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
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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
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
Case study............................................................................................................................................ 36 Seasonality in the euro zone..................................................................................................................................................36 US seasonality .......................................................................................................................................................................38
Overview............................................................................................................................................... 41 From inflation bonds to inflation swaps .................................................................................................................................41 From inflation swaps to inflation volatility ..............................................................................................................................43
Description and conventions ...........................................................................................................................................45
Lag and indexation ..........................................................................................................................................................47
Key pricing and valuation concepts.......................................................................................................................................48
Inflation Market Handbook
Inflation Market Handbook – January 2008 4
Invoice price and quotation .............................................................................................................................................48
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
Real swaps ......................................................................................................................................................................63
Building a CPI forward curve .................................................................................................................................................65
Par/par and proceeds asset swaps.................................................................................................................................70
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
Why another model?..............................................................................................................................................................95 Model definition .....................................................................................................................................................................95 A possible improvement: inflation ratio as a state variable....................................................................................................97
Which model for which purpose? ...................................................................................................... 99
Index ..................................................................................................................................... 113
Executive Summary
Inflation Market Handbook – January 2008 6
Executive Summary
Executive Summary
Inflation Market Handbook – January 2008 7
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
Inflation Market Handbook – January 2008 8
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.
Market Review History
Inflation Market Handbook – January 2008 10
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
Market Review History
Inflation Market Handbook – January 2008
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
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
Market Review History
Inflation Market Handbook – January 2008 12
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
Inflation Market Handbook – January 2008
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
Inflation Market Handbook – January 2008 14
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
Market Review Market participants
Inflation Market Handbook – January 2008
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
Source: SG Quantitative Strategy
Since 2003, the number of investors willing to receive inflation has increased significantly and the focus
has switched from traditional bond products to more sophisticated structured products. EMTN
issuance activities have helped this trend by offering investors access to the inflation market through
structured bonds. This has forced retail banks to hedge themselves, increasing volumes of swaps and
Market Review Market participants
Inflation Market Handbook – January 2008 16
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
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
Measuring Inflation Introduction
Inflation Market Handbook – January 2008 19
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.
Measuring Inflation Introduction
Inflation Market Handbook – January 2008 20
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
Measuring Inflation Introduction
Inflation Market Handbook – January 2008 21
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
Source: SG Quantitative Strategy
Real and nominal interest rates are sometimes compared to the (nominal) interest rates paid by two
different currencies. The inflation index (CPI) plays the role of an exchange rate that translates the
�value� of assets in one currency (the real economy) into the other currency (the nominal economy). The
former is a basket of goods and services, the latter is the nominal value of this basket. The inflation rate
is the growth of this �exchange rate�.
The relationship between real and nominal rates is also known as the Fisher equation (see technical box
on page 88).
Measuring Inflation Calculation of indices
Inflation Market Handbook – January 2008 22
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
Measuring Inflation Calculation of indices
Inflation Market Handbook – January 2008 23
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
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.
Seasonality Measurement
Inflation Market Handbook – January 2008 34
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.
Seasonality Measurement
Inflation Market Handbook – January 2008 35
HICP Inflation and YoY seasonality for the 96-06 period Estimation residuals
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%
J F M A M J J A S O N D
MoM Adjustments (%) Jan 96 - Dec 00
-1.5%
-1.0%
-0.5%
0.0%
0.5%
1.0%
1.5%
J F M A M J J A S O N D
MoM Adjustments (%) Jan 01 - Dec 06
-0.3%
-0.2%
-0.1%
0.0%
0.1%
0.2%
0.3%
J F M A M J J A S O N D
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;
Seasonality Case study
Inflation Market Handbook – January 2008 38
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
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.
Inflation Products Overview
Inflation Market Handbook – January 2008 42
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
Source: SG Quantitative Strategy
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.
Inflation Products Overview
Inflation Market Handbook – January 2008
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
Exotic Option Prices
Source: SG Quantitative Strategy
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
Inflation Products Overview
Inflation Market Handbook – January 2008 44
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 Market Handbook – January 2008 45
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;
Inflation Products Inflation-linked bonds
Inflation Market Handbook – January 2008 46
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
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
Inflation Products Inflation-linked bonds
Inflation Market Handbook – January 2008 47
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:
Inflation Products Inflation-linked bonds
Inflation Market Handbook – January 2008 48
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
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:
Inflation Products Inflation-linked bonds
Inflation Market Handbook – January 2008 49
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.
Inflation Products Inflation-linked bonds
Inflation Market Handbook – January 2008 50
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.
Inflation Products Inflation-linked bonds
Inflation Market Handbook – January 2008 51
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.
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.
Inflation Products Inflation-linked bonds
Inflation Market Handbook – January 2008 52
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.
Inflation Products Inflation-linked bonds
Inflation Market Handbook – January 2008 53
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
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 Market Handbook – January 2008 58
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
Source: SG Quantitative Strategy
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.
Inflation Products Inflation Swaps
Inflation Market Handbook – January 2008 59
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
Source: SG Quantitative Strategy
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.
Inflation Products Inflation Swaps
Inflation Market Handbook – January 2008 60
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
Inflation Products Inflation Swaps
Inflation Market Handbook – January 2008 61
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=
Inflation Products Inflation Swaps
Inflation Market Handbook – January 2008 62
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
Every year until maturity
Inception
Source: SG Quantitative Strategy
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.
Inflation Products Inflation Swaps
Inflation Market Handbook – January 2008 63
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 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.
Inflation Products Inflation Swaps
Inflation Market Handbook – January 2008 69
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 Market Handbook – January 2008 70
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
Inflation Products Inflation-linked asset swaps
Inflation Market Handbook – January 2008 71
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
IL Coupon =CPI(Ti)/CPI(0)*R
Asset SwapSeller
Par or Proceeds amount
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
Par or Proceeds amount
Asset SwapSeller
Libor +/- spread on par or proceeds amount
Asset SwapBuyer
IL Coupon =CPI(Ti)/CPI(0)*R
IL Coupon =CPI(Ti)/CPI(0)*R
Asset SwapSeller
Par or Proceeds amount
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
Source: SG Quantitative Strategy
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
Inflation Products Inflation-linked asset swaps
Inflation Market Handbook – January 2008 72
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
Source: SG Quantitative Strategy
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
Inflation Products Inflation-linked asset swaps
Inflation Market Handbook – January 2008 73
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:
Source: SG Fixed Income Strategy Research – Bloomberg Source: SG Fixed Income Strategy Research - Bloomberg
Inflation Products Inflation-linked options
Inflation Market Handbook – January 2008 78
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)
Source: SG Quantitative Strategy
The strike is expressed in annual average inflation growth so that the pay-off of a zero coupon cap is
defined as follows:
( ) ( )( ) ( ) ⎟⎟
⎠
⎞⎜⎜⎝
⎛+−= 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 Products Inflation-linked options
Inflation Market Handbook – January 2008 79
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.
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.
Inflation Products Inflation-linked futures
Inflation Market Handbook – January 2008 84
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
Inflation Products Inflation-linked futures
Inflation Market Handbook – January 2008 85
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)
- 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
Source: SG Inflation Trading Desk
Pricing Inflation Derivatives Short-Rate Models
Inflation Market Handbook – January 2008
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
Pricing Inflation Derivatives Short-Rate Models
Inflation Market Handbook – January 2008
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.
Pricing Inflation Derivatives Short-Rate Models
Inflation Market Handbook – January 2008
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.
Pricing Inflation Derivatives Short-Rate Models
Inflation Market Handbook – January 2008
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?
Inflation Market Handbook – January 2008
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?
Inflation Market Handbook – January 2008
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
Exotic Option Prices
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
Exotic Option Prices
Year on Year
ZC Volatility
Short Rate Model
Source: SG Quantitative Strategy
Structured Products Catalogue
Inflation Market Handbook – January 2008 101
Structured Products Catalogue
Structured Products Catalogue
Inflation Market Handbook – January 2008 102
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
Structured Products Catalogue
Inflation Market Handbook – January 2008 103
10Y EUR Livret A swap
Target Clients Risk Profile Currency Maturity Format
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).
STRUCTURE DESCRIPTION
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
Structured Products Catalogue
Inflation Market Handbook – January 2008 104
10Y EUR TFR swap
Target Clients Risk Profile Currency Maturity Format
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.
STRUCTURE DESCRIPTION
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
Structured Products Catalogue
Inflation Market Handbook – January 2008 105
10Y EUR swap spread France/Europe
Target Clients Risk Profile Currency Maturity Format
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.
STRUCTURE DESCRIPTION
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%
Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07
Statistics Since January 1991
Average spread: 0.528%
Maximum/minimum spread: 1.415% / -0.567%
Structured Products Catalogue
Inflation Market Handbook – January 2008 106
10Y EUR swap switch (spread France/Europe)
Target Clients Risk Profile Currency Maturity Format
Liability EUR 20Y Swap
Client receives
Quarterly, Act/360
Y1 to Y20: Euribor 12M p.a.
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.
STRUCTURE DESCRIPTION
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.
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