SG Handbook Inflation 2008

Important Notice: In relation to European MIF directive, this publication could not be characterised as independent investment research. Please refer to disclaimer on last page.

Inflation Market Handbook

January 2008

Analyst

With contributions from

Sandrine Ungari Vincent Chaigneau – Head of Fixed Income & Forex Strategy (33) 1 42 13 43 02 (44) 20 7676 7707

[email protected]

[email protected]

Stéphane Salas – Head of Inflation Trading (33) 1 42 18 05 39

[email protected]

Julien Turc – Head of Quantitative Strategy (33) 1 42 13 40 90

[email protected]

Quantitative Strategy


Inflation Market Handbook – January 2008 2



Table of Contents Executive Summary.................................................................................................................. 6

Market Review.......................................................................................................................... 8

History..................................................................................................................................................... 9

Volumes ................................................................................................................................................ 13

Market participants ............................................................................................................................. 14

Measuring Inflation ................................................................................................................. 17

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 des prix à la consommation, IPC) ............................................................................................................27 UK RPI (Retail Price Index).....................................................................................................................................................27 Further information ................................................................................................................................................................28

Seasonality ............................................................................................................................. 29

Definition .............................................................................................................................................. 31

Measurement ....................................................................................................................................... 32

Case study............................................................................................................................................ 36 Seasonality in the euro zone..................................................................................................................................................36 US seasonality .......................................................................................................................................................................38

Inflation Products ................................................................................................................... 40

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

Pricing Inflation Derivatives.................................................................................................... 86

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

Structured Products Catalogue............................................................................................ 101

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

Index ..................................................................................................................................... 113

Executive Summary


Executive Summary

Executive Summary


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.

Market Review


Market Review

Market Review History

Inflation Market Handbook – January 2008

9

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 asset-swap 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



11

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.

Inflation market timeline: from market infancy to the structured product age

1998< 2000 2007200620052004200320022001

Cash Market infancy Derivatives Market Birth Structured Inflation Age

Infla

tio

n li

nke

d B

on

ds

Infla

tio

n li

nke

d S

wap

sO

ther

Der

ivat

ives

British,

Australian,

Sweden,

Canadian

Linkers

UK RPI

swaps

US TIPS,

French

OAT

50cts Bid Ask on 2.5M

25cts Bid Ask on 10M



Bonds quote in break-even

10Y,30Y on

French CPI

First HICP

10Y bond

(France),

Fixing July

30Y HICP

(France)

5Y (Italy),

20Y (Greece),

Fixing Sep.

10Y on

French CPI

7Y on

French CPI

15Y,

10Y HICP

First

Japanese ILBOutstanding amounts

$200b. $230b. $260b. $340b. $450b. $680b. $850b. <$1000b.

5Y, 10Y, 30Y

HICP

12Y on

French CPI

10Y

(Germany),

11Y HICP

Fixing March

30Y, 32Y,

33Y, 49Y, 50Y

HICP

30Y on

French CPI

First

Structured for

EMTN desks

Inflation future attempts in the US

CME future on US CPI

CME future on HICP

Eurexfuture

Asset swap

packages on

BTPe08

Inter-bank

asset swaps

on other

issues

Rate/Inflation

hybrids

Range

Accruals

Customized

structured for LDI

First options

on European

inflation

UK LPI

options

>€1b. >€1b.

First swaps

on European

inflation

>€1b. >€2b. €20b. €45b.

Bond ASW are

calculated from

swaps break-even

Market consensus on seasonality

Increased

liquidity

up to 10Y

Swap prices

are calculated

from bonds

First annual

0% floors

Long term

inflation swaps

(up to 30Y)

Liquidity on all

the curve

$50b.

Secondary market volumes (Euro zone only)

€56b. €66b. €74b.

1998< 2000 2007200620052004200320022001

Cash Market infancy Derivatives Market Birth Structured Inflation Age

Infla

tio

n li

nke

d B

on

ds

Infla

tio

n li

nke

d S

wap

sO

ther

Der

ivat

ives

British,

Australian,

Sweden,

Canadian

Linkers

UK RPI

swaps

US TIPS,

French

OAT

50cts Bid Ask on 2.5M




Bonds quote in break-even

10Y,30Y on

French CPI

First HICP

10Y bond

(France),

Fixing July

30Y HICP

(France)

5Y (Italy),

20Y (Greece),

Fixing Sep.

10Y on

French CPI

7Y on

French CPI

15Y,

10Y HICP

First

Japanese ILBOutstanding amounts

$200b. $230b. $260b. $340b. $450b. $680b. $850b. <$1000b.

5Y, 10Y, 30Y

HICP

12Y on

French CPI

10Y

(Germany),

11Y HICP

Fixing March

30Y, 32Y,

33Y, 49Y, 50Y

HICP

30Y on

French CPI

First

Structured for

EMTN desks

Inflation future attempts in the US

CME future on US CPI

CME future on HICP

Eurexfuture

Asset swap

packages on

BTPe08

Inter-bank

asset swaps

on other

issues

Rate/Inflation

hybrids

Range

Accruals

Customized

structured for LDI

First options

on European

inflation

UK LPI

options

>€1b. >€1b.

First swaps

on European

inflation

>€1b. >€2b. €20b. €45b.

Bond ASW are

calculated from

swaps break-even

Market consensus on seasonality

Increased

liquidity

up to 10Y

Swap prices

are calculated

from bonds

First annual

0% floors

Long term

inflation swaps

(up to 30Y)

Liquidity on all

the curve

$50b.

Secondary market volumes (Euro zone only)

€56b. €66b. €74b.

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.

Market Review Volumes


13

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 inflation-

linked 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.

Outstanding amount of inflation-linked government bonds Secondary market volumes in the euro zone

-100200300400500600700800900

1,000

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

USD EUR CAD SEK JPY GBP

Outstanding amount $bn

M€ / Month

-

10,000

20,000

30,000

40,000

50,000

60,000

Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07

OATe/i Inflation Swaps (10y equivalent)

Source: SG Fixed Income Research – AFT Source: SG Fixed Income Research - ICAP

Market Review Market participants


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



15

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 Market participants: payers and receivers

Inflation Payers Inflation Receiver

DMO, AFT, UST, AUD

Sovereigns

CADES, CNA

Supra and agencies

RFF, NRI, NG

Corporate

Italy, Greece

Sovereigns

Japan, Germany

SovereignsInfrastructure

Project Finance

Tax revenues

Regions/Municipality

Rents

Real Estate holder

Mortgages

Bank ALM

Reduce cost of funding vol

Active Debt Managers

Asset diversification

Asset Managers

Hedge IL Liabilities

Pension Funds/Life Ins.

Hedge for IL swap

Bank ALMCarry, alpha strategy

Alternative Investments

Structured notes

Regional BanksItaly, Swiss retail

Regional Banks

Relative Value

Inflation Linked Funds

Benchmark replication

Inflation Linked FundsHedge for Livret A

Bank ALM

Pension funds

LDI Funds

Hedge inflation claims

Non Life Insurance

2000

2003

2008

Bond Market Derivatives Market Bond Market Derivatives Market

RV and diversification

Prop desks

Inflation Payers Inflation Receiver

DMO, AFT, UST, AUD

Sovereigns

CADES, CNA

Supra and agencies

RFF, NRI, NG

Corporate

Italy, Greece

Sovereigns

Japan, Germany

SovereignsInfrastructure

Project Finance

Tax revenues

Regions/Municipality

Rents

Real Estate holder

Mortgages

Bank ALM

Reduce cost of funding vol

Active Debt Managers

Asset diversification

Asset Managers

Hedge IL Liabilities

Pension Funds/Life Ins.

Hedge for IL swap

Bank ALMCarry, alpha strategy

Alternative Investments

Structured notes

Regional BanksItaly, Swiss retail

Regional Banks

Relative Value

Inflation Linked Funds

Benchmark replication

Inflation Linked FundsHedge for Livret A

Bank ALM

Pension funds

LDI Funds

Hedge inflation claims

Non Life Insurance

2000

2003

2008

Bond Market Derivatives Market Bond Market Derivatives Market

RV and diversification

Prop desks


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.

Flows in the inflation market: from assets to liabilities

Real Income

ASSETS LIABILITIES

Inflation Market

Inflation Payers:UtilitiesProject FinanceReal EstateRetailersSovereignsAgencies

IL Income IL Coupon

Financing

Real Payment

Inflation Receivers:

Pension fundsInsurance

Mutual fundsCorporate ALM

IL Coupon IL Payment

Libor

Inflation Receivers:Retail Banks

Inflation Payers

Receivers:Proprietary

tradersInvestment

banksHedge funds

Investors

IL Coupon Financing

IL Coupon

Financing

Real Income

ASSETS LIABILITIES

Inflation Market

Inflation Payers:UtilitiesProject FinanceReal EstateRetailersSovereignsAgencies

IL Income IL Coupon

Financing

Real Payment

Inflation Receivers:

Pension fundsInsurance

Mutual fundsCorporate ALM

IL Coupon IL Payment

Libor

Inflation Receivers:Retail Banks

Inflation Payers

Receivers:Proprietary

tradersInvestment

banksHedge funds

Investors

IL Coupon Financing

IL Coupon

Financing


Measuring Inflation


17

Measuring Inflation

Measuring Inflation Introduction


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.

How to measure inflation? 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 Communities � September 2001

BLS Handbook of Methods, Chapter 17 - The Consumer Price Index



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

consumer price index.

Index rebasing

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:

19962005

20052005

BaseDec

BaseDec

RB IEIEC =

20052005

BaseDecIE is the Eurostat index of December 2005 expressed in the new 2005 = 100 base (i.e. 101.1).

19962005

BaseDecIE 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:

RBbasemd

basemd CIRIR 1996

,2005

, =

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 index calculation

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: ∑∑= 0010nnnnL pwpwP

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:

( ) ∑∑= 0010nnnnL EppEP

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:

∑∑= 0111nnnnP pwpwP .

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:

10010005/0506/06

07/0707

CJanDec

CJanDecC

JanAugCAug

Px

PxPP = .

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: PLF PPP =

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 similar results:

( ) ( )∑∑ ++= 001101nnnnnnME pwwpwwP



Introducing real interest rates 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.

From real to nominal economy, via the inflation ratio

$100

$100 x (1+r)T

$100 x R0

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

Inflation Ratio at 0: R0=CPI0/CPI0=1

Inflation Ratio at T: RT=CPIT/CPI0

Real Economy Nominal Economy

Time 0

Time T

r : real interest rate n : nominal interest rate

$100

$100 x (1+r)T

$100 x R0

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

Inflation Ratio at 0: R0=CPI0/CPI0=1

Inflation Ratio at T: RT=CPIT/CPI0

Real Economy Nominal Economy

Time 0

Time T

r : real interest rate n : nominal interest rate


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).

Measuring Inflation Calculation of indices


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.

US CPI 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%.

US CPI-U constituents (January 2007) US CPI-U historical annual inflation ratio and regional maximum/minimum (the US is split in four regions South, Mid West, North East and West)

15%

43%4%

17%

6%

6%6% 3%

Food and Beverages Housing

Clothing and Footwear Transport

Medical Care Recreation

Education and Communication Other Goods and Services

0%

1%

2%

3%

4%

5%

6%

7%

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

US CPI-U Regional Minimum Regional Maximum

Source: SG Quantitative Strategy - Bloomberg Source: SG Quantitative Strategy - Bloomberg

Maximum inflation history: Inflation has been higher in the Northeast and West region in the last 20 years

Spread between maximum and minimum inflation in the four main urban regions

0%

1%

2%

3%

4%

5%

6%

7%

88 90 92 94 96 98 00 02 04 06

South Urban Mid West Urban North East Urban West Urban

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

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




Euro HICP 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 euro-

denominated 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 comprehensive methodological documents on HICP calculations and

methodologies4.

4 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.

Euro HICP excluding tobacco constituents (January 2007) Euro HICP annual inflation (revised series)

20%

15%

22%

10%

10%

8%

7%4% 4%

Food and beverages Transport

Housing Recreation and culture

Restaurants and hotels Misc. goods and services

Clothing and footwear Health

Education and Communications

0%

1%

2%

3%

4%

5%

6%

7%

97 99 01 03 05 07

HICP Maximum on the 9 first inflations Minimum

Source: SG Quantitative Strategy - Eurostat Source: SG Quantitative Strategy - Bloomberg

5 Inflation convergence and divergence within the European Monetary Union � F.Busetti, L.Forni, A.Harvey,

F.Venditti � January 2006



EU members maximum inflation: over the past 10 years inflation has been highest in Greece, Spain and the Netherlands

Spread between highest and the lowest inflation for the nine main countries in the HICPxT

0%

1%

2%

3%

4%

5%

6%

7%

97 98 99 00 01 02 03 04 05 06 07

Spain Greece Netherland Ireland Portugal

0%

1%

2%

3%

4%

5%

6%

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.

Average inflation over the five past years (10/02-10/07) Inflation history in the sectors with max/min inflation

-3% -2% -1% 0% 1% 2% 3% 4% 5% 6%

HICP

Education

Alcohol and tobacco

Housing

Health

Transport

Restaurant and Hotel

Misc.

Food and non alc

Household

Clothing

Recreation and culture

Communication

-10%

-8%

-6%

-4%

-2%

0%

2%

4%

6%

8%

10%

97 98 99 00 01 02 03 04 05 06 07

Education Food and non alc Clothing

Communication HICP




French CPI (Indice des prix à la consommation, IPC) 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.

French IPC excluding tobacco constituents (January 2007) French IPC annual inflation

16%

5%

21%

10%17%

3%

9%

7%

12%

Food and Beverages Clothing and Footwear

Housing and energy Healthcare

Transport Education and Communication

Recreation and culture Restaurants and Hotels

Other goods and services

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%


French ICPFrench HICP

Source: SG Quantitative Strategy - Eurostat Source: SG Quantitative Strategy - Bloomberg

UK RPI (Retail Price Index) 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%.

UK RPI excluding tobacco constituents (2007) UK RPI historical annual inflation

20%

5%

24%

17%

13%

4%

4%2%

11%

Food, Beverages and Tobacco Catering

Housing Motoring and Energy

Household goods and services Clothing and Footwear

Personal goods and services Fares and other travel costs

Leisure

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

4.5%

5.0%


UK RPIUK HICP

Source: SG Quantitative Strategy – UK ONS Source: SG Quantitative Strategy - Bloomberg

Further information National statistics office web site On Bloomberg

US inflation http://www.bls.gov CPURNSA <INDEX>

European inflation http://ec.europa.eu/eurostat CPXTEMU <INDEX> (revised

data)

CPTFEMU <INDEX> (non revised)

UK inflation http://www.statistics.gov.uk UKRPI <INDEX> (UK RPI)

CPXTUKI <INDEX> (Harmonised

CPI)

French inflation http://www.insee.fr FRCPXTOB <INDEX>

CPXTFRI <INDEX>

Seasonality


Seasonality

Seasonality


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.

Seasonality Definition


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.

MoM versus YoY seasonality adjustments

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.

%)100%...%(%... 11 ++++∗=∗∗∗∗=∗= −− JannnnJannnnnnSAn MoMMoMMoMIMoMMoMMoMIYoYII

SAnI is the seasonally-adjusted CPI index at time n and nI 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%.

Averaged seasonal adjustments for the major inflation indices calculated over the period Jan 1996 – Dec 2006 using X-12 ARIMA methodology

%MoM MoM YoY %MoM MoM YoY %MoM MoM YoY %MoM MoM YoY

J -0.35% 99.65% 99.65% -0.28% 99.72% 99.72% -0.59% 99.41% 99.41% 0.20% 100.20% 100.20%

F 0.16% 100.16% 99.81% 0.25% 100.25% 99.97% 0.20% 100.20% 99.62% 0.17% 100.17% 100.36%

M 0.23% 100.23% 100.04% 0.23% 100.23% 100.20% 0.16% 100.16% 99.78% 0.24% 100.24% 100.61%

A 0.22% 100.22% 100.26% 0.11% 100.11% 100.31% 0.53% 100.53% 100.31% 0.16% 100.16% 100.76%

M 0.06% 100.06% 100.32% 0.08% 100.08% 100.39% 0.12% 100.12% 100.43% 0.00% 100.00% 100.76%

J -0.09% 99.91% 100.23% -0.06% 99.94% 100.33% -0.12% 99.88% 100.32% -0.09% 99.91% 100.67%

J -0.19% 99.81% 100.04% -0.29% 99.71% 100.04% -0.40% 99.60% 99.92% -0.04% 99.96% 100.63%

A -0.05% 99.95% 99.99% 0.08% 100.08% 100.12% 0.09% 100.09% 100.01% -0.01% 99.99% 100.62%

S 0.07% 100.07% 100.06% 0.08% 100.08% 100.21% 0.23% 100.23% 100.24% 0.07% 100.07% 100.70%

O -0.04% 99.96% 100.01% -0.04% 99.96% 100.17% -0.14% 99.86% 100.10% -0.02% 99.98% 100.68%

N -0.15% 99.85% 99.86% -0.15% 99.85% 100.02% -0.14% 99.86% 99.96% -0.34% 99.66% 100.33%

D 0.14% 100.14% 100.00% -0.02% 99.98% 100.00% 0.04% 100.04% 100.00% -0.33% 99.67% 100.00%

HICPxT French CPIxT UK RPI US CPI-U


Seasonality Measurement


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 dummies 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 in-

depth 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.



Averaged seasonality adjustments for HICP for the Jan 96 – Dec 06 period

Averaged seasonality adjustments in US for the Jan 96 – Dec 06 period

-0.4%

-0.3%

-0.2%

-0.1%

0.0%

0.1%

0.2%

0.3%

0.4%

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

X12-ARIMA TRAMO/SEATS Dummies

MoM adjustments

-0.4%

-0.3%

-0.2%

-0.1%

0.0%

0.1%

0.2%

0.3%

0.4%


X12-ARIMA TRAMO/SEATS Dummies

MoM adjustments

Source: SG Quantitative Strategy Source: SG Quantitative Strategy

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 widely-

used 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.

AR, MA, ARMA and ARIMA models

An Auto-Regressive (AR), Integrated (I) Moving Average (MA) – 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:

ttttt XXXX εθθθ +++= −−− 332211

An MA model represents a time 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:

11 −−= tttX εαε

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:

11332211 −−−− −=−−− tttttt XXXX εαεθθθ

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 derivative of the X process:

( ) ( )211 −−− −−−= ttttt XXXXY

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 12−− tt XX . 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.



HICP Inflation and YoY seasonality for the 96-06 period Estimation residuals

80

85

90

95

100

105

Jan-96 Jan-98 Jan-00 Jan-02 Jan-04 Jan-06-0.6%

-0.4%

-0.2%

0.0%

0.2%

0.4%

0.6%

HICPxT Seasonality

-0.5%

-0.3%

-0.1%

0.1%

0.3%

0.5%



HICP versus HICP ex-tobacco seasonality January and December MoM seasonal per sector

-0.40%

-0.30%

-0.20%

-0.10%

0.00%

0.10%

0.20%

0.30%


HICPxT HICP

-0.6%

-0.4%

-0.2%

0.0%

0.2%

HIC

PxT

Food andBev.

Transport

Housing

Recreation& C

ult.

Resto &H

otel

Misc.

Household

Clothing

Other

January MoM (%)

-0.1%

0.0%

0.1%

0.2%

HIC

PxT

Food andBev.

Transport

Housing

Recreation& C

ult.

Resto &H

otel

Misc.

Household

Clothing

Other

December MoM (%)

Source: SG Quantitative Strategy Source: SG Quantitative Strategy – The seasonality adjustments for each

sector are multiplied by the sector weights in the HICP index.

Seasonality Case study


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.

Seasonality in the euro zone 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.

European countries weighting in the HICP 4 countries account for up to 80% of the inflation index

Germany France ItalySpain Netherlands BelgiumAustria Greece PortugalFinland Ireland Luxembourg&Slovenia

-0.6%

-0.4%

-0.2%

0.0%

0.2%

0.4%

0.6%


Basket GER, FRA, ITA, ESP Others

HICPxT MoM


Italy’s seasonality pattern changed completely 96-00 / 01-07 European countries exhibit similar seasonal patterns

-0.2%

-0.1%

0.0%

0.1%

0.2%


MoM Adjustments (%) Jan 96 - Dec 00

-1.5%

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%


MoM Adjustments (%) Jan 01 - Dec 06

-0.3%

-0.2%

-0.1%

0.0%

0.1%

0.2%

0.3%


Germany France Italy Spain

Source: SG Quantitative Strategy 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.

Evolution of January and March MoM seasonality

Evolution of the seasonal maximum magnitude

-0.80%

-0.60%

-0.40%

-0.20%

0.00%

0.20%

0.40%

0.60%

97 98 99 00 01 02 03 04 05 06

jan mar

MoM adjustments (%)

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%

1.2%

97 98 99 00 01 02 03 04 05 06


US seasonality 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.



Evolution of seasonal maximum magnitude in the US

Clothing and transport exhibit typical seasonal patterns

0.0%

0.1%

0.2%

0.3%

0.4%

0.5%

0.6%

0.7%

0.8%

0.9%

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

-4.00%

-3.00%

-2.00%

-1.00%

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%


Transport Clothing


US versus EU seasonality Housing, clothing and transport sectors explain US seasonality

-0.80%

-0.60%

-0.40%

-0.20%

0.00%

0.20%

0.40%

0.60%


HICP uscpi

-0.60%

-0.50%

-0.40%

-0.30%

-0.20%

-0.10%

0.00%

0.10%

0.20%

0.30%

0.40%


USCPIHousing, Clothing and Transport


Inflation Products


40

Inflation Products

Inflation Products Overview


41

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.

From inflation bonds to inflation swaps 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.



From real to nominal and from bonds to swaps

Real BondYield

NominalBondYield

Gov

ernm

ent

RealSwapRate

NominalSwapRate

Inte

r-ba

nkReal Economy Nominal Economy

NominalSwapSpread

RealSwap Spread

BondBreak-Even

SwapBreak-Even

Real BondYield

NominalBondYield

Gov

ernm

ent

RealSwapRate

NominalSwapRate

Inte

r-ba

nkReal Economy Nominal Economy

NominalSwapSpread

RealSwap Spread

BondBreak-Even

SwapBreak-Even


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 inflation-

linked bonds rate.



43

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.

From inflation swaps to inflation volatility 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.

Volatility Pricing Mechanism

Option Prices

InflationVolatility

InflationZC swap -

annual points

Model

InflationForward

Year on Year

Option Prices

ZC and Year onYear

Interpolation:Seasonality issue

No productRisk Premium

InflationForward

Zero Coupon

Exotic Option Prices

Option Prices

InflationVolatility

InflationZC swap -

annual points

Model

InflationForward

Year on Year

Option Prices

ZC and Year onYear

Interpolation:Seasonality issue

No productRisk Premium

InflationForward

Zero Coupon



As shown in the graph above, a consistent pricing framework needs to tackle the following 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 Products Inflation-linked bonds


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.

Product Mechanism 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.

Description and conventions 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.

Bond market conventions

UK Australia Sweden Canada TIPS OATi OATei Greece Italy Japan Germany

First Issuance 1981 1983 1994 1991 1997 1998 2001 2003 2003 2004 2006

Maturity 2006-2055 2010-2020 2008-2028 2021-2036 2007-2032 2009-2029 2012-2040 2025-2030 2008-2035 2014-2017 2013-2016

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 CPI quarterly CPI monthly CPI monthly CPI-U monthlyCPI France

ex-tobacco

HICP EMU ex-

tobacco

HICP EMU ex-

tobacco

HICP EMU ex-

tobacco

CPI ex-fresh

food

HICP EMU ex-

tobacco

Bloomberg ticker UKRPI Index AUCPI Index SWCPI Index CACPI IndexCPURNSA

Index

FRCPXTOB

IndexCPTFEMU Index

CPTFEMU

Index

CPTFEMU

Index

JCPNJGBI

Index

CPTFEMU

Index

Bloomberg ILB

pageUKTI Govt SAFA Govt SGB Govt CAN Govt TII Govt FRTR Govt FRTR Govt GGB Govt BTPS Govt JGBI Govt

DBRI Govt

OBLI Govt

Coupon

Semi-annual

(pre-fixed for

8-month lag)

Quarterly (pre-

fixed)Annual Semi-annual Semi-annual Annual Annual Annual Semi-annual Semi-annual Annual

Principal

3M (after

2005) and 8M

lag

6M lag 3M lag 3M lag 3M lag 3M lag 3M lag 3M lag 3M lag 3M lag 3M lag

Repayment of

principalNo Floor

Coupon and

pricipal

protected

Floor at par No Floor Floor at par Floor at par Floor at par Floor at par Floor at par No Floor Floor at par

Source: SG inflation trading desk – SG Fixed Income Research



Lag and indexation 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:

( ) ( )mysInMonthNumberOfDadCPICPICPIDIR mmmmd

1323,

−−+= −−−

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:

( ) ( ) 25129.104312405.10431.10405.104

31125

,25 =−+=−

−+= AprMayAprJul CPICPICPIDIR

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:

BaseIndexDIRIR mdmd ,, =

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%.

Interpolated daily inflation reference and unrevised HICPxT Daily inflation reference calculations for the OATei 25 July 2012

99

100

101

102

103

104

105

106

Nov-05 May-06 Nov-06 May-07 Nov-07

DIR

HICP ex tobacco

1 May 1 June 1 July

Payment in July

1 August 1 Sep 1 Oct

Coupon payment schedule

March CPI release

April CPI release

May CPI release

30 April 31 May 30 June 31 July

CPI release schedule

June CPI release

Accrued coupon in

August

1 May 1 June 1 July

Payment in July

1 August 1 Sep 1 Oct

Coupon payment schedule

March CPI release

April CPI release

May CPI release

30 April 31 May 30 June 31 July

CPI release schedule

June CPI release

Accrued coupon in

August


Key pricing and valuation concepts 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.

Invoice price and quotation 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:

Coupontt

ttAI

DateLastCouponDateNextCoupon

DateLastCoupont ×

−

−=

2) Calculate the unadjusted dirty price (UDP), the sum of the unadjusted clean price and the

accrued interest:

tUCPt

UDPt AIPP +=

3) Multiply the unadjusted dirty price by the index ratio to get the adjusted dirty price (ADP) or

invoice price:

UDPtt

ADPt PIRP =

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.

Linkers yield, inflation breakeven 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:

( ) ( ) Ni TR

N

iT

R

UDPt yy

cP+

++

= ∑= 1

10011

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:

( )( )TN

N yTB

+=

11,0 , ( )

( ) 0inf 1

1,0II

yTB T

TR

la+

=

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: ( )TT iII

+= 10

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: yN = 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

Risk premium 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.

OATei BEIR term structure compared with ECB target inflation.

BEIR history on the OATei 2012

150

170

190

210

230

250

270

2010 2015 2020 2025 2030 2035 2040

OATei Curve @ Nov 07

ECB Target

OATei Curve @ Jan 04

OATei 3% 2012

OATei 1.6% 2015

OATei 2.25% 2020

OATei 3.15% 2032

OATei 1.8% 2040

150

160

170

180

190

200

210

220

230

240

250

Nov-02 Nov-03 Nov-04 Nov-05 Nov-06 Nov-07

BEIR OATei 2012 vs OAT Apr 2012

ECB Target


Duration and beta 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-to-

maturity 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.



Bond convexity and duration.

0

10

20

30

40

50

60

70

80

90

100

0% 1% 2% 3%

Yield

Price

Decrease in yield

Gain due to durationBond price as a function of yield

Gain due to convexity

0

10

20

30

40

50

60

70

80

90

100

0% 1% 2% 3%

Yield

Price

Decrease in yield

Gain due to durationBond price as a function of yield

Gain due to convexity

Source: SG Quantitative Strategy - Bloomberg

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:

BEIRyy RN +=

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:

( ) ( ) ( ) ( )bevyCoVarbevVaryVaryVar RRN ,2++=

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 inflation-

linked 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.

Beta of the OATei 2012 versus OAT April 2012, measured on weekly yield variations over a 10-week and a one-year period

Beta of the OATei 09, 12 and 29 versus their most similar nominal bond; weekly return, one-year horizon

-

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Jan-03 Jan-04 Jan-05 Jan-06 Jan-07

1 Year data (weekly return)

10 Weeks data (weekly return)

10 weeks data (daily return)

0%

20%

40%

60%

80%

100%

120%

140%

160%

05/02 05/03 05/04 05/05 05/06 05/07

OATei 2009 OATei 2012 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:

( ) ( )∑=

−+

++

=N

iTT

quotedmd erestAccruedInt

yycP

Ni1

, 1100

1

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 and forward price 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.

Carry of the OATei July 2009 versus the OAT April 2009 BEIR spot and forward on the OATei July 2009

-40

-30

-20

-10

0

10

20

30

Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06

OATei 3% 25-Jul-09

OAT 4% 25-Apr-09

98

118

138

158

178

198

218

238

Mar-01 Mar-02 Mar-03 Mar-04 Mar-05 Mar-06 Mar-07

beir beir fwd


Example of carry measurement and BEIR forward for main euro zone inflation-linked bonds Bond type Description Real yield BEIR 1Mth Carry ILB 3Mth Carry ILB 6Mth Carry ILB 1Mth Fwd BEIR 3Mth Fwd BEIR 6Mth Fwd BEIROATei 3% Jul 2012 1.51 220.09 6.16 -9.49 6.82 212.94 226.38 206.51OATei 1.6% Jul 2015 1.68 222.37 3.86 -4.93 5.18 218.19 226.23 214.96OATei 2.25% Jul 2020 1.91 224.74 2.68 -2.61 4.36 222.08 227.32 220.29OATei 3.15% jul 2032 2.09 242.59 1.72 -1.33 3.14 241.10 244.53 240.65OATei 1.8% jul 2040 2.08 244.11 1.26 -0.99 2.27 243.04 245.61 242.85BTANei 1.25% Jul 2010 1.40 220.41 10.29 -18.23 10.18 207.86 231.06 193.16OATi 3% Jul 2009 1.54 218.59 17.55 0.15 22.41 197.92 207.03 164.77OATi 1.6% Jul 2011 1.55 210.79 7.39 0.24 7.91 201.96 205.82 192.71OATi 2.5% Jul 2013 1.61 216.14 4.92 0.38 5.43 210.53 213.53 206.03OATi 1% Jul 2017 1.87 218.36 3.05 0.96 4.51 215.22 217.01 213.05OATi 3.4% Jul 2029 2.17 232.30 1.89 1.03 3.48 230.64 231.90 230.06BUNDei 1.5% April 2016 1.75 215.99 3.63 -4.32 6.83 212.09 219.40 207.19BTPei 1.65% Sep 2008 1.71 213.61 49.19 -82.77 235.96 160.21 275.35 -137.70BTPei 0.95% Sep 2010 1.56 218.25 10.31 -15.45 13.02 206.64 229.34 194.96BTPei 1.85% Sep 2012 1.66 217.32 6.18 -8.06 8.19 210.41 223.04 203.98BTPei 2.15% Sep 2014 1.77 219.63 4.56 -5.23 6.68 214.82 224.03 211.15BTPei 2.1% Sep 2017 2.00 219.77 3.47 -2.98 6.03 216.46 223.16 214.53BTPei 2.6% Sep 2023 2.27 227.10 2.53 -1.47 5.07 224.91 229.52 223.94BTPei 2.35% Sep 2035 2.39 245.77 1.65 -0.79 3.48 244.52 247.73 244.65GGBei 2.9% Jul 2025 2.30 239.58 2.34 -1.34 4.85 237.60 241.92 236.75GGBei 2.3% Jul 2030 2.41 247.23 1.91 -0.91 4.13 245.71 249.24 245.31CADESi 3.4% Jul 2011 1.58 208.29 7.66 0.39 8.45 199.19 203.17 189.67CADESi 3.15% Jul 2013 1.66 211.45 5.08 0.63 6.04 205.68 208.58 200.73CADESi 1.85% Jul 2019 1.95 221.19 2.72 1.03 4.30 218.49 220.13 216.81 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 Products Inflation Swaps


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.

Real, inflation and standard swap markets 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 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:

Which swaps for which market?

IR Market Inflation Market Real Market

YoY

ZCIL

B

Standard IRS

ZC Swap

YoY Swap

ZC Real swap

Real Swap

ZC IRS

Good Liquidity

Poor Liquidity

No Liquidity

ZC YoY swap

IR Market Inflation Market Real Market

YoY

ZCIL

B

Standard IRS

ZC Swap

YoY Swap

ZC Real swap

Real Swap

ZC IRS

Good Liquidity

Poor Liquidity

No Liquidity

ZC YoY swap


In the nominal market, the most liquid swap is the standard vanilla Libor 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 whose underlying is the CPI index), the most liquid swap is the

zero coupon 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. 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.

Inflation and real swaps: characteristics and mechanisms 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.

Zero coupon swaps 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 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 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).

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

maturity.

Flows in a zero coupon swap

InflationSeller

InflationBuyer

InflationSeller

CPI(T)/CPIbase – 1

InflationBuyer

(1+BEV)T-1

maturity

InceptionInflation

SellerInflationBuyer

InflationSeller

CPI(T)/CPIbase – 1

InflationBuyer

(1+BEV)T-1

maturity

Inception


When the CPI base value is not known at inception, the swap is a forward starting inflation swap.

When the CPI base value is known, it is a spot starting inflation 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 mid-

September 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 document8. The table below summarises the conventions for the main markets.

Zero coupon swap market conventions

UK Australia Sweden Canada US France Europe Greece Italy Japan Germany Spain

Swap type ZC basedZC

Interpolated

ZC

Interpolated

ZC

Interpolated

ZC

Interpolated

ZC

interpolatedZC Based ZC Based ZC Based

ZC

interpolatedZC Based ZC Based

Swap Lag 2M lag 6M lag 3M lag 3M lag 3M lag 3M lag 3M lag 3M lag 3M lag 3M lag 3M lag 3M lag

Swap reference

index (ISDA Def)

Non-revised

All ItemsRPI

Non-revised

AUD CPI

SEK Non

revised CPI

Non-revised

CAD CPI

US Non

revised CPI-U

Non revised

FRC CPI

Unrevised HICPxT

or all items or

Revised All items

GRD non

revised HICP

or non

revised CPI

NICxT or NIC

or FOIxT or

FOI

JPY non

revised CPI

excl. Fresh

food

DEM Non

revised CPIITCPI

Liquidity in swap

marketVery good Low Low Low Good Very good Very good Low Average Average 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



Zero coupon swap valuation

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,

0CPICPITtBEtZCInflaLeg T

NNt

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:

( )[ ] ( )[ ]TtBCPIETtBCPIE NTNtR

Rt ,,0 =

RtE is the real economy expectation at time t and N

tE 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:

( ) ( ) ( )TtBTtBtZCInflaLeg NR ,, −=

The other leg (non inflation-linked) is given by:

( ) ( ) ( )( )( )11, −+= TN TBEIRTtBtZCFixedLeg

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:

( ) ( )( ) ( )TtBTBEIRTtB NT

R ,1, +=

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:

( ) ( ) [ ]TTNtNR CPIETtBTtBCPI ,

0 ,, =

So that the expected value of the CPI index at maturity is given by dividing the real by the nominal discount factor:

( ) ( )( )TtBTtBITCPI

N

R

,,

0=



Year-on-Year inflation swaps 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.

Flows in a YoY swap

InflationSeller

InflationBuyer

InflationSeller

CPI(Ti)/CPI(Ti-1)-1

InflationBuyer

Libor – spreador Fixed rate

Every year until maturity

InceptionInflation

SellerInflationBuyer

InflationSeller

CPI(Ti)/CPI(Ti-1)-1

InflationBuyer

Libor – spreador Fixed rate


Inception


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.



From YoY to ZC swaps: convexity adjustment

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,,,

11

i

iiN

Ntii TI

TITtBETTtgYoYInflaLe

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

( ) ( ) ( ) ( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−−−−− 1,,,,

11111

i

iiiN

NiiN

Ntii TI

TITTBETtBETTtgYoYInflaLe

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:

( ) ( ) ( )[ ] ( )iNiiRiNNtii TtBTTBTtBETTtgYoYInflaLe ,,,,, 111 −= −−−

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 nominal

and real diffusion, and therefore on their volatility and correlation. This adjustment is known as 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 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.

Flows in a real swap

InflationSeller

InflationBuyer

InflationSeller

X% x CPI(Ti)/CPIbase

InflationBuyer

Libor


Inception

InflationSeller

CPI(Ti)/CPI(Ti-1)

InflationBuyer

Par

At maturity

InflationSeller

InflationBuyer

InflationSeller

X% x CPI(Ti)/CPIbase

InflationBuyer

Libor


Inception

InflationSeller

CPI(Ti)/CPI(Ti-1)

InflationBuyer

Par

At maturity


Real swap valuation

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:

( ) ( ) ( ) ( ) ( ) ( ) ( )MR

M

iiRT

MNN

M

i

iiNT

Nt TtBTtBR

ITITtB

ITITtBREtaLegalSwapInfl ,,,,Re

101 0

+=⎥⎦

⎤⎢⎣

⎡+= ∑∑

==

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:

( ) ( ) ( ) ( ) ( ) 1,,,,1

11 =+−=∑=

−− MN

M

iiiiNii TtBTTtLTtBtttLiborLeg

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

( )( )∑

=

−= N

iiR

MRT

TtB

TtBR

1

,

,1

Moreover, quotes can be found in the market for the 1Y, 2Y � 30Y real swap breakeven. Using these quotes, the real zero

coupon prices can be calculated recursively:

Year 1 Year 2 (�) Year m

( )Y

R RYtB

1111,

+= ( ) ( )( )YtBR

RYtB RY

YR 1,1

112, 2

2

−+

= (�) ( ) ( )⎟⎠

⎞⎜⎝

⎛−

+= ∑

−

=

1

1,1

11,

m

iRmY

mYR iYtBR

RmYtB



Building a CPI forward curve 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,

EU Inflation-linked ZC swap. The BEV term structure… … can be translated into a forward reference CPI curve

1.5

1.7

1.9

2.1

2.3

2008 2011 2014 2017 2020 2023 2026 2029 2032 2035

80

100

120

140

160

180

200

220

2008 2011 2014 2017 2020 2023 2026 2029 2032 2035

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

Seasonally-adjusted EU interpolated swap breakeven Seasonally-adjusted EU interpolated annualised forward inflation rate

1.7

1.8

1.9

2

2.1

2.2

2.3

Aug-09 Aug-14 Aug-19 Aug-24 Aug-29 Aug-34 Aug-39

Unadjusted BEV Seasonality Adjusted BEV

-6.0%

-4.0%

-2.0%

0.0%

2.0%

4.0%

6.0%

8.0%

Aug-08 Aug-10 Aug-12 Aug-14 Aug-16 Aug-18

Adjusted Fwd Infla Yield Unadjusted Fwd Infla Yield

Source: SG Quantitative Strategy – SG Inflation Trading Source: SG Quantitative Strategy - Bloomberg

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:

( ) ( )TBEVITCPI += 10 , 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.



CPI projections as of the 21 November 2007

Break-even Maturity

Maturity

Reference

Fixing Base CPI

CPI

Projection

Inflation

forward

yield

1Y 2.119 22-Nov-08 Aug-08 104.19 106.40 2.10%

2Y 1.972 22-Nov-09 Aug-09 104.19 108.34 1.81%

3Y 1.925 22-Nov-10 Aug-10 104.19 110.32 1.81%

4Y 1.914 22-Nov-11 Aug-11 104.19 112.40 1.86%

5Y 1.917 22-Nov-12 Aug-12 104.19 114.57 1.91%

6Y 1.934 22-Nov-13 Aug-13 104.19 116.88 2.00%

7Y 1.957 22-Nov-14 Aug-14 104.19 119.33 2.07%

8Y 1.983 22-Nov-15 Aug-15 104.19 121.91 2.14%

9Y 2.016 22-Nov-16 Aug-16 104.19 124.69 2.25%

10Y 2.049 22-Nov-17 Aug-17 104.19 127.62 2.32%


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 ) = 108.34, 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.



CPI Projections as of 21 November 2007; adjusted reference and unadjusted reference over the year 2008.

Seasonality

Adjustment

MoM

Cumulated

Seasonality

August

Based

Unadjusted

Reference

Seasonal

Factor

Adjusted

Reference

31-Jan -0.35% -0.34% 31-Aug-09 108.34 100.00% 108.34

28-Feb 0.16% -0.18% 30-Sep-09 108.50 100.07% 108.58

31-Mar 0.23% 0.05% 31-Oct-09 108.67 100.03% 108.70

30-Apr 0.22% 0.27% 30-Nov-09 108.83 99.88% 108.70

31-May 0.06% 0.33% 31-Dec-09 109.00 100.01% 109.01

30-Jun -0.09% 0.25% 31-Jan-10 109.16 99.66% 108.79

31-Jul -0.19% 0.05% 28-Feb-10 109.33 99.82% 109.13

31-Aug -0.05% 0.00% 31-Mar-10 109.49 100.05% 109.55

30-Sep 0.07% 0.07% 30-Apr-10 109.66 100.27% 109.96

31-Oct -0.04% 0.03% 31-May-10 109.82 100.33% 110.19

30-Nov -0.15% -0.13% 30-Jun-10 109.99 100.25% 110.26

31-Dec 0.14% 0.01% 31-Jul-10 110.16 100.05% 110.21


CPI interpolation

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:

( ) ( ) ( ) ( )∫×∫×=TT

duusduuieeCPITCPI 000,0

In this expression, i is the inflation rate and s 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:

( ) ( )( ) ( )

( ) ( )11

1

11

1,00,0 −−

−

=− −

−

−+−

×=∑

×= jjjjjj

j

kkkk TTi

j

TTiTTi

j eTCPIeCPITCPI

( )njjT ..0=

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 t such as [ ]jj TTt ,1−∈ , where t is in the nth month after the base month and is the dth day of the

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

seasonal adjustment):

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑××=

∫××=

−

=−−−×+

−−

−−

1

111111 ,0,0,0

n

kjj

t

jTjjNdnsks

Ttij

duusTtij eeTCPIeeTCPItCPI

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 Products Inflation-linked asset swaps


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.

Asset swaps definitions Par/par and proceeds asset swaps 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.

Asset Swap Mechanism

Asset SwapSeller

Asset SwapBuyer

Par or Proceeds amount

Asset SwapSeller

Libor +/- spread on par or proceeds amount

Asset SwapBuyer

IL Coupon =CPI(Ti)/CPI(0)*R


Asset SwapSeller


Asset SwapBuyer

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

IL Bond

IL Bond

IL Bond

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

maturity

Coupon payment

date

InceptionAsset Swap

SellerAsset Swap

Buyer


Asset SwapSeller

Libor +/- spread on par or proceeds amount

Asset SwapBuyer



Asset SwapSeller


Asset SwapBuyer

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

IL Bond

IL Bond

IL Bond

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

maturity

Coupon payment

date

Inception


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).

Swap flows in the asset swap package (par/par) for the OATi July 2012. Upward arrows represent positive cash flows for the asset swap seller and downward arrows represent negative cash flows for the asset swap seller.

At

inception,

par is

received

�

� And

par is

paid back

At maturity,

inflated

notional

is received�

� And IL

bond is

delivered

Throughout the life of the asset swap, fixed notional is

paid on the Libor leg

Throughout the life of the asset swap, notional inflates as

inflation increases

Jul08

L – 16bp L – 16bp L – 16bp L – 16bp

3%xI25Jul01

I25Jul083%x

I25Jul01

I25Jul093%x

I25Jul01

I25Jul103%x

I25Jul01

I25Jul113%x

I25Jul01

I25Jul12

I25Jul01 = 92.98

Oct08

L – 16bp L – 16bp L – 16bp L – 16bp

Jul09 Oct09 Jul10 Oct10 Jul11 Oct11 Jul12

100

100Pmkt=104.8

max( , 1)I25Jul01

I25Jul12

At

inception,

par is

received

�

� And

par is

paid back

At maturity,

inflated

notional

is received�

� And IL

bond is

delivered

Throughout the life of the asset swap, fixed notional is

paid on the Libor leg

Throughout the life of the asset swap, notional inflates as

inflation increases

Jul08

L – 16bp L – 16bp L – 16bp L – 16bp

3%xI25Jul01

I25Jul083%x

I25Jul01

I25Jul083%x

I25Jul01

I25Jul093%x

I25Jul01

I25Jul093%x

I25Jul01

I25Jul103%x

I25Jul01

I25Jul103%x

I25Jul01

I25Jul113%x

I25Jul01

I25Jul113%x

I25Jul01

I25Jul123%x

I25Jul01

I25Jul12

I25Jul01 = 92.98

Oct08

L – 16bp L – 16bp L – 16bp L – 16bp

Jul09 Oct09 Jul10 Oct10 Jul11 Oct11 Jul12

100

100Pmkt=104.8

max( , 1)I25Jul01

I25Jul12


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. Inflation-

linked 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.

Calculating par/par and proceeds asset swap spread

In a par/par asset swap, the two counterparties exchange par (assumed to be equal to 100%) for the dirty market 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:

( ) ( )( ) ( ) ( )NN

N

iiNiiiIMPLIEDMKT TBTBsTTLibtPPLiborLegBondLegmentUpfrontPay ,0,0,1

11 −+∆−+−=++ ∑

=−

With ( ) ( )( )

( )( ) ( )∑

=

+=M

jMN

MijNIMPLIED TB

CPITCPI

CPITCPI

TBRP1

,000

,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:

01PVPP

s MKTIMLIED −=

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:

( ) ( )( ) ( ) ( )NNMKT

N

iiNiiiMKTIMPLIEDMKTMKT TBPTBsTTLibtPPPPLiborLegBondLegmentUpfrontPay ,0,0,

11 −+∆−+−=++ ∑

=−

Simplifying in the same way as above leads to the following spread value for the proceeds swap:

01PVPPP

sMKT

MKTIMLIED −=

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.

Asset swap cash flows for the OATi July 2012: example of schedule and calculations.

Date

Swap break-evens

Index Ratio (1)

Nominal Discount (2)

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

Libor Discounted Cash Flows

25-Oct-07 104.19 1.12 186,040 25-Jan-08 0.989 4.24% 10,682 25-Jul-08 106.37 1.144 0.967 33,180 4.59% 22,141

25-Jan-09 0.946 4.30% 20,535 25-Jul-09 108.29 1.165 0.928 32,429 4.07% 18,740

25-Jan-10 0.908 4.26% 19,483 25-Jul-10 110.75 1.192 0.889 31,792 4.29% 18,904

25-Jan-11 0.870 4.29% 18,820 25-Jul-11 113.32 1.220 0.852 31,171 4.34% 18,352

25-Jan-12 0.834 4.38% 18,395 25-Jul-12 116.00 1.248 0.816 1,048,700 4.42% 833,550

Bond Value (A) 1,177,272 Upfront payment (B) 186,040 Libor Leg Value (C) 999,603 PV01 (D) 4.26 Spread (A-B-C)/D (19.66)


Accreting asset swaps 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.

Calculating accreting asset swap spread

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.

( ) ( ) ( )( ) ( ) ( ) ( ) 0,01,0,111

1 =+−+∆+−+−=++ ∑=

− NNN

N

iiNiii

iIMPLIEDMKT TBrTBsTTLibtrPPLiborLegBondLegmentUpfrontPay

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

( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( )∑

∑

=

=−

∆+

+−−+−+−= N

iiNi

i

NNN

N

iiNiN

iIMPLIEDMKT

TBtr

TBrTBTBrPPs

1

11

,01

,01,0,011

Early redemption asset swaps 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:

( ) ( )( ) Notional

CPITCPI

TtionalInflatedNo MM ⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛−= 1

0

The total inflated notional can also be viewed as the sum of the increments of the inflated notional at

two subsequent payment dates:

( ) ( ) ( ) ( )01

1 TNotionalTtionalInflatedNoTtionalInflatedNoTtionalInflatedNoM

iiiM +−= ∑

=−

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:

( )( )

( )( ) ( )MiNii TTBNotional

CPITCPI

CPITCPI

,00

1 ⋅⋅⎟⎟⎠

⎞⎜⎜⎝

⎛− −

Another asset swap measure for bonds: Z-spread 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.



Indicative levels of Z-spreads, accreting, proceeds and par/par asset swaps

Bond type Description ZSpread Accreting Proceeds Par/ParNominal ZSpread

Delta Zspread

Clean Price

Real Yield

BTPe 1.65% 15-Sep-2008 -9.00 -9.00 -8.60 -9.50 -19.10 10.00 99.54% 2.016BTPe 0.95% 15-Sep-2010 -6.30 -6.40 -6.40 -6.50 -15.60 9.30 96.25% 2.139BTPe 1.85% 15-Sep-2012 -3.90 -4.10 -4.10 -4.00 -13.00 9.10 98.34% 2.194BTPe 2.15% 15-Sep-2014 -2.20 -2.40 -2.30 -2.50 -11.40 9.20 99.53% 2.231BTPe 2.10% 15-Sep-2017 2.50 2.30 2.70 2.80 -7.00 9.50 98.07% 2.324GGBe 2.90% 25-Jul-2025 7.00 6.50 7.60 9.00 0.30 6.70 107.12% 2.411GGBe 2.30% 25-Jul-2030 11.50 11.00 14.00 14.00 11.50 0.00 97.21% 2.460BTPe 2.35% 15-Sep-2035 14.50 13.80 18.00 18.70 10.50 4.00 98.05% 2.461

BTANe 1.25% 25-Jul-2010 -10.70 -10.70 -10.80 -11.00 -22.70 12.00 97.63% 2.028OATe 3.00% 25-Jul-2012 -9.30 -9.40 -9.30 -11.30 -21.80 12.50 104.43% 2.087OATe 1.60% 25-Jul-2015 -10.20 -10.40 -11.10 -11.30 -22.00 11.80 96.03% 2.134BUNDe 1.50% 15-Apr-2016 -13.00 -13.20 -14.40 -13.80 -25.50 12.50 94.43% 2.196OATe 2.25% 25-Jul-2020 -11.60 -11.90 -12.80 -14.00 -22.80 11.20 100.61% 2.196OATe 3.15% 25-Jul-2032 -14.40 -15.10 -16.90 -21.80 -18.90 4.50 118.48% 2.187OATe 1.80% 25-Jul-2040 -12.50 -13.40 -17.10 -15.80 -17.80 5.30 91.71% 2.152

OATi 3.00% 25-Jul-2009 -10.00 -10.00 -9.50 -12.80 -23.10 13.10 101.69% 2.192OATi 1.60% 25-Jul-2011 -10.00 -10.10 -10.20 -10.80 -21.40 11.40 97.46% 2.244OATi 2.50% 25-Jul-2013 -9.50 -9.60 -9.70 -11.00 -20.60 11.10 101.48% 2.239OATi 1.00% 25-Jul-2017 -11.00 -11.20 -12.40 -11.30 -22.00 11.10 88.49% 2.280OATi 3.40% 25-Jul-2029 -15.20 -15.70 -17.20 -23.80 -19.70 4.50 119.63% 2.264

Source: SG Inflation Trading Desk

OATei 2012 versus OAT 2012 Z-spreads history Z-spread difference between the OAT and the OATei 2012

-50-45-40-35-30-25-20-15-10-50Feb-07 Apr-07 Jun-07 Aug-07 Oct-07

OAT 5% Apr-12 OATei 3% Jul 2012

-20

-18

-16

-14

-12

-10

-8

-6

-4Feb-07 Apr-07 Jun-07 Aug-07 Oct-07

OATei/OAT spread

Source: SG Fixed Income Strategy Research – Bloomberg Source: SG Fixed Income Strategy Research - Bloomberg

Inflation Products Inflation-linked options


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.

Standard options Inflation zero coupon caps and floors 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�.

Combined flows of a zero coupon swap and a zero coupon cap.

InflationSeller

CPI(T)/CPIbase-1

InflationBuyer

(1+BEV)T-1

Option Seller

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

InflationSeller

CPI(T)/CPIbase-1

InflationBuyer

(1+BEV)T-1

Option Seller

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


The strike is expressed in annual average inflation growth so that the pay-off of a zero coupon cap is

defined as follows:

( ) ( )( ) ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛+−= 0,1

0max,, TK

CPITCPIKTTZCCap

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.



Inflation year-on-year caps and floors 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.

Inflation YoY volatility surface Inflation YoY volatility smile

0.0%0.9%

1.8%2.7%

3.6%

1 6 11 16 21 26

0

0.5

1

1.5

2

2.5

3

3.5

4 bp vol /

Maturity Strike

1

1.5

2

2.5

3

3.5

4

4.5

0% 1% 2% 3% 4%1Y 2Y 5Y

30Y 10Y 20Y

bp vol/day

Source: SG Inflation Trading Desk Source: SG Inflation Trading Desk

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.



Combined flows of a YoY swap and a YoY cap

InflationSeller

InflationBuyer

Libor – spreador fixed rate

Option Seller

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

InflationSeller

InflationBuyer

Libor – spreador fixed rate

Option Seller

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


Real rate swaptions 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.

An example of swaption smile

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0.0% 1.0% 2.0% 3.0%

bp vol / day




Strategies with caps and floors 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.



Application for caps and floors: hedging or investment strategy?

Hedge Strategy / Buy Option Investment Strategy / Sell Option

Yo

YC

ap

Yo

YF

loo

r

Inflation Buyer

Inflation Seller Inflation Buyer

Inflation Seller

-2%

0%

2%

4%

0% 1% 2% 3% 4% 5%

-2%

0%

2%

4%

6%

0% 1% 2% 3% 4% 5%

-6%

-4%

-2%

0%

2%

0% 1% 2% 3% 4% 5%

Buy the floor

Receive Inflation

Net Pay-off: floored inflation leg

Sell the cap

Receive InflationNet Pay-off: capped

inflation leg

Buy the cap

Pay InflationNet Pay-off: capped

inflation leg

-6%

-4%

-2%

0%

2%

0 0.01 0.02 0.03 0.04 0.05

Premium

Premium

Premium

Premium

Sell the floor


Pay Inflation

Hedge Strategy / Buy Option Investment Strategy / Sell Option

Yo

YC

ap

Yo

YF

loo

r

Inflation Buyer

Inflation Seller Inflation Buyer

Inflation Seller

-2%

0%

2%

4%

0% 1% 2% 3% 4% 5%

-2%

0%

2%

4%

6%

0% 1% 2% 3% 4% 5%

-6%

-4%

-2%

0%

2%

0% 1% 2% 3% 4% 5%

Buy the floor

Receive Inflation


Sell the cap

Receive InflationNet Pay-off: capped

inflation leg

Buy the cap

Pay InflationNet Pay-off: capped

inflation leg

-6%

-4%

-2%

0%

2%

0 0.01 0.02 0.03 0.04 0.05

Premium

Premium

Premium

Premium

Sell the floor


Pay Inflation


Inflation Products Inflation-linked futures


Inflation-linked futures

CME future 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.



Market expectation for the YoY inflation rate for the HICP ex-tobacco, as implied by the CME euro zone future (end October 2007).

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

Nov-07

Dec-07

Jan-08

Feb-08

Mar-08

Apr-08

May-08

Jun-08

Jul-08

Aug-08

Sep-08

Oct-08

midbidask

YoY(%)

Source: SG Quantitative Strategy – Bloomberg

CME HICP future – Contract features. Prices for the current CME HICPxT contract and later contracts are available on Bloomberg (code AAA <Index>).

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 12 consecutive calendar months Trading venue and hours Available for trading on CME® Globex® from 8:00 a.m. to

4:00 p.m. (London time) on Mondays to Fridays.Minimum price fluctuation 0.01 Index points or 100.00 (this renders the contract

equivalent to 1,000,000 notional) Last trading day 4:00 p.m. (London time) on the business day preceding

the scheduled day the HICP announcement is made in thecontract month.

Final settlement price By cash settlement on the day the HICP announcement is made.The final settlement price shall be calculated as 100 less theannual % change in HICP over past 12-months, rounded to fourdecimal places, or: 100 – [ 100 * ( (HICP(t) ÷ HICP(t-12)) -1 ) ]

E.g., for the March 2005 contract, the applicable HICP figures arethose 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 the12-month period.)

Source: CME



Eurex future 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.

Eurex HICP future – Contract features. Launch date 21 January 2008. Underlying Unrevised Harmonised Index of Consumer Prices of the Eurozone Exclusing

Tobacco (HICP) Contract Value EUR 1,000.000

Settlement Cash settlement, payable on the first exchange trading day after the final settlement day

Price quotation in percent, with two decimal places based on 100 minus the annual inflation rate based on the HICP

Minimum price change 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 Futurescontract 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

Daily settlement price 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.

Final settlement price Will be determined by Eurex on the final settlement day. Relevant is theunrevised Harmonised Consumer Price Index of the eurozone excludingtobacco 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 (102.38released on August 17, 2006). The final settlement price is calculatedaccordingly:100-[100*(104.14/102.38-1)] = 98.2809

The final settlement price is calculated on the last trading day after Eurostat’s publication of the latest index (approx. 11:15 CET)

Trading Hours 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-Trading 17:00-17:30 (CET)

Source: EUREX

Pricing Inflation Derivatives


Pricing Inflation Derivatives

Pricing Inflation Derivatives Background to Pricing Models


87

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.

Pricing Inflation Derivatives Background to Pricing Models


88

The Fisher equation

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:

( ) ( )( )πµ ++≈+ 111 i

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:

rin +=

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

Pricing Inflation Derivatives Foreign Currency Analogy


89

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 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

10 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



90

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.

US nominal term structure from Treasury bills and TIPS (bid yield October 2007)

US real term structure from Treasury bills and TIPS (bid yield October 2007)

3.8

4

4.2

4.4

4.6

4.8

5

5.2

Oct-07 Oct-12 Oct-17 Oct-22 Oct-27 Oct-32 Oct-37

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Oct-07 Oct-12 Oct-17 Oct-22 Oct-27 Oct-32


� Second, calculation of the volatility terms:

o 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).

o 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.

o 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.



91

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 Jarrow-Yildirim (JY) Model

The JY assumes an HJM diffusion for real and nominal forward rates, under the risk-neutral measure

( ) ( ) ( )( ) ( ) ( ) rate forward real for the

rate forward nominal for the

,

,

RtRRR

NtNNN

dWTtdtTtTtdf

dWTtdtTtTtdf

,,,

,,,

σα

σα

+=

+=

The inflation CPI is lognormally distributed, and all Brownian motion is correlated:

,,,,,,

,

dtdWdWdtdWdWdtdWdW

rridWdtiIdI

RIRt

ItNI

It

NtRN

Rt

Nt

Rt

Ntt

ItIt

t

t

ρρρ

σ

===

−=+= CPI inflationfor ,

The arbitrage-free (HJM) condition gives the drift terms in function of volatility functions and under the risk-neutral measure

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )∫∫

−=

=T

t RIIRRRR

T

t NNN

tTtduutTtTt

duutTtTt

ρσσσσα

σσα

,,,,

,,,

And a volatility structure is chosen - for example, the classical Hull-White volatility function: ( ) ( ) ( )tTaXX

XetTt −−= σσ ,

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:

( )( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ⎟

⎠⎞

⎜⎝⎛ Γ−Γ−Γ−Γ−=

−=ΓΓ+=

∫∫

∫t N

uNN

t

NNN

NN

T

t XNNtN

Nt

N

N

dWtuTudutuTutBTB

TtB

duutTtdWTtdtrTtBTtdB

00

22 ,,,,21exp

,0,0

,

,,,,,,

σ

And the real zero coupon diffusion and prices are given by:

( )( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ⎟

⎠⎞

⎜⎝⎛ Γ−Γ−Γ−Γ+Γ−Γ−=

−=ΓΓ+Γ+=

∫∫∫

∫t R

uRR

t

RRIRI

t

RRR

RR

T

t RRRtRRIRI

Rt

R

R

dWtuTudutuTuudutuTutBTBTtB

duutTtdWTtdtTttrTtBTtdB

000

22 ,,,,,,21exp

,0,0,

,,,,,,,

σρ

σσρ

Pricing Inflation Derivatives Market Models


92

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, δ;

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 � N. Belgrade, E.

Benhamou � August 2004.



93

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.

Inflation volatility cube: most common options are ZC and YoY options

First fixing date, T

Time span between the two fixing dates, δ

Strike K

T=0

δ=1ZC V

ol, V

ol(0

,δ,K

)

YoY Vol, Vol(T,1,K)

TK

YoY

vol,

Vol(T

,1,K

)

δK

ZC V

ol, V

ol(0

,δ,K

)

First fixing date, T

Time span between the two fixing dates, δ

Strike K

T=0

δ=1ZC V

ol, V

ol(0

,δ,K

)

YoY Vol, Vol(T,1,K)

TK

YoY

vol,

Vol(T

,1,K

)

δK

ZC V

ol, V

ol(0

,δ,K

)




94

A market model

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:

( )( ) ( ) ( ) i

tiii

i dWTtdtTtTtCPITtdCPI

,,,,

σµ +=

The nominal zero coupon price is also lognormally distributed according to a standard HJM type of model:

( )( ) ( ) t

Nt dZTtdtr

TtBTtdB ,

,,

Γ+=

All the Brownian motions driving the diffusions are correlated. There are two types of correlation:

- nominal/inflation correlation: dtdZdW Nitit ,, ρ=

- correlation between the different CPI forwards: dtdWdW jijt

it ,, ρ=

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:

( ) ( ) ( ) ( ) ( ) ⎟⎠⎞⎜

⎝⎛ +−++= ∫∫∫

+ iii T

iiji

T

O i

T

O i dsTsTsdsTsdsTsTVol0 ,

222 ,,2,,1, δσσρδσσδ

δδ

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 volatilities:

( ) ( ) ( ) ( ) ( ) ⎟⎠⎞⎜

⎝⎛ −+

−=+= ∫

iT

jiijjZCjiZCiij

ijiYoY dsTsTsTVolTTVolTTT

TTTVol0

222 ,,211, σσρ

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.

YoY convexity adjustment

-25

-20

-15

-10

-5

0

5

0 5 10 15 20 25 30

bp


Pricing Inflation Derivatives Short-Rate Models


95

Short-Rate Models

Why another model? 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 mean-

reverting 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.

Model definition 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;

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

September 2006



96

� 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.

A short-rate model

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

( ) It

Itttt

ISt

IStt

t

t

dWdtidi

dWdtiIdI

σθλ

σ

+−=

+=

The nominal short rate follows a standard HJM diffusion process.

( )( ) ( ) N

tNNt

N

N dWTtdtrTtBTtdB

,,,

Γ+=

Correlations are defined as follows, between each Brownian motion:

dtdWdWdtdWdWdtdWdW IISIt

IStNIS

ISt

NtNI

It

Nt ,,, ,,,,, ρρρ ===

The index expression is easily expressed in function of its yield: ( ) ( )∫=T

duuieITI 0

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.



97

A possible improvement: inflation ratio as a state variable 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.



98

Defining a state variable

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:

( ) tttt

tt

t

dWdtikdi

dtiYoYdYoY

σθ +−=

=

The nominal short-term rate can be defined by a mean-reverting process, traditionally by the Hull-White model:

( ) tntt

ntt dZdtradr σθ +−=

The diffusions are correlated:

dtdZdW tt ρ=,

The YoY ratio is expressed directly from the model parameters at time 0.

( ) ( ) ⎟⎠⎞⎜

⎝⎛ +−+= ∫ −−−− T

uutk

ukTkT

T dWeeeiYoY00 1exp σθ

This

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

( ) [ ] ( ) ( ) ⎟⎠⎞

⎜⎝⎛ +−+== ∫ −−−− T utk

ukTkT

TQYoY

T dueeeiYoYEF0

220 2

11exp0 σθ

and

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

( )∫ −−=T

uutk

uT dWex0σ

We obtain:

( ) ( )⎟⎠⎞

⎜⎝⎛ −= TT

YoYTT xVarxFYoY

21exp0

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:

( ) ( ) ( ) ( )( )

( ) ( )∫ −−==Σ

Σ−=Σ+⎟⎟⎠

⎞⎜⎜⎝

⎛

Σ=

−=−

T uTkuT

YoYT

YoYTN

dueT

xVarT

TddTKF

Td

dKNdNFTBTTKCapletYoY

0

222

121

21

11

,21ln1

,01,,

σ

Pricing Inflation Derivatives Which model for which purpose?


99

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;

Pricing Inflation Derivatives Which model for which purpose?


100

� 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.

From zero coupon prices to YoY volatility: pricing inflation exotics.

Option Prices

Long Term ZC

InflationZC swap -

annual points

Market ModelInflationForward

Year on Year

Option Prices

Year on Year Vol and smile

InflationForward

Zero Coupon


Year on Year

ZC Volatility

Short Rate Model

Option Prices

Long Term ZC

InflationZC swap -

annual points

Market ModelInflationForward

Year on Year

Option Prices

Year on Year Vol and smile

InflationForward

Zero Coupon


Year on Year

ZC Volatility

Short Rate Model


Structured Products Catalogue





20Y EUR revenue swap

Target Clients Risk Profile Currency Maturity Format

Clients with revenue linked to inflation

EUR 20Y Swap

Client receives

Y1 to Y20: (1 + X)n - 1

Client pays

Y1 to 20:

inflation = ( ) 1−0

)(HICPxTEuro

nHICPxTEuro

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 HICPxT for the month

preceding the inception date

STRUCTURE DESCRIPTION

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.

PRODUCT OVERVIEW



10Y EUR Livret A swap


French bank ALMs EUR 10Y Swap

Client receives

Quarterly, Act/360

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

Client pays

Semiannual, 30/360

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

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 = 1)12(

)(−

−nCPIxTFrenchnCPIxTFrench

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).


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.

PRODUCT OVERVIEW



10Y EUR TFR swap


French bank ALMs EUR 10Y Swap

Client receives

Quarterly, Act/360

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

Client pays

Annual

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

Italian YoY = 1)12(

)(−

−nCPIxTItaliannCPIxTItalian

Italian CPIxT (n) = Value of the Italian national price index

excluding tobacco, (FOIxT) and as published on the Bloomberg

page ITCPI.


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.

PRODUCT OVERVIEW



10Y EUR swap spread France/Europe


Asset or Liability EUR 20Y Swap

Client receives

Quarterly, Act/360

Y1 to Y20: Euribor 3M p.a.

Client pays

Annual Act / 360

Y1 – Y2 X1% - Unconditional

Y3 – Y20 X2%- 5 x spread

With

spread = YoY Euro inflation � YoY French inflation

YoY Euro inflation = 1)12(

)(−

−nHICPxTEuronHICPxTEuro

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 = 1)12(

)(−

−nCPIxTFrenchnTFrenchCPIx

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.

Spread France/Europe

-0.4%

-0.2%

0.0%

0.2%

0.4%

0.6%

0.8%

1.0%


Statistics Since January 1991

Average spread: 0.528%

Maximum/minimum spread: 1.415% / -0.567%



10Y EUR swap switch (spread France/Europe)


Liability EUR 20Y Swap

Client receives

Quarterly, Act/360


Client pays

Annual Act / 360

Y1 – Y 10 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 = 1)12(

)(−

−nHICPxTEuronHICPxTEuro

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

Euribor rate environment.

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

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

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

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

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

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

as long as the inflation spread is higher than 0.216% (note that over the past decade it

has averaged 0.528%).

Risk: The most substantial risk is a sharp increase in French inflation in absolute terms

or relative to levels in other European countries.

French and Euro Inflation

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%


French HICP Euro HICP

France/Europe Spread vs 6M Euribor

-1.000%

-0.500%

0.000%

0.500%

1.000%

1.500%

2.000%

0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00%

Correl Infla Spread € -Fr & Euribor3M Linear (Correl Infla Spread € -Fr & Euribor3M)



5Y EUR range accrual


Investment EUR 5Y EMTN

Maturity: 5 years

Coupon: 30/360 annually

(Euribor 3M + X%) * n/N

Where:

n = number of observations during the interest period when YoY CPTFEMU1 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%.

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.

Historical evolution of YoY relative

to limits

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%




10Y EUR swap corridor



Client Receives

Quarterly, Act/360


Client pays

Annual Act / 360

Y1: X1%

Y2 �Y10: X1% + X2% * n / N

With

n= number of months where YoY European inflation1 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.

Mechanism: The client receives Euribor 3M every quarter. He pays a fixed

rate of X1% and an extra X2% p.a. for every month European inflation is

observed to be lower than 1.00% or higher than 2.60%.

Advantages:

Guaranteed unconditional Euribor - X1% 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 X1% + X2%)

Historical & forward EUR inflation

0.000%

0.500%

1.000%

1.500%

2.000%

2.500%

3.000%

Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07 Jan-09 Jan-11 Jan-13 Jan-15 Jan-17

Histo Eur Inflation Fwd Eur Inflation Barrier



20Y EUR Kheops



Client receives

Quarterly, Act/360


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 CPTFEMU index.

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.


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%.

Historical & forward EUR inflation

2%

3%

4%

5%

6%

1.0% 1.4% 1.8% 2.2% 2.6% 3.0%

KHEOPS Profile

Current 20Y



10Y EUR HICP index-linked leverage slope


Liability EUR 10Y EMTN

Maturity: 10 years

Coupon: 30/360 annually

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.

Historical evolution of YoY relative

to limits

0

50

100

150

200

Jan-00 Jan-02 Jan-04 Jan-06 Jan-08

spread (bp)



Hybrid inflation/rate performance swap (HIRPS)

Target Investors Risk Profile Currency Maturity Format

LDI/AM EUR 5Y to 20Y EMTN

Fear of rate increase

Capping French inflation

French Inflation Performance Swap

Receive Euribor 3M

Pay

Years 1-2 X1% without condition

Years 3-20 Lev * French inflation if Euribor 12M < X2%Euribor 12M if Euribor 12M > X2%Inflation capped at 2.50%

French Inflation Bear Performance Swap

Receive Euribor 3M

Pay

Years 1-2 Fixed Rate without condition

Years 3-20 Market Rate if Euribor 12M < X1%Leverage * French inflation if X1% < Euribor 12M < X2%Market Rate if X2% < Euribor 12M < X3%Market Rate + Fixed Rate if Euribor 12M > X3%

French Inflation Carry Swap

Receive

Product Independent of Euribor condition

Pay Lev * French inflationInflation capped at 2.30%

Capped French Inflation Bear Performance Swap

Receive Euribor 3M

Pay


Years 3-20 Market Rate if Euribor 12M < X1%Leverage * French inflation if X1% < Euribor 12M < X2%Market Rate if X2% < Euribor 12M < X3%Market Rate + Fixed Rate if Euribor 12M > X3%Inflation capped at 2.50%

Euribor 3MFear of rate

increase

Capping French inflation

French Inflation Performance Swap

Receive Euribor 3M

Pay

Years 1-2 X1% without condition

Years 3-20 Lev * French inflation if Euribor 12M < X2%Euribor 12M if Euribor 12M > X2%Inflation capped at 2.50%

French Inflation Bear Performance Swap

Receive Euribor 3M

Pay


Years 3-20 Market Rate if Euribor 12M < X1%Leverage * French inflation if X1% < Euribor 12M < X2%Market Rate if X2% < Euribor 12M < X3%Market Rate + Fixed Rate if Euribor 12M > X3%

French Inflation Carry Swap

Receive

Product Independent of Euribor condition

Pay Lev * French inflationInflation capped at 2.30%

Capped French Inflation Bear Performance Swap

Receive Euribor 3M

Pay


Years 3-20 Market Rate if Euribor 12M < X1%Leverage * French inflation if X1% < Euribor 12M < X2%Market Rate if X2% < Euribor 12M < X3%Market Rate + Fixed Rate if Euribor 12M > X3%Inflation capped at 2.50%

Euribor 3M


The range of products available has widened dramatically and our specialists advise clients on how best to protect them-

selves 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



Client receives

Quarterly, Act/360


Client pays

Annual Act / 360

Y1 – Y 2 X1% unconditional

Y3 – Y 20 leverage * inflation if Euribor 12M < X2%

Euribor 12M – 0.02% if Euribor 12M > X2%

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.

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.

Historical correlation between French

inflation and Euribor 12M

0%

1%

2%

3%

4%

0% 2% 4% 6% 8% 10% 12%

Euribor

French inflation

0%

1%

2%

3%

4%

0% 2% 4% 6% 8% 10% 12%

Euribor

French inflation


INDEX

A

Accreting asset swaps ................................................... 74

AR, MA, ARMA and ARIMA models ............................... 34

B

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

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

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

C

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

Carry and forward price.................................................. 54

Chained index ................................................................. 20

CME future ...................................................................... 83

Collar ............................................................................... 81

Convexity ........................................................................ 51

Convexity adjustment..................................................... 63

CPI forward curve........................................................... 65

CPI interpolation ....................................................... 66, 68

D

Dummies method ........................................................... 32

Duration........................................................................... 51

E

Early redemption asset swaps ....................................... 75

Eurex future..................................................................... 85

Euro HICP ....................................................................... 24

Euro inflation derivatives .................................................. 9

European Inflation Convergence.................................... 25

F

Fisher equation ............................................................... 88

Fisher index..................................................................... 20

Foreign Currency Analogy.............................................. 89

French CPI (Indice des prix à la consommation, IPC)... 27

H

History ............................................................................... 9

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

I

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

inflation breakeven ..........................................................49

Inflation forecasting.........................................................10

Inflation payers................................................................14

Inflation Products ............................................................40

Inflation receivers ............................................................14

Inflation Swaps................................................................58

Inflation year-on-year caps and floors ...........................79

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

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

Inflation-linked bonds .....................................................45

Inflation-linked futures ....................................................83

Inflation-linked options ...................................................78

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

J

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

L

Lag and indexation..........................................................47

Laspeyres index price .....................................................19

M

Market Models.................................................................92

Market participants .........................................................14

Marshall-Edgeworth index..............................................20

Measuring Inflation..........................................................17

N

Nominal economy ...........................................................21

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

P

Paasche’s price index.....................................................20

Par/par and proceeds asset swap spread .....................73

Par/par and proceeds asset swaps................................70

Price index calculation....................................................19

Pricing Models.................................................................87

R

Range accruals................................................................12

Real economy..................................................................21

Real interest rates ...........................................................21

Real rate swaptions.........................................................80

Real swap valuation ........................................................64

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


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

S

Seasonality...................................................................... 30

Seasonality in the euro zone .......................................... 36

Short-Rate Models ......................................................... 95

T

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

U

UK RPI (Retail Price Index)............................................. 27

US CPI............................................................................. 22

US seasonality ................................................................ 38

X

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

Y

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

Z

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

Zero coupon swaps ........................................................59

Z-spread ..........................................................................75

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