Treasury TIPS programme. Only 22 contracts were traded in 1997, as the TIPS issuance programme was too young and the market not mature enough to trade this sort of instrument (following the success of the inflation market, exchanges are today trying to find a format which could satisfy investors and enhance liquidity ). 1998-2002: Infancy of the cash market and birth of the derivatives market

Inflation derivatives really came into existence between 1998 and 2002. This is when the real asset market - i.e. the inflation-linked bond market - contained too few points to construct a liquid curve and develop an efficient swap market. Market makers running bond books hedged their exposure with nominal bonds. Hedge ratios were based on a priori 50% correlation assumptions: the real market was assumed to move by 0.5bp when the nominal market moved 1bp. This means that market makers were exposed on this correlation assumption in a period when the statistical beta between nominal and real bonds was fairly volatile  an approach which proved costly for many market making books. Moreover, bid/ask spreads were very wide by todays standards - 50 cents in 2.5 Mio EUR on 10Y maturity, for example. In the late 1990s bonds were the only liquid instruments. Inflation swaps started to trade progressively around 2001, especially in the UK. 2003: Big Bang in the euro zone inflation market

2003 saw a big development in euro inflation derivatives, thanks to a series of issuance of European inflation-linked bonds corresponding to missing maturities on the longer-term segment of the curve. France, for example, issued the OATei 2032 in October 2002; Greece and Italy launched their first inflation-linked bond with the GGBei 2025 in March 2003 and the BTPSei 2008 in September 2003.

Increased outstanding amounts available in the market meant more liquidity and tighter bid/ask spreads. Bid/asks were reduced to 25 cents in 10 Mio EUR on 10Y maturity. At this time at least three points became available to construct an inflation curve (5Y, 10Y, 30Y) and associated CPI projections. For the first time, inflation-linked bonds started to trade in breakeven terms, i.e. in spread against the closest nominal bond. At the same time, as more data became available EMTN desks started to issue structured inflation-linked products. Dealers bought inflation hedge to balance the flows coming from this structuring activity. This was the real turning point for the inflation swap market. Dealers hedging flows considerably increased the volumes of inflation swaps on maturities up to 10 years. The inflation derivatives market really took hold and people started to move away from real yield trading to embrace inflation trading. At this time swaps were still priced from bonds, as the latter were more liquid than the former. And most banks kept their market making bonds activities separate from their inflation swap trading desk. 2004: Asset swaps on euro zone ILBs

Going into 2004 and after the big wave of EMTN issuance in 2003, inflation swap desks were left long inflation-linked coupons, and in an effort to reduce their exposure they started to sell bonds in assetswap packages. A lot of interest was generated by the BTPei 2008 issued in September 2003 (most structured products issued in 2003 had a five-year maturity). The Italian bond was the ideal hedge for inflation swap desks. During 2004, the asset swap on BTPei 2008 traded as cheap as Euribor + 8bp due to mispricing by some dealers and an oversized offer in the market. With the structured issuance desks development of custom-made profiles, inflation exposure did not necessarily coincide with the coupon payment date of available bonds. In this case, swaps became the preferred hedge instrument. Simultaneously, seasonality due to monthly inflation irregularities became more of an issue. Liquidity kept increasing on the bond and swap markets (up to 10Y maturity), with the bid/ask spread reduced to 10 cents in 50 Mio EUR on 10Y maturity. 2004 was also marked by a new attempt to launch an inflation future. The Chicago Mercantile Exchange (CME) launched a future on the US CPI in September. Its success was relatively moderate and the monthly volumes decreased progressively. This is mainly because this future was based on a three-month fixing whereas the inflation market works on year-on-year fixings. Finally, Japan joined the pool of inflation issuers with three new bonds: the JGBi March 2014, the JGBi June 2014 and the JGBi December 2014. 2005: Inflation forecasting

In 2005 the focus was on inflation forecasting: as structured desks were offering highly customised structures, dealers were increasingly at risk regarding their seasonality and inflation forecasts. A better understanding of the seasonal effects intrinsic to inflation started to spread in the market. In particular, this marked the end of carry-mispricing arbitrage1. The risks of CPI fixing - due either to seasonality effects or inaccurate economic forecasts - were especially relevant, as volumes in the structured market decreased and real yields in Europe reached historical lows.

1

See Inflation Products – Inflation-linked bonds, page 45

In terms of products and liquidity, asset swaps also started to be quoted on other underlyings, on the interbank market, up to 30 years and in tighter bid/ask prices (2bp). The bid/ask spread on the 10Y bonds was reduced to 10 cents for a standard ticket size of 100 Mio EUR. Competition between banks increased and most clients managed to get mid-prices. In September 2005, the Chicago Mercantile Exchange (CME) launched a future on the European price index (Harmonised Index of Consumer Prices, HICP). This was more of a success than the previous year s attempt using American inflation, mainly thanks to its monthly fixing.

Cash Market infancy 1998< 2000 British, Australian, s d Sweden, n Canadian o BLinkers d e k n US TIPS, i l n French o i t OAT a l f n I

Derivatives Market Birth 2001 2002

10Y,30Y on French CPI First HICP 10Y bond (France), Fixing July

30Y HICP (France)

$230b.

$260b.

s p UK RPI a w swaps S d e k n i l n o i t a Secondary market volumes (Euro zone only) l f n I >€1b. >€1b. >€1b.

s e v i t a v i r e D r e h t O

Inflation future attempts in the US

2005

10cts Bid Ask on 100M

7Y on French CPI

12Y on French CPI

5Y (Italy), 20Y (Greece), Fixing Sep.

15Y, 10Y HICP

5Y, 10Y, 30Y HICP

10Y (Germany), 11Y HICP Fixing March

$680b.

$850b.

$340b. First swaps on European inflation

First Japanese ILB $450b. Increased liquidity up to 10Y

Long term inflation swaps (up to 30Y)

Swap prices are calculated from bonds >€2b.

Structured Inflation Age 2006 2007

10Y on French CPI

€20b.

First Structured for EMTN desks UK LPI options

2004

25cts Bid Bonds quote 10cts Bid Ask on 10M in break-even Ask on 50M

50cts Bid Ask on 2.5M

Outstanding amounts $50b. $200b.

2003

First annual 0% floors

30Y on French CPI 30Y, 32Y, 33Y, 49Y, 50Y HICP

<$1000b.

Liquidity on all the curve

Bond ASW are calculated from swaps break-even €45b.

€56b.

CME future on US CPI

CME future on HICP

Asset swap packages on BTPe08

Inter-bank asset swaps on other issues

€66b.

€74b.

Eurex future Rate/Inflation Range hybrids Accruals First options on European inflation

Customized structured for LDI

Market consensus on seasonality Source: SG Quantitative Strategy

2006: Spread France-Europe and hybrid structures

In 2006 the market was ready for its first optional products. Structured desks launched optional features with hybrid structures mixing Libor and inflation fixings. For example, some banks issued structures paying the Libor minimum plus a margin and year-on-year inflation rates multiplied by a lever. This type of structure is sensitive to inflation/interest rate correlation and was probably a way for some dealers to unwind correlation exposures. In Europe, this was the time of the French/European spread, the first example of an imbalance between inflation in Europe and that in one of its member countries. 2006 saw an increase in demand for the French Livret A. The Livret A is one of France s most popular savings accounts, whose remuneration formula has been based on the year-on-year fixing of the French inflation index for December and June

since August 2004. As the demand on the Livret A rose, the banks offering this product needed to buy more OATi as an inflation hedge. The pressure on the OATi (French bonds indexed to French inflation) was higher than on the OATei (French bonds indexed to European inflation), leading to higher relative value for the French inflation bonds. Also in 2006, Germany issued its first inflation-linked bond for a ten-year maturity, the DBRI 2016. 2007: Inflation range accruals and LDI on Eurozone market

2007 was the year of inflation range accruals and of the Liability Driven Investment (LDI). Range accruals are fairly common products in the standard interest rate world. Increasing inflationary pressures on the central banks generated interest for these products over the year. They pay Euribor plus a margin, multiplied by the number of times year-on-year inflation falls within a given range, divided by twelve. This is a way for investors to get enhanced yields if the ECB manages to contain inflation at around 2%. When dealers sell inflation range accruals they are long volatility, so they sell caps and floors as the offsetting hedge position. In 2006, inflation desks saw about one option per week, while in 2007 volumes increased to four per week. Although these volumes are lower than those of the standard interest rate market, they have increased significantly. The second development in 2007 was the Liability Driven Investment (LDI). This investment framework appeared following recent developments in regulations for pension funds in the UK, the Netherlands, Sweden and Denmark. In these countries, regulators required pension funds to change the way they reported their discounted liabilities on their balance sheets. Encouraged by the new rules and in an effort to avoid inflation exposure on their liabilities, pension funds are looking to invest more in inflation-linked bonds and inflation swaps. LDIs largely benefit the global liquidity of the inflation swap market. Driven by this appetite for long term to very long term inflation protection, Italy and Greece issued each a 50Y bond linked to European inflation as a private placement. 2008: More innovations on the way?

So what comes next? What innovations will the inflation market see in 2008? First, Eurex launched its new European inflation future in January. This should enhance the liquidity of the European inflation futures market, as it will be subject to a compulsory daily auction. Second, the underlying swap market seems to be liquid enough to obtain a daily consensus on five and ten-year swap fixing. If market makers are successful in defining a daily inflation swap fixing, market transparency will be greatly improved and more investors will be attracted to inflation derivatives. A successful daily fixing should also provide the basis for a dynamic inflation swaption market. For the short term range, inflation options should probably be one of the markets next developments, as the underlying breakeven market is extremely liquid. Finally, increased regulation pressure on the pension funds industry should help the development of products designed for asset liability management. Inflationary pressures might continue to develop in 2008, so pension fund managers and ALM desks will be increasingly interested in investing in instruments based on real rates. This will be the time for real swaps, real Bermudan swaption, and hybrid equity/inflation products.

Volumes With the growing interest in inflation products, the trading volumes in circulation of both cash and derivative products have increased significantly. Firstly, sovereigns such as France, the UK and the US launch issuance programs at regular intervals to fund their internal budgets. Issuing inflation linkers offers sovereigns a way to source cheaper funding. It also sends positive signals to the market, confirming the governments confidence in regulators capacity to keep inflation under control. The graph of cumulated outstanding amounts below shows the exponential growth in the linkers market. At the end of the 1990s, prior to the American TIPS programme, the global market size was approximately $70 billion, mainly from UK inflation-linked treasuries. By 2000, US issuance had increased the market size to $200 billion. And with the contributions of new European issuance, there was over $1000 billion outstanding in 2007.. Swap market volumes have increased sharply over recent years, from almost zero in 2001 to over $110 billion in 2007. However, inflation swaps trading volumes are still much lower than those of inflationlinked bonds on the secondary market. This might appear counterintuitive. Inflation-linked swaps are the best inflation hedge for asset liability management - their flexibility makes cash-flow matching much easier than with inflation linked bonds, for instance. The reason for the difference in volumes lies in the newness of the swap market. Investors are reluctant to invest in instruments whose mechanisms do not seem fully transparent. One issue is the price of seasonality. Although the market is converging towards a seasonality consensus, it is still not clear whether this consensus is optimal or not. And the absence of a really liquid futures market and swap rate fixings does not improve pricing transparency. Moreover, each government usually issues inflation-linked bonds in the same month of the year. An ILB book therefore has limited exposure to seasonality, which corresponds to the month where the bonds pay their coupon. An inflation swap book, on the other hand, will have almost as many different fixing dates as there are instruments in the book. So the cost of fixing and seasonality risk limits the tightening of the bid/ask spread on inflation swaps. Despite that and as the demand for inflation protection grows, inflation swap trading volumes should continue to increase.

1,000 900 800 700 600 500 400 300 200 100 -

60,000

Outstanding amount $bn

M€ / Month

50,000 40,000 30,000 20,000 10,000 -

82 84 86 88 90 92 94 96 98 00 02 04 06

USD

EUR

CAD

Source: SG Fixed Income Research – AFT

SEK

JPY

GBP

Jan-02 OATe/i

Jan-03

Jan-04

Jan-05

Jan-06

Jan-07

In fl ation Swaps (10y equi vale nt)

Source: SG Fixed Income Research - ICAP

Market participants Participants in the inflation markets have very different profiles because of the diversity of their activities, needs and goals. Inflation payers receive inflation-linked revenues from their business line and want to exchange it to better match their non-inflation linked expenses and resources. Inflation receivers want to hedge themselves against a rise in inflation that could adversely affect their future income. And payers/receivers seek opportunities in the lack or excess of flows in the core market. Inflation payers

Inflation payers are sovereigns or institutions whose income is linked to inflation, such as utilities, real estate companies and project finance businesses. The value of payments they receive from their customers depend on inflation figures. And they need a fair amount of short-term liquidity to finance their investments in material and equipment. In England, for example, a lot of water and waste companies issue inflation-linked bonds so that they can transfer their revenues directly onto their liabilities. Sovereigns and regional agencies are among the biggest inflation payers. Bonds are generally one of their main sources of financing. As taxes (income or indirect taxes) are expressed in percentage terms, their income is also indexed to inflation. Paying inflation to the market is therefore a way to match income with liabilities. Until 2000, only a few sovereigns issued inflation-linked bonds. These included the UK Debt Management Office (DMO), the Agence France Trésor (AFT), the US Treasury and the Canadian, Australian and Swedish governments. From 2000 to 2003, the number of sovereigns issuing inflation linkers increased as Italy and Greece joined in. Supranational institutions and corporates started to issue inflation-linked debt at this stage as well, for example the CADES (Caisse d amortissement de la dette sociale) and RFF (Réseau Ferré de France) in France and the National Grid and Network Rail in the UK. Japan and Germany joined the pool of inflation issuers from 2003. Other activities also started to use the inflation derivatives market from 2003 onwards: project finance for infrastructure financing, regions and municipalities to manage their tax revenues, real estate brokers to balance their income from rents and mortgage lenders ALM desks and debt managers to reduce their funding costs. Issuing inflation-linked bonds is an attractive way of sourcing cheaper financing. Buying inflation-linked bonds rather than ordinary fixed coupon bonds buys a hedge against inflation. The coupon paid on the inflation-linked instrument benefits from this. The issuer saves the inflation risk premium2. Also, the coupon is very low at issue date and increases as time goes by. Linkers are therefore efficient instruments for obtaining cheaper financing upfront and delaying higher payments until a time when revenues have increased. Inflation receivers

Inflation receivers are generally financial companies whose liabilities are linked to inflation. Pension funds are the prime consumers of inflation-linked coupons. They traditionally try to minimise the risk of shortfall - the risk of their assets being less than their liabilities. Buying inflation-linked bonds is a way of reducing this risk, as their assets move in line with their liabilities. Changing regulations in some European countries have reinforced this need for inflation-linked products. In the UK, a change in accounting rules in 2000 (FRS17) forced the pensions industry to report liabilities mark-to-market, discounted with an AA curve. This regulation also stated that liabilities 2

See Inflation Products, inflation-linked bonds page 45

should be valued using market-implied forward inflation rates. As pensions in the UK are linked to the LPI index (Retail Price Index floored at 0% and capped at 5%), the new regulation has significantly increased hedging activities on UK RPI and LPI swaps. Other European countries followed this policy and are now trying to regulate the way pension funds manage risk. In the Netherlands, a new regulatory framework, the FTK, was introduced in 2007. The same year in France saw the implementation of the IAS19,under which employers must pay additional pension reserves before the end of 2008. The Italian government has also reformed its pension system (TFR), forcing pension funds to guarantee the principal plus some return linked to Italian inflation. And Swedish and Danish regulators have set up stress tests to detect funds which would suffer in case of highly distressed markets.

Inflation Payers Bond Market

Inflation Receiver

Derivatives Market

Bond Market

Sovereigns

Asset Managers

DMO, AFT, UST, AUD

Asset diversification

Derivatives Market

Pension Funds/Life Ins.

Hedge IL Liabilities

2000 Supra and agencies

Alternative Investments

Bank ALM

CADES, CNA

Carry, alpha strategy

Hedge for IL swap

Corporate

RFF, NRI, NG Sovereigns

Italy, Greece

2003 Sovereigns

Project Finance

Regional Banks

Regional Banks

Japan, Germany

Infrastructure

Italy, Swiss retail

Structured notes

Regions/Municipality

Inflation Linked Funds

Bank ALM

Tax revenues

Benchmark replication

Hedge for Livret A

Real Estate holder

LDI Funds

Rents

Pension funds

Bank ALM

Non Life Insurance

Mortgages

Hedge inflation claims

Active Debt Managers

Reduce cost of funding vol Inflation Linked Funds

Relative Value

2008

Prop desks

RV and diversification Source: SG Quantitative Strategy

Since 2003, the number of investors willing to receive inflation has increased significantly and the focus has switched from traditional bond products to more sophisticated structured products. EMTN issuance activities have helped this trend by offering investors access to the inflation market through structured bonds. This has forced retail banks to hedge themselves, increasing volumes of swaps and

options. The development of some national characteristics such as saving accounts indexed to inflation (typically the Livret A in France) has encouraged the use of inflation derivatives as a hedge. All these flows have contributed to increased liquidity in the market. Relative-value players have started to appear, seeking to take advantage of occasional market tensions. Investors in quest of diversification are nowadays also looking increasingly at inflation-linked products. All these investors, whether they be relative value funds or proprietary traders, opportunistically receive or pay inflation in the market. They act as regulators in the inflation market and contribute to the increase in liquidity.

Inflation Payers Receivers: Proprietary traders Investment banks Hedge funds

Inflation Payers:

ASSETS

Real Income

IL Income

Utilities Project Finance Real Estate Retailers Sovereigns Agencies

IL Coupon

Inflation Market

Financing

IL Coupon Libor

IL Coupon

Inflation Receivers: Pension funds Insurance Mutual funds Corporate ALM

LIABILITIES

IL Payment

Real Payment

Financing

Inflation Receivers: Retail Banks

IL Coupon Investors Financing

Source: SG Quantitative Strategy

Introduction Inflation is a measure of price increases. It cannot be observed directly but is estimated using various types of price index, each of which aims to measure the cost of living in a certain part of the world, and each based on different criteria. Building a price index is a daunting task, for two main reasons: first, indices are based on subjective baskets of goods and services; second, these baskets evolve over time, as prices, products offered on the market and consumers interests change. This section introduces inflation indices and details the calculations used to account for changes in their composition. We use these inflation indices to define real interest rates as nominal rates adjusted by inflation. The rest of this handbook frequently refers to the real and nominal economies, depending on whether money is considered by its nominal value or by the amount of goods and services that it can buy. The calculation procedures section gives further details of the different types of index frequently referred to on the market and explains which types of goods and services are included in these indices.

Inflation is perceived in widely differing ways, so its measurement is a key issue for inflation products and derivatives. National statistics offices define a reference basket of goods and services whose value is recalculated and published every month. Known as the CPI (Consumer Price Index) in the US, the HICP (Harmonised Index of Consumer Prices) in Europe and the RPI (Retail Price Index) in the UK, these measure the average monthly change in the nominal price of the reference basket. The inflation indices are based at 100 on an arbitrary chosen date. From time to time, national statistics offices decide to rebase their price index, choosing a new date on which the reference basket of goods and services is worth 100. One of the raisons for this rebasing is to prevent the index diverging too far from the 100 reference value. For example, the European Statistics office Eurostat rebased the HICP All Items ex Tobacco in July 2005. Inflation indices are usually calculated on a monthly basis and published two to three weeks after the end of the month in question. The composition of the reference basket is fixed at a given time, but can be changed by the national statistics institute. This happens either when the reference basket no longer corresponds to the populations spending or on a regular basis, depending on the country. New weights are calculated to reflect changes in lifestyle and consumption habits. Changes in the reference basket lead to two series of inflation indices, the revised and unrevised series. The unrevised series contains the index values as originally published by the national statistics institutes. The revised series contains modified values, reflecting changes in the reference basket. When a revision takes place, new weights are estimated for the reference basket, reflecting the populations expenditure since the previous survey. These are then used to recompute the price index backwards. The values are only re-estimated between two revision dates. More documentation on revision policy is available from the national statistics institutes3. For example, European harmonised

3

Minimum Standard for revision  Journal of the European BLS Handbook of Methods , Chapter 17 - The Consumer Price Index

Communities



September

2001

indices are revised on a regular basis, while for the US BLS advises against revisions of the urban consumer price index.

In July 2005, Eurostat decided to rebase all HICP indices. The previous reference year was 1996. Whenever the base changes, a rebasing key is calculated and published by regulators. But there is a problem with existing contracts such as inflation-linked bonds: if the terms of the contract are not changed, there is a risk of discrepancy between the value used to calculate coupon fixings and the reference index used to calculate the inflation rate. If no adjustment is made, the inflation rate used in existing products will not reflect the realised price increase. In the case of the HICP rebasing in 2005, the International Swaps and Derivatives Association (ISDA) published market practice guidelines advising on the best way to rescale existing pay-offs. The rebasing key was defined by the ISDA as: C RB =

Base 2005 IE Dec 2005 Base1996 IE Dec 2005

Base 2005 is the Eurostat index of December 2005 expressed in the new 2005 = 100 base (i.e. 101.1). IE Dec 2005 Base1996 IE Dec 2005 is the Eurostat index of December 2005 expressed in the old 1996 = 100 base (i.e. 118.5).

Eurostat

index refers to any index or sub-index published by Eurostat (HICP all items, HICPxT, French HICP etc).

With the rebasing key it is possible to rebase any index value or daily reference: base 2005

IRd ,m

1996 = IRd base C RB ,m

The index time series can therefore be calculated backwards and any daily reference index used in a contract can be recalculated.

Calculation methods can differ from one national statistics office to another and even from one national index to another. In the UK, for example, there are major differences between the RPI national index and the European harmonised index, the HICP. The baskets of goods and services can differ widely, both according to different consumption styles in different countries and the methodology used to calculate the baskets. The price aggregation method can also vary from one index to another: see the technical box below for a review of the most popular methods.

Price indices aim to objectively measure the change in cost of living from one period to another (typically on a monthly basis). But the weights in the basket can change from month to month. This effect should not affect price measurement. Several methods are available: Base-weighted index or Laspeyres index price This method calculates the change in price relative to a base date, assuming constant weights in the basket of goods and services. The change in price level is given by: P L = wn0 p 1n wn0 p n0

∑

∑

where w are the weights in the basket and p the prices. A 100% Laspeyres index means that purchasing power did not change from one period to another.

This index systematically overstates inflation as it does not account for the fact that consumers adapt their consumption to price changes by buying less when prices increase and more when they go down. Expenditure data is sometimes more readily available than weights. Expenditure data is the total sum of money used by consumers to buy one particular item, i.e. weight multiplied by price. In this case, the calculation formula (which leads to the same results as the formula above) is:

P L =

∑ E ( p 0 n

1 n

p n0 )

∑ E

0 n

where E is expenditure. End-year weighted index or Paasche’s price index This method is similar to Laspeyres, except that that the weights are taken from the latest available period. The change in price level is expressed as:

P P =

∑ w p ∑ w 1 n

1 n

1 n

0 pn .

A 100% Paasche index means that consumption over the latest period is the same as before. Because consumers tend to increase the quantity they buy when prices go down, the denominator tends to be higher than reality and the Paasche index tends to understate inflation. From a practical point of view, this index requires a monthly update of the weights or expenditure data. Chained index Each year, an index is calculated with the base value in January at 100%. The resulting chained index over several years is defined by: C Aug 07

P

= P

C Aug 07 / Jan 07

x

C P Dec 06 / Jan 06

100

x

C P Dec 05 / Jan 05

.

100

Most of the time the Laspeyres index is used to calculate the index value within the same year. Using the chained index avoids revising the index series each time there is a change in weights. This is particularly useful when the weights are changed on a regular basis. Rebasing can occur on a different time basis. Fisher index The Fisher index aims to solve the problem of understatement or overstatement posed by the two previous indices. It is calculated as the geometric average of the Laspeyres and Paasche indices: P F = P L P P It has the same disadvantage as the Paasche index - monthly calculation of weights, which is much more difficult than computation of price levels. Marshall-Edgeworth index This index is another alternative to the Fisher index. It is an arithmetic average of prices, weighted by the quantities in the current and base periods. In practice, it provides si milar results: P ME =

∑ (w

1 n

+ wn0 ) p 1n

∑ (w

1 n

+ wn0 )p n0

In financial markets, traders and market players are used to considering investments by their nominal value. But in everyday life, people tend to focus on what is directly relevant to them - the amount of goods and services that can be acquired with a specific amount of money. Hence the distinction between the nominal and real economy: 

In the nominal economy, investments are gauged according to their nominal value;

In the real economy, the value of an investment is related to the actual amount of goods and services that can be bought. 

This distinction matters when considering the value of an investment over time. Price increases reduce the amount of goods and services that can be bought with a given amount of money, so the real rate of return of an investment is its nominal rate of return minus the inflation rate. By this definition, real rates are not directly observable but can be deduced from nominal rates by using inflation, defined as the growth rate of inflation indices.

Real Economy r : real interest rate

Nominal Economy n : nominal interest rate

Inflation Ratio at 0: R 0=CPI0 /CPI0=1

Time 0

$100

$100 x R0

Time T

$100 x (1+r)T

$100 x (1+n)T = $100 x (1+r)T x RT

Inflation Ratio at T: RT=CPIT /CPI0 Source: SG Quantitative Strategy

Real and nominal interest rates are sometimes compared to the (nominal) interest rates paid by two different currencies. The inflation index (CPI) plays the role of an exchange rate that translates the value of assets in one currency (the real economy) into the other currency (the nominal economy). The former is a basket of goods and services, the latter is the nominal value of this basket. The inflation rate is the growth of this exchange rate. The relationship between real and nominal rates is also known as the Fisher equation (see technical box on page 88).

Calculation of indices Measuring prices is a complex task, as different calculations may be used and different choices made as to which data to include in the reference basket. Inflation can differ widely from one country to another because of the inclusion or exclusion of particular reference basket items. In this section we review the calculation procedures for the main national indices (US, Europe, France and UK). We also highlight the regional and sectoral differences in Europe and the US.

The US CPI index is calculated by the United States Department of Labor Bureau of Labor Statistics (BLS), which publishes: The CPI for all Urban Consumers (CPI-U), which covers approximately 87% of the total US population (in the 1990 census). It is available both at country level and at some lower levels such as census regions, certain metropolitan areas classified by population size and 26 local areas. It is published in the second week of the month with a one-month lag. This is the index commonly used by inflation markets and US Treasury Inflation-Protected Securities (TIPS); 

The CPI for Urban Wage Earners and Clerical Workers (CPI-W) covers 32% of the total population. It represents a subset of the urban population and is published for the same areas as the CPI-U; 

The Chained CPI for All Urban Consumers (C-CPI-U) also covers the urban population, but uses different formulae and weights in the reference basket. It is a new index and has been published since August 2002 with data starting in 2000. 

Monthly movement in the CPI is calculated from the weighted average of price changes for the items in the reference basket. The reference basket is constructed to reflect the cost of living of a preselected (urban) population. The items in the basket and their weights are chosen in line with spending reported in the Consumer Expenditure Survey. There are eight main categories of item, the most important of which are house prices, transport costs and food prices which together contribute 75% (see pie chart below). Investment items (stocks, life insurance, changes in interest rates), income and other direct taxes are excluded, but taxes on consumer products (sales and excise taxes) are included. The set of goods and services is subdivided into 211 categories, resulting in 8018 basic indices. The urban areas of the United States comprise 38 geographic areas. The CPI is calculated in two stages. First, the basic indices are calculated from a monthly survey carried out by BLS field representatives who gather prices for each individual item from selected businesses. The BLS calculates basic indices from these prices, using a weighted geometric average or a Laspeyres index. The quantities used in the calculation come from sampling data and statistical analysis. Then aggregated indices are produced across geographic areas and sectors. The all-items, all-geographical areas CPI-U index is an aggregate of all the basic indices. The BLS provides the calculation methodology in detail in one of its publications ( BLS Handbook of Methods, Chapter 17 The Consumer Price Index). There can be big differences in inflation between the USs 38 urban geographic areas, as is shown by looking at the four main urban regions (South urban, Midwest urban, Northeast urban and West urban). Over the last 20 years, US CPI-U annual inflation has oscillated between 6% (maximum value in the 90s) and 1.5% (minimum value in 2002). During this period, the spread between maximum and

minimum regional inflation was as low as 0.1% in 2000 and as much as 2% in 2007. Inflation was generally higher in the Northeast and West regions: goods or services worth $100 in 1998 would in 2007 be worth $186.3 in the Northeast urban region and $182 in the West urban region compared with $176 in South urban and $175 in Midwest urban. These disparities are visible within a single population group (urban population) and would be much higher in the case of a mixed (urban and non-urban) population. The price indices at the urban zone level show that annual inflation is highest in Miami and Seattle (3.65% and 3.05% respectively) and lowest in Detroit and Boston (0.55% and 0.80% respectively) for an average CPI-U index level of 2.36%.

3%

6%

7%

15%

6%

6%

6%

5% 4% 17%

3% 43% 4%

2%

Food and Beverages

Housing

Clothing and Footwear

Transport

1%

Medical Care

Recreation

Education and Communication

Other Goods and Services

0% 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07 US CPI-U

Source: SG Quantitative Strategy - Bloomberg

Regional Minimum

Regional Maximum


3.0%

7% 6%

2.5%

5% 2.0% 4% 1.5%

3% 2%

1.0%

1% 0.5% 0% 88

90

South Urban

92

94

96

Mid West Urban

98

00

02

North East Urban


04

06 West Urban

0.0% 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07


The Harmonised Indices of Consumer Prices for EU countries are published by Eurostat using data issued by EU member states statistics offices. They provide a unified framework to calculate and compare inflation data. HICPs and national CPIs can be significantly different as national CPIs are mostly based on methodologies chosen prior to the creation of the HICP. Some of the differences are: Subsidised healthcare and education: the HICPs include the net price paid by consumers, while some national indices either exclude these purchases altogether or record the gross price; 

Owner-occupied housing: the HICPs currently exclude the cost to owners of financing their property (interest and credit charges), while some national indices include these costs. The problem with including these items is that it introduces a direct dependency on nominal interest rates into the measurement of inflation; 

Aggregation formulae: the HICPs are Laspeyres-type indices. National methods may be somewhat different; 

Geographical and population coverage: the HICPs cover expenditure by residents and visitors in each country, while some national CPIs cover expenditure by domestic residents within and outside the country. 

The HICPs are published by Eurostat every month, generally 17 to 19 days after the end of the month measured. The main areas covered are housing, food and beverages, transport, recreation and culture, restaurants and hotels, which together account for 77%. They include all costs faced by consumers and so include sales taxes such as Value Added Tax (VAT). In addition to the aggregate index and the sectoral indices, some special aggregates are provided such as the HICP excluding tobacco and the HICP excluding energy. The unrevised HICP excluding tobacco is the reference for all eurodenominated inflation-linked bonds. Like the BLS in the US, representatives of member states statistics offices collect prices from local retailers and service providers. When needed, the local offices make price adjustments to account for potential changes in products quality. Eurostat imposes minimum standards of quality adjustment, but there is as yet no harmonised calculation method, although one is being developed. The coverage of the reference basket is the same from one country to the next. However, sector weights are defined at a country level based on local expenditure to preserve the consumption characteristics of each member state. All countries use the same computation and aggregation methods and the same calculation formulae. The final HICP index is compiled as a weighted average of the countries in the euro zone. The country weights are derived from national accounts data for household final monetary consumption expenditure. Eurostat provides methodologies4.

4

comprehensive

methodological

documents

on

HICP

calculations

and

Harmonised Indices of Consumer Prices (HICPs) – A short guide for Users  European Commission  March

2004

European Inflation Convergence

The European Economic Community (EEC) was founded in 1957 with the signing of the Treaty of Rome. In 1979, the Jenkins Commission set up the European Monetary System (EMS), whereby EEC member states (with the exception of the UK) agreed to link their currencies to the European Currency Unit (ECU) through the Exchange Rate Mechanism (ERM). From this point in time, the inflation rates of ERM members have tended to converge. In a recent study5, the ECB analyses inflation convergence since the introduction of the ERM and its evolution since the introduction of the euro. Its main conclusions are as follows: The ERM was essential to achieve inflation convergence by 1997. By this year, under the Maastricht treaty, any country wishing to adopt the euro had to have fulfilled a certain number of criteria. 

Countries that joined the ERM at an early stage showed strong convergence until 1997, while countries that joined the programme later experienced higher inflation rates; 

Since 1998, the report has found evidence of diverging behaviour between two main groups: a low inflation group comprising Germany, France, Belgium, Austria and Finland and a high inflation group including the Netherlands, Ireland, Spain, Greece, Portugal and Ireland. Italy stands in between the two groups. Inflation convergence seems to have been achieved within each group. 

The graphs below illustrate this convergence: we computed the maximum and minimum inflation levels across European countries since 1996 and the spread between these two values. We looked at Germany, France, Italy, Spain, the Netherlands, Belgium, Austria, Greece and Portugal. The high/low spread has followed a decreasing trend over the past ten years. Similarly, the standard deviation of annual inflation rates has decreased over the same period from 1.54% in 1996 to 0.63% in 2007. The graph in the bottom left-hand corner shows the highest annual European inflation over time. Greece had the highest inflation over the 96-99 period and alternated with Spain in the 04-07 period. The Netherlands had the highest inflation in 2001-2002 and Ireland in 2000-2001 and 2002-2004.

7%

4%

4%

7%

20%

6%

8%

5% 4% 15%

0%

3% 2%

10% 22%

Food and beverages

Transport

Housing

Recreation and culture

R estaurants and hotels

Mis c. goods and s er vice s

Clothing and footwear

Health

1% 0% 97

99 HICP

01

03

05

Maximum on the 9 first inflations

07 Minimum

Education and Communications

Source: SG Quantitative Strategy - Eurostat

5


Inflation convergence and divergence within the European Monetary Union

F.Venditti  January 2006



F.Busetti, L.Forni, A.Harvey,

7%

6%

6%

5%

5% 4%

4% 3%

3%

2%

2%

1% 1%

0% 97

98 Spain

99

00

Greece

01

02

03

Netherland

04

05

Ireland

06

07

Portugal


0% 97

99

01

03

05

07


Inflation rates vary greatly between sectors. In the past five years, the communications, recreation and culture sectors have gone through a period of disinflation. These sectors include all computer, audio, video and telephone expenditure, plus all goods and services for personal leisure (indoor and outdoor recreational equipments, toys and gardening). There was almost no inflation for clothing, notably due to cheap imports from Asia. At the other end of the spectrum, education expenses inflation has always been very high and jumped even higher recently. The housing sector is also a main contributor to the final inflation figure. And the food and non-alcoholic beverage sector recently saw an increase in inflation due to a rise in commodity prices.

Communication

10%


8%

Clothing

6%

Household

4%

Food and non alc

2%

Misc.

0%

Restaurant and Hotel

-2%

Transport Health Housing Alcohol and tobacco Education HICP -3% -2% -1% 0% 1% 2% 3% 4% 5% 6% Source: SG Quantitative Strategy - Bloomberg

-4% -6% -8% -10% 97

98

99

00

01

02

03

04

Education

Food and non alc

Communication

HICP


05

06

Clothing

07

The French CPI is close to its European HICP counterpart both in terms of composition and methodology. It is published monthly by the National Institute for Statistics and Economic Studies (INSEE) in the Official Gazette ( Journal Officiel) around the 13th day of the following month. It covers all sectors except private hospital services, certain kinds of insurance policy such as life insurance (considered as financial products) and gambling. Seasonal effects such as sales periods are taken into account. The INSEE produces different types of aggregate, such as the all-items IPC and the IPC excluding tobacco. The latter is used in the inflation-linked bond market. Like the other price indices, the CPI is measured using a reference basket and after a monthly survey of more than 110 000 elementary products and services in 96 different urban areas including the French overseas regions. The prices gathered by the INSEE surveyors are corrected by a quality coefficient which depends on the evolution of the quality of each product. The reference baskets components and weights are updated every year according to changes in French consumption. Aggregation is first carried out geographically and then by sector. The IPC is an annual Laspeyres-type price index.

12%

3.0%

16%

7% 5%

2.5%

9%

2.0% 1.5%

3%

21% 17%

1.0% 10%

Food and Beverages


Housing and energy

Healthcare

Transport

Education and Communication


Restaurants and Hotels

Other goods and services

Source: SG Quantitative Strategy - Eurostat

French ICP

0.5% 0.0% Jan-97

French HICP

Jan-99

Jan-01

Jan-03

Jan-05

Jan-07


The UK RPI is published every month by the Office for National Statistics (ONS). It has been the standard domestic measure of inflation in the UK since June 1947 and is used especially to calculate state pensions and benefits and for inflation-linked gilts. It covers all private households on the mainland but excludes the Channel Islands and the Isle of Man. It also excludes pensioner households (those which derive more than three-quarters of their income from state pensions and benefits) and high-income households (total income in the top 4% of all households). In terms of expenditure items, it covers all consumption goods and services but excludes spending linked to financial and investment products (credit and investment expenditure, income taxes and other direct taxes, property purchased for investment and gambling). However, council tax and mortgage interest rate payments are included, as these represent a big part of the cost of housing in the UK. So movement in interest rates can have a direct impact on the inflation index. The RPI is drawn up using similar methods to those of other national statistics offices. Specific items are chosen in accordance with the spending reports and surveyors collect prices countrywide every month. Prices are then aggregated from the lowest level (geographic area, single item) to the highest (national, all items) using an annually Laspeyres-type chain-linked index. As the index is chained, there

is no need for rebasing or revising the series each time the reference basket s composition changes (in such cases, rebasing is done purely for scaling purposes). The current index is based on 1987 prices. The RPI reference basket is very different from that of the harmonised indices, especially in terms of the treatment of interest and mortgages linked to owner-occupied houses. For example, the annual inflation rate measured by the RPI was 4.1% at the end of August 2007 while the national harmonised CPI was only 1.8%.

11% 4%

5.0%

20%

2%

4.5%

4% 5%

4.0% 3.5% 3.0% 2.5%

13% 24%

2.0% 1.5%

17%

F oo d, B ev er ag es an d T ob ac co

C at er in g

Housing

Motoring and Energy

Household goods and services


Pe rs onal go od s an d s er vic es

F ar es an d o the r tr av el co st s

Leisure

Source: SG Quantitative Strategy – UK ONS

1.0%

UK RPI

0.5%

UK HICP

0.0% Jan-97

Jan-99

Jan-01

Jan-03

Jan-05

Jan-07


National statistics office web site

On Bloomberg

US inflation

http://www.bls.gov

CPURNSA

European inflation

http://ec.europa.eu/eurostat

CPXTEMU data)

(revised

CPTFEMU (non revised) UK inflation

http://www.statistics.gov.uk

UKRPI (UK RPI) CPXTUKI (Harmonised CPI)

French inflation

http://www.insee.fr

FRCPXTOB CPXTFRI

Seasonality is a change in prices or business patterns at given times of the year. For example, if annual inflation is 2% over a year, this means that goods or services worth 100 in January will be worth 102 the following January. However, this price increase is not uniform and is subject to monthly or seasonal variations. The possible causes for seasonal variations include natural factors (seasons, the weather), legal measures (administered price increases, tax regime changes) and sociocultural traditions (Christmas, summer holidays). Seasonality accounts for fluctuations of up to 0.3%-0.4% in inflation. Over the last 10 years, inflation in euro zone Europe (the HICP) has been maintained at between 1% and 3%, which means that seasonal adjustments represent up to 10-30% of inflation itself - a significant proportion. Seasonality matters when it comes to building curves of forward or zero coupon inflation. There is a liquid market for zero coupon inflation swaps with maturities expressed in number of years from inception. Outside this range of standardised maturity dates, there is no product that directly prices zero coupon inflation, so interpolation techniques need to be used. However, the size of seasonal adjustments is such that linear interpolation cannot be relied on, and past estimates of seasonality are used to build forward inflation curves (see Building a CPI forward curve  in the inflation swaps section, page 65). In turn, these zero coupon curves play a key role when valuing inflation options, even on standardised dates. Therefore, before introducing the main inflation products traded on the market we will take a break to discuss seasonality and the main statistical techniques used to measure it.

Definition Seasonality is defined as a change in a given variable which is entirely due to events at a specific time of the year. For example, the European HICPxT index (HICP excluding Tobacco) usually decreases by an impressive 0.35% in January, probably due to the winter sales. Seasonality is measured on a monthly basis using the inflation indices time series, and can be expressed as a Month-on-Month (MoM) or Year-on-Year (YoY) correction. See the technical box below for more details on definition and calculation. Seasonality measurement is linked to inflation measurement: seasonal economic cycles are reflected in the time series for the standard consumer prices indices. The European harmonised index (HICP) is the weighted sum of the individual national composite indices, which are themselves the weighted sum of the price index components (goods, energy, services etcetera). Seasonality at composite level can be explained by looking at the subcomponents. Seasonality measurement is not only crucial to be able to remove seasonal effects from a time series and to better understand inflation dynamics. It is also important for inflation-linked derivative pricing and strategies.

Seasonal adjustments are calculated every month and can be expressed as YoY (Year on Year) or MoM (Month on Month), or % MoM. MoM gives the change imputed to seasonality from one month to the next. YoY adjustment cumulates the monthly changes from the month of January. % MoM is the difference between two consecutive YoY adjustments. I nSA = I n ∗ YoY n = I n ∗ MoM n ∗ MoM n −1 ∗ ... ∗ MoM Jan = I n ∗ (% MoM n + % MoM n−1 + ... + % MoM Jan + 100%) I nSA is the seasonally-adjusted CPI index at time n and I n is the non-adjusted CPI index at time n. For example, using the table below, the seasonality adjustment in euros for the month of February is x99.81%.

J F M A M J J A S O N D

%MoM -0.35% 0.16% 0.23% 0.22% 0.06% -0.09% -0.19% -0.05% 0.07% -0.04% -0.15% 0.14%

HICPxT MoM 99.65% 100.16% 100.23% 100.22% 100.06% 99.91% 99.81% 99.95% 100.07% 99.96% 99.85% 100.14%

YoY 99.65% 99.81% 100.04% 100.26% 100.32% 100.23% 100.04% 99.99% 100.06% 100.01% 99.86% 100.00%

French CPIxT %MoM MoM YoY -0.28% 99.72% 99.72% 0.25% 100.25% 99.97% 0.23% 100.23% 100.20% 0.11% 100.11% 100.31% 0.08% 100.08% 100.39% -0.06% 99.94% 100.33% -0.29% 99.71% 100.04% 0.08% 100.08% 100.12% 0.08% 100.08% 100.21% -0.04% 99.96% 100.17% -0.15% 99.85% 100.02% -0.02% 99.98% 100.00%

%MoM -0.59% 0.20% 0.16% 0.53% 0.12% -0.12% -0.40% 0.09% 0.23% -0.14% -0.14% 0.04%

UK RPI MoM 99.41% 100.20% 100.16% 100.53% 100.12% 99.88% 99.60% 100.09% 100.23% 99.86% 99.86% 100.04%

YoY 99.41% 99.62% 99.78% 100.31% 100.43% 100.32% 99.92% 100.01% 100.24% 100.10% 99.96% 100.00%

%MoM 0.20% 0.17% 0.24% 0.16% 0.00% -0.09% -0.04% -0.01% 0.07% -0.02% -0.34% -0.33%

US CPI-U MoM 100.20% 100.17% 100.24% 100.16% 100.00% 99.91% 99.96% 99.99% 100.07% 99.98% 99.66% 99.67%

YoY 100.20% 100.36% 100.61% 100.76% 100.76% 100.67% 100.63% 100.62% 100.70% 100.68% 100.33% 100.00%


Measurement Statistical seasonality measurement has been studied for a long time. Several methods have been developed and thoroughly tested. Three have emerged over the past years: the dummies method, the TRAMO/SEATS and the X-12 ARIMA . The first method is fairly straightforward. It makes use of 12 dummies , which are functions equal to one if the index is observed, say, in January (or February, March etc.) and zero elsewhere. The regression of the index return time series against the dummie s gives an estimate of seasonal adjustment. The results found with the dummies and the averaged results found using more advanced methods are similar. However, the dummies do not capture differences in seasonality from one year to the next, whilst the more sophisticated methods mentioned below can show the evolution of seasonality over several years. Also, the dummies method cuts the historical data into twelve separate time series, which greatly reduces the accuracy of the estimation process. More sophisticated techniques try to estimate all seasonality adjustments at the same time using all the available data. Although the dummies method can be a useful instrument for a quick estimate of seasonality parameters, it cannot replace more indepth statistical analysis. The other methods are more elaborate and use widely-tested statistical models: TRAMO/SEATS (Time Series Regression with ARIMA noise, Missing value and Outliers  Signal Extraction in ARIMA Time Series) was developed by the Bank of Spain. See the technical section later in this section for more details on seasonality in ARIMA models. 

X12-ARIMA (experimentation 12  Auto Regressive Integrated Moving Average). This algorithm was developed and has been extensively used by the US Census Bureau. 

These methods have both been implemented by Eurostat in an application called Demetra, a tool which can be downloaded from the Eurostat website. The statistical methods available in Demetra decompose time series of returns into three components: 

a trend, which can be purely stochastic or linked to macro-economic variables;

a seasonal factor, which is a constant monthly factor reflecting the impact of seasonal behaviour on the time series; 



white noise, which contains all the effects not captured by the other components.

The procedure for calculating seasonal adjustments starts with preliminary treatment of data6 in both these methods.

6

Data is first weighted by the number of working days in each month, in order to be able to work with comparable quantities. Because the analysis can be done either on normal returns (difference of the index between two dates) or on lognormal returns (difference of the log-index between two dates), a lognormality test is then run. Normal returns are used when seasonal fluctuation is independent of the index level, and lead to additive factors and additive adjustments. Lognormal returns are used when the size of the seasonal fluctuation is related to the level of the index and the calculations lead to multiplicative adjustments. Outliers are then identified and removed from the time series. In the case of inflation, this should happen very rarely since the series are fairly stable.

0.4%

0.4%

MoM adjustments

0.3%

0.3%

0.2%

0.2%

0.1%

0.1%

0.0%

0.0%

-0.1%

-0.1%

-0.2%

-0.2%

-0.3%

MoM adjustments

-0.3% X12-ARIMA

TRAMO/SEATS

Dummies

-0.4%

X12-ARIMA

TRAMO/SEATS

Dummies

-0.4% J

F

M

A


M

J

J

A

S

O

N

D

J

F

M

A

M

J

J

A

S

O

N

D


The extraction of the trend and the computation of the seasonal adjustment depend on the statistical method: X12-ARIMA is a non-parametric procedure which successively estimates moving average filters. Validation of initial assumptions (no autocorrelation, white noise residuals) after several iterations allows for retention of the best filter. 

TRAMO/SEATS is a parametric approach based on a fitted ARIMA model. It uses this filter to extract trends and seasonality from the time series. A parametric model is usually slightly less flexible than a non-parametric one like X12-ARIMA, but it also requires less historical data. The technical box on ARIMA models provides more details on the estimation of seasonality. 

Eurostat conducted a study to investigate which method was better. TRAMO/SEATS appeared to be robust and efficient for evaluating a specific statistical model. X12 ARIMA does not depend on the choice of statistical model and is in that sense more flexible. It is older and seems to be more widelyused in the industry. Because there is no particular reason to choose one method rather than the other, Demetra provides a battery of statistical tests to evaluate the quality of an approach over another one. Once the question of calculation methodology is solved, there are still practical issues to address. The ECB highlighted these issues and offered answers for the euro zone in some of its publications: One of the first issues which springs to mind is the revision of seasonal estimates, i.e. the frequency of calculation. Inflation indices are usually published on a monthly basis and it could be argued that the seasonal calculation should be re-run every month to incorporate the latest available information. The ECBs study of standard monetary statistics ( Criteria to determine the optimal revision policy: a case study based on euro zone monetary aggregates data  L.Martin  ECB), divides its revision policy into three steps: identification of the model, estimation of its parameters and the seasonality forecast. It concludes that optimal frequency depends on the data themselves and that in most cases systematic re-estimation of the model and its coefficients does not improve the quality of the estimates. It finally recommends annual revision of seasonal adjustments. 

The second issue is the aggregation of seasonality between inflation indices. Seasonal adjustments are usually calculated for the more synthetic series, i.e. the composite index series. A composite index is not only the aggregate of basket prices over different sectors, but is also averaged over several geographical areas or even several countries, as is the case for the European composite. Seasonality can then be calculated over each sector and/or each area and aggregated. This method is known as the indirect approach. It can also be computed directly for the composite series using the direct approach. The ECBs 2003 paper Seasonal adjustment of European aggregates: direct versus indirect approach  D.Ladiray and G.Luigi Mazzi  ECB) concludes on this matter that for European inflation, there are no significant differences between the direct and indirect approach, using either the TRAMO/SEATS or X-12 ARIMA methodology. For pure seasonality measurement, the direct approach is therefore preferable as it is simple to implement. But the indirect approach can still provide some additional information in terms of analysis of seasonal phenomena. 

An Auto-Regressive ( ), Integrated ( ) Moving Average (

) – ARIMA - model aims to explain the realisation of a

variable at a given time using past values of the same variable. This is equivalent to regressing a time series against a lagged version of itself. For example, the following model is an AR order 3 (AR(3)) model:

X t = θ 1 X t −1 + θ 2 X t − 2 + θ 3 X t −3 + ε t An MA model represents a tim e series moving randomly around its average. The randomness is generated by white noise elements. The number of white noise elements used to reconstruct the time series gives the order of the model. For example, the following model is an MA(1) model:

X t = ε t −

ε

1 t −1

An ARMA model combines an AR and an MA model. It represents a time series generated by its past values and its past errors. It is characterised by the order of the underlying AR and MA processes. The following example is an ARMA(3,1) model:

X t − θ 1 X t −1 − θ 2 X t − 2 − θ 3 X t −3 = ε t − α 1ε t −1 An ARMA model can be fitted to a time series using the Box Jenkins method, provided that the time series is stationary. In reality, very few time series are directly stationary. However, by looking at their derivative, a stationary derived time series can be isolated. An ARIMA model is an ARMA model fitted to the nth derivative of the underlying process. For example, the following expression defines the second derivativ e of the X process:

Y t = ( X t − X t −1 ) − ( X t −1 − X t − 2 ) And an ARMA(3,1) applied to Y defines an ARIMA(3,1,2). Seasonality is taken into account by applying an ARIMA model to changes over the period in question. For example, when analysing seasonality throughout the year, a traditional ARIMA model is estimated on X t − X t −12 . These models are used to decompose X into the sum of two components - a seasonal component plus a seasonally-adjusted series. The seasonal component can be forecast by applying a specific filter to past data

105

0.6% 0.4%

100

0.5%

0.3%

0.2% 95

0.1% 0.0%

90

-0.1% -0.2%

85

-0.4% HICPxT

Seasonality

80

-0.6%

Jan-96

Jan-98

-0.3%

Jan-00

Jan-02

Jan-04

-0.5%

Jan-06

Jan-96


Jan-98

Jan-00

Jan-02

Jan-04

Jan-06


0.2% 0.30%

0.0%

0.20%

-0.2%

0.10%

-0.4%

0.00%

-0.6%

-0.10%

0.2%

January MoM (%) H I C P x T

F o B o e d v a . n d

T r a n s p o r t

H o u s i n g

R R H e & e c o s r t C e o e t u a l & l t t i . o n

M i s c .

H o u s e h o l d

C l o t h i n g

O t h e r

H o u s i n g

R R H e & e c o s r t t C e e o u a l & l t t i . o n

M i s c .

H o u s e h o l d

C l o t h i n g

O t h e r

December MoM (%)

-0.20%

0.1% -0.30%

0.0%

-0.40% J

F

M

A

HICPxT Source: SG Quantitative Strategy

M

J

J

A

S

O

HICP

N

D

-0.1%

H I C P x T

F o B o d e v a . n d

T r a n s p o r t

Source: SG Quantitative Strategy – The seasonality adjustments for each sector are multiplied by the sector weights in the HICP index.

Case study In this section we concentrate on inflation in Europe and in the US. We show that seasonality in Europe increases over time, due both to growth in international competition and to inflation convergence. Seasonality has also augmented in the US over the past 10 years, although the level is lower than in Europe. We identify the most seasonal sectors and highlight seasonal pattern differences between the two zones.

In this section we analyse the effect of seasonality on European inflation, using the HICP ex-tobacco (CPXTEMU), the composite indices for the European countries and Europe-level sectoral indices. This will help us to understand where European seasonality effects come from, both in terms of economic activities and geographical area. Inflation in the euro zone is typically lower than average in January, July and November and significantly above average in March and December. This is due to economic patterns which can be highlighted using sectoral analysis: Clothing and footwear: In 2007 this sector was ranked 8th-highest in terms of weights in the HICP ex-tobacco. But it is an extremely seasonal sector, with sales periods in January and July and the arrival of new collections in March and September. MoM seasonality adjustments range from -7.2% for January and July to +5.2% in March and September - by far the widest low season/high season range. 

Food and non-alcoholic beverages: This is the most heavily-weighted sector in the HICP. So it has a negative impact in summer (July, August) when fresh food prices are low and a (relatively moderate) positive impact in winter when prices are high. However, its total effect on the aggregate index is moderate, ranging from -0.6% in summer to +0.35% in winter. 

Recreation and culture: It is no surprise that this sector contributes the most to European inflation in December, during the festive season. 

Transport: This is also worth mentioning as it is the second most heavily-weighted sector in the HICP. Its seasonality peaks positively in April at +1.1% and negatively in October at -1%. 

Price controls and regulations: This is particularly sensitive for all items whose prices are regulated or highly taxed, such as tobacco and alcoholic beverages. It is the main reason for the difference between the inflation index excluding tobacco and its all items counterpart. For example, in January the seasonality adjustment for the ex-tobacco composite is lower due to the increase in regulated prices which usually occurs at the beginning of the year. 

We can make the following comments concerning European countries  contributions to the HICP: Four countries account for up to 80% of the European inflation index and its seasonality: Germany (28.7%), France (20.3%), Italy (19%) and Spain (12%). These four countries have similar characteristics, which are those mentioned above (strong negative seasonality in January and July, positive seasonality over the spring months). 

Germany has strong positive seasonality over the month of December. A sector analysis run on Germany shows that this is due to the combined effect of the Restaurants & Hotels and the Recreation & Culture sectors and is probably explained by Germanys strong Christmas traditions. 

Like inflation levels, seasonal patterns are converging under EU influence. For example, the seasonal adjustments for Italy differed widely between the 1996-2000 period and the 2001-2007 period. This is partly explained by the harmonisation of the methods used to calculate inflation in the euro zone. For example, Italy started to include sales price reductions in its CPI in 2001. 

0.6%

HICPxT MoM

0.4% 0.2% 0.0% -0.2% -0.4%

Germany

France

Italy

Spain

Netherlands

Belgium

Austria

Greece

Portugal

Finland

Ireland

Luxembourg&Slovenia

-0.6% J


0.2%

F

M

A

M

J

J

A

S

Basket GER, FRA, ITA, ESP

O

N

D

N

D

Others


MoM Adjustments (%) Jan 96 - Dec 00 0.3%

0.1% 0.0%

0.2%

-0.1%

0.1%

-0.2% J

F

M

1.5%

A

M

J

J

A

S

O

N

D

0.0%

-0.1%

MoM Adjustments (%) Jan 01 - Dec 06

1.0% 0.5%

-0.2%

0.0% -0.3%

-0.5%

J

-1.0%

F

M

A

G erm any

-1.5% J

F

M

A


M

J

J

A

S

O

N

M

J

J

France

A Ital y

S

O Spai n

D Source: SG Quantitative Strategy – The sector seasonality adjustments for each country are multiplied by the country weights in the HICP index.

Not only has seasonality in the different European countries tended to show the same pattern, but the magnitude of seasonal changes (difference between the highest seasonal adjustment and the lowest) has also increased: Since the launch of the euro and the introduction of the open European market, trade between European countries has become much easier, increasing competition between manufacturers. More competition favours bigger swings in prices; 

Competition has also increased in services and transports, leading to bigger seasonal changes in these sectors; 

The acceleration of international and European competition, new joiners in the harmonised euro zones and reinforcement of harmonisation policy will all probably continue to contribute to growth in seasonal magnitude. 

0.60%

1.2%

MoM adjustments (%) 0.40%

1.0%

0.20%

0.8%

0.00%

0.6% -0.20%

0.4% -0.40%

0.2%

-0.60% jan

mar

0.0%

-0.80% 97

98

99

00

01

02

03


04

05

06

97

98

99

00

01

02

03

04

05

06


In this section we analyse seasonal effects on US inflation. We ran analyses on the CPI-U excluding tobacco and its main sectoral sub-indices. Several points can be highlighted: Increase in seasonality : The US market is naturally impacted by international competition, as US prices are exhibiting bigger swing movements over time. For example, in 1996 the maximum difference between two monthly inflation rates was 0.4%, while in 2006 this difference had increased to 0.82%. Although the increase in price swing is less than in Europe, the internationalisation of the economy still has a noticeable impact; 

Importance of textiles, housing and transport: The US textiles sector follows the classical seasonality pattern  the US sales periods are around the months of January, June and July. Housing represents more than 40% of total US expenditure, and prices in the housing sector tend to be lower at the end of the year and higher at the beginning of the year. Transport is more expensive in April and cheaper in November. 

US versus European inflation: The main differences between US and EU seasonality patterns occur during the month of January and from July to December. This can be essentially explained by looking at the composition of the two indices. On the one hand, clothing and footwear accounts for a very small proportion of the US inflation index (3.8%), while in Europe it accounts for a larger portion in inflation measurement (7%). And textiles are much more seasonal in Europe than in the US, with a maximum spread of 12.4% in Europe compared with 7.6% in the US. On the other hand, the transportation sector represents 17.4% in the US and 16% in Europe and there is a higher seasonality adjustment in the US in October and November. 

0.9%

5.00%

0.8%

4.00%

0.7%

3.00%

0.6%

2.00%

0.5%

1.00%

0.4%

0.00%

0.3%

-1.00%

0.2%

-2.00%

0.1%

-3.00%

0.0% 94 95 96 97 98 99 00 01 02 03 04 05 06

-4.00% J

F

M

A

M

J

J

Transport Source: SG Quantitative Strategy

A

S

O

N

D

O

N

D

Clothing


0.40%

0.60%

0.30%

0.40%

0.20% 0.20%

0.10% 0.00%

0.00%

-0.10%

-0.20%

J

F

M

A

M

J

J

A

S

-0.20% -0.40%

-0.30% -0.40%

-0.60%

-0.50% -0.80% J

F

M

A

HICP Source: SG Quantitative Strategy

M

J

J

A

uscpi

S

O

N

D

-0.60%

USCPI Housing, Clothing and Transport


Overview Before we look at inflation-linked products in detail, let us take a step back and quickly review the different types of product and how they relate to each other. We will focus on the link between bonds and swaps and that between swaps and options.

We can consider that financial products are distributed along two axes: Nominal vs. real economy : As explained in the inflation indices section, the economy is nominal or real, depending on whether market players look at the nominal value of financial investments or the amount of goods and services they can buy. The inflation market aims to create and trade products which have fixed features in the real economy - for example a fixed coupon - which in practice means indexing cash flows on inflation indices. 

Credit risk: sovereign vs. interbanking: Inflation derivatives are essentially used by sovereigns, via bond issuance, or in the interbanking system7, with the recent development of inflation swaps. The issuance of inflation-indexed products by other bodies (mainly long-term financials and corporate issuers) is beyond the scope of this publication. 

This gives us four kinds of product and relative value opportunity plus indicators for measuring relative value. These four categories are: Government issuance in the real economy : As explained in greater detail in the Inflation-linked bonds section (page 45), it is in sovereigns  interest to issue bonds which guarantee the notional at maturity in real terms. This means that the bond holder will have the same purchasing power at maturity as at inception. This kind of bond pays a real coupon, which also guarantees the bond holder s purchasing power. These products are commonly called inflation-linked bonds. As with any bond, a real yield can be calculated to reflect the bond yield in real terms. 

Government issuance in the nominal economy : This is traditional government bond issuance. It is useful to mention this kind of bond here to provide an overall picture of the links between the nominal and real economies. The difference between the usual nominal yield and the real yield is the bond breakeven, which is the main relative value indicator for inflation-linked versus nominal bond strategies. 

Interbank products in the nominal economy : All the traditional interest rate products fall into this category. Standard vanilla swaps are particularly interesting as they are the equivalent of inflation swaps. The difference between the nominal swap rate and the nominal bond yield is the swap spread. This is a relative value measure of sovereign and interbank risk: the higher the swap spread, the more expensive is funding for banks compared to sovereigns and therefore the riskier the banks credit signature. 

7

Used as a generic term covering banks and other institutional investors such as pension funds.

Real Economy

k n a b r e t n I

Real Swap Rate

Nominal Economy

Swap Break-Even

Nominal Swap Spread

Real Swap Spread t n e m n r e v o G

Real Bond Yield

Nominal Swap Rate

Bond Break-Even

Nominal Bond Yield


Interbank products in the real economy : To be perfectly consistent with the existing products in the nominal economy, this category should be represented by the real swap, a product which in the nominal economy exchanges a fixed (nominal) rate for an inflation-indexed (real) rate, with an exchange of nominals at the maturity date. But there is unfortunately no liquid market for real swaps. 

The interbank inflation market is instead based on inflation swaps, which exchange future realised inflation for nominal rates. Zero coupon inflation swaps exchange realised inflation for a fixed nominal rate on a specific date, whilst year-on-year (YoY) swaps annually exchange realised yearly inflation for a fixed nominal rate. If future inflation is constant on all payment dates, this fixed rate prices an inflation breakeven level or swap breakeven. Both zero coupon inflation swaps and swap breakevens provide an indirect valuation of real rates, because implied inflation can always be interpreted as nominal minus real rate. In the case of zero coupon swaps this relationship is straightforward, and these swaps are the most liquid of all inflation derivative products. However, YoY swaps price forward inflation, and given that future inflation is unknown, inflation volatility and convexity adjustments also need to be taken into account. Pricing a YoY swap is therefore no easy task and requires some degree of knowledge about inflation volatility. These technicalities are explained in more detail in the Inflation Swaps subsection (page 58). Similarly, the equivalent of the nominal swap spread in the real economy, the real swap spread, is not quoted directly but can be deduced from existing market data (nominal swap spread, inflation bond breakeven and inflation swap breakeven). However, if the real swap rate develops further, the real swap spread could be priced directly as the differential between the real swap rate and the real inflationlinked bonds rate.

Non-optional products can be classified either in terms of credit risk or type of economy, while options are a different type of product whose importance is increasing and which provide a way of pricing inflation volatility. In the next section we take a look at the link between non-optional (swaps) and optional instruments.

One of the main reasons for the development of the inflation swap market is to provide an alternative way to synthetically hedge the flows usually associated with inflation-linked bonds. These are best reproduced by zero coupon swaps. Zero coupons are therefore the reference instrument for the inflation swap market. Similarly, two types of underlying are possible for optional contracts, leading to zero coupon options and year-on-year options. A zero coupon option pays the buyer the difference between an inflation rate and a fixed strike as long as this is positive. As with the swap, the inflation rate is measured between the expiry date and the inception date, as the ratio of the reference price index between these two dates. A year-on-year option pays the buyer the same difference, except that the inflation rate is measured by a rolling one-year ratio of price index values. In the inflation options market, the predominant liquid instruments are year-on-year contracts, making it much more difficult to obtain a consistent pricing framework. While the swap market prices the zero coupon forward, the options market requires the year-on-year forward. The difference between the forwards is the convexity adjustment, which depends on the options volatility.

Inflation ZC swap annual points

Interpolation:

Seasonality issue No product Risk Premium

Option Prices

Inflation Forward

Inflation Forward

Zero Coupon

Year on Year

ZC and Year onYear

Inflation Volatility

Exotic Option Prices

Model

Option Prices


As shown in the graph above, a consistent pricing framework needs to tackle the foll owing points: Deduction of the zero coupon forwards or CPI projections from the zero coupon swaps prices. CPI projections are known at the dates corresponding to the annual market quotes; 

A complete curve of zero coupon forwards requires an interpolation procedure, especially to handle the issue of seasonal adjustment. As there are no products to exactly price the seasonality risk at 

intermediate points, this procedure relies on statistical methods and/or a risk premium associated with the markets appetite to take on this additional risk. Some option prices will provide volatility information to calibrate the volatility function of some chosen models; 

A model should be calibrated from the information provided by zero coupon forwards and option prices. This then provides all the information for pricing other inflation derivatives: 



The year-on-year forward curve;



The calibrated volatility function.

Once the model is calibrated, we can calculate the year-on-year forward, the prices of non-quoted options, exotic options and structured products. The following sections describe these different inflation-related products. The first part deals with inflation-linked bonds, their mechanisms and main relative value indicators (page 45). The second focuses on inflation swaps, the different types quoted in the market and how to calculate CPI forwards from zero coupon swap prices (page 58). In the third part we develop the issue of inflation-linked asset swaps (page 70). The fourth details inflation-linked options (page 78) and the final part provides a brief introduction to inflation-linked futures (page 83). We leave the question of inflation modelling briefly mentioned above - for a later section.

Inflation-linked bonds In this section we start by looking at bond cash flows and conventions. We also show how CPI fixing is calculated and how to handle the publication lag for inflation indices. We then examine the differences between dirty, clean and invoice prices, explain how to calculate the real yield, define the beta between nominal and real bonds and the real duration and finally detail the specificity of calculating carry for inflation-linked bonds.

In this subsection we focus on the mechanism of inflation-linked bonds: their cash flows, market conventions for the major currencies, their specificities compared to nominal bonds and their link with the inflation reference price index.

Inflation-linked bonds are bonds whose notional is linked to a reference index measuring the inflation level. This means that coupons are paid in real rather than nominal terms, providing protection against inflation risk. Inflation-linked bonds were issued for the first time in the UK in 1981, followed closely by Australia in 1983. 1991 marked a big step in the development of inflation linkers with the first Canadian issue. The Canadian bond had an innovative structure, and its format is now the benchmark convention for all linkers. Unlike the usual fixed-rate bonds, the future cash flows for inflation-linked bonds are not known at the time of purchase, as they depend on the future values of the reference index at the fixing date. As the reference index rises, the notional of the bond rises proportionally. The investor is paid the fixed real coupon multiplied by the inflated notional. At maturity, the bond usually reimburses either the inflated notional or par, whichever is greater. In real money terms, the investor is always paid the coupon and is therefore hedged against inflation risk. It is in sovereigns interest to issue inflation-linked bonds rather than fixed-coupon bonds. In all developed linker markets, the central bank is responsible for keeping inflation under control (although not all central banks have an official inflation target). The European Central Bank (ECB), for example, has publicly committed to maintain inflation around a reference level of 2%. However, market expectations are often higher than the reference level. As we will see later, this depends on the risk premium the market implicitly prices in the bond prices. As governments are more inclined to believe in their scenario, they can benefit from cheaper financing by issuing bonds with a substantially lower coupon to start with. Moreover, issuing inflation-linked bonds gives the market the signal that the government or central banks are committed to respecting their inflation targets. This helps to keep market expectations in line with published inflation targets. Lastly, linkers offer investors an embedded inflation hedge for which they compensate the government by accepting lower coupons. The inflation-linked bonds issued by sovereigns have converged towards the same benchmark convention defined by Canada in 1991. In broad terms, conventions generally include the following elements: Measurement of inflation using the national reference inflation index, as described in the previous section; 

Calculation of index fixing - usually with a three-month lag because inflation indices for a month m are published in the middle of the following month m+1; 

Coupons constant in real terms. On the payment date, the notional is multiplied by the inflation index ratio. The index ratio is the value of the index at payment date (reference index) divided by the value of the index at issue date (base index); 

In most countries - excluding Canada, the UK and Japan - flooring of the notional at 100 at maturity as protection against a prolonged period of deflation. In Japan the notional has no floor because of historically low inflation levels: inclusion of a floor would change the bonds valuation by too much; 

No protection of the coupon against deflation, except in Australia where both the notional and the coupon are protected; 

Payment of coupons is annual in Germany, Greece, the euro zone and Sweden. Coupons are paid semi-annually in the UK, Canada, Italy and the US. 

All conventions are summarised in the table below.

UK

Australia

Sweden

Canada

TIPS

OATi

OATei

Greece

First Issuance

1981

1983

1994

1991

1997

1998

2001

2003

2003

2004

Maturity

2006-2055

2010-2020

2008-2028

2021-2036

2007-2032

2009-2029

2012-2040

2025-2030

2008-2035

2014-2017

Amount Outstanding (local currency)

78

6

215

24

418

63

58

10.7

74

7917

15

Amount Outstanding (USD)

155

5.2

34

24

418

93

85

15.7

109

73

22

Reference Index

RPI monthly

HICP EMU extobacco

Bloomberg ticker

UKRPI Index AUCPI Index SWCPI Index CACPI Index

CPURNSA Index

CPI France ex-tobacco FRCPXTOB Index

CAN Govt

TII Govt

FRTR Govt

FRTR Govt

Semi-annual

Semi-annual

Annual

Annual

Annual

Semi-annual

Semi-annual

Annual

3M lag

3M lag

3M lag

3M lag

3M lag

3M lag

3M lag

3M lag

No Floor

Floor at par

Floor at par

Floor at par

Floor at par

Floor at par

No Floor

Floor at par

Bloomberg ILB page Coupon

Principal Repayment of principal

UKTI Govt

CPI quarterly CPI monthly

SAFA Govt

SGB Govt

Semi-annual Quarterly (pre(pre-fixed for Annual fixed) 8-month lag) 3M (after 2005) and 8M 6M lag 3M lag lag Coupon and No Floor pricipal Floor at par protected

CPI monthly CPI-U monthly

CPTFEMU Index

Italy

Japan

Germany 2006 2013-2016

HICP EMU ex HICP EMU ex- CPI ex-fresh HICP EMU ex tobacco tobacco food tobacco CPTFEMU CPTFEMU JCPNJGBI CPTFEMU Index Index Index Index DBRI Govt GGB Govt BTPS Govt JGBI Govt OBLI Govt

Source: SG inflation trading desk – SG Fixed Income Research

Inflation-linked bonds use a reference index published by national statistical institutes. The publication of this price index follows a long monthly process of measuring expenditure and prices at regional and national levels. The index value for month m is finally published during the second part of month m+1 (for example, the European HICP for the September is published in mid-October). This is the index publication lag, which needs to be addressed when calculating the current value of the CPI fixing. Knowing the CPI fixings precisely is particularly important in two instances: when calculating the amount to be paid to the bond holder on the coupons payment date. This date rarely corresponds to an index publication date; 

if the bond is bought or sold on the secondary market between two coupon payment dates. The bond-holder then receives an accrued coupon, which is proportional to the time the bond holder held the bond before selling it. 

Using the Canadian format, a CPI fixing is calculated as the interpolated value of the unrevised CPI index three months and two months prior to the coupon payment date. The interpolated CPI value is called the daily inflation reference (DIR) or daily CPI. By convention, the daily reference index and index ratios are rounded to the fifth decimal place. Let us look at an example. The OATei 2012 pays its coupons on 25 July each year. The July CPI is not known on this date. Moreover, the June CPI is only known only by the middle of July. So in July, the most recent HICP fixings known throughout July are those published mid-June and mid-May, i.e. the May and April unrevised CPIs. So the interpolation is done using the May and April numbers. In general terms, the daily inflation reference for any day in the month m is an interpolated value of the price index for the months m  2 and m  3: d − 1 DIRd , m = CPI m −3 + (CPI m− 2 − CPI m −3 ) NumberOfDa ysInMonth (m )

Using this convention, the reference price index for the first day of the month m is the price index for the month m  3. For instance, the reference price for July 1 is the price index for the month of April. If we go back to our example of the OATei 2012, the calculation of the coupon paid on July 25 2007 is: DIR 25, Jul = CPI Apr + (CPI May − CPI Apr )

25 − 1 31

= 104.05 + (104.31 − 104.05 )

24 31

= 104.25129

When a CPI number is released, usually by the middle of the month, the daily reference index can be calculated until the end of the following month. So in our example, on the price index release date in mid-July, the daily reference index can be calculated until end of August. The base reference index is calculated when the bond is issued. It gives the level at which the inflation rate measurement for this particular bond starts. Calculation of the base reference index is subject to the same interpolation principles as the daily reference. The index ratio (IR) - the ratio between the current daily inflation reference and the base reference index - gives the accretion rate to apply to the notional at the current date: IRd ,m = DIRd ,m BaseIndex

Once the index ratio is known, the coupon calculation is straightforward and follows standard procedure:

The coupon to be paid to the bond holder (at payment date) is the bond s real fixed coupon multiplied by the inflated notional. The inflated notional is the notional multiplied by the index ratio at payment date; 

The accrued coupon is calculated in real terms using the proportion of the time the bond holder held the bond between the last coupon payment date before selling and the following one. This is then multiplied by the inflation notional, which is equal to the notional multiplied by the inflation ratio on the date of the transaction. 

Lets return to our example. In the case of the OATei 12 issued on 25 July 2001, the base index is 92.98393, calculated as the interpolated value between the unrevised CPI (base year 1996) in April 01 (108.6) and May 01 (109.1) and multiplied by the rebasing key (see pages 18-20 for more information on rebasing). The annual coupon paid on 25 July 2001 is the real rate (3%) multiplied by the inflation ratio: 3% x inflation ratio = 3% x 104.25129 / 92.9839 = 3.36%.

106 105

March CPI release

DIR

April CPI release

May CPI release

June CPI release

HICP ex tobacco

104

30 April

31 May

30 June

CPI release schedule

31 July

103 Coupon payment schedule

102 101

1 May

100

1 June

1 July

1 August Payment in July

99 Nov-05

May-06

Nov-06


May-07

1 Sep

1 Oct

Accrued coupon in August

Nov-07 Source: SG Quantitative Strategy - Bloomberg

We start this section with the concept of invoice price, which is closely related to the dirty price calculation for a standard bond. We then define real yield, inflation breakeven and risk premium. We highlight the differences between linker duration and standard nominal duration, and finally we introduce the notion of carry and forward price in the linker world.

Once issued, in normal market conditions inflation-linked bonds are very liquid in the secondary market and quotes can easily be found. The linkers face value is expressed as the unadjusted clean price (UCP). This is the price of the bond excluding inflation and interest accrued since the last coupon. This price is obviously different from the final price billed to the investor buying the bond. The invoice price is calculated in the following way:

1) Calculate the accrued real coupon with the usual calculations for a nominal bond. This accrued interest (AI) is the interest due to the bond holder, corresponding to the time since the last coupon date and before the bond transfer: AI t =

t − t LastCoupon Date t NextCoupon Date − t LastCoupon Date

× Coupon

2) Calculate the unadjusted dirty price (UDP), the sum of the unadjusted clean price and the accrued interest: P t UDP = P t UCP + AI t

3) Multiply the unadjusted dirty price by the index ratio to get the adjusted dirty price (ADP) or invoice price: P t ADP = IRt P t UDP

Of course, calculation of the invoice price from the quoted price is particularly relevant when trading inflation-linked bonds, but it is also important when calculating asset swap spread, as we will see in the asset swap section (page 70). To illustrate this calculation, lets consider that we buy the OATei 2012 on 5 November 2007 (settlement date 8 November 2007). The price quoted on Bloomberg is 105.706. The inflation ratio is 1.12152, calculated as the current daily reference index (104.28333) divided by the base reference index as of 25 July 2001 (92.98393). The time between the last coupon payment date and the next one is 0.28962 year. So the accrued coupon is 0.86885 (3 x 0.28962). The unadjusted dirty price is 106.5749 (= 105.706 + 0.86885). The invoice price is 119.5258, calculated as 106.5749 x 1.12152.

A bond yield is a generic concept used for all bonds and is the return paid if the bond is held until maturity. It depends on the bond coupon and market price. If the yield and the coupon are equal the bond is at par. For an inflation-linked bond, the yield to maturity is calculated in real terms and gives the yield of the bond in the real economy. It is therefore expressed in constant monetary terms and is deduced from the unadjusted dirty price as follows: N

c

∑ (1 + y

P t UDP =

i =1

R

)

T i

+

100

(1 + y R )T

N

The difference between the yield of a nominal and an inflation-linked bond of equivalent maturity issued by the same government is commonly called the breakeven inflation rate (BEIR). This gives an idea of the inflation rate that needs to be realised over the life of the bond for the inflation-linked bond to outperform the nominal one. If we return to our example of the OATei 2012, the yield is 1.727% while that on the OAT October 2012 is 4.075% on 5 November 2007. BEIR is 4.075%-1.727% = 234.8bp.

In order to better understand the concept of inflation breakeven, lets look at a nominal zero coupon bond which matures at a given time T. Its value today is simply given by its yield to maturity. The nominal value of an inflation-linked zero coupon bond maturing on the same date is the value of the real zero coupon times the inflation ratio: B N (0, T ) =

1 T

(1 + y N )

, Binf la (0, T ) =

I T

1 T

(1 + y R )

I 0

Two investment strategies are possible: buying the inflation-linked bond or buying the nominal bond. An investment of 100 in the nominal zero coupon will result in a final value of 100 x (1+yN )T, while investing 100 in the inflation-linked zero coupon will produce a final value of 100 x (1+yR )T x IT /I0. IT and I0 are the values for the inflation reference index at maturity and at issue date respectively. The expected inflation rate, i is: I T = (1 + i )T I 0

The investor will have no preference for either strategy if the realised inflation rate is such that: 100

x (1+yN )T = 100 x (1+yR )T x (1+i)T

Or in other terms: (1+yN ) = (1+yR ) x (1+i) This is the Fisher relationship for the bond yields. As the yields are relatively small, the relationship can be approximated to the first order by dropping the crossed terms: y N = yR + i. The two strategies (buying the nominal or the inflation-linked bond) are equally effective if the realised inflation rate reaches its target: BEIR = yN - yR

The inflation breakeven tradable in the market can theoretically be broken down into two components: Inflation expectations: There is no exact way of calculating inflation expectations. A first approximation might involve central banks inflation targets. However, market inflation expectations can be lower or higher than these targets depending on current market conditions and macroeconomic factors. A second idea might be to use the economists consensus. This is the average of a pool of economists forecasts for the following year. But there is no guarantee that this forecast is up to date or that it properly reflects market expectations. And there is no consensus forecast for the long term beyond two years. 

Inflation risk premium: this is the term generally used to define investors preferences. If demand for inflation-linked bonds is higher than that for nominal bonds, the real yield tends to be lower and the breakeven tends to rise. So as long as inflation expectations remain constant, an increase in the demand for inflation-linked bonds will increase the inflation risk premium. 

In general, the inflation risk premium depends on investors appetite for inflation-linked bonds, which depends on their risk aversion. Investors can be willing to take on inflation risk or not, depending on their portfolio profile or market views. For example, long-term investors care about the real value of money and like to secure their assets in real terms. Long-term nominal bonds are riskier in real terms, as their final real value depends on the inflation rate. So the difference between the nominal yield and the real yield needs to be higher to compensate the nominal bond holder for this additional risk.

Conversely, demand for linkers might be lower than that for sovereign issuance, at least in the short term: short-horizon investors (such as hedge funds) set their targets in nominal terms. In this case, the BEIR value would be pushed down and could possibly be lower than inflation expectations. The two graphs below provide examples of OAT BEIR compared with the ECB inflation target. BEIR have recently been well above central bank targets, reflecting an increase in both market inflation expectations and inflation risk premium.

250

270 250

OATei 1.6% 2015

210

230 OATei 1.8% 2040

230 OATei 3% 2012

240

OATei 3.15% 2032

210 OATei 2.25% 2020

200 190

190

180

OATei Curve @ Nov 07 170 150 2010

220

BEIR OATei 2012 vs OAT Apr 2012

170

ECB Target

ECB Target

160

OATei Curve @ Jan 04

150

2015

2020

2025


2030

2035

2040

Nov-02

Nov-03

Nov-04

Nov- 05

Nov-06

Nov- 07


The standard duration or Macaulay duration of a nominal bond is defined as the average maturity of a bonds cash flows weighted by their net present value. This indicator is homogenous to time-tomaturity and provides intuitive information on the bonds average life. Alternatively, the modified or effective duration is the sensitivity of the bond price to a small change in bond yield. The modified duration is also the ratio of standard duration to 100% plus the bond yield. If the yield of a bond increases from 4% to 4.1%, the price decreases by 0.1% multiplied by the modified duration. Similarly, the convexity of a nominal bond is defined as the second derivative of a security price with respect to its yield. Positive convexity means that the security s price decreases less if its yield goes up than it increases in a downward move of the same size.

100

Price

90 80 70

Gain due to convexity

60 50

Gain due to duration Bond price as a function of yield

40 Decrease in yield

30 20 10

Yield

0 0%

1%

2%

3%


The real duration of an inflation-linked bond is calculated in the same way as the duration of a nominal bond and is the sensitivity of the bond price to the real bond yield. Inflation-linked bonds usually have a higher real duration and real convexity than nominal bonds of same maturity. This is because the coupon and yield of a linker are likely to be lower than the coupon and yield of a nominal of similar maturity. For example, at time of writing the real effective duration of the OATei 2032 in November 2007 is 17.3, while the duration of the OAT October 2032 is 13.9. Likewise, a linkers real convexity is calculated as the second derivative of the bond price with respect to its real yield. The real convexity of the OATei 2032 is 3.8 and the convexity of the OAT 3032 is 2.8. The real duration is not an accurate measure of nominal duration, i.e. the sensitivity of a linker s price to the nominal yield. In the linkers yield, inflation breakeven section (page 49), we explained that the nominal yield is the sum of the breakeven and the real yield: y N = y R + BEIR

If the breakeven was constant, the real and nominal durations of a linker would be exactly the same. However, in reality a 1bp move in nominal yield comes partly from a movement of the real yield and partly from a movement of the inflation breakeven. The relationship between the nominal and real variance can easily be calculated from the previous equation: Var ( y N ) = Var ( y R ) + Var (bev ) + 2CoVar ( y R , bev )

Provided that the correlation between the real yield and inflation is not negative, this implies that the nominal yield is more volatile than the real yield. This means that the real yield will tend to move less than the nominal yield and when the nominal yield moves by 1 bp, the real yield moves by less than 1 bp. The average amount the real rate moves when the nominal yield moves 1bp is called the beta. By definition, the nominal duration of a linker is the real duration multiplied by the beta. Similarly, nominal convexity is the real convexity multiplied by the square of the beta. Calculation of the nominal duration of a linker therefore depends entirely on accurate measurement of its beta.

Accurate estimation of nominal duration is fundamental for a mixed portfolio of nominal and inflationlinked bonds. This is one way of having a consistent duration report across the whole portfolio. How can this number be estimated? Market standards usually assume a beta of 50%, but this may seem somewhat arbitrary, as the statistics can differ widely. Beta can also be measured historically using an estimator. One possible estimator is the regression coefficient of the variations of a linker real yield time series versus the variations of an equivalent nominal bond yield time series. However, the beta also remains sensitive to other assumptions - the length of the time series and the frequency of the data. The graph below illustrates this. We calculated the beta between the OATei2012 and the OAT 2012 on a daily basis over a 10-week time period and on a weekly basis over a 10-week and a one-year time period. Beta is more stable measured over a year. In 2003, average beta was around 50%, consistent with the standard market assumption. It has now increased to levels around 80% for the OATei2012.

1.8

1 Year data (weekly return)

160%

1.6

10 Weeks data (weekly return)

140%

1.4

10 weeks data (daily return)

120%

1.2

100%

1.0

80%

0.8

60%

0.6

40%

0.4

20% 0.2 Jan-03

Jan-04

Jan-05

Jan-06

Jan-07


0% 05/02

05/03 OATei 2009

05/04

05/05 OATei 2012

05/06

05/07 OATei 2029


What can Bloomberg tell us? Bloombergs YA function provides a wide range of references. The screen is split into four boxes: Yield Calculations: this shows the real yield of the bond, also called the street real yield on the Bloomberg screen. It is the bond yield y corresponding to the market price using the usual formula: 

P d quoted = ,m

N

c

∑ (1 + y) i =1

T i

+

100

(1 + y )T

N

− AccruedInt erest

Note that Bloombergs equivalent 2/yr compound is the US version of the equivalent semi-annual yield. Sensitivity Analysis includes duration and convexity calculations. On the one hand, due to its lower fixed coupon an inflation-linked bond has higher duration and convexity than a nominal bond with the same maturity. On the other hand, real yields are less volatile than nominal yields. Standard calculations applied to inflation-linked bonds can thus be misleading. In the sensitivity analysis box, the investor can choose a beta between nominal and real yields to calculate effective duration and convexity. The effective duration is the standard duration multiplied by the beta, and a linkers convexity is standard convexity multiplied by the square of the beta. 

Economic Factors provide information on the CPI fixings. It gives the base index value for the bond, the last coupon value, the two latest CPI fixings and the current daily inflation reference. 

Payment Invoice: this details the payment for a transaction on the secondary market. The quoted price multiplied by the index ratio is the gross amount . The accrued interest is calculated and the total is given by the net amount . This is the invoice payment that would be paid for the bond at that point in time. 

Carry is a measure of how much an investor would gain or lose over a short horizon by holding an asset rather than investing the corresponding amount in the money market. Generally defined for a bond (nominal or inflation-linked), it can be measured by estimating the return on the following strategy: 1) Buy the bond and hold it over a given period (typically one to six months). If a coupon is paid, reinvest it at the money market rate. 2) Finance the bond with a secured loan (using the bond as collateral) on the repo market. The nominal repo amount will be equal to the invoice price when initiating the transaction. 3) At the end of the period, sell the bond and unwind the repo transaction. The amount to be reimbursed is the initial repo nominal amount plus the accrued interest over the holding period. This strategy breaks even for a given yield change between now and the end of the period. The difference between this breakeven yield and the yield at inception gives the carry in yield terms. We will look at two examples: one on a nominal bond, the OAT April 2012, and one on an inflation-linked bond, the OATei 2012.

Calculating carry for a nominal bond Lets apply the above strategy to the nominal OAT 5% April 2012: 

We buy the bond on 6 July 2007 (settlement date 11 July). The clean price quoted on the market is 101.466 (yield 4.647%). The accrued interest is 1.0519. Buying the bond on this date costs 102.5179 (101.466 + 1.0519).



We finance the bond with a repo at a given rate (say 4.11%).





After one month, we reimburse the repo and sell the bond. The loan has a total value of 102.8758. The accrued interest on the bond is 1.4754 and the clean price of the bond should be 101.40 (102.8758  1.4754) for the strategy to break even. The corresponding yield is 4.656%. The carry in yield terms is 0.9bp (4.656% - 4.647%).

Calculating carry for an inflation-linked bond We take the OATei 3% July 2012 as our example: 

On 5 November 2007 (settlement date 8 November), the market price is 105.706, corresponding to a real yield of 1.727%. The index ratio is 1.12152 and the accrued 0.86885, so that the invoice price is 119.5258 (1.12152 x (105.706 + 0.86885) ). The bond is bought at this price.



The bond is financed by a repo at 4.14%. After one month, the cash due to reimburse the loan is 119.9325 and the index ratio is 1.12611. The accrued interest in real terms at this date would be 1.11475. So the unadjusted clean bond price for the strategy to break even is 105.3868 (119. 9325 / 1.12611  1.11475).



The yield for a breakeven strategy is 1.776%. The carry in yield terms is 4.9bp (1.776% 1.727%).

As explained above, the carry of an inflation-linked bond depends on the current index ratio and the index ratio at the end of the period. This ratio is a function of the past values of the index, through the lagging system (see Lag and indexations section on page 47 above). In most cases, the index ratio for a one-month carry will be fully known. For a carry over a longer period, the index ratio will depend on index forecasts, calculated either from market quotes or from economic forecasts. Dealers usually prefer to use economic consensus for short-term forecasts. The methodology used to calculate inflation forecasts from market prices will be explained in the section on calculating the CPI forward curve (page 65). A last point to note is that the index ratio is not constant over time and can change significantly due to seasonal effects. This has a large impact on linkers  carry, which is significantly more volatile than that of nominal bonds. To illustrate this, we show the carry of the OATei 09 and the OAT July 09 historically in the left-hand graph below. The nominal carry moves between 3 and -3 bp. The linker carry oscillates between 26 and -30bp. The size of the oscillations increases as the maturity of the bond shortens, meaning that the shorter the bond, the more important the seasonality effect on the bond carry (as defined previously in yield terms). The seasonal impact on the carry defined in yield terms is therefore less significant on long-dated issues.

These seasonality effects also significantly impact the BEIR forward, especially for short-term bonds. We illustrate this effect in the right-hand graph below. The table underneath provides some examples of carry and forward BEIR for some inflation-linked euro zone bonds.

30 238

20 218

10 198

0

178

-10

158

-20

138

-40 Dec-01

beir

OATei 3% 25-Jul-09

-30

OAT 4% 25-Apr-09

98

Dec-02

Dec-03

Dec-04

Dec-05

Dec-06


Bond type Description

OATei OATei OATei OATei OATei BTANei OATi OATi OATi OATi OATi BUNDei BTPei BTPei BTPei BTPei BTPei BTPei BTPei GGBei GGBei CADESi CADESi CADESi

beir fwd

118

3% Jul 2012 1.6% Jul 2015 2.25% Jul 2020 3.15% jul 2032 1.8% jul 2040 1.25% Jul 2010 3% Jul 2009 1.6% Jul 2011 2.5% Jul 2013 1% Jul 2017 3.4% Jul 2029 1.5% April 2016 1.65% Sep 2008 0.95% Sep 2010 1.85% Sep 2012 2.15% Sep 2014 2.1% Sep 2017 2.6% Sep 2023 2.35% Sep 2035 2.9% Jul 2025 2.3% Jul 2030 3.4% Jul 2011 3.15% Jul 2013 1.85% Jul 2019

Mar-01

Mar-02

Mar-03

Mar-04

Mar-05

Mar-06

Mar-07


Real yield

1.51 1.68 1.91 2.09 2.08 1.40 1.54 1.55 1.61 1.87 2.17 1.75 1.71 1.56 1.66 1.77 2.00 2.27 2.39 2.30 2.41 1.58 1.66 1.95

BEIR

220.09 222.37 224.74 242.59 244.11 220.41 218.59 210.79 216.14 218.36 232.30 215.99 213.61 218.25 217.32 219.63 219.77 227.10 245.77 239.58 247.23 208.29 211.45 221.19

1Mth Carry ILB 3Mth Carry ILB 6Mth Carry ILB 1Mth Fwd BEIR 3Mth Fwd BEIR 6Mth Fwd BEIR

6.16 3.86 2.68 1.72 1.26 10.29 17.55 7.39 4.92 3.05 1.89 3.63 49.19 10.31 6.18 4.56 3.47 2.53 1.65 2.34 1.91 7.66 5.08 2.72

-9.49 -4.93 -2.61 -1.33 -0.99 -18.23 0.15 0.24 0.38 0.96 1.03 -4.32 -82.77 -15.45 -8.06 -5.23 -2.98 -1.47 -0.79 -1.34 -0.91 0.39 0.63 1.03

6.82 5.18 4.36 3.14 2.27 10.18 22.41 7.91 5.43 4.51 3.48 6.83 235.96 13.02 8.19 6.68 6.03 5.07 3.48 4.85 4.13 8.45 6.04 4.30

212.94 218.19 222.08 241.10 243.04 207.86 197.92 201.96 210.53 215.22 230.64 212.09 160.21 206.64 210.41 214.82 216.46 224.91 244.52 237.60 245.71 199.19 205.68 218.49

226.38 226.23 227.32 244.53 245.61 231.06 207.03 205.82 213.53 217.01 231.90 219.40 275.35 229.34 223.04 224.03 223.16 229.52 247.73 241.92 249.24 203.17 208.58 220.13

206.51 214.96 220.29 240.65 242.85 193.16 164.77 192.71 206.03 213.05 230.06 207.19 -137.70 194.96 203.98 211.15 214.53 223.94 244.65 236.75 245.31 189.67 200.73 216.81

Source: SG Fixed income Research

What can Bloomberg tell us? Many investors use Bloomberg to analyse inflation-linked bonds. The FPA function calculates the forward price and carry in terms of yield. We can input the settlement date (usually three working days hence), the current market price, the repo or financing rate and the termination date or horizon of the carry. Assumptions concerning the CPI fixing at termination can be specified and the index ratio at the horizon (term index ratio) is calculated.

The bottom field summarises all the results: the forward price (unadjusted clean price), the full forward price (adjusted dirty price or forward invoice price), the drop in price (gain or loss due to the passage of time or carry in monetary amount), the YYIELD field (forward yield to maturity calculated to cancel the P&L of the strategy) and yield drop (difference between the initial and the forward yield).

Inflation Swaps In this section we concentrate on interbank products in the real economy. We first answer some questions about different swap products: what are the similarities between a nominal swap, an inflation swap and a real swap? Which are liquid and why? We then take a detailed look at the mechanisms and characteristics of real and inflation swaps. And finally we explain how the quotes of the most liquid swaps (zero coupon inflation swaps) can be used to estimate forward values for the CPI Index.

The inflation swap market, like other inflation-linked instruments, has developed at a fast pace over the past few years. Inflation swaps can be an effective alternative to inflation-linked bonds for pension funds and liability managers: they are not limited by issuance levels and are more flexible in terms of matching duration. Unfortunately, they still suffer from relatively lower liquidity and less transparent pricing than inflation-linked bonds. Some investors feel that they may find it difficult to mark to market an inflation swap book or to evaluate the additional swap counterparty risk. Despite this, the interbank market has boomed and volumes in the Euromarket have skyrocketed since 2002. A fixed-rate swap (nominal, real or inflation-linked) is a transaction in which a predefined p redefined floating cash flow is exchanged for a fixed one. Such transactions are generally entered into with no exchange of money upfront, as the fixed rate is adjusted to price the fair value of the transaction. In the fixed income world, there are various ways of structuring a swap, depending on the chosen underlying and the calculation method used to obtain the floating rate. The diagram below offers a synthetic view of the different possible swaps:

Y o Y

IR Market

Inflation Market

Real Market

Standard IRS

YoY YoY Swap Swap

ZC YoY YoY swap

Good Liquidity Poor Liquidity

C Z

ZC IRS

ZC Swap

ZC Real Real swap No Liquidity

B L I

Real Swap


In the nominal market, market, the most liquid swap is the standard vanilla Libor swap. swap. This can be seen as a year-on-year swap. The floating rate used is the Libor index, which is the ratio of two discount factors. It is paid at regular intervals. 

In the inflation market (the market (the market whose underlying is the CPI index), the most liquid swap is the zero coupon swap. swap. The year-on-year swap based on regular payment of the CPI ratio exists, but is much less liquid. However - as we will show below - the inflation options market is much more advanced in the year-on-year space. The main advantage of YoY swaps is their suitability as a hedge for inflation-linked options. 

In the real market (i.e. market (i.e. the market based directly on real rates), the most liquid swap is the real swap, whose mechanism we will also explain below (pages 63-4). Zero coupon real swaps are starting to generate some interest among investors and are quoted by some dealers. A YoY real swap would be based on a real Libor rate, defined as a ratio of real discount factors. Although it is attractive in terms of real exposure, this kind of transaction remains very rare for now. 

We will now look at the mechanisms of the most liquid inflation and real swaps and show how these instruments can be used to construct a projection curve for the CPI indices.

In this section we focus on the two main types of inflation swaps: zero coupon and year-on-year. We also explain the mechanism of the real swap.

The transaction is similar to a standard swap transaction. At inception, two counterparties agree to exchange the following cash flows at maturity: The inflation seller or payer payer agrees to pay at maturity the inflation return over the holding. The inflation return is defined as the ratio of the CPI index at maturity to the CPI index at a start date called the base. 

The inflation buyer or receiver receiver agrees to pay at maturity a fixed rate accrued over the holding period. The fixed rate is calculated in such a way that there is no exchange of cash flows at the inception of the transaction. It is usually called the swap breakeven (BEV). (BEV). 

This transaction is a way for the inflation buyer to index his investment profile to inflation for a given maturity.

Inception

Inflation Seller

Inflation Buyer

CPI(T)/CPIbase – 1 maturity

Inflation Seller

Inflation Buyer (1+BEV)T-1


When the CPI base value is not known at inception, the swap is a forward starting inflation swap. swap. When the CPI base value is known, it is a spot starting inflation swap. swap . Dealers use spot starting inflation swaps to quote prices in the market. By convention, payment occurs in the same month and on the same day as the value date. Quotes are given for an exact number of years (2Y, 5Y, 10Y etc). For example, a 10Y swap starting on 25 November 2007 will mature on 25 November 2017.

Calculation of the CPI values generally follows the lagging conventions of the related cash market - for example, in the European market the reference index is subject to a three-month lag. As explained in the bonds section, this is due to the index publication lag: the August number is known only by midSeptember and the September number is known by mid-October. Because two numbers are necessary to calculate the daily reference index (base for the accrued coupon calculation for inflation-linked bonds), the August and September numbers are used in November. There are two conventions for fixing the CPI base for inflation swaps, depending on geographical location: The fixed base convention: This convention considers that the base is set for one whole month. This is the case for European and UK inflation. For example, for any HICPxT swap starting in November, the basis is the August HICP number (m - 3). This also means that when payment occurs at maturity in November, the August CPI fixing will be used to calculate the final cash flow. The main advantage of this convention is that all the swaps trading within the same month have exactly the same final pay-off. This simplifies inflation swap book management. 

The interpolated base convention: This consists of interpolating the reference index, in a way similar to that used to calculate the accrued interest for inflation-linked bonds. This is the convention used for French and US inflation. An inflation swap starting on 25 November and linked to French inflation would have a base index value calculated as the interpolation between the August and September fixings. By convention, the same calculation is made at maturity. 

All conventions and calculations are defined by the International Swaps and Derivatives Association (ISDA) in a reference document 8. The table below summarises the conventions for the main markets.

UK

2M lag

Australia ZC Interpolated 6M lag

Sweden ZC Interpolated 3M lag

Swap reference index (ISDA Def)

Non-revised All ItemsRPI

Non-revised AUD CPI

SEK Non revised CPI

Liquidity in swap market

Very good

Low

Low

Swap type

ZC based

Swap Lag

Canada ZC Interpolated 3M lag

US ZC Interpolated 3M lag

France ZC interpolated 3M lag

Europe

Greece

Italy

ZC Based

ZC Based

ZC Based

3M lag 3M lag 3M lag GRD non Unrevised HICPxT NICxT or NIC Non-revised US Non Non revised revised HICP or all items or or FOIxT or CAD CPI revised CPI-U FRC CPI or non Revised All items FOI revised CPI Low

Good

Very good

Very good

Low

Average

Japan ZC interpolated 3M lag JPY non revised CPI excl. Fresh food Average

Germany

Spain

ZC Based

ZC Based

3M lag

3M lag

DEM Non revised CPI

ITCPI

Low

Average

Source: SG Inflation Desk and SG Fixed Income Strategy

The zero coupon breakeven quoted by the market is useful for obtaining meaningful information on market expectations. As we will explain in one of the following subsections, zero coupon breakeven can be used to calculate either CPI forward values or real zero coupon term structure from the market quotes.

8

2006 – ISDA Inflation Derivatives Definitions

Valuing zero coupon swaps is much easier than valuing their YoY counterparts and can be done using a simple non-arbitrage argument. The inflation leg of a zero coupon swap can be written in function of the CPI fixing at maturity:

⎡

⎞⎤ − 1⎟⎟⎥ ⎝ CPI 0 ⎠⎦ ⎛ CPI T

ZCInflaLeg (t ) = E t N ⎢ B N (t , T )⎜⎜

⎣

In this expression, BN is the nominal discount factor or zero coupon price. CPIT is the CPI value at maturity and CPI0 is the CPI value at the start date. As we will explain in more detail in the Pricing Inflation Derivatives section, the real and nominal economies are analogous to the foreign and domestic economies for FX products. In virtue of this analogy, the relationship between nominal end real zero coupon bond prices and the CPI (analogous to the FX rate) is: E t R [CPI 0 B R (t , T )] = E t N [CPI T B N (t , T )] E t R is the real economy expectation at time t and E t N is the expectation in the nominal economy at the same time. BN is the nominal discount factor or zero coupon price and BR is the real discount factor. This leads to the following simplified expression for the inflation leg of the zero coupon swap: ZCInflaLeg (t ) = B R (t , T ) − B N (t , T ) The other leg (non inflation-linked) is given by:

(

)

ZCFixedLeg (t ) = B N (t , T ) (1 + BEIR (T )) − 1 T

Zero coupon swaps can be valued without a model, using a non-arbitrage argument. This result is essential, as it allows the real curve term structure to be deduced from zero coupon swap market prices and the nominal structure. In practice, the market quotes the breakeven at the level where the transaction is zero-cost at inception. This is equivalent to equating the fixed leg and the inflation leg above. After a little algebra, we can find the zero coupon price for maturity T in the real economy: B R (t , T ) = (1 + BEIR (T )) B N (t , T ) T

This is obtained for each maturity quoted by the market. For intermediate maturities, the real discount factors can be inferred, taking seasonal effects into account. Another way to exploit the above relationship is to write the real expectation in the forward measure, T: CPI 0 B R (t , T ) = B N (t , T ) E t N ,T [CPI T ] So that the expected value of the CPI index at maturity is given by dividing the real by the nominal discount factor: CPI (T ) = I 0

B R (t , T ) B N (t , T )

In the current inflation market, YoY swaps are not yet liquid. This is mainly because zero coupon swaps came first and are matching all of investors  flexibility needs. However, it is interesting to understand the definition, specificities and nature of YoY transactions because the inflation options market is mainly based on YoY ratios. A YoY swap is a transaction engaging two counterparties in a bilateral contract: 

The inflation seller pays the inflation ratio over the past year at regular intervals. In Europe, payments are usually annual.



The inflation buyer pays either a constant rate or the Libor minus a spread. The fixed rate or margin is calculated so that the transaction is zero-cost at inception.

The YoY swap allows the inflation buyer to receive regular payments indexed to inflation. YoY swaps can be replicated by a series of forward starting zero coupon swaps. For a spot starting transaction, the first inflation payment is exactly the same as that for a 1Y zero coupon swap. For the other payments, the base value of the index is unknown. Intuitively, the forward starting CPI ratio should depend not only on the volatility of the final CPI fixing (as in the zero coupon swap case), but also on the volatility of the CPI fixing at the beginning of the period. This could lead to the simplistic conclusion that the forward CPI ratio is the ratio of the projected CPIs as calculated from the zero coupon swap prices. This is not true, especially because of this extra volatile component. In general, the forward CPI ratio will be the ratio of the two CPI projections plus a correction term, the convexity adjustment.

Inception

Inflation Seller

Inflation Buyer

CPI(T )/CPI(T i i-1 )-1 Every year until maturity

Inflation Seller

Inflation Buyer Libor – spread or Fixed rate


As YoY swaps are over-the-counter instruments with no particular fixed conventions, they come in several different flavours. For example, payment can be spread out over the year, so that the inflation leg is still based on the YoY ratio but is paid on a semi-annual, quarterly or monthly basis. The YoY ratio can also be replaced by a month-on-month ratio, where the inflation leg pays the ratio of the CPI over one month. However, this type of swap is exposed to seasonal variations, which need to be taken into account in the pricing.

The YoY swap pays the CPI ratio over a one-year period. If we concentrate on a single cash flow from the inflation leg, its value before the first fixing date is given by:

⎡

⎞⎤ − 1⎟⎟⎥ ⎝ I (T i −1 ) ⎠⎦ ⎛ I (T i )

YoYInflaLe g (t , T i −1 , T i ) = E t N ⎢ B N (t , T i )⎜⎜

⎣

This expression can be rewritten as an expectation at the time of the first fixing, Ti-1:

⎡

⎡

⎢⎣

⎣

⎞⎤ ⎤ − 1⎟⎟⎥ ⎥ ⎝ I (T i −1 ) ⎠⎦ ⎥⎦ ⎛ I (T i )

YoYInflaLe g (t , T i −1 , T i ) = E t N ⎢ B N (t , T i −1 ) E i N −1 ⎢ B N (T i −1 , T i )⎜⎜

The expectation inside the first set of brackets has exactly the same value as a zero coupon swap at the time of the first fixing. Replacing this by its value (see the previous box, Zero Coupon Swap Valuation ) and doing some elementary algebra leads to the final expression: YoYInflaLe g (t , T i −1 , T i ) = E t N [ B N (t , T i −1 ) B R (T i −1 , T i )] − B N (t , T i ) The first term is the price of a derivative paying the value of a real zero coupon bond at time Ti-1. If the real rates were deterministic, this would be the present value of this real zero coupon bond paid in the nominal economy. Unfortunately, real rates are stochastic, and the remaining expectation is not simple to calculate: it depends on the assumptions made on nom inal and real diffusion, and therefore therefore on their volatility and correlation. This adjustment is known as the YoY the YoY convexity correction and is fundamental for obtaining the correct price of YoY swaps and YoY options, as we will show in the section on Pricing Models page 87).

Real swaps are designed to synthetically replicate the flows of inflation-linked bonds. Two counterparties sign up to the following kind of contract: 

The inflation seller will pay annually a fixed real rate X, applied to an inflated notional. As with inflation-linked bonds, the notional is multiplied by the inflation ratio, whose reference is the price index at inception date. At maturity the inflation seller pays back the total inflated notional.



In exchange, the inflation buyer pays a Libor rate, typically the 6M Euribor. At maturity, the inflation buyer pays the non-inflated notional.

The fixed real rate X is calculated so that the transaction is zero-cost at inception. This product offers a synthetic way of transforming a floating rate note into an inflation-linked one. Moreover, combined with a standard vanilla swap, a fixed-rate bond can be synthetically changed into an inflation-linked one. These swaps are increasingly popular. Dealers are now quoting real rates on screen and the number of transactions is increasing substantially. They offer constant revenue in real terms and as such are an attractive tool for asset and liability management. Pricing details are given in the technical box below. Real swaps offer an alternative way to obtain the real discount term structure, as they are expressed in pure real terms at inception. Over the life of the transaction, a real swap receiver will essentially be exposed to real rates, as the sensitivity of the Libor leg to the nominal curve is marginal.

Other kinds of real swap could be envisaged. As it is possible to construct a real discount curve, one could imagine defining a real Libor rate as the cost of borrowing money over a short period in real terms. Swapping a fixed rate against this real Libor would be equivalent to a standard vanilla swap, but expressed in real terms. Alternatively, one could envisage a transaction where there would be only one real payment and one Libor payment. This would be the equivalent transaction to the zero coupon inflation swap, but in the real economic space.

Inflation Seller

Inception

Inflation Buyer

X% x CPI(T )/CPI i base Inflation Seller

Every year until maturity

Inflation Buyer Libor CPI(T )/CPI(T i i-1 )

Inflation Seller

At maturity

Inflation Buyer Par


Valuing a real swap is very similar to zero coupon valuation and we can use an analogy to foreign currency exchange to simplify the expression: M I (T ) I (T )⎤ RT B N (t , T i ) i + B N (t , T N ) M ⎥ = RT B R (t , T i ) + B R (t , T M ) I 0 I 0 ⎦ i =1 ⎣ i =1

⎡ M

Re alSwapInfl aLeg (t ) = E t N ⎢

∑

∑

In the previous expression, BR is the real zero coupon price and RT is the fixed real rate associated with the real swap of maturity T. The Libor leg expression is: LiborLeg (t ) =

M

∑ (t − t i

i −1

) B N (t , T i ) L (t , T i−1 , T i )+ B N (t , T M ) = 1

i =1

The real swap breakeven is calculated in such a way that the real swap is entered at zero cost: RT =

1 − B R (t , T M ) N

∑ B

R

(t ,T i )

i =1

Moreover, quotes can be found in the market for the 1Y, 2Y coupon prices can be calculated recursively: Year 1 B R (t ,1Y ) =

 30Y

Year 2 1 1 + R1Y

B R (t ,2Y ) =

real swap breakeven. Using these quotes, the real zero (  )

1 1 + R2Y

(1 − R2Y B R (t ,1Y )) (  )

Year m

m −1 ⎛ ⎞ B R (t , mY ) = ⎜1 − RmY ∑ B R (t , iY )⎟ 1 + RmY ⎝ i =1 ⎠

1

A CPI forward curve is calculated in two steps: 1. Using the most liquid swap instruments, we calculate the CPI forwards for the dates the market quotes the transactions,

2.3

220 200

2.1

180 160

1.9

140 120

1.7

100 80

1.5 2008 2011 2014 2017 2020 2023 2026 2029 2032 2035

Source: SG Quantitative Strategy – SG Inflation Trading, November 2007

2008 2008 2011 2011 2014 2014 2017 2017 2020 2020 2023 2023 2026 2026 2029 2032 2035 Source: SG Quantitative Strategy – SG Inflation Trading, November 2007

2. Once the CPI forwards are known for a certain date, we choose an interpolation method to calculate intermediary points. The difficulty here lies in integrating seasonal adjustments.

8.0%

2.3

6.0%

2.2

4.0%

2.1

2.0%

2

0.0% -2.0%

1.9

-4.0%

1.8

-6.0%

Unadjus djustted BE BEV 1.7 Aug-09

Aug-14

Aug-19

Aug-24

Seaso asonality lity Adjus justed BEV BEV Aug-29

Aug-34

Aug-39

Source: SG Quantitative Strategy – SG Inflation Trading

Aug-08

Aug-10

Aug-12

Adjusted Fwd Infla Infla Yield

Aug-14

Aug-16

Aug-18

Unadjusted Unadjusted Fwd Infla Yield Yield


Lets look at these two steps in more detail: CPI forwards

Zero coupon swaps are the most liquid instruments on the inflation derivatives market. They are quoted for annual maturity, with the maturity date corresponding to the reference month, which changes every month. In the technical box zero coupon swap valuation (page 61), we explained the link between zero coupon breakevens and CPI forwards. Using these relationships, the forward price index can be simply deduced from the swap breakevens: CPI (T ) = I 0 (1 + BEV )

T

, where T is the maturity of the swap and I0 the price index reference value.

Let us take a numerical example. On 22 November, the mid breakeven for the 10Y zero coupon swap on European inflation is 205.3bp, on an August 2007 fixed basis. In August, the unrevised HICP fixing is 104.19 and the 10Y nominal discount factor is 0.64. As explained in the technical box, the value of the fixed leg is given by: Fixed Leg = BN(21/11/07, 21/11/2017) x [ (1+ BEV(10Y) )10  1 ] = 0.64 x [ (1+0.02053)10  1 ] = 0.1443

However, the expected CPI value at maturity is also unknown. The inflation leg can be expressed as a function of this number. Taking the indexation lags into account, this gives: Inflation Leg = BN(21/11/07, 21/11/2017) [ CPI(31/08/2017) / CPI(31/08/2007) - 1 ] = 0.1443

The CPI projection for the month of August 2017 is therefore 127.6. The zero coupon swap market is therefore the standard market way of obtaining the CPI projection curve. However, it gives the CPI projection for one particular month (August in our example). In most cases the CPI forwards are also needed for some intermediary dates, so it is vital to find an adequate interpolation method. Such a method should incorporate some seasonal adjustment to account for inflation variations over a year. Let us now define this interpolation method: CPI interpolation

Once the CPI forwards have been calculated from zero coupon prices, intermediate values need to be interpolated. A simple approach would involve the linear interpolation of CPI values estimated from the zero coupon price. But this approach would completely ignore monthly seasonal variations and would severely misprice some inflation-linked products. A better alternative is to consider that the CPI reference numbers are the product of three components: 1) A reference level, which is the base level used to price current zero coupon breakevens in the market; 2) An exponential inflation factor calculated from quoted zero coupon breakevens. The inflation rate is assumed to be piecewise constant; 3) An exponential seasonal adjustment, which equals 100% on the fixing date of the base index. The seasonality yield is also assumed to be piecewise constant. To return to our example, the table below gives the summary of the zero coupon (fixed basis) breakevens, as well as implied CPI projections at maturity for the reference fixing date of 21 November 2007. In November, the August fixings are completely known from market quotes.

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y

Break-even 2.119 1.972 1.925 1.914 1.917 1.934 1.957 1.983 2.016 2.049

Maturity 22-Nov-08 22-Nov-09 22-Nov-10 22-Nov-11 22-Nov-12 22-Nov-13 22-Nov-14 22-Nov-15 22-Nov-16 22-Nov-17

Maturity Inflation Reference CPI forward Fixing Base CPI Projection yield Aug-08 104.19 106.40 2.10% Aug-09 104.19 108.34 1.81% Aug-10 104.19 110.32 1.81% Aug-11 104.19 112.40 1.86% Aug-12 104.19 114.57 1.91% Aug-13 104.19 116.88 2.00% Aug-14 104.19 119.33 2.07% Aug-15 104.19 121.91 2.14% Aug-16 104.19 124.69 2.25% Aug-17 104.19 127.62 2.32% Source: SG Quantitative Strategy

The projected fixing in August 2008 can be found the table above. Assuming that on the last day of August the seasonal adjustment is 100%, we can deduce the constant inflation rate over the first year: CPI(31Aug08) = CPI(31Aug07) x exp( i(0, 1Y) x 1 ) = 106.40, so that i(0,1Y) = 2.1%

Similarly, the August 2009 fixing is known and can be expressed in function of the August 2008 fixing. The constant inflation rate for the 1Y to 2Y period can be calculated from this: CPI(31Aug09) = CPI(31Aug08) x exp( i(1Y, 2Y) x 1 ) = 1 08.34, so that i(1Y,2Y) = 1.81%

We can calculate the whole term structure of the forward inflation rate recursively. The non-adjusted CPI reference can then be calculated from this inflation rate curve. The seasonal components are calculated using the seasonal factors, which are either found using statistical software or provided by market consensus. Using the seasonality factors given in the Seasonality section (page 29) as an example, we rebase the seasonal adjustments in the table on the left below. The month of August is taken as a reference and its seasonal adjustment is therefore zero. The unadjusted reference index for a given month is calculated as the product of the previous months reference index and the monthly exponential yield. For example, CPI on 30 September 2009 is calculated as follows: CPIU(30Sep09) = CPIU(31Aug09) x exp( 1.81% / 12 ) = 108.5

The adjusted reference index is calculated as the unadjusted index multiplied by the corresponding seasonal adjustment. With the previous example, this gives: CPI(30Sep09) = CPIU(30Sep09) x exp(0.07%) = 108.58

The right-hand table shows calculations for a whole year. It is slightly more complicated to calculate a date in the middle of the month. The exact formula is given in the technical box on the next page.

Cumulated Seasonality Seasonality Adjustment August MoM Based 31-Jan -0.35% -0.34% 28-Feb 0.16% -0.18% 31-Mar 0.23% 0.05% 30-Apr 0.22% 0.27% 31-May 0.06% 0.33% 30-Jun -0.09% 0.25% 31-Jul -0.19% 0.05% 31-Aug -0.05% 0.00% 30-Sep 0.07% 0.07% 31-Oct -0.04% 0.03% 30-Nov -0.15% -0.13% 31-Dec 0.14% 0.01%

31-Aug-09 30-Sep-09 31-Oct-09 30-Nov-09 31-Dec-09 31-Jan-10 28-Feb-10 31-Mar-10 30-Apr-10 31-May-10 30-Jun-10 31-Jul-10

Unadjusted Seasonal Adjusted Reference Factor Reference 108.34 100.00% 108.34 108.50 100.07% 108.58 108.67 100.03% 108.70 108.83 99.88% 108.70 109.00 100.01% 109.01 109.16 99.66% 108.79 109.33 99.82% 109.13 109.49 100.05% 109.55 109.66 100.27% 109.96 109.82 100.33% 110.19 109.99 100.25% 110.26 110.16 100.05% 110.21 Source: SG Quantitative Strategy

The projected CPI reference index is assumed to be the product of the initial CPI reference, a ‘discount’ function to represent accreting inflation and a seasonality adjustment: T

T

0

0

∫ i (u )du × e ∫ s( u )du CPI (0, T ) = CPI (0 )× e In this expression, is the inflation rate and

the seasonal adjustment.

Both the inflation rate and the seasonal component are generally assumed to be piecewise. The inflation rate is calibrated using the available swap breakevens and seasonality can either be calculated using statistical analysis or market consensus. The inflation rate is calibrated every year in the same month. This is the base month for the breakeven quotations. During this month the seasonal adjustment is assumed to be null, so that: j −1

∑= i (T −T − )+i (T −T − )

CPI (0, T j ) = CPI (0 )× e k 1

(T ) =

j j 0..n

k

k

k 1

j

j 1

j

= CPI (0, T j −1 )× e

(

i j T j −T j −1

)

are the dates on which the breakevens are known from dealers in the market, assuming that they fall in the same

month of the year. For any date between two market points, the CPI is calculated using the formula above and the calibrated inflation rates. For example, at a time such as t ∈ T j −1 , T j , where t is in the n th month after the base month and is the day of the month,

is the number of days in the month and ( sk )k =1..12 is the MoM rebased vector of seasonality adjustment (monthly

seasonal adjustment):

CPI (0, t ) = CPI (0, T j −1 )× e j

(

i t −T j −1

)

×e

∫

t

n −1

s (u ) du

T j −1

= CPI (0, T j −1 )× e

(

i j t −T j −1

)

×e

∑= s ( k )+ s( n)×d N k 1

This formula allows us to calculate the forward CPI for any date. In its construction, it is consistent with all market swap breakevens.

The seasonal component in the swap breakevens tends to even out over time. This is because the same seasonal adjustment is applied every year, while the swap breakevens are annualised. The seasonal factor is mechanically reduced as the maturity of the swap increases. This can be seen in the graph on the bottom left-hand side on page 68.

Inflation-linked asset swaps Asset swaps have been available in the market for some time, allowing investors to turn a fixed rate bond into floating rate structures. Although they started to develop at a later stage than simple interest rate swaps, they are now very popular among investors interested in the nominal bond market. Similarly, when sovereigns started to issue inflation-linked bonds, inflation-linked asset swap products appeared in the market. These can serve various purposes, from balance-sheet management to relative value strategies. In this section we first review the different asset swaps offered by the market, then cover the relative value indicators and strategies available within the asset swap space.

There are many different kinds of inflation-linked asset swap, but the two main ones are the par/par asset swap and the proceeds asset swap. Par/par is more common in the euro zone, while proceeds asset swaps are favoured in the UK and the US. Entering into either kind of asset swap is a two-step process. From the asset swap buyer s point of view, this involves: Buying an inflation-linked bond, at par in the case of par/par asset swap or at dirty market price in the case of a proceeds asset swap. The bond pays the asset swap buyer the real coupon multiplied by the inflated notional until maturity; 

Entering into a swap transaction, where the inflation-linked coupon paid by the bond is swapped against a floating nominal index (typically Libor or Euribor) plus or minus the asset swap spread. The notional amount for this floating leg is par or proceed (i.e. 100 or the dirty market price of the bond). At maturity, the inflated notional of the bond is swapped against par for a par/par asset swap, or against the initial dirty price for the proceeds asset swap. 

In terms of cash flows, this means that: 1) At inception, the bond is bought either at par or at its market dirty price; 2) The swap is initiated at the same time. For a proceeds asset swap, the net value of the transaction at inception is zero: the cash paid by the asset swap seller in exchange for the bond is exactly the dirty market price. In the case of a par/par asset swap, this amount is the difference between the bond invoice price and par; 3) Over the life of the trade, the asset swap buyer receives a floating Libor payment plus or minus a spread. The inflated coupons paid by the bond to the asset swap buyer are transferred to the asset swap seller; 4) At maturity, the asset swap buyer is paid back either par or the initial dirty price. The inflated notional paid by the bond is transferred to the asset swap seller. In the nominal world, differences between par/par and proceeds asset swaps are irrelevant, but this is not the case in the inflation world. Par/par swaps do not take into account the fact that the notional of a linker is potentially already inflated - for example, buying 100mn of OATei July 2012 in October 2007 corresponds to 112mn of inflated notional. The date on which the par/par asset swap is entered therefore has an impact on the spread. However in a proceeds asset swap, the notional on the swap is

equal to the bond invoice price. In this case, the spread level does not depend on the inflated notional. This methodology is therefore more consistent with asset swap calculations in the nominal world.

Inception

Asset Swap Seller

IL Bond

Asset Swap Buyer

Par or Proceeds amount

Libor +/- spread on par or proceeds amount Coupon payment date

Asset Swap Seller

Asset Swap Buyer IL Coupon = CPI(T )/CPI(0)*R i

IL Bond IL Coupon = CPI(T )/CPI(0)*R i

Par or Proceeds amount maturity

Asset Swap Seller

Asset Swap Buyer IL Redemption = max(CPI(TN )/CPI(0),1)

IL Bond IL Redemption = max(CPI(T)/CPI(0),1)


An inflation-linked asset swap spread is calculated in a similar way to a traditional nominal asset swap spread. The difference lies in the initial calculation of the inflation index fixing. The bonds future payments depend on the realised values of the CPI fixings, which are not known in advance. Fortunately, the inflation swap market gives market projections of the future fixings. As explained in the previous subsection, the CPI fixings are easily calculated from the zero coupon swap breakevens. Once the inflation index projections have been estimated, the asset swap spread is calculated in a similar way to that for nominal bonds. Two elements are required for this task - the bond market price and a discount curve: 

Data providers or brokers provide the bond market price;



The discount curve is simply the nominal zero coupon curve. It is bootstrapped from the money market instruments and the nominal interest rate swaps. It contains an implicit interest rate risk linked to macroeconomic expectations, and a counterparty risk linked to the default risk of the swap counterparty. As the counterparty is usually a bank or a financial institution, the credit risk is considered to be that of an average AA counterparty.

Armed with the CPI projections and the discount curve, we can calculate the bonds implied value as the discounted value of its cash flows. Comparing this implied value with the market price and dividing

by the bond PV019 produces the asset swap spread. For a proceeds asset swap, the spread is equal to the par/par asset swap spread divided by the bonds dirty price. In reality, the bond-holders capital is protected from several years of consecutive deflation thanks to the implicit floor at par on the notional at maturity. This floor is assumed to have no value or at most a negligible value in the calculation of the asset swap spread (above).

max( 100 At inception, par is received



Throughout the life of the asset swap, notional inflates as inflation increases 3%x

I25Jul08 I25Jul01

3%x

I25Jul09 I25Jul01

3%x

I25Jul10 I25Jul01

3%x

I25Jul11 I25Jul01

3%x

I25Jul12 I25Jul01

, 1)

At maturity, inflated notional is received

I25Jul12 I25Jul01

I25Jul01 = 92.98 Jul08

 And IL

Oct08

Jul09

Oct09

Jul10

Oct10

Jul11

Jul12

 And L – 16bp

L – 16bp

L – 16bp

L – 16bp

L – 16bp

L – 16bp

L – 16bp

bond is delivered

Pmkt=104.8

Oct11

Throughout the life of the asset swap, fixed notional is paid on the Libor leg

L – 16bp

par is paid back 100


Who buys asset swaps? With the increasing demand for inflation-linked swaps, dealers have to pay the inflation-linked flows. To hedge their book as a whole, they buy inflation-linked bonds and sell the associated asset swaps. By doing this, they still receive the inflation-linked coupon versus a nominal floating index, but reduce their exposure on the nominal part of the transaction. Inflation-linked asset swaps are primarily used by dealers to manage their balance sheet exposure. Some funds are also willing to invest in asset swaps, purely as instruments of speculation. For example, Libor funds are kinds of hedge funds funded at Libor and which invest at Libor plus a margin. Inflationlinked asset swaps are usually negative. However, long-term bonds on riskier sovereigns can offer positive rewards. The BTPSi 2035 issued by Italy, for instance was offering Libor +18.7bp (Oct 2007), while the OATi 2029 was quoted at 24.7bp on the same day. Other investors are willing to invest directly in the asset swap package. This was the case for example when Greece recently issued an inflation-linked bond (GGBi 2030). Some relative value opportunities between inflation-linked and nominal bonds can also be found, as explained in more detail at the end of this section. Inflation-linked asset swap pricing is impacted by: Seasonality : its effect is strong when the bond fixing does not correspond to the current base month for the quoted swaps. This is because the bond is hedged with quoted instruments which have a different seasonal risk, and there is more uncertainty on the fixings. 

9

Variation of the bond price to 1bp change in yield

Distortion due to non accretion on the nominal leg: in a par/par or proceeds asset swap, the notional on the Libor leg is constant, while the notional on the real leg is inflated by the inflation ratio. So the accreting notional can diverge substantially from par. This increases the counterparty risk for the asset swap seller. Most of the time, collateral agreement can be set up to mitigate this risk, although this is not always possible. This partly explains the fact that inflation-linked bonds are cheaper on an asset-swap basis than nominal bonds. 

The nominal structure of standard asset swaps can be changed to mitigate distortion and counterparty risks. Possibilities include changing the notional on the nominal leg and earlier payment of the inflated notional due at maturity. This leads to the other kinds of asset swap, which we will look at shortly.

In a par/par asset swap, the two counterparties exchange par (assumed to be equal to 100%) for the dirty m arket price (i.e. the price at which the bond gets bought on the market) upfront. The net upfront cash-flow is not null. In addition, the two counterparties are considered to have an AA counterparty risk, so the usual nominal swap curve can be used for discounting. The bond cash flow and the Libor cash flow are discounted with this curve. The total present value for the transaction is:

UpfrontPayment + BondLeg + LiborLeg = (1 − P MKT ) + P IMPLIED −

N

∑ ∆t ( Lib(T

i −1

i

, T i ) + s ) B N (0, T i ) − B N (0, T N )

i =1

With P

M

IMPLIED

= R ∑ B N (0, T j ) j =1

CPI (T i ) CPI (0 )

+

CPI (T M ) B N (0, T M ) CPI (0 )

The spread is calculated so that this expression equals 0. If there is no accrued payment, i.e. the valuation is done on a fixing date, the Libor leg is equivalent to a single upfront payment which is equal to 100%. The spread is then simply: s =

P IMLIED − P MKT PV 01

where PV01 is the value of a 1bp move on the Libor leg. In a proceeds asset swap, the upfront payment is cancelled out and the notional applied on the Libor leg is the bond market price at inception. The total cash flows can be represented as: UpfrontPay ment + BondLeg + LiborLeg = ( P MKT − P MKT ) + P IMPLIED − P MKT

N

∑ ∆t ( Lib (T i

i −1

, T i ) + s ) B N (0, T i ) − P MKT B N (0, T N )

i =1

Simplifying in the same way as above leads to the foll owing spread value for the proceeds swap: s =

P IMLIED − P MKT P MKT PV 01

The seasonal pattern is implicitly taken into account in the pricing above: the CPI estimates are derived from inflation swaps (see Inflation-linked options below) and include seasonal effects.

Below we show the cash flows of the OATi 2012 as at end-October 2007. The real annual coupon is 3% and the underlying inflation index is HICPxT - based on 25 July 2001 - equal to 92.98. The current market price is 104.8 and the current asset swap spread is -19.8bp. This is a par/par asset swap. The notional paid on the Euribor leg is therefore 100 for the whole life of the transaction. The net cash flow at inception favours the asset swap buyer as the dirty market price is generally above par.

Date 25-Oct-07 25-Jan-08 25-Jul-08 25-Jan-09 25-Jul-09 25-Jan-10 25-Jul-10 25-Jan-11 25-Jul-11 25-Jan-12 25-Jul-12

Swap breakevens

Index Nominal Ratio Discount (1) (2) 1.12 104.19 0.989 106.37 1.144 0.967 0.946 108.29 1.165 0.928 0.908 110.75 1.192 0.889 0.870 113.32 1.220 0.852 0.834 116.00 1.248 0.816

Inflation Discounted Cash Flows (3)=(1)x(2)

33,180

32,429

31,792

31,171

1,048,700 1,177,272 186,040 999,603 4.26 (19.66)

Libor Rate 4.24% 4.59% 4.30% 4.07% 4.26% 4.29% 4.29% 4.34% 4.38% 4.42%

Libor Discounted Cash Flows 186,040 10,682 22,141 20,535 18,740 19,483 18,904 18,820 18,352 18,395 833,550


One of the characteristics of the par/par asset swap - and to a lesser extent the proceeds asset swap is the accreting notional on the inflation leg and the fixed notional on the nominal leg. This produces a distortion of counterparty risk for the asset swap seller: at maturity the asset swap seller pays par or at best the initial market dirty price, while he receives the inflated notional, potentially much higher than par. The counterparty exposure of the asset swap seller increases over time. Posting collateral can solve this issue, and the Credit Support Annex (CSA) of the standard ISDA swap contract can be used. However, posting collateral is not necessarily convenient for all investors. Another way to solve this issue would be to structure asset swaps with an accreting notional on the nominal leg. Several possibilities are readily available, including: 

Fixing the accretion rate at a predefined ratio. Even if realised inflation cannot be calculated exactly, this technique can significantly reduce counterparty risk. Once the accretion rate is fixed, the asset swap valuation is very simple.



Linking the accretion rate to the inflation fixings in the same way as in the inflation leg. This is an ideal solution in terms of cash-flow matching, guaranteeing the same notional on the inflation and nominal legs. However, the nominal leg also depends on the inflation index. This makes pricing much more complicated, as the correlation between the inflation and the Libor fixings is needed as an input. As this type of asset swap is unusual and its valuation is more complicated, a risk premium is usually paid when entering this kind of transaction.

Risk mitigation can be achieved most simply with a fixed accretion rate, which in most cases will significantly reduce the counterparty risk. More complicated structures may introduce some other risks which are not necessarily well understood. The asset swap spread calculation in the fixed accretion case is fairly simple to calculate, as explained below.

The assumptions here are the same as in the par/par and proceeds asset swap case: the two counterparties are considered to have an AA counterparty risk, so the usual nominal swap curve can be used for discounting. Accretion is assumed to be constant: every 6 months, the notional on the Libor leg is multiplied by the accretion ratio, 1+r. The spread is calculated so that the total flows cancel out. UpfrontPay ment + BondLeg + LiborLeg = (1 − P MKT ) + P IMPLIED −

N

∑ (1 + r ) ∆t ( Lib (T i

i

i −1

, T i ) + s ) B N (0, T i ) − (1 + r ) B N (0, T N ) = 0 N

i =1

So the spread can easily be calculated from the relationship above: N

(1 − P MKT ) + P IMPLIED − ∑ (1 + r )i ( B N (0, T i−1 ) − B N (0, T i )) − (1 + r ) N B N (0, T N ) i =1

s =

N

∑ (1 + r ) ∆t B i

i

N

(0, T i )

i =1

Another counterparty risk mitigation technique uses an early redemption asset swap. The idea is to prepay some of the total inflated notional prior to maturity. The total inflated notional is the initial notional multiplied by the CPI return between the issue and maturity dates: ⎛ CPI (T M )

⎞ − 1 ⎟⎟ ⋅ Notional ⎝ CPI (0 ) ⎠

InflatedNo tional (T M ) = ⎜⎜

The total inflated notional can also be viewed as the sum of the increments of the inflated notional at two subsequent payment dates: InflatedNo tional (T M ) =

M

∑ InflatedNo tional (T ) − InflatedNo tional (T i

i −1

) + Notional (T 0 )

i =1

In an early redemption asset swap, part of the notional is repaid at each coupon date. The amount of notional repaid is proportional to the notional accretion and is multiplied by the nominal discount factor until maturity: ⎛ CPI (T i ) CPI (T i −1 ) ⎞ ⎜⎜ ⎟ ⋅ Notional ⋅ B N (T i , T M ) − CPI (0 ) ⎠⎟ ⎝ CPI (0 )

Z-spread can be used as an alternative relative value. Z-spread is the quantity by which the discount curve needs to be shifted so that the market value of the bond and the present value of its discounted cash flows are equal. The discount curve is in this case calculated from the vanilla swap prices. Z-spread can be calculated for any type of bond and as such can be a useful indicator to compare one bond with another. It shows the risk associated with the bond in terms of yield: if the Z-spread is

positive and large, the bond is substantially riskier than the reference Libor curve, usually associated with an AA counterparty risk. Conversely, if the Z-spread is negative, the bond is less risky than the usual swap AA counterparty. This is the case for most government bonds from G8 countries, though the long-term Italian bonds are an exception. An inflation-linked bond and a similar nominal bond (same maturity, same issuer) do not have the same cash flows, especially as the notional of an inflation-linked bond increases over time if inflation remains positive. As inflation-linked coupons are smaller than nominal ones, the credit risk for linkers is essentially concentrated at maturity. Intuitively, the Z-spread for inflation linked bonds should be higher than that for their nominal counterparts, as the total long-term credit risk is more important with the accreting notional. Z-spread can also be a measure of how much the swap and cash market diverge from each other. To understand this point, we can imagine that the discount curve is calculated from the reference government bond. With such a reference, the nominal Z-spread would always be zero. Moreover, the government credit risk would be directly priced in the discount curve. In this case, we can argue that the inflation linked-bond Z-spread is also zero when measured with the government discount curve. However in reality, the inflation-linked bond would have a positive Z-spread. This is due to the CPI fixing used in pricing the bond. As we already explained, the standard market practice when calculating inflation fixings is to use swap market breakevens. As inflation swaps are increasingly popular especially for asset liability management - swap breakevens are becoming more expensive than bond breakevens and CPI fixings calculated from the swap market are slightly higher than those calculated from the bond market. Pricing the inflation-linked bonds with CPI fixings from swap breakevens makes the bond price higher than the market price. To compensate for this, the Z-spread calculated to match the market price is positive. Measuring Z-spread on inflation bonds and comparing it with that on nominal bonds gives a relative measure of the bond market versus the swap market: the bigger the difference between nominal and inflation Z-spread, the more expensive swap breakevens are compared with bond breakevens. The table below gives some indicative levels of Z-spread, accreting, proceeds and par/par asset swap for comparison and illustration purposes.

Bond type

Description

Nominal ZSpread Accreting Proceeds Par/Par ZSpread

Delta Zspread

Clean Price

Real Yield

BTPe BTPe BTPe BTPe BTPe GGBe GGBe BTPe

1.65% 15-Sep-2008 0.95% 15-Sep-2010 1.85% 15-Sep-2012 2.15% 15-Sep-2014 2.10% 15-Sep-2017 2.90% 25-Jul-2025 2.30% 25-Jul-2030 2.35% 15-Sep-2035

-9.00 -6.30 -3.90 -2.20 2.50 7.00 11.50 14.50

-9.00 -6.40 -4.10 -2.40 2.30 6.50 11.00 13.80

-8.60 -6.40 -4.10 -2.30 2.70 7.60 14.00 18.00

-9.50 -6.50 -4.00 -2.50 2.80 9.00 14.00 18.70

-19.10 -15.60 -13.00 -11.40 -7.00 0.30 11.50 10.50

10.00 9.30 9.10 9.20 9.50 6.70 0.00 4.00

99.54% 96.25% 98.34% 99.53% 98.07% 107.12% 97.21% 98.05%

2.016 2.139 2.194 2.231 2.324 2.411 2.460 2.461

BTANe OATe OATe BUNDe OATe OATe OATe

1.25% 25-Jul-2010 3.00% 25-Jul-2012 1.60% 25-Jul-2015 1.50% 15-Apr-2016 2.25% 25-Jul-2020 3.15% 25-Jul-2032 1.80% 25-Jul-2040

-10.70 -9.30 -10.20 -13.00 -11.60 -14.40 -12.50

-10.70 -9.40 -10.40 -13.20 -11.90 -15.10 -13.40

-10.80 -9.30 -11.10 -14.40 -12.80 -16.90 -17.10

-11.00 -11.30 -11.30 -13.80 -14.00 -21.80 -15.80

-22.70 -21.80 -22.00 -25.50 -22.80 -18.90 -17.80

12.00 12.50 11.80 12.50 11.20 4.50 5.30

97.63% 104.43% 96.03% 94.43% 100.61% 118.48% 91.71%

2.028 2.087 2.134 2.196 2.196 2.187 2.152

OATi OATi OATi OATi OATi

3.00% 25-Jul-2009 1.60% 25-Jul-2011 2.50% 25-Jul-2013 1.00% 25-Jul-2017 3.40% 25-Jul-2029

-10.00 -10.00 -9.50 -11.00 -15.20

-10.00 -10.10 -9.60 -11.20 -15.70

-9.50 -10.20 -9.70 -12.40 -17.20

-12.80 -10.80 -11.00 -11.30 -23.80

-23.10 -21.40 -20.60 -22.00 -19.70

13.10 11.40 11.10 11.10 4.50

101.69% 97.46% 101.48% 88.49% 119.63%

2.192 2.244 2.239 2.280 2.264

Source: SG Inflation Trading Desk

Feb-07 0

Apr-07

Jun-07

Aug-07

Oct-07

-5

Feb-07 -4

Apr-07

Jun-07

Aug-07

-6

-10

-8

-15 -20

-10

-25

-12

-30 -35

-14

-40

-16

-45 -50

-18 OAT 5% Apr-12

OATei 3% Jul 2012

Source: SG Fixed Income Strategy Research – Bloomberg

-20

OATei/OAT spread

Source: SG Fixed Income Strategy Research - Bloomberg

Oct-07

Inflation-linked options Inflation options are the next step for inflation market makers. As demand for custom structured products increases, dealers will increasingly need to hedge their inflation volatility exposure. Relative value players will probably have a role to play here to take advantage of market distortion in the volatility space. In this section we review the most common inflation options and look at some of the strategies which can be played through them.

The natural underlying for an inflation option is the CPI index. The most natural option would be a call or put on the inflation rate over a predefined period. This would exactly match the flows on a zero coupon inflation swap. This kind of option might for example pay the difference between the CPI ratio and the strike if the difference is positive and nothing otherwise. A long position on a zero coupon call and a paying position on a zero coupon swap would be strictly equivalent to a position in a capped paying zero coupon swap. This is why we will use the terms cap and floor rather than call and put.

max(CPI(T)/CPIbase-(1+K)T,0) Option Seller

CPI(T)/CPIbase-1 Inflation Seller

Inflation Buyer (1+BEV)T-1 Source: SG Quantitative Strategy

The strike is expressed in annual average inflation growth so that the pay-off of a zero coupon cap is defined as follows: ⎛ CPI (T )

T ⎞ − (1 + K ) ,0 ⎟⎟ ⎝ CPI (0 ) ⎠

ZCCap (T , T , K ) = max ⎜⎜

Some inflation linked bonds (depending on conventions) have an embedded floor at zero on the principal at maturity. This floor guarantees the bond holder at least recovers par at maturity. If the inflation rate is sufficiently low for the floor to have a significant price, the price of the bond will be increased, as it will contain the option premium. Zero coupon caps and floors are the options which are most in line with the underlying liquid swap market, as they share the same underlying. However, in practice zero coupon options are not quoted as frequently as YoY options and are therefore less useful for estimating inflation volatility.

The most liquid options are YoY caps and floors. These transactions are similar to standard caps and floors in the nominal market. In a YoY inflation cap contract the cap seller agrees to pay the cap buyer annually (at each fixing date of the reference inflation rate) either zero or the difference between the YoY CPI ratio and the strike, whichever is greater, in return for a premium paid upfront.

4

bp vol/day

4.5

bp vol /

4

3.5 3

3.5

2.5

3

2

2.5

1.5 1

2

0.5

1.5

0 1

6

Maturity

11

16


21

26

0.0% 0.9% 1.8% 2.7% Strike 3.6%

1 0%

1%

2%

3%

4%

1Y

2Y

5Y

30Y

10Y

20Y


The market quotes YoY option prices in terms of implied volatility, as illustrated in the graph above. There is one volatility number per strike and per maturity. The strikes are usually quoted on an absolute scale (1%, 2%, 3%...) and the maturity for a round number of years (1Y, 2Y, 10Y  ). The one-year YoY cap is special in that it has only one payment and the first fixing is already known. The one-year YoY cap is therefore strictly equivalent to the 1Y zero coupon cap. From the implied volatility number, we can calculate the option price. The market usually quotes volatility in terms of Black volatility. This means that the option premium is calculated by inserting market volatility and option characteristics into the Black formula (see Models section, page 98). The main problem here lies in calculating the YoY forward value. In Building a CPI forward curve  (page 65) we explained how to calculate implied CPI forward values from the zero coupon breakeven. A simplistic view would be to calculate the YoY forward ratio as the ratio of these CPI projections. By doing this we implicitly assume that the forward value of a ratio is the ratio of the forward values. This is generally not true and is not so in this particular case. The correct forward to use is the convexity adjusted one, which is model-dependent. We show how the convexity adjustment can be calculated from some models in the section on pricing inflation derivatives.

max(CPI(T )/ i CPI(Ti-1)-(1+K),0)

CPI(T )/ i CPI(Ti-1)-1 Inflation Seller

Option Seller

Inflation Buyer Libor – spread or fixed rate Source: SG Quantitative Strategy

Like in the standard interest rate market, a real rate swaption provides the option of entering into a real swap at expiry. As explained in the sections above, a real rate swap is a transaction in which two counterparties exchange a floating nominal rate (6M Libor for example) for a real fixed rate multiplied by the accrued inflation ratio over the period. Like with the real rate swap, a real rate swaption can be used to hedge future payments linked to inflation. However, the swaption also provides the flexibility to choose whether it is worth entering into the swap at maturity or not. For example, an investor might want to hedge future real income in one years time. He can choose to wait a year and then enter a into a real swap rate agreement. However, if real rates are expected to decrease over the coming year, it is worth him entering into a real rate swaption. This will lock in the current level as the future real rate to be received by the investor. A real rate swaption can also be used to express a view on the evolution of real rates. For example, an investor who expects real rates to increase can sell a real rate swap (i.e. pay the leg linked to inflation). However, he can also express this view through real rate swaptions, selling a real rate receiver swaption or buying a real rate payer swaption. Depending on the level of volatility, playing a view through swaptions may be more attractive than plain swaps. The whole range of speculative strategies using options can also be expressed through real rate swaptions. For example, if an investor expects the real rate to move sharply, he may initiate a straddle or strangle. However, the problem with real swaptions is their lack of liquidity, which means that swaption price/volatility has to be inferred from YoY caps and floors volatility.

2.4

bp vol / day

2.2 2 1.8 1.6 1.4 1.2 1 0.0%

1.0%


2.0%

3.0%

Inflation caps and floors can be used as building blocks to construct customised pay-off or to play out strategic views on the inflation market. Here are some simple strategies which can be implemented by buying or selling inflation options. YoY caps and floors can enhance the inflation profile of YoY swaps. For example, the inflation seller in a swap transaction can decide to limit his outflows by buying an inflation cap. If the YoY inflation ratio hits the cap level, the amount paid by the inflation seller will be limited to the cap strike. In compensation for the payment of the option premium up front, the inflation seller is hedged against a high inflation scenario. Another possible strategy is to enhance the yield of an inflation swap by selling a cap or a floor. For example, the inflation buyer can cap his gains on the inflation leg by selling a cap. If inflation stays below the cap level, the inflation buyer will have earned the option premium. The four different possibilities are illustrated in the graph below. Other classical strategies include the collar spread and the butterfly, both of which are linear combinations of caps and floors. A collar is a position where the investor buys and finances the cap by selling the floor. The transaction is made at zero cost. It is particularly appropriate for an investor wanting to bet on an increase in inflation, and can also be advantageous in a situation where caps are cheaper than floors. A butterfly is a bet that inflation will remain within a given range. It can be implemented by buying two options on the central strike and selling two options on either side of the strike.

Hedge Strategy / Buy Option

Investment Strategy / Sell Option

Inflation Seller 2%

Inflation Buyer 6%

Buy the cap Premium

p a C Y o Y

0%

4%

-2%

2%

-4%

Net Pay-off: capped inflation leg

Receive Inflation Net Pay-off: capped inflation leg

0%

Premium

Pay Inflation

Sell the cap

-2%

-6% 0%

1%

2%

3%

4%

0%

5%

1%

Inflation Buyer

2%

3%

4%

5%

Inflation Seller 2%

4%

Sell the floor

Receive Inflation r o o l F Y o Y

0% 2%

Net Pay-off: floored inflation leg

-2%

Premium

-4%

Premium Net Pay-off: floored inflation leg

0%

Pay Inflation

Buy the floor -2%

-6% 0%

1%

2%

3%

4%

5%

0

0.01

0.02

0.03

0.04

0.05


Inflation-linked futures The Chicago Mercantile Exchange (CME) launched a US CPI in September 2004 and an HICPxT future in September 2005. Prior to that, other US exchanges had made a few attempts to list exchange traded futures. Although it seems to have attracted some interest, the HICPxT future has only been modestly successful. It was designed to offer the investor maximum flexibility. It tracks the annual changes in HICPxT and represents the inflation on a 1,000,000 notional for 12 consecutive months. Twelve contracts are quoted at any one time, maturing on the business day before the HICPxT announcement is made and for 12 consecutive months. The future is quoted as 100 minus the inflation rate the market expects when the contract expires. For example, if the market expects the annual inflation rate to be 2.22% as of end of November, the future quote is 97.78. The graph below gives the market expectation for the YoY ratio, calculated from the future prices. The bid-ask spread is still wide (20 to 40bp), denoting poor liquidity on this instrument. However, there are several advantages in having an efficient market for inflation futures. First, it provides a tool for short-term hedging and liability management. A strip of 12 futures is available at any time, so that matching short-term exposure is very easy. Second, it allows counterparty risk mitigation: with the system of daily margin calls, the counterparty risk associated with futures is almost zero, compared with the AA counterparty risk associated with inflation swaps. Finally, as the futures are quoted for 12 subsequent months, they can be used to hedge seasonal risk. However, investors can only take advantage of all this if the market is sufficiently liquid, and the liquidity comes with investors using the instruments. Liquidity is therefore the main issue for this instrument to succeed. This might happen in the next few years, as the swap market continues to develop rapidly. The US CPI future launched in 2004 has not been as successful as its European cousin. This is mainly due to some of its features. It is very similar to the Eurodollar future in that it is based on CPI-U changes over a three-month period. The contracts mature every three months (in March, June, September, and December) as do the Eurodollar futures. This contradicts the way the inflation market is structured, as YoY ratios are favoured and seasonal effects occur on a monthly basis. Having a quarterly contract provides exposure to only four months for seasonality hedging. Moreover, seasonality runs over three months so that interpolation of the CPI fixing is fairly complicated. In its current form, the US CPI future does not appear sustainable and is less and less frequently exchanged on the market. As with any listed future instrument, inflation futures are subject to daily margin calls. This process guarantees final payment of the inflation rate. Unfortunately, it also makes the valuation task slightly more complicated. If inflation increases, the margin calls are paid to the future holder daily and the resulting cash can be invested in the money market. In addition, the future matures as soon as the CPI fixing is known, while the zero coupon swap matures with a lag similar to that used for calculating the bond fixings. This triggers a correction (usually called convexity) which depends on the volatility of the inflation ratio and the correlation between inflation and nominal rates. Developing a highly liquid inflation futures market would be extremely beneficial to inflation derivatives in general, providing increased hedging capabilities in the short term and on bond fixings, transparent consensus measuring seasonality and more tools for short term liability management.

YoY(%)

2.5 2.4 2.3 2.2 2.1 2 1.9 1.8

mid

1.7

bid

1.6

ask

1.5 Nov- Dec- Jan- Feb- Mar- Apr- May- Jun07 07 08 08 08 08 08 08

Jul- Aug- Sep- Oct08 08 08 08

Source: SG Quantitative Strategy – Bloomberg

Contract size

Contract valued at 100,000 times reference HICP ex-tobacco future Index

Reference HICP futures index

100 - annual inflation rate in the 12-month preiod preceding the contract month based on the unrevised Eurozone harmonised index of consumer prices excluding tobacco (HICP) published by Eurostat

Contract months Trading venue and hours

12 consecutive calendar months Available for trading on CME® Globex® from 8:00 a.m. to 4:00 p.m. (London time) on Mondays to Fridays. 0.01 Index points or 100.00 (this renders the contract equivalent to 1,000,000 notional) 4:00 p.m. (London time) on the business day preceding the scheduled day the HICP announcement is made in the contract month. By cash settlement on the day the HICP announcement is made. The final settlement price shall be calculated as 100 less the annual % change in HICP over past 12-months, rounded to four decimal places, or: 100 – [ 100 * ( (HICP(t) ÷ HICP(t-12)) -1 ) ]

Minimum price fluctuation Last trading day

Final settlement price

E.g., for the March 2005 contract, the applicable HICP figures are those for February 2005 (115.1, released on March 16, 2005) and February 2004 (113.5, released on March 17, 2004). The final settlement price shall be: 98.2379 = 100 – [ 100 * ( (115.5 ÷ 113.5) – 1 ) ] (Note that a price of over 100.00 suggests deflation during the 12-month period.) Source: CME

Eurex launched a new HICP future on 21 January 2008. As with the CME HICP future, the underlying is a one-year rolling ratio of the HICPxT. The future is settled the day after publication of the Eurostat index and has two main advantages over the CME future. First, it is traded on 20 consecutive maturities rather than 12. And more importantly, a pool of market makers will provide daily bid and ask quotes during two auction periods at the start and at the end of the trading day.

Underlying Contract Value Settlement Price quotation Minimum price change

Unrevised Harmonised Index of Consumer Prices of the Eurozone Exclusing Tobacco (HICP) EUR 1,000.000 Cash settlement, payable on the first exchange trading day after the final settlement day in percent, with two decimal places based on 100 minus the annual inflation rate based on the HICP 0. 01 percent; equivalent to a value of EUR 100

Contract months

The next twenty successive calendar months. Relevant for the futures contract is the annual inflation rate of the twelve-month period receding the maturity month (e.g. Feb08 maturity month refers to the annual inflation rate measured in the time period between January 2007 and January 2008)

Last trading day

Last trading day and final settlement day of the Euro Inflation Futures contract is the day Eurostat announces the HICP index, if this is trading day; otherwise, the next exchange trading day. Close of trading for the maturing contract month is 10:00 CET The daily settlement price is the closing price fixed in the closing uction. If it is not possible to fix a closing price within the closing auction, or if the price thus fixed does not reflect the actual market conditions, Eurex Clearing AG will determine the settlement price by means of a theoretical pricing model.

Daily settlement price

Final settlement price

Will be determined by Eurex on the final settlement day. Relevant is the unrevised Harmonised Consumer Price Index of the eurozone excluding tobacco published by Eurostat on this day. The final settlement price of a Euro Inflation Futures contract calculated in percentage with four decimal places based on annual inflation rate of the twelve months period of the HICP maturity month (also rounded to four decimal places). The for the calculation of the maturing contract month (t) is: FSPt = 100 – [100 * (HICPt-1/HICPt-13 – 1)] E.g., for the August 2007 contract, the applicable HICP figures are those for July 2007 (104.14 released on August 16, 2007) and July 2006 (102.38 released on August 17, 2006). The final settlement price is calculated accordingly: 100-[100*(104.14/102.38-1)] = 98.2809

Trading Hours

Source: EUREX

The final settlement price is calculated on the last trading day after Eurostat’s publication of the latest index (approx. 11:15 CET) Pre-Trading 09:00-09:45 (CET) Opening Auction 09:45-10:00 (CET) Continuous Trading 10:00-16:45 (CET) Closing Auction 16:45-17:00 (CET) Post-Tradin 17:00-17:30 CET

Background to Pricing Models The inflation market is relatively new and fast-growing, with a promising future thanks to pressure from some countries regulators and especially with the development of pension funds. This has meant that inflation pricing models have recently come under the spotlight. 2007 saw a big development in the options market, driven by hedging flows from structured products and inflation range accruals in particular. So option liquidity has greatly improved and it is getting easier to find market prices for a wider range of options. This is where the modelling expertise developed on standard interest rate products comes into play and can be applied to inflation derivatives. Most inflation models have so far been derived from interest rate models. Here we will review these models and look at how far the interest rate world has been applied to the inflation world. The inflation market is the combination of three different elements: first, the nominal economy, which is the world where we live and price financial products; second, the real economy, which is a hypothetical inflation-free world; and finally, the CPI, which converts an asset in the real economy into an asset in the nominal economy. The Fisher equation relates nominal economy yield to real economy and inflation index yields. This equation provides an economics-based framework for pricing models (see more detailed explanation below). In the light of the Fisher equation, the analogy between the inflation market and the FX market is striking and provides the base for the first inflation model we look at. One of the difficulties in inflation modelling is that a good framework needs to take at least two elements into account - the CPI index and nominal interest rate dynamics. The real interest rate can also be modelled, but calibration will be difficult as there is no market instrument trading real prices directly. The challenge is to come up with pricing which is consistent with all the prices observed in the market, either in the nominal economy or the inflation market. Inflation derivatives also have some specific risks and are relatively unexplored in terms of theoretical pricing. The risks include correlation risk (the correlation between inflation and the nominal market), fixing risk, seasonal effects, the inflation options market-specific smile risk and the term structure of inflation volatility. The most common market models tackle some but not all of these issues and there is still great potential for new research. As far as we know, there are two models which market participants recognise and on which articles have been published: 

The first is Jarrow and Yildirims model, which is based on the analogy between the inflation market and the foreign exchange market.



The second is inspired by nominal-world market models and has been proposed independently by both Benhamou et al. and Brigo and Mercurio.

We will provide a general description of these two models, along with another model which is based on a short-rate approach and is particularly well-adapted to year-on-year (YoY) pricing.

Irving Fisher (1867-1947) was an American economist who developed a price level theory. He was the first economist to make a clear distinction between the real and nominal economy. The Fisher equation is based on a modern version of the quantity theory of money and the equation of exchange. This theory states that the product of the total amount of money in circulation in an economy ( M ) and the speed of spending ( V ) is equal to the product of the price level ( P ) and a given index of the real value of expenditure (Q): MV = PQ The real value of expenditure is mostly measured by gross domestic product (GDP). Assuming that the spending speed is stable in the long term, this relationship can be re-written using the yields:

(1 + µ ) ≈ (1 + i )(1 + π ) Where µ is the yield of the monetary mass, i is the price-level yield and π GDP yield. This gives the basis of the Fisher equation. GDP can be interpreted as a measure of the real economy, the amount of money in circulation reflects the nominal economy and the price level is simply the inflation rate. Moreover, the yields are usually relatively small so the first order of the previous relationship allows us to obtain the Fisher equation:

n = i + r where n is the nominal yield, r is the real yield and i the inflation rate.

Foreign Currency Analogy An FX derivative is based on the exchange rate between two currencies. Its pricing depends on the two economies (foreign and domestic) and on the exchange rate. The exchange rate makes the transition between the foreign and domestic prices and the exchange rate yield is the difference between the foreign and domestic yields. Similarly, an inflation derivative depends on two economies, the nominal and the real economy, and on the consumer price index. So it is very tempting to make an analogy. Jarrow and Yildirim (JY) proposed a model along these lines in 200310 - one of the first inflation models that emerged in academic literature. The paper was originally written to price TIPS and bond options but has been extended to price other derivatives. In the original paper, the authors propose a model based on a short-rate approach for both economies, similar to a three-factor model in the FX world: 

The nominal economy corresponds to the domestic economy. It has its own interest rate and term structure. The nominal interest rate is based on an HJM (Heath Jarrow Merton) diffusion model.



The real economy corresponds to the foreign economy, with the real economy s rate term structure following a one-factor HJM diffusion model.



The spot inflation index (CPI) corresponds to the exchange rate. Like the foreign exchange framework, it is assumed to be lognormal (Black-Scholes type). The trend component of the spot inflation process is the difference between the nominal and the real short rates, consistent with the Fisher equation.

The HJM framework, first introduced by Heath, Jarrow and Merton in the late 1980s, has proved very useful for the pricing of pure interest rate derivatives and is more or less universally used. The key point in the model is the so-called HJM drift condition. This states that, assuming there is no arbitrage opportunity, the dynamic of the underlying variables (forward rate, zero coupon bond prices) is completely defined by their volatility. In other words, no drift estimation is required. And if the volatility function is well-chosen, the model becomes Markovian (meaning that the state of the underlying variables at a given time does not depend on their past values but only on the current one). The Markov property makes the numerical implementations of the model particularly user-friendly. In addition, zero coupon bond prices are lognormal martingales under a well-chosen probability. The analytical formula for zero coupon prices can be derived easily, which makes the pricing of vanilla instruments (such as caps, floors and swaptions) much easier. This tractability is particularly useful when calibrating the nominal market to swap and swaption prices. The model can be calibrated to market parameters in several steps: 

First, calibration of the initial term structures (initial zero coupon prices): o

10

Nominal term structure: In theory and as presented in the original JY paper, this should be calculated from government bond prices. In practice it is more frequently

Pricing Treasury Inflation Protected Securities and Related Derivatives using an HJM Model  R.A. Jarrow, Y.

Yildirim  Journal of Financial and Quantitative Analysis  June 2003

calculated from the money market (deposit and futures) and swap market instruments, using traditional bootstrapping routines. o

Real term structure: Similarly, this should be calculated from government-issued inflation-linked bond prices. However, as the inflation swap market is now fairly liquid, one alternative is to use the initial nominal term structure and quoted zero coupon swaps. In the previous (swap) section, we explained how to deduce the real zero coupon term structure from zero coupon swap prices and the zero coupon nominal term structure.

4.5

5.2

4

5

3.5 4.8

3

4.6

2.5

4.4

2 1.5

4.2

1 4

0.5

3.8 Oct-07

Oct-12

Oct-17

Oct-22

Oct-27

Oct-32

Oct-37




Oct-12

Oct-17

Oct-22

Oct-27

Oct-32


Second, calculation of the volatility terms: o

o

o



0 Oct-07

Volatility term structure of nominal interest rate forwards: In their paper, Jarrow and Yildirim calibrate the nominal term structure historically using nominal bond prices. However, this volatility term structure can also be calibrated to nominal market instruments, as it would be in a standard one-factor model for the nominal interest rate curve. The instruments used for this purpose would typically be standard swap options (swaptions) or the options on standard Libor (caps and floors). Volatility term structure of real interest rate forwards: Calibration of this parameter is much more difficult, as no market instruments trade the real curve directly. It is usually done on a historical basis. Volatility of CPI spot process: This is usually a single number with no term structure. So it could be calibrated either on ATM YoY options or historically, using the CPI time series. Although several YoY option prices are known, the model does not have enough parameters to reprice them all.

Finally, calculation of the various correlations - between the real and nominal economies, between the inflation index and the real economy and between the inflation index and the nominal economy. These are usually calibrated historically using CPI time series and the calibrated term structures of real and nominal zero coupon prices.

After defining the model and parameterising all its coefficients, we can calculate the forward value of the YoY ratio and an equivalent Black volatility. Pricing the YoY option is then only a question of applying the Black formula. The main problems of the Jarrow-Yildirim approach are its over-parameterisation and the number of a priori assumptions, particularly with respect to the real economy. And as the real and nominal rates are Gaussian, there is a higher than zero probability of rates becoming negative, which can be another limiting factor. Also, smile effect is not taken into account.

The JY assumes an HJM diffusion for real and nominal forward rates, under the risk-neutral measure df N (t , T ) = α N (t , T )dt + σ N (t , T )dW t N , for the nominal forward rate df R (t , T ) = α R (t , T )dt + σ R (t , T )dW t R , for the real forward rate The inflation CPI is lognormally distributed, and all Brownian motion is correlated: dI t I t

= i t dt + σ I dW t I , i t = r t N − r t R , for inflation CPI

dW t N , dW t R = ρ RN dt , dW t N , dW t I = ρ NI dt , dW t I , dW t R = ρ RI dt , The arbitrage-free (HJM) condition gives the drift terms in function of volatility functions and under the risk-neutral measure T

∫ (t , T )∫ σ

α N (t , T ) = σ N (t , T ) σ N (t , u )du t

α R (t , T ) = σ R

T

t

R

(t , u )du − σ R (t , T )σ I (t ) ρ RI

And a volatility structure is chosen - for example, the classical Hull-White volatility function: σ X (t , T ) = σ X (t )e − a X (T −t ) Using this model, the bond prices and the value of the index can easily be calculated. This is the starting point for finding analytically tractable expressions for normal vanilla products. For nominal zero coupon prices under the risk-neutral probability measure, zero coupon diffusion and its price can be written as: dB N (t , T ) B N (t , T ) B N (t , T ) =

T

= r t N dt + Γ N (t , T )dW t N , Γ N (t , T ) = − ∫ σ X (t , u )du t

B N (0, T ) B N (0, t )

t ⎛ 1 t 2 2 u , T u , t du ( ) ( ) ( ) (Γ N (u , T ) − Γ N (u, t ))dW u N ⎞⎟ Γ − Γ − N N ∫ ∫ 0 0 ⎝ 2 ⎠

exp⎜ −

And the real zero coupon diffusion and prices are given by: dB R (t , T ) B R (t , T ) B R (t , T ) =

T

= (r t R + ρ RI σ I (t )Γ R (t , T ))dt + Γ R (t , T )dW t R , Γ R (t , T ) = − ∫ σ R (t , u )du t

B R (0, T ) B R (0, t )

t t ⎛ 1 t 2 2 u , T u , t du ρ σ u u , T u , t du ( Γ − Γ ) + Γ − Γ − ( ) ( ) ( ) ( ( ) ( ) ) (Γ R (u, T ) − Γ R (u, t ))dW u R ⎞⎟ R R RI I R R ∫ ∫ ∫ 0 0 0 ⎝ 2 ⎠

exp⎜ −

Market Models The models presented in this section derive from the so-called Libor nominal market models. When Vasicek, Hull-White and others introduced short-rate models in the late 1980s and early 1990s, these were efficient in terms of calibration. Unfortunately, because they had too few parameters and the diversity of instruments on the market was growing, the models (at least the one-factor version with deterministic volatility) quickly reached their limits. Brace et al. (1997), Mitersen et al. (1997) and Jamshidian (1997) presented a new approach using observable market variables (forward Libor rates) as underlying model variables. The inflation market models are based on this approach. First, the assumptions about the real economy are dropped. Second, instead of considering the CPI fixings as the same variable observed at different times, the market models assume each fixing is a different stochastic variable observed at one point in time. For example, Benhamou et al. (2004)11 take a set of CPI fixings and assume that each follows a lognormal diffusion process. This model takes two main types of uncertainty into account: The nominal curve: nominal zero coupon bond prices are driven by one-dimensional Brownian motion. This is usually an HJM type of diffusion. 



A set of CPI forwards: Each CPI forward is lognormally distributed with its own uncertainty source.

Contrary to a Jarrow-Yildirim-type multi-currency model, the real curve is not used as an input. It is enough to know all the CPI forwards in order to completely determine the value of any inflation-linked derivatives. For example, real cash flow will always be valued in nominal terms and multiplied by an inflation ratio. The model parameters are also more restricted: The nominal zero coupon term structure: calibrated on nominal money market and swap prices, as in the Jarrow-Yildirim model; 

The volatility structure of the nominal curve: calculated using optional instruments from the nominal market (swaptions, caps and floors); 



The volatility structure of the CPI fixings.

In this model, the CPI volatility structure is particularly well-adapted to the available instruments. Generally speaking, inflation options are written on a consumer price index ratio. This CPI ratio can generically be defined by: 1. The first fixing date (denominator fixing date), T; 2. The time span between the two fixing dates, ; 3. The option strike K . These three elements form a volatility cube.

11

Reconciling year-on-year and zero-coupon inflation swap: a market model approach

Benhamou  August 2004.



N. Belgrade, E.

Here is an example: A 10Y zero coupon option is defined on the ratio between the CPI index fixing in ten years and at inception date. This corresponds to a first fixing date at 0 and a time span of 10Y. Similarly, a 10Y YoY option is defined on the ratio between the CPI index fixing in 10 years and in 11 years. This corresponds to a first fixing date in 10Y and a time span of 1Y. The most common options correspond to two planes in the volatility cube (see graph below): zero coupon options are represented by the plane T=0 and YoY options are represented by the plane δ =1. If the volatility cube is fully defined in the future, this model will contain all the necessary information. In addition, one of the market model s main advantages is that it shows a natural relationship between YoY and zero coupon market implied volatilities. YoY volatility depends on two zero coupon volatilities and a covariance term, which in turn depends on the CPI local volatility function. Conversely, the convexity adjustment - between the YoY forward and the CPI forward ratio - is a function of the nominal and inflation volatilities and covariance terms (see technical box below). To sum up, this model benefits from: 

its definition, directly compatible with market observable data (the consumer price index);



a direct relationship between ZC and YoY volatilities.

However, as explained in the technical box below, the relationship between YoY and ZC volatilities depends on: 

how CPI volatility is specified;



the correlations chosen between the different CPI fixing dates. Estimating these correlations is a fairly difficult task as the fixings are not known a priori.

The inflation volatility market is currently orientated towards YoY products, with the smile in particular defined in term of YoY option prices. Although the CPI fixings are market observables, the price index does not seem to be the natural underlying variable to use. A more natural state variable would be either the inflation rate or the YoY CPI ratio, as explained in the following section.

Time span between the two fixing dates, ) K , , 0 ( l o V , l o V C Z

T=0

K

) K , , 0 ( l o V , l o V C Z

) K , 1 , T ( l o V , l o v Y o Y

T

K

YoY Vol, Vol(T,1,K)

=1

First fixing date, T

Strike K Source: SG Quantitative Strategy

In the model proposed by E. Benhamou et al, each CPI forward follows a lognormal diffusion, with its own driving Brownian motion, drift and local volatility process: dCPI (t , T i ) CPI (t , T i )

= µ (t , T i )dt + σ (t , T i )dW t i

The nominal zero coupon price is also lognormally distributed according to a standard HJM type of model: dB (t , T ) B (t , T )

= r t N dt + Γ(t , T )dZ t

All the Brownian motions driving the diffusions are correlated. There are two types of correlation: - nominal/inflation correlation: dW t i , dZ t = ρ i , N dt - correlation between the different CPI forwards: dW t i , dW t j = ρ i , j dt Using this approach, the implied volatilities of the most common market instruments can be derived from CPI local volatility and the various correlations. Generically, the terminal (market volatility) in the model and for any ratio is given by: Vol 2 (T , δ ) =

1 ⎛

T T +δ T 2 2 ⎜ ∫O σ ( s, T i )ds + ∫O σ ( s , T i + δ )ds − 2 ∫0 ρ i , jσ ( s, T i )σ ( s, T i + δ )ds ⎞⎟ ⎠ δ ⎝ i

i

i

This pricing formula contains the necessary information to interpolate any point in the volatility cube as defined in the text or graph. Moreover, in the particular case of YoY and zero coupon options, this formula relates YoY and zero coupon vo latilities: 2 Vol YoY (T i , T j = T i + 1) =

1

T 2 2 ⎛ (T j ) − 2 ρ ij ∫0 σ ( s, T i )σ ( s, T j )ds ⎞⎟ (T i ) + T jVol ZC ⎜ T iVol ZC ⎠ T j − T i ⎝ i

In this model, YoY convexity adjustment can be expressed as a function of zero coupon volatilities and a covariance term. The YoY convexity adjustment is the difference between the ratio of the CPI forward and the ratio forward. It is crucial to get this convexity adjustment right to correctly price options on YoY inflation rates.

5 0 0

5

10

-5 -10 -15 -20 -25

bp


15

20

25

30

Short-Rate Models As highlighted above, the JY model and the market models are not ideal for pricing inflation derivatives. The main disadvantages of the JY model are its over-parameterisation and its dependence on the real economy. In the current inflation market, real economy variables are not observable. The problem is that market models are well-adapted to pricing zero coupon options, but are not as good for YoY options. Another approach popular among practitioners involves absorbing real-economy diffusion into the inflation rate drift12 so that the real economy no longer appears in the definition of the model. The rationale for this model stems from the observation that the inflation rate is made up of two components: 

An annual inflation rate, which changes in function of monetary policy and inflation volatility;



An idiosyncratic component, reflecting uncertainty on index fixing - for example linked to seasonality uncertainty.

Another key factor to be taken into account in the construction of a realistic model is the meanreverting property of inflation. The inflation level and central banks monetary policy are intimately related. Central banks are usually committed to controlling inflation levels and GDP, and seek to keep them in line with a pre-defined target. The Taylor rule provides policy-makers with guidance on what to do in various economic situations. It says that short-term interest rates should be adjusted in response to deviations of inflation and GDP from their targets. If the inflation level is above the target level, or if the economy is doing better than expected, policy-makers should increase short-term nominal interest rates. The reverse is also true. And then sometimes - in a stagflation situation for example - inflation and GDP numbers conflict, and though inflation pressures increase, the economy enters a recession cycle. In terms of inflation modelling, the Taylor rule is the main reason behind mean-reverting behaviour by inflation.

The purpose of short-rate models is to account for these two key observations. The following assumptions are therefore made: 

The price index is lognormally distributed. Its drift term corresponds to the inflation rate and its volatility to the idiosyncratic component;



For purposes of consistency with central bank policies, the annual inflation rate is assumed to be mean-reverting. It follows a Hull-White type of diffusion process;



The nominal economy is driven by an HJM-type diffusion;

See for example Inflation-Linked Derivative  Matthew Dogson and Dherminder Kainth  Risk Training Course  September 2006 12



All sources of uncertainty are correlated, and the main correlation is between the inflation rate and the nominal short-term rate.

Calibration of the nominal part of this model is commonly carried out using the nominal money market and swap instruments. The inflation rate can be calibrated in two steps: 1. The mean reversion term structure is defined by zero coupon swap prices and the HJM drift condition; 2. Its volatility term structure can be defined to match option prices. The volatility of the idiosyncratic component is more difficult to estimate, as no observable market variable corresponds to this value. But this idiosyncratic component can initially be ignored. The underlying dynamic in this model is that of the CPI index. In a context where the most liquid instruments are YoY options, and where the smile is defined in YoY terms, it is tempting to model the YoY ratio directly, as seen in the next section.

The short-rate model assumes a stochastic drift for the inflation index: dI t I t

IS = it dt + σ IS t dW t

di t = λ (θ t − it )dt + σ I t dW t I The nominal short rate follows a standard HJM diffusion process. dB N (t , T ) B N (t , T )

= r t N dt + Γ N (t , T )dW t N

Correlations are defined as follows, between each Brownian motion:

dW t N , dW t I = ρ I , N dt , dW t N , dW t IS = ρ IS , N dt , dW t IS , dW t I = ρ IS , I dt T

i (u )du The index expression is easily expressed in function of its yield: I (T ) = I 0 e ∫ 0

In this model, the YoY caplets can be calculated using the Black formula, using a convexified forward and modified volatility, expressed in function of the model parameters.

In the current inflation market, the natural underlying variable in the inflation volatility space is the YoY ratio (defined as the ratio of two CPI indices, one year apart). It is therefore natural to define a pricing model in terms of YoY ratio. The short-rate model approach especially can be adapted to the YoY ratio. As this is still a subject in development, there is no market consensus on the exact definition of this model. As previously highlighted, inflation is historically mean-reverting and this needs to be taken into account. Again, the following assumptions are made: 

The YoY ratio is lognormally distributed. Its drift term corresponds to the annual inflation rate and its optional volatility to an idiosyncratic component.



The annual inflation rate follows a mean-reverting diffusion process (Vasicek type).



The nominal economy is modelled by a one-factor Hull-White model.



Inflation rate and nominal short-term rate are correlated.

In terms of calibration, this model is flexible enough to integrate market prices as they are quoted: 

Nominal-world volatility is calibrated on the vanilla swap term structure and chosen swaption prices;



Inflation volatility is calculated in a fairly straightforward manner from YoY option prices. The main assumption is the functional form given to yearly inflation rate volatility.



The correlation between nominal and inflation diffusions can be calculated historically using previous YoY inflation rates and chosen swap rates.

In addition, as the YoY underlying is modelled directly, the addition of YoY smile is fairly easy and can use established techniques such as displaced diffusion techniques and stochastic volatility modelling. This handbook does not cover such techniques.

In practice, it is convenient to use the YoY ratio as the options model underlying variable, beginning with a short-rate model specification on the YoY ratio: dYoY t

= it dt

YoY t

dit = k (θ − it )dt + σ t dW t The nominal short-term rate can be defined by a mean-reverting process, traditionally by the Hull-White model:

(

)

dr t = a θ t n − r t dt + σ t ndZ t The diffusions are correlated: dW t , dZ t = ρ dt The YoY ratio is expressed directly from the model parameters at time 0. YoY T = exp⎛ ⎜ i0 e − kT + θ 1 − e − kT +

(

⎝

) ∫

T

0

σ u e −k (t −u ) dW u ⎞⎟

⎠

This 1) defines YoY future value at time 0 for maturity T

⎛ ⎝

(

)

F T YoY (0 ) = E Q [YoY T ] = exp⎜ i0 e −kT + θ 1 − e − kT +

1

T

σ e 2∫ 0

2 u

− 2 k (t −u )

⎞ ⎠

du ⎟

and

2) introduces a new process x corresponding to the stochastic part of the YoY xT =

T

∫ σ e

− k (t −u )

u

0

dW u

We obtain: 1 ⎛ ⎞ YoY T = F T YoY (0 )exp⎜ xT − Var (xT )⎟ 2 ⎝ ⎠ So the proposed model is entirely defined by knowledge of the YoY future ratio and an integrated Hull-White type of process. The price of a YoY caplet of strike K and maturity T in this model is simply given by the Black formula, with the appropriate forward and volatility:

(

)

CapletYoY ( K , T , T − 1) = B N (0, T ) F T YoY N (d 1 ) − KN (d 2 ) d 1 =

Σ2 =

1

Σ T 1 T

⎛ F T YoY ⎞ 1 ⎟⎟ + Σ T , d 2 = d 1 − Σ T ⎝ K ⎠ 2

ln⎜⎜

Var ( xT ) =

1

T ∫

T

0

σ u2 e

− 2 k (T − u )

du

Which model for which purpose? As highlighted in the previous sections, the difficulties of constructing a consistent inflation model are manifold. Let us summarise the modelling challenges: 

The swap market and the options market have taken different directions: while the swap market is based on a zero coupon underlying, and prices the price index forward directly, the option market is based on the YoY underlying, whose forward depends on a convexity adjustment which itself depends on volatility. A good pricing model should therefore ensure consistency between its volatility structure and the YoY forward.



Parameterisation of the model itself: which type of volatility should be chosen? How to model the volatility term structure? How to include volatility smile, if necessary? The answers to these questions are highly dependent on the type of model chosen. Some statistical properties should provide hints on how to parameterise the model: o

Nominal-rate volatilities are higher than real-rate volatilities and breakeven volatility;

o

Real and nominal rates have historically tended to be exhibit similar behaviour;

o



Volatilities are fairly stable over time.

The third difficulty lies in estimating correlation parameters. Three observable correlations can be used as a model consistency check: the real/nominal correlation is high, historically greater than 80%; the inflation/nominal correlation is close to 35% historically and the inflation/real correlation is usually negative.

Another prickly point is the development level of the inflation market. Inflation options are relatively new, and investors preferences for one or the other kind of model may vary with product innovations or market conditions. So it is worthwhile keeping all available models in mind, as each may be useful at a particular stage: The Jarrow-Yildirim model is over-parameterised for now. However, it is the only model which proposes an explicit definition of the real economy. If real-rate products develop, this model will be well-adapted. 

The market models can currently be used to calculate the convexity adjustment between YoY forward and CPI forward ratios. This usually uses a couple of liquid at-the-money points in the zero coupon option space. However, it is difficult to add smile effect in this model, as the market defines the YoY smile and the model is build on CPI diffusion. If the zero coupon options market develops, especially across strikes, this would be the reference model of choice. 

Of the two possible short-rate approaches, the first has the same drawbacks as the market model approach, as it is based on CPI diffusion. The second is innovative in that it is defined using the YoY ratio and exhibits a synthetic state variable. This approach can easily be extended to include some YoY smile effect. Also, the state variable can be defined as a multivariate state variable. 

At the moment, the following steps should be used to price an exotic inflation derivative: 

Calculate the CPI forwards using zero coupon inflation swap prices;



Calculate long-term zero coupon volatility using liquid quotes for at-the-money zero coupon options;



Calculate the convexity adjustment between YoY forwards and the forward CPI ratio using a market model;



Calibrate a short-rate model on the annual inflation rate, using liquid quotes for YoY options (on-the money or out-of-the money);



Price any exotic inflation derivative on YoY ratio.

In conclusion, there is no optimal model choice. This field is evolving constantly and innovation can change the exotic products landscape from one month to the next.

Inflation ZC swap annual points

Option Prices

Inflation Forward

Year on Year Vol and smile

Zero Coupon

Market Model

Inflation Forward

Short Rate Model

Year on Year

Option Prices Long Term ZC

ZC Volatility

Exotic Option Prices Year on Year


20Y EUR revenue swap STRUCTURE DESCRIPTION

Clients with revenue linked to inflation

Client receives Y1 to Y20:

(1 + X)n - 1

EUR

20Y

Swap

Client pays Y1 to 20: inflation =

Euro HICPxT ( n) Euro HICPxT (0 )

−1

Euro HICPxT (n) = Monthly index value of the non-seasonally adjusted euro zone Harmonised Index of Consumer Prices excluding tobacco for the month preceding the end of interest period n and published on Bloomberg page CPTFEMU Euro HICPxT (0) = Monthly euro preceding the inception date

HICPxT for the month

PRODUCT OVERVIEW Mechanism: The revenue swap is a series of zero coupon swaps with annually increasing maturities. Economic rationale: This structure represents a hedge for a stream of future cash flows, each of which is linked to the total realised inflation between its start date and its pay date. It replicates the payout profile of a stream of revenues linked to inflation, where each annual inflation rate not only affects the payout for that specific year but also has an impact on all future cash flow projections. Risks and advantages: This structure is a hedging instrument used to decrease the volatility of the net present value of a project  for example a real estate investment with a stream of future rental income linked to inflation.

10Y EUR Livret A swap STRUCTURE DESCRIPTION

French bank ALMs

EUR

10Y

Client receives

Client pays

Quarterly, Act/360

Semiannual, 30/360

Euribor 3M +/- X% p.a.

0.25% + 0.5 x ( average Euribor 3M + French YoY)

Swap

Roll dates : 1 Feb and 1 Aug Average Euribor 3M is the average of the Euribor fixings for the month of June (roll date August) or the month of December (roll date February) French YoY =

French CPIxT ( n) French CPIxT (n − 12)

−1

French CPIxT (n) = Value of the French national price index ( Indice des prix à la consommation ) excluding tobacco, measured either in June (for the August roll date) or December (for the February roll date). PRODUCT OVERVIEW Economic rationale: The decision to link the Livret A French public sector savings rate to inflation from August 2004 increased activity levels in the French CPIxT. The Livret A is one of France s most popular saving accounts and is exclusively distributed by three banks in France (Banque Postale, Caisse dEpargne and Crédit Mutuel under the name of Livret bleu). This should change soon following the European Regulators injunction in May 2007. Mechanism: The Livret A swap is a structure to hedge cash flows linked to the Livret A savings account. The savings account offers a rate of half the YoY CPI ratio, plus half the 3M Euribor average, plus 0.25%. In exchange for a Livret A type of rate, the swap offers Euribor plus or minus a funding margin. Risks and advantages: This structure is a pure hedging instrument offered to managers who do not want to bear inflation risk.

10Y EUR TFR swap STRUCTURE DESCRIPTION

French bank ALMs

EUR

10Y

Client receives

Client pays

Quarterly, Act/360

Annual

Euribor 3M +/- X% p.a.

1.5% + 3/4 x max( Italian YoY, 0% ) Italian YoY =

Italian CPIxT ( n) Italian CPIxT ( n − 12)

Swap

−1

Italian CPIxT (n) = Value of the Italian national price index excluding tobacco, (FOIxT) and as published on the Bloomberg page ITCPI. PRODUCT OVERVIEW Economic rationale: In Italy, corporates are required to give employees a payoff of about 7% of their total wages when they leave the company. This is called the TFR payment. Recent reforms encourage employees to put this TFR payment into a pension scheme rather than keeping it as a lump sum paid when they leave. Mechanism: The TFR payment is increased every year by a 1.5% capitalisation rate plus ¾ of the Italian inflation YoY, floored at 0%. The reference index used for the YoY calculation is the FOI ( Famiglie di Operai e Impiegati ) index, which measures the purchasing power of blue-collar workers and employees. Risks and advantages: This structure is a pure hedging instrument offered to managers who do not want to bear inflation risk.

10Y EUR swap spread France/Europe STRUCTURE DESCRIPTION

Asset or Liability

EUR

Client receives

Client pays

Quarterly, Act/360

Annual Act / 360

Y1 to Y20:

Euribor 3M p.a.

Y1 – Y2

X % - Unconditional

Y3 – Y20

X %- 5 x spread

20Y

Swap

1

2

With spread = YoY Euro inflation  YoY French inflation YoY Euro inflation =

Euro HICPxT ( n) Euro HICPxT (n − 12)

−1

French inflation is not floored at 0% Euro HICPxT (n) = Monthly index value of the non-seasonally adjusted euro zone Harmonised Index of Consumer Prices excluding tobacco for the month preceding the end of interest period n and published on Bloomberg page CPTFEMU Euro HICPxT (n-12) = Monthly euro HICPxT for the month preceding the end of interest period n-12 YoY French Inflation =

FrenchCPIxT ( n) FrenchCPIxT (n − 12)

−1

French CPIxT: defined in same way as Euro HICPxT, and published on Bloomberg page FRCPXTOB.

PRODUCT OVERVIEW Market view: This structure is aimed at clients who consider that French inflation will remain low in coming years, and lower than European inflation. Economic rationale: This trade is based on the idea that YoY French inflation has over time been lower than European inflation, and that this situation is expected to continue. Advantages: Benefits from low French inflation. Risk: The most substantial risk is a sharp increase in French inflation, either in absolute terms or relative to inflation in other European countries.

1.0% 0.8% 0.6% 0.4% 0.2% 0.0% -0.2% -0.4% Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07

Since January 1991 Average spread: 0.528% Maximum/minimum spread: 1.415% / 0.567%

10Y EUR swap switch

(spread France/Europe)

STRUCTURE DESCRIPTION

Liability

EUR

Client receives

Client pays

Quarterly, Act/360

Annual Act / 360 Y1 – Y 10

Y1 to Y20:

Euribor 12M p.a.

20Y

Swap

Lev1 * French Inflation If Euribor 12M ≤ 6.0% X% - Lev2 * Spread If Euribor 12M > 6.0%

With spread = YoY Euro inflation  YoY French inflation YoY Euro inflation =

Euro HICPxT ( n) Euro HICPxT (n − 12)

−1

French inflation is not floored at 0.00% Euro HICPxT (n) = Monthly index value of the non-seasonally adjusted euro zone Harmonised Index of Consumer Prices excluding tobacco for the month preceding the end of interest period n, published on Bloomberg page CPTFEMU Euro HICPxT (n-12) = Monthly Euro HICPxT for the month preceding the end of interest period n-12 YoY French Inflation: defined in same way as Euro HICPxT, and published on

Bloomberg page FRCPXTOB.

PRODUCT OVERVIEW Market view: This structure is aimed at clients who consider that French inflation will remain low in the coming years, and lower than European inflation - especially in a high

3.0%

Euribor rate environment.

2.5%

Economic rationale: This trade is based on the idea that YoY French inflation has been lower than European inflation over time and is expected to remain so (see chart on right). This structure indexes client payments to French inflation in a low-to-normal Euribor rate environment. In addition, when the Euribor 12M rate was fixed at high levels

to cool inflationary pressures in the European block, the Europe



France inflation

2.0% 1.5% 1.0% 0.5% 0.0% Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07 French HICP

spread was at its historical maximum level (see chart below right). This structure indexes

Euro HICP

client payments to the Europe - France inflation spread when Euribor rates (and spread) are high.

Advantages: Benefits from a low French inflation rate in a normal-to-low Euribor rate environment. The client will have a positive carry compared to EUR 10Y IRS as long as

2.000%

French inflation is below 1.88% (note that over the past decade, French inflation

1.500%

averaged 1.47%). In a high Euribor rate environment, where the inflation spread has

1.000%

historically been greatest, the client will have a positive carry compared to EUR 10Y IRS

0.500%

as long as the inflation spread is higher than 0.216% (note that over the past decade it has averaged 0.528%).

0.000% 0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

-0.500%

Risk: The most substantial risk is a sharp increase in French inflation in absolute terms or relative to levels in other European countries.

-1.000% Correl In fla Spread € -Fr & Euribor3M

Linear (Correl Infla Spread € -F r & Euribor3M)

5Y EUR range accrual STRUCTURE DESCRIPTION

Investment

Maturity:

5 years

Coupon:

30/360 annually

EUR

5Y

EMTN

(Euribor 3M + X%) * n/N Where: n = number of observations during the interest period when YoY CPTFEMU 1 is observed between 1.10% and 2.60% N = total number of monthly observations during the interest period (= 12) 1

YoY CPTFEMU refers to the ratio of the CPI 3 months before the observation date/15 months before the observation

date, minus 1.

PRODUCT OVERVIEW Market view: This note is aimed at investors who expect European inflation to remain close to the ECB inflation target of  below, but close to, 2% over the medium term. Mechanism: This 5-year structure pays a semi-annual coupon equivalent to Euribor 6m + X% for each monthly observation of the YoY inflation rate between 1.10% and 2.60%.

3.0% 2.5% 2.0% 1.5% 1.0%

Advantages: Historical: Since the creation of the euro, more than 73% of monthly fixings have been within this range. Over the past five years, the spread has been outside of this range on only one monthly fixing date. X% carry & comfortable barriers: By receiving a floating rate (Euribor) in the current increasing rate environment, the client benefits from improvements in the notes MTM value. Risk: The most substantial risk is a sharp move in inflation, either up or down. This seems quite unlikely considering the strong ECB commitment to keeping inflation low.

0.5% 0.0% Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07

10Y EUR swap corridor STRUCTURE DESCRIPTION

Liability

Client Receives Quarterly, Act/360 Y1 to Y20:

Euribor 3M p.a.

EUR

20Y

Swap

Client pays Annual Act / 360 Y1:

X 1%

Y2 Y10:

X 1% + X 2% * n / N

With n= number of months where YoY European inflation 1 is observed outside of the range [1.00% to 2.60%] N = total number of monthly observations during the interest period (= 12) 1

YoY CPTFEMU refers to the ratio of the CPI 3 months before the observation date/15 months before the observation date, minus 1.

PRODUCT OVERVIEW Market view: This structure is aimed at investors who consider that the ECB will continue its hawkish monetary policy and continue to monitor inflation in the coming years.

3.000%

2.500%

2.000%

Mechanism: The client receives Euribor 3M every quarter. He pays a fixed rate of X1% and an extra X 2% p.a. for every month European inflation is observed to be lower than 1.00% or higher than 2.60%. Advantages:

1.500%

1.000%

0.500%

0.000% J an -9 7 J an -9 9 J an -0 1 J an -0 3 J an -0 5 J an -0 7 J an -0 9 J an -1 1 J an -1 3 J an -1 5 J an -1 7 Hi st o E ur Infl at io n

Guaranteed unconditional Euribor - X 1% carry for the first year. After the first year, the client can continue to benefit from the same carry, conditional on YoY European inflation fixing. A comfortable range of YoY inflation evolution. The ECB has attained a high level of credibility, especially by implementing a clear and efficient monetary policy based on controlled inflation. Risk: The most substantial risk is a sharp move up or down in inflation. The structure is capped (at X 1% + X 2%)

F wd Eur In fl at io n

B arr ier

20Y EUR Kheops STRUCTURE DESCRIPTION

Liability

Client receives Quarterly, Act/360 Y1 to Y20:

Euribor 3M p.a.

EUR

20Y

Swap

Client pays Annual Act / 360 Y1 – Y 20

X%

If inflation < 1.5%

X % + lev * (1.5% - inflation ) If inflation < 2.0% and inflation > 1.5% X % + lev * ( inflation – 2.30%) If inflation < 2.5% and inflation > 1.5% X %

If inflation > 2.5%

With Inflation = YoY Euro inflation, measured using the CP TFEMU index. Each YoY CPTFEMU observation refers to the ratio of the CPI 3 months before the observation date/15 months before the observation date, m inus 1.

PRODUCT OVERVIEW Market view: This structure is aimed at investors who consider that the ECB will take action to control the inflation level. Mechanism: This structure pays X% minus leverage multiplied by the value of a butterfly. The butterfly is the result of a long position in two call options at strike 1.5% and 2.5% and a short position in two call options struck at 2%. Economic rationale: The ECB is committed to maintaining inflation levels around its 2% target. Advantages: As long as inflation remains below 2.3% and above 1.7%, the structure benefits from a higher rate than the current 20Y swap rate (4.90%). Risk: If inflation stays well above the 2% target, the client will pay a higher rate than the current 20Y swap rate. However, it remains capped at X%.

6% 5% 4% KHEOPS Profile

3%

Current 20Y 2% 1.0%

1.4%

1.8%

2.2%

2.6%

3.0%

10Y EUR HICP index-linked leverage slope STRUCTURE DESCRIPTION

Liability

EUR

Maturity:

10 years

Coupon:

30/360 annually

10Y

EMTN

Y1

X%

Y2 to 10

YoY European inflation + leverage * max( CMS10Y  CMS2Y, 0.00%)

Where: Each YoY CPTFEMU observation refers to the ratio of the CPI 3 months before the observation date/15 months before the observation date, minus 1. CMS10Y and CMS2Y refer to the 10Y and 2Y swap rates on the fixing date, reference Reuters ISDAFIX2.

PRODUCT OVERVIEW Market view: This note is aimed at investors who forecast a steepening of the euro swap curve and want coupons also indexed on consumer price evolution. Mechanism: This 10-year structure pays an annual coupon equal to YoY European inflation plus the EUR CMS10Y  CMS2Y spread, multiplied by a predefined leverage. Advantages: This is excellent timing to enter strategies indexed on the swap curve slope: the curve is flat on the 2Y-10Y segment and the proposed leverage is relatively high. The two underlyings are complementary : compared to the current monetary cycle, the curve will steepen when the market forecasts the end of the tightening cycle. Central banks stop raising rates when they consider inflationary pressures are no longer threatening. Risk: The most substantial risk is a flat 10Y-2Y curve over the next few years, combined with low inflation. This seems quite unlikely if the ECB is forced to cut rates in the coming years, and as inflationary pressure is increasing.

200

spread (bp)

150 100 50 0 Jan-00

Jan-02

Jan-04

Jan-06

Jan-08

Hybrid inflation/rate performance swap (HIRPS) STRUCTURE DESCRIPTION

LDI/AM

EUR

5Y to 20Y

EMTN

French Inflation Performance Swap

Years 1-2

X1% without condition

Years 3-20

if Euribor 12M < X2% Lev * French inflation Euribor 12M if Euribor 12M > X2% Inflation capped at 2.50%

French Inflation Carry Swap

Lev * French inflation Inflation capped at 2.30%

French Inflation Bear Performance Swap

Years 1-2

Fixed Rate without condition

Years 3-20

Market Rate Leverage * French inflation Market Rate Market Rate + Fixed Rate

if Euribor 12M < X1% if X1% < Euribor 12M < X2% if X2% < Euribor 12M < X3% if Euribor 12M > X3%

Capped French Inflation Bear Performance Swap

Years 1-2

Fixed Rate without condition

Years 3-20

Market Rate Leverage * French inflation Market Rate Market Rate + Fixed Rate Inflation capped at 2.50%

if Euribor 12M < X1% if X1% < Euribor 12M < X2% if X2% < Euribor 12M < X3% if Euribor 12M > X3%

STRUCTURE DESCRIPTION The range of products available has widened dramatically and our specialists advise clients on how best to protect themselves against inflation or maximise returns. The attached decision tree gives a good example of how clients can fine-tune investment decisions according to their risk appetite and macro views. Clients usually trade performance swaps on nominal interest rates. This hybrid rate/inflation product allows them to benefit from the correlation smile structure. This is a very versatile structure that can be tailored according to clients  expectations, leading to HIRPS variations such as the Bear Performance Swap.

20Y EUR Hybrid performance swap STRUCTURE DESCRIPTION

Liability

Client receives Quarterly, Act/360 Y1 to Y20:

Euribor 3M p.a.

EUR

20Y

Swap

Client pays Annual Act / 360 Y1 – Y 2 X 1% unconditional Y3 – Y 20 leverage * inflation

if Euribor 12M < X 2%

Euribor 12M – 0.02%

if Euribor 12M > X 2%

Inflation capped at 2.5% With Inflation = yoy French Inflation PRODUCT OVERVIEW Market view: This strategy on debt is based on the fact that the French inflation rate is consistently found to be low, both in relation to the rest of Europe and in absolute terms. It also takes the ECBs strict monetary policy into account.

4%

French inflation

3%

Economic rationale: In its inflation control policy, the ECB tends to increase the nominal rate in response to an increase in inflation. In addition, the Euribor rate and French inflation tend to be correlated: when inflation is high, the Euribor rate is high and vice versa. The graph on the right hand side shows the regression coefficient of French inflation versus the Euribor 12M. Note that French inflation and the Euribor 12M are positively correlated. The average French inflation rate since 1997 has been 1.414%. The top left hand corner of the graph, in grey, corresponds to a situation where inflation is high and Euribor is below the X2% level. This risk remains historically remote. Advantages: Benefits from a guaranteed 100 bps carry gain for the first two years. The carry gain remains higher than 80 bps every month when the ECB achieves its objectives. Risk: highest when the inflation rate goes beyond the range.

2%

1% Euribor 0% 0%

2%

4%

6%

8%

10%

12%

inflation breakeven..........................................................49 Inflation forecasting.........................................................10 Accreting asset swaps ................................................... 74 AR, MA, ARMA and ARIMA models ............................... 34

Inflation payers................................................................14 Inflation Products ............................................................ 40 Inflation receivers ............................................................ 14 Inflation Swaps................................................................58 Inflation year-on-year caps and floors ...........................79

Beta ........................................................... ...................... 51

Inflation zero coupon caps and floors............................78

Bloomberg ................................................................ 53, 57

Inflation-linked asset swaps ........................................... 70

Butterfly............................. .............................................. 81

Inflation-linked bonds ..................................................... 45 Inflation-linked futures .................................................... 83 Inflation-linked options ................................................... 78

Calculation of indices ..................................................... 22

Invoice price and quotation ............................................ 48

Carry and forward price.................................................. 54 Chained index ................................................................. 20 CME future ............................................................. ......... 83

Jarrow-Yildirim (JY) Model..............................................91

Collar ............................................................. .................. 81 Convexity ................................................................ ........ 51 Convexity adjustment..................................................... 63 CPI forward curve ........................................................... 65 CPI interpolation ....................................................... 66, 68

Dummies method ........................................................... 32 Duration............................................ ............................... 51

Lag and indexation..........................................................47 Laspeyres index price ..................................................... 19

Market Models.................................................................92 Market participants ......................................................... 14 Marshall-Edgeworth index .............................................. 20 Measuring Inflation..........................................................17

Early redemption asset swaps ....................................... 75 Eurex future............................................................. ........ 85 Euro HICP ........................................................ ............... 24 Euro inflation derivatives .................................................. 9

Nominal economy ........................................................... 21 Nominal vs. real economy...............................................41

European Inflation Convergence.................................... 25 Paasche’s price index ..................................................... 20 Fisher equation ............................................................... 88 Fisher index......................................................... ............ 20 Foreign Currency Analogy........... ................................... 89 French CPI (Indice des prix à la consommation, IPC)... 27

History ............................................................. .................. 9 Hybrid inflation/rate performance swap (HIRPS) ........ 111

Par/par and proceeds asset swap spread .....................73 Par/par and proceeds asset swaps................................70 Price index calculation....................................................19 Pricing Models.................................................................87

Range accruals................................................................12 Real economy..................................................................21 Real interest rates ........................................................... 21 Real rate swaptions.........................................................80 Real swap valuation ........................................................ 64

Index rebasing ................................................................ 19

Real swaps .......................................................... ............63

Risk premium .................................................................. 50

Zero coupon swaps ........................................................ 59 Z-spread ........................................................... ...............75

Seasonality........................... ........................................... 30 Seasonality in the euro zone .......................................... 36 Short-Rate Models ......................................................... 95

TRAMO/SEATS ............................................................... 32

UK RPI (Retail Price Index)......... .................................... 27 US CPI...................................................................... ....... 22 US seasonality ................................................................ 38

X12-ARIMA...................................................................... 32

Year-on-Year inflation swaps ......................................... 62

Zero coupon swap valuation.......................................... 61

SG Handbook Inflation 2008

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