Download - SG Handbook Inflation 2008
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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
Stéphane Salas – Head of Inflation Trading (33) 1 42 18 05 39
Julien Turc – Head of Quantitative Strategy (33) 1 42 13 40 90
Quantitative Strategy
Inflation Market Handbook
Inflation Market Handbook – January 2008 2
Inflation Market Handbook
Inflation Market Handbook – January 2008 3
Table of Contents Executive Summary.................................................................................................................. 6
Market Review.......................................................................................................................... 8
History..................................................................................................................................................... 9
Volumes ................................................................................................................................................ 13
Market participants ............................................................................................................................. 14
Measuring Inflation ................................................................................................................. 17
Introduction.......................................................................................................................................... 18 How to measure inflation? .....................................................................................................................................................18 Introducing real interest rates ................................................................................................................................................21
Calculation of indices.......................................................................................................................... 22 US CPI ...................................................................................................................................................................................22 Euro HICP ..............................................................................................................................................................................24 French CPI (Indice des prix à la consommation, IPC) ............................................................................................................27 UK RPI (Retail Price Index).....................................................................................................................................................27 Further information ................................................................................................................................................................28
Seasonality ............................................................................................................................. 29
Definition .............................................................................................................................................. 31
Measurement ....................................................................................................................................... 32
Case study............................................................................................................................................ 36 Seasonality in the euro zone..................................................................................................................................................36 US seasonality .......................................................................................................................................................................38
Inflation Products ................................................................................................................... 40
Overview............................................................................................................................................... 41 From inflation bonds to inflation swaps .................................................................................................................................41 From inflation swaps to inflation volatility ..............................................................................................................................43
Inflation-linked bonds ......................................................................................................................... 45 Product Mechanism...............................................................................................................................................................45
Description and conventions ...........................................................................................................................................45
Lag and indexation ..........................................................................................................................................................47
Key pricing and valuation concepts.......................................................................................................................................48
Inflation Market Handbook
Inflation Market Handbook – January 2008 4
Invoice price and quotation .............................................................................................................................................48
Linkers yield, inflation breakeven.....................................................................................................................................49
Risk premium...................................................................................................................................................................50
Duration and beta............................................................................................................................................................51
Carry and forward price...................................................................................................................................................54
Inflation Swaps .................................................................................................................................... 58 Real, inflation and standard swap markets............................................................................................................................58 Inflation and real swaps: characteristics and mechanisms....................................................................................................59
Zero coupon swaps.........................................................................................................................................................59
Year-on-Year inflation swaps ..........................................................................................................................................62
Real swaps ......................................................................................................................................................................63
Building a CPI forward curve .................................................................................................................................................65
Inflation-linked asset swaps............................................................................................................... 70 Asset swaps definitions .........................................................................................................................................................70
Par/par and proceeds asset swaps.................................................................................................................................70
Accreting asset swaps ....................................................................................................................................................74
Early redemption asset swaps.........................................................................................................................................75
Another asset swap measure for bonds: Z-spread .........................................................................................................75
Inflation-linked options ....................................................................................................................... 78 Standard options ...................................................................................................................................................................78
Inflation zero coupon caps and floors .............................................................................................................................78
Inflation year-on-year caps and floors.............................................................................................................................79
Real rate swaptions .........................................................................................................................................................80
Strategies with caps and floors .............................................................................................................................................81
Inflation-linked futures ........................................................................................................................ 83 CME future.............................................................................................................................................................................83 Eurex future............................................................................................................................................................................85
Pricing Inflation Derivatives.................................................................................................... 86
Background to Pricing Models........................................................................................................... 87
Foreign Currency Analogy .................................................................................................................. 89
Market Models ..................................................................................................................................... 92
Short-Rate Models .............................................................................................................................. 95
Inflation Market Handbook
Inflation Market Handbook – January 2008 5
Why another model?..............................................................................................................................................................95 Model definition .....................................................................................................................................................................95 A possible improvement: inflation ratio as a state variable....................................................................................................97
Which model for which purpose? ...................................................................................................... 99
Structured Products Catalogue............................................................................................ 101
20Y EUR revenue swap ..................................................................................................................... 102
10Y EUR Livret A swap...................................................................................................................... 103
10Y EUR TFR swap............................................................................................................................ 104
10Y EUR swap spread France/Europe ............................................................................................ 105
10Y EUR swap switch (spread France/Europe) .............................................................................. 106
5Y EUR range accrual ....................................................................................................................... 107
10Y EUR swap corridor ..................................................................................................................... 108
20Y EUR Kheops................................................................................................................................ 109
10Y EUR HICP index-linked leverage slope .................................................................................... 110
Hybrid inflation/rate performance swap (HIRPS) ........................................................................... 111
20Y EUR Hybrid performance swap ................................................................................................ 112
Index ..................................................................................................................................... 113
Executive Summary
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
Infla
tio
n li
nke
d B
on
ds
Infla
tio
n li
nke
d S
wap
sO
ther
Der
ivat
ives
British,
Australian,
Sweden,
Canadian
Linkers
UK RPI
swaps
US TIPS,
French
OAT
50cts Bid Ask on 2.5M
25cts Bid Ask on 10M
10cts Bid Ask on 50M
10cts Bid Ask on 100M
Bonds quote in break-even
10Y,30Y on
French CPI
First HICP
10Y bond
(France),
Fixing July
30Y HICP
(France)
5Y (Italy),
20Y (Greece),
Fixing Sep.
10Y on
French CPI
7Y on
French CPI
15Y,
10Y HICP
First
Japanese ILBOutstanding amounts
$200b. $230b. $260b. $340b. $450b. $680b. $850b. <$1000b.
5Y, 10Y, 30Y
HICP
12Y on
French CPI
10Y
(Germany),
11Y HICP
Fixing March
30Y, 32Y,
33Y, 49Y, 50Y
HICP
30Y on
French CPI
First
Structured for
EMTN desks
Inflation future attempts in the US
CME future on US CPI
CME future on HICP
Eurexfuture
Asset swap
packages on
BTPe08
Inter-bank
asset swaps
on other
issues
Rate/Inflation
hybrids
Range
Accruals
Customized
structured for LDI
First options
on European
inflation
UK LPI
options
>€1b. >€1b.
First swaps
on European
inflation
>€1b. >€2b. €20b. €45b.
Bond ASW are
calculated from
swaps break-even
Market consensus on seasonality
Increased
liquidity
up to 10Y
Swap prices
are calculated
from bonds
First annual
0% floors
Long term
inflation swaps
(up to 30Y)
Liquidity on all
the curve
$50b.
Secondary market volumes (Euro zone only)
€56b. €66b. €74b.
1998< 2000 2007200620052004200320022001
Cash Market infancy Derivatives Market Birth Structured Inflation Age
Infla
tio
n li
nke
d B
on
ds
Infla
tio
n li
nke
d S
wap
sO
ther
Der
ivat
ives
British,
Australian,
Sweden,
Canadian
Linkers
UK RPI
swaps
US TIPS,
French
OAT
50cts Bid Ask on 2.5M
25cts Bid Ask on 10M
10cts Bid Ask on 50M
10cts Bid Ask on 100M
Bonds quote in break-even
10Y,30Y on
French CPI
First HICP
10Y bond
(France),
Fixing July
30Y HICP
(France)
5Y (Italy),
20Y (Greece),
Fixing Sep.
10Y on
French CPI
7Y on
French CPI
15Y,
10Y HICP
First
Japanese ILBOutstanding amounts
$200b. $230b. $260b. $340b. $450b. $680b. $850b. <$1000b.
5Y, 10Y, 30Y
HICP
12Y on
French CPI
10Y
(Germany),
11Y HICP
Fixing March
30Y, 32Y,
33Y, 49Y, 50Y
HICP
30Y on
French CPI
First
Structured for
EMTN desks
Inflation future attempts in the US
CME future on US CPI
CME future on HICP
Eurexfuture
Asset swap
packages on
BTPe08
Inter-bank
asset swaps
on other
issues
Rate/Inflation
hybrids
Range
Accruals
Customized
structured for LDI
First options
on European
inflation
UK LPI
options
>€1b. >€1b.
First swaps
on European
inflation
>€1b. >€2b. €20b. €45b.
Bond ASW are
calculated from
swaps break-even
Market consensus on seasonality
Increased
liquidity
up to 10Y
Swap prices
are calculated
from bonds
First annual
0% floors
Long term
inflation swaps
(up to 30Y)
Liquidity on all
the curve
$50b.
Secondary market volumes (Euro zone only)
€56b. €66b. €74b.
Source: SG Quantitative Strategy
2006: Spread France-Europe and hybrid structures
In 2006 the market was ready for its first optional products. Structured desks launched optional
features with hybrid structures mixing Libor and inflation fixings. For example, some banks issued
structures paying the Libor minimum plus a margin and year-on-year inflation rates multiplied by a
lever. This type of structure is sensitive to inflation/interest rate correlation and was probably a way for
some dealers to unwind correlation exposures.
In Europe, this was the time of the French/European spread, the first example of an imbalance between
inflation in Europe and that in one of its member countries. 2006 saw an increase in demand for the
French Livret A. The Livret A is one of France�s most popular savings accounts, whose remuneration
formula has been based on the year-on-year fixing of the French inflation index for December and June
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
Real Income
ASSETS LIABILITIES
Inflation Market
Inflation Payers:UtilitiesProject FinanceReal EstateRetailersSovereignsAgencies
IL Income IL Coupon
Financing
Real Payment
Inflation Receivers:
Pension fundsInsurance
Mutual fundsCorporate ALM
IL Coupon IL Payment
Libor
Inflation Receivers:Retail Banks
Inflation Payers
Receivers:Proprietary
tradersInvestment
banksHedge funds
Investors
IL Coupon Financing
IL Coupon
Financing
Real Income
ASSETS LIABILITIES
Inflation Market
Inflation Payers:UtilitiesProject FinanceReal EstateRetailersSovereignsAgencies
IL Income IL Coupon
Financing
Real Payment
Inflation Receivers:
Pension fundsInsurance
Mutual fundsCorporate ALM
IL Coupon IL Payment
Libor
Inflation Receivers:Retail Banks
Inflation Payers
Receivers:Proprietary
tradersInvestment
banksHedge funds
Investors
IL Coupon Financing
IL Coupon
Financing
Source: SG Quantitative Strategy
Measuring Inflation
Inflation Market Handbook – January 2008
17
Measuring Inflation
Measuring Inflation Introduction
Inflation Market Handbook – January 2008 18
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
0%
1%
2%
3%
4%
5%
6%
7%
88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07
US CPI-U Regional Minimum Regional Maximum
Source: SG Quantitative Strategy - Bloomberg Source: SG Quantitative Strategy - Bloomberg
Maximum inflation history: Inflation has been higher in the Northeast and West region in the last 20 years
Spread between maximum and minimum inflation in the four main urban regions
0%
1%
2%
3%
4%
5%
6%
7%
88 90 92 94 96 98 00 02 04 06
South Urban Mid West Urban North East Urban West Urban
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06 07
Source: SG Quantitative Strategy - Bloomberg Source: SG Quantitative Strategy - Bloomberg
Measuring Inflation Calculation of indices
Inflation Market Handbook – January 2008 24
Euro HICP The Harmonised Indices of Consumer Prices for EU countries are published by Eurostat using data
issued by EU member states� statistics offices. They provide a unified framework to calculate and
compare inflation data. HICPs and national CPIs can be significantly different as national CPIs are
mostly based on methodologies chosen prior to the creation of the HICP. Some of the differences are:
Subsidised healthcare and education: the HICPs include the net price paid by consumers, while
some national indices either exclude these purchases altogether or record the gross price;
Owner-occupied housing: the HICPs currently exclude the cost to owners of financing their
property (interest and credit charges), while some national indices include these costs. The problem
with including these items is that it introduces a direct dependency on nominal interest rates into the
measurement of inflation;
Aggregation formulae: the HICPs are Laspeyres-type indices. National methods may be somewhat
different;
Geographical and population coverage: the HICPs cover expenditure by residents and visitors in
each country, while some national CPIs cover expenditure by domestic residents within and outside the
country.
The HICPs are published by Eurostat every month, generally 17 to 19 days after the end of the month
measured. The main areas covered are housing, food and beverages, transport, recreation and culture,
restaurants and hotels, which together account for 77%. They include all costs faced by consumers
and so include sales taxes such as Value Added Tax (VAT). In addition to the aggregate index and the
sectoral indices, some special aggregates are provided such as the HICP excluding tobacco and the
HICP excluding energy. The unrevised HICP excluding tobacco is the reference for all euro-
denominated inflation-linked bonds.
Like the BLS in the US, representatives of member states� statistics offices collect prices from local
retailers and service providers. When needed, the local offices make price adjustments to account for
potential changes in products� quality. Eurostat imposes minimum standards of quality adjustment, but
there is as yet no harmonised calculation method, although one is being developed.
The coverage of the reference basket is the same from one country to the next. However, sector
weights are defined at a country level based on local expenditure to preserve the consumption
characteristics of each member state. All countries use the same computation and aggregation
methods and the same calculation formulae. The final HICP index is compiled as a weighted average of
the countries in the euro zone. The country weights are derived from national accounts data for
household final monetary consumption expenditure.
Eurostat provides comprehensive methodological documents on HICP calculations and
methodologies4.
4 Harmonised Indices of Consumer Prices (HICPs) – A short guide for Users � European Commission � March
2004
Measuring Inflation Calculation of indices
Inflation Market Handbook – January 2008 25
European Inflation Convergence
The European Economic Community (EEC) was founded in 1957 with the signing of the Treaty of Rome.
In 1979, the Jenkins Commission set up the European Monetary System (EMS), whereby EEC member
states (with the exception of the UK) agreed to link their currencies to the European Currency Unit
(ECU) through the Exchange Rate Mechanism (ERM). From this point in time, the inflation rates of ERM
members have tended to converge.
In a recent study5, the ECB analyses inflation convergence since the introduction of the ERM and its
evolution since the introduction of the euro. Its main conclusions are as follows:
The ERM was essential to achieve inflation convergence by 1997. By this year, under the Maastricht
treaty, any country wishing to adopt the euro had to have fulfilled a certain number of criteria.
Countries that joined the ERM at an early stage showed strong convergence until 1997, while
countries that joined the programme later experienced higher inflation rates;
Since 1998, the report has found evidence of diverging behaviour between two main groups: a low
inflation group comprising Germany, France, Belgium, Austria and Finland and a high inflation group
including the Netherlands, Ireland, Spain, Greece, Portugal and Ireland. Italy stands in between the two
groups. Inflation convergence seems to have been achieved within each group.
The graphs below illustrate this convergence: we computed the maximum and minimum inflation levels
across European countries since 1996 and the spread between these two values. We looked at
Germany, France, Italy, Spain, the Netherlands, Belgium, Austria, Greece and Portugal. The high/low
spread has followed a decreasing trend over the past ten years. Similarly, the standard deviation of
annual inflation rates has decreased over the same period from 1.54% in 1996 to 0.63% in 2007. The
graph in the bottom left-hand corner shows the highest annual European inflation over time. Greece
had the highest inflation over the 96-99 period and alternated with Spain in the 04-07 period. The
Netherlands had the highest inflation in 2001-2002 and Ireland in 2000-2001 and 2002-2004.
Euro HICP excluding tobacco constituents (January 2007) Euro HICP annual inflation (revised series)
20%
15%
22%
10%
10%
8%
7%4% 4%
Food and beverages Transport
Housing Recreation and culture
Restaurants and hotels Misc. goods and services
Clothing and footwear Health
Education and Communications
0%
1%
2%
3%
4%
5%
6%
7%
97 99 01 03 05 07
HICP Maximum on the 9 first inflations Minimum
Source: SG Quantitative Strategy - Eurostat Source: SG Quantitative Strategy - Bloomberg
5 Inflation convergence and divergence within the European Monetary Union � F.Busetti, L.Forni, A.Harvey,
F.Venditti � January 2006
Measuring Inflation Calculation of indices
Inflation Market Handbook – January 2008 26
EU members maximum inflation: over the past 10 years inflation has been highest in Greece, Spain and the Netherlands
Spread between highest and the lowest inflation for the nine main countries in the HICPxT
0%
1%
2%
3%
4%
5%
6%
7%
97 98 99 00 01 02 03 04 05 06 07
Spain Greece Netherland Ireland Portugal
0%
1%
2%
3%
4%
5%
6%
97 99 01 03 05 07
Source: SG Quantitative Strategy - Bloomberg Source: SG Quantitative Strategy - Bloomberg
Inflation rates vary greatly between sectors. In the past five years, the communications, recreation and
culture sectors have gone through a period of disinflation. These sectors include all computer, audio,
video and telephone expenditure, plus all goods and services for personal leisure (indoor and outdoor
recreational equipments, toys and gardening). There was almost no inflation for clothing, notably due to
cheap imports from Asia. At the other end of the spectrum, education expenses inflation has always
been very high and jumped even higher recently. The housing sector is also a main contributor to the
final inflation figure. And the food and non-alcoholic beverage sector recently saw an increase in
inflation due to a rise in commodity prices.
Average inflation over the five past years (10/02-10/07) Inflation history in the sectors with max/min inflation
-3% -2% -1% 0% 1% 2% 3% 4% 5% 6%
HICP
Education
Alcohol and tobacco
Housing
Health
Transport
Restaurant and Hotel
Misc.
Food and non alc
Household
Clothing
Recreation and culture
Communication
-10%
-8%
-6%
-4%
-2%
0%
2%
4%
6%
8%
10%
97 98 99 00 01 02 03 04 05 06 07
Education Food and non alc Clothing
Communication HICP
Source: SG Quantitative Strategy - Bloomberg Source: SG Quantitative Strategy - Bloomberg
Measuring Inflation Calculation of indices
Inflation Market Handbook – January 2008 27
French CPI (Indice des prix à la consommation, IPC) The French CPI is close to its European HICP counterpart both in terms of composition and
methodology. It is published monthly by the National Institute for Statistics and Economic Studies
(INSEE) in the Official Gazette (Journal Officiel) around the 13th day of the following month. It covers all
sectors except private hospital services, certain kinds of insurance policy such as life insurance
(considered as financial products) and gambling. Seasonal effects such as sales periods are taken into
account. The INSEE produces different types of aggregate, such as the all-items IPC and the IPC
excluding tobacco. The latter is used in the inflation-linked bond market.
Like the other price indices, the CPI is measured using a reference basket and after a monthly survey of
more than 110 000 elementary products and services in 96 different urban areas including the French
overseas regions. The prices gathered by the INSEE surveyors are corrected by a quality coefficient
which depends on the evolution of the quality of each product. The reference basket�s components and
weights are updated every year according to changes in French consumption. Aggregation is first
carried out geographically and then by sector. The IPC is an annual Laspeyres-type price index.
French IPC excluding tobacco constituents (January 2007) French IPC annual inflation
16%
5%
21%
10%17%
3%
9%
7%
12%
Food and Beverages Clothing and Footwear
Housing and energy Healthcare
Transport Education and Communication
Recreation and culture Restaurants and Hotels
Other goods and services
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07
French ICPFrench HICP
Source: SG Quantitative Strategy - Eurostat Source: SG Quantitative Strategy - Bloomberg
UK RPI (Retail Price Index) The UK RPI is published every month by the Office for National Statistics (ONS). It has been the
standard domestic measure of inflation in the UK since June 1947 and is used especially to calculate
state pensions and benefits and for inflation-linked gilts. It covers all private households on the
mainland but excludes the Channel Islands and the Isle of Man. It also excludes pensioner households
(those which derive more than three-quarters of their income from state pensions and benefits) and
high-income households (total income in the top 4% of all households). In terms of expenditure items, it
covers all consumption goods and services but excludes spending linked to financial and investment
products (credit and investment expenditure, income taxes and other direct taxes, property purchased
for investment and gambling). However, council tax and mortgage interest rate payments are included,
as these represent a big part of the cost of housing in the UK. So movement in interest rates can have a
direct impact on the inflation index.
The RPI is drawn up using similar methods to those of other national statistics offices. Specific items
are chosen in accordance with the spending reports and surveyors collect prices countrywide every
month. Prices are then aggregated from the lowest level (geographic area, single item) to the highest
(national, all items) using an annually Laspeyres-type chain-linked index. As the index is chained, there
Measuring Inflation Calculation of indices
Inflation Market Handbook – January 2008 28
is no need for rebasing or revising the series each time the reference basket�s composition changes (in
such cases, rebasing is done purely for scaling purposes). The current index is based on 1987 prices.
The RPI reference basket is very different from that of the harmonised indices, especially in terms of the
treatment of interest and mortgages linked to owner-occupied houses. For example, the annual inflation
rate measured by the RPI was 4.1% at the end of August 2007 while the national harmonised CPI was
only 1.8%.
UK RPI excluding tobacco constituents (2007) UK RPI historical annual inflation
20%
5%
24%
17%
13%
4%
4%2%
11%
Food, Beverages and Tobacco Catering
Housing Motoring and Energy
Household goods and services Clothing and Footwear
Personal goods and services Fares and other travel costs
Leisure
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
4.0%
4.5%
5.0%
Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07
UK RPIUK HICP
Source: SG Quantitative Strategy – UK ONS Source: SG Quantitative Strategy - Bloomberg
Further information National statistics office web site On Bloomberg
US inflation http://www.bls.gov CPURNSA <INDEX>
European inflation http://ec.europa.eu/eurostat CPXTEMU <INDEX> (revised
data)
CPTFEMU <INDEX> (non revised)
UK inflation http://www.statistics.gov.uk UKRPI <INDEX> (UK RPI)
CPXTUKI <INDEX> (Harmonised
CPI)
French inflation http://www.insee.fr FRCPXTOB <INDEX>
CPXTFRI <INDEX>
Seasonality
Inflation Market Handbook – January 2008 29
Seasonality
Seasonality
Inflation Market Handbook – January 2008 30
Seasonality is a change in prices or business patterns at given times of the year. For example, if annual
inflation is 2% over a year, this means that goods or services worth 100 in January will be worth 102 the
following January. However, this price increase is not uniform and is subject to monthly or seasonal
variations. The possible causes for seasonal variations include natural factors (seasons, the weather),
legal measures (administered price increases, tax regime changes) and sociocultural traditions
(Christmas, summer holidays).
Seasonality accounts for fluctuations of up to 0.3%-0.4% in inflation. Over the last 10 years, inflation in
euro zone Europe (the HICP) has been maintained at between 1% and 3%, which means that seasonal
adjustments represent up to 10-30% of inflation itself - a significant proportion.
Seasonality matters when it comes to building curves of forward or zero coupon inflation. There is a
liquid market for zero coupon inflation swaps with maturities expressed in number of years from
inception. Outside this range of standardised maturity dates, there is no product that directly prices
zero coupon inflation, so interpolation techniques need to be used. However, the size of seasonal
adjustments is such that linear interpolation cannot be relied on, and past estimates of seasonality are
used to build forward inflation curves (see �Building a CPI forward curve� in the inflation swaps section,
page 65). In turn, these zero coupon curves play a key role when valuing inflation options, even on
standardised dates.
Therefore, before introducing the main inflation products traded on the market we will take a break to
discuss seasonality and the main statistical techniques used to measure it.
Seasonality Definition
Inflation Market Handbook – January 2008 31
Definition Seasonality is defined as a change in a given variable which is entirely due to events at a specific time
of the year. For example, the European HICPxT index (HICP excluding Tobacco) usually decreases by
an impressive 0.35% in January, probably due to the winter sales.
Seasonality is measured on a monthly basis using the inflation indices time series, and can be
expressed as a Month-on-Month (MoM) or Year-on-Year (YoY) correction. See the technical box below
for more details on definition and calculation.
Seasonality measurement is linked to inflation measurement: seasonal economic cycles are reflected in
the time series for the standard consumer prices indices. The European harmonised index (HICP) is the
weighted sum of the individual national composite indices, which are themselves the weighted sum of
the price index components (goods, energy, services etcetera). Seasonality at composite level can be
explained by looking at the subcomponents.
Seasonality measurement is not only crucial to be able to remove seasonal effects from a time series
and to better understand inflation dynamics. It is also important for inflation-linked derivative pricing
and strategies.
MoM versus YoY seasonality adjustments
Seasonal adjustments are calculated every month and can be expressed as YoY (Year on Year) or MoM (Month on Month), or
% MoM. MoM gives the change imputed to seasonality from one month to the next. YoY adjustment cumulates the monthly
changes from the month of January. % MoM is the difference between two consecutive YoY adjustments.
%)100%...%(%... 11 ++++∗=∗∗∗∗=∗= −− JannnnJannnnnnSAn MoMMoMMoMIMoMMoMMoMIYoYII
SAnI is the seasonally-adjusted CPI index at time n and nI is the non-adjusted CPI index at time n.
For example, using the table below, the seasonality adjustment in euros for the month of February is x99.81%.
Averaged seasonal adjustments for the major inflation indices calculated over the period Jan 1996 – Dec 2006 using X-12 ARIMA methodology
%MoM MoM YoY %MoM MoM YoY %MoM MoM YoY %MoM MoM YoY
J -0.35% 99.65% 99.65% -0.28% 99.72% 99.72% -0.59% 99.41% 99.41% 0.20% 100.20% 100.20%
F 0.16% 100.16% 99.81% 0.25% 100.25% 99.97% 0.20% 100.20% 99.62% 0.17% 100.17% 100.36%
M 0.23% 100.23% 100.04% 0.23% 100.23% 100.20% 0.16% 100.16% 99.78% 0.24% 100.24% 100.61%
A 0.22% 100.22% 100.26% 0.11% 100.11% 100.31% 0.53% 100.53% 100.31% 0.16% 100.16% 100.76%
M 0.06% 100.06% 100.32% 0.08% 100.08% 100.39% 0.12% 100.12% 100.43% 0.00% 100.00% 100.76%
J -0.09% 99.91% 100.23% -0.06% 99.94% 100.33% -0.12% 99.88% 100.32% -0.09% 99.91% 100.67%
J -0.19% 99.81% 100.04% -0.29% 99.71% 100.04% -0.40% 99.60% 99.92% -0.04% 99.96% 100.63%
A -0.05% 99.95% 99.99% 0.08% 100.08% 100.12% 0.09% 100.09% 100.01% -0.01% 99.99% 100.62%
S 0.07% 100.07% 100.06% 0.08% 100.08% 100.21% 0.23% 100.23% 100.24% 0.07% 100.07% 100.70%
O -0.04% 99.96% 100.01% -0.04% 99.96% 100.17% -0.14% 99.86% 100.10% -0.02% 99.98% 100.68%
N -0.15% 99.85% 99.86% -0.15% 99.85% 100.02% -0.14% 99.86% 99.96% -0.34% 99.66% 100.33%
D 0.14% 100.14% 100.00% -0.02% 99.98% 100.00% 0.04% 100.04% 100.00% -0.33% 99.67% 100.00%
HICPxT French CPIxT UK RPI US CPI-U
Source: SG Quantitative Strategy
Seasonality Measurement
Inflation Market Handbook – January 2008 32
Measurement Statistical seasonality measurement has been studied for a long time. Several methods have been
developed and thoroughly tested. Three have emerged over the past years: the dummies method, the
TRAMO/SEATS and the X-12 ARIMA.
The first method is fairly straightforward. It makes use of 12 dummies, which are functions equal to one
if the index is observed, say, in January (or February, March etc.) and zero elsewhere. The regression of
the index return time series against the dummies gives an estimate of seasonal adjustment. The results
found with the dummies and the averaged results found using more advanced methods are similar.
However, the dummies do not capture differences in seasonality from one year to the next, whilst the
more sophisticated methods mentioned below can show the evolution of seasonality over several
years. Also, the dummies method cuts the historical data into twelve separate time series, which greatly
reduces the accuracy of the estimation process. More sophisticated techniques try to estimate all
seasonality adjustments at the same time using all the available data. Although the dummies method
can be a useful instrument for a quick estimate of seasonality parameters, it cannot replace more in-
depth statistical analysis.
The other methods are more elaborate and use widely-tested statistical models:
TRAMO/SEATS (Time Series Regression with ARIMA noise, Missing value and Outliers � Signal
Extraction in ARIMA Time Series) was developed by the Bank of Spain. See the technical section later
in this section for more details on seasonality in ARIMA models.
X12-ARIMA (experimentation 12 � Auto Regressive Integrated Moving Average). This algorithm was
developed and has been extensively used by the US Census Bureau.
These methods have both been implemented by Eurostat in an application called Demetra, a tool
which can be downloaded from the Eurostat website. The statistical methods available in Demetra
decompose time series of returns into three components:
a trend, which can be purely stochastic or linked to macro-economic variables;
a seasonal factor, which is a constant monthly factor reflecting the impact of seasonal behaviour on
the time series;
white noise, which contains all the effects not captured by the other components.
The procedure for calculating seasonal adjustments starts with preliminary treatment of data6 in both
these methods.
6 Data is first weighted by the number of working days in each month, in order to be able to work with comparable
quantities. Because the analysis can be done either on normal returns (difference of the index between two dates)
or on lognormal returns (difference of the log-index between two dates), a lognormality test is then run. Normal
returns are used when seasonal fluctuation is independent of the index level, and lead to additive factors and
additive adjustments. Lognormal returns are used when the size of the seasonal fluctuation is related to the level
of the index and the calculations lead to multiplicative adjustments. Outliers are then identified and removed from
the time series. In the case of inflation, this should happen very rarely since the series are fairly stable.
Seasonality Measurement
Inflation Market Handbook – January 2008 33
Averaged seasonality adjustments for HICP for the Jan 96 – Dec 06 period
Averaged seasonality adjustments in US for the Jan 96 – Dec 06 period
-0.4%
-0.3%
-0.2%
-0.1%
0.0%
0.1%
0.2%
0.3%
0.4%
J F M A M J J A S O N D
X12-ARIMA TRAMO/SEATS Dummies
MoM adjustments
-0.4%
-0.3%
-0.2%
-0.1%
0.0%
0.1%
0.2%
0.3%
0.4%
J F M A M J J A S O N D
X12-ARIMA TRAMO/SEATS Dummies
MoM adjustments
Source: SG Quantitative Strategy Source: SG Quantitative Strategy
The extraction of the trend and the computation of the seasonal adjustment depend on the statistical
method:
X12-ARIMA is a non-parametric procedure which successively estimates moving average filters.
Validation of initial assumptions (no autocorrelation, white noise residuals) after several iterations allows
for retention of the best filter.
TRAMO/SEATS is a parametric approach based on a fitted ARIMA model. It uses this filter to extract
trends and seasonality from the time series. A parametric model is usually slightly less flexible than a
non-parametric one like X12-ARIMA, but it also requires less historical data. The technical box on
ARIMA models provides more details on the estimation of seasonality.
Eurostat conducted a study to investigate which method was better. TRAMO/SEATS appeared to be
robust and efficient for evaluating a specific statistical model. X12 ARIMA does not depend on the
choice of statistical model and is in that sense more flexible. It is older and seems to be more widely-
used in the industry. Because there is no particular reason to choose one method rather than the other,
Demetra provides a battery of statistical tests to evaluate the quality of an approach over another one.
Once the question of calculation methodology is solved, there are still practical issues to address. The
ECB highlighted these issues and offered answers for the euro zone in some of its publications:
One of the first issues which springs to mind is the revision of seasonal estimates, i.e. the frequency
of calculation. Inflation indices are usually published on a monthly basis and it could be argued that the
seasonal calculation should be re-run every month to incorporate the latest available information. The
ECB�s study of standard monetary statistics (Criteria to determine the optimal revision policy: a case study based on euro zone monetary aggregates data � L.Martin � ECB), divides its revision policy into
three steps: identification of the model, estimation of its parameters and the seasonality forecast. It
concludes that optimal frequency depends on the data themselves and that in most cases systematic
re-estimation of the model and its coefficients does not improve the quality of the estimates. It finally
recommends annual revision of seasonal adjustments.
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
80
85
90
95
100
105
Jan-96 Jan-98 Jan-00 Jan-02 Jan-04 Jan-06-0.6%
-0.4%
-0.2%
0.0%
0.2%
0.4%
0.6%
HICPxT Seasonality
-0.5%
-0.3%
-0.1%
0.1%
0.3%
0.5%
Jan-96 Jan-98 Jan-00 Jan-02 Jan-04 Jan-06
Source: SG Quantitative Strategy Source: SG Quantitative Strategy
HICP versus HICP ex-tobacco seasonality January and December MoM seasonal per sector
-0.40%
-0.30%
-0.20%
-0.10%
0.00%
0.10%
0.20%
0.30%
J F M A M J J A S O N D
HICPxT HICP
-0.6%
-0.4%
-0.2%
0.0%
0.2%
HIC
PxT
Food andBev.
Transport
Housing
Recreation& C
ult.
Resto &H
otel
Misc.
Household
Clothing
Other
January MoM (%)
-0.1%
0.0%
0.1%
0.2%
HIC
PxT
Food andBev.
Transport
Housing
Recreation& C
ult.
Resto &H
otel
Misc.
Household
Clothing
Other
December MoM (%)
Source: SG Quantitative Strategy Source: SG Quantitative Strategy – The seasonality adjustments for each
sector are multiplied by the sector weights in the HICP index.
Seasonality Case study
Inflation Market Handbook – January 2008 36
Case study In this section we concentrate on inflation in Europe and in the US. We show that seasonality in Europe
increases over time, due both to growth in international competition and to inflation convergence.
Seasonality has also augmented in the US over the past 10 years, although the level is lower than in
Europe. We identify the most seasonal sectors and highlight seasonal pattern differences between the
two zones.
Seasonality in the euro zone In this section we analyse the effect of seasonality on European inflation, using the HICP ex-tobacco
(CPXTEMU), the composite indices for the European countries and Europe-level sectoral indices. This
will help us to understand where European seasonality effects come from, both in terms of economic
activities and geographical area.
Inflation in the euro zone is typically lower than average in January, July and November and significantly
above average in March and December. This is due to economic patterns which can be highlighted
using sectoral analysis:
Clothing and footwear: In 2007 this sector was ranked 8th-highest in terms of weights in the HICP
ex-tobacco. But it is an extremely seasonal sector, with sales periods in January and July and the
arrival of new collections in March and September. MoM seasonality adjustments range from -7.2% for
January and July to +5.2% in March and September - by far the widest low season/high season range.
Food and non-alcoholic beverages: This is the most heavily-weighted sector in the HICP. So it has
a negative impact in summer (July, August) when fresh food prices are low and a (relatively moderate)
positive impact in winter when prices are high. However, its total effect on the aggregate index is
moderate, ranging from -0.6% in summer to +0.35% in winter.
Recreation and culture: It is no surprise that this sector contributes the most to European inflation in
December, during the festive season.
Transport: This is also worth mentioning as it is the second most heavily-weighted sector in the
HICP. Its seasonality peaks positively in April at +1.1% and negatively in October at -1%.
Price controls and regulations: This is particularly sensitive for all items whose prices are regulated
or highly taxed, such as tobacco and alcoholic beverages. It is the main reason for the difference
between the inflation index excluding tobacco and its �all items� counterpart. For example, in January
the seasonality adjustment for the ex-tobacco composite is lower due to the increase in regulated
prices which usually occurs at the beginning of the year.
We can make the following comments concerning European countries� contributions to the HICP:
Four countries account for up to 80% of the European inflation index and its seasonality: Germany
(28.7%), France (20.3%), Italy (19%) and Spain (12%). These four countries have similar characteristics,
which are those mentioned above (strong negative seasonality in January and July, positive seasonality
over the spring months).
Germany has strong positive seasonality over the month of December. A sector analysis run on
Germany shows that this is due to the combined effect of the Restaurants & Hotels and the Recreation
& Culture sectors and is probably explained by Germany�s strong Christmas traditions.
Seasonality Case study
Inflation Market Handbook – January 2008 37
Like inflation levels, seasonal patterns are converging under EU influence. For example, the seasonal
adjustments for Italy differed widely between the 1996-2000 period and the 2001-2007 period. This is
partly explained by the harmonisation of the methods used to calculate inflation in the euro zone. For
example, Italy started to include sales price reductions in its CPI in 2001.
European countries weighting in the HICP 4 countries account for up to 80% of the inflation index
Germany France ItalySpain Netherlands BelgiumAustria Greece PortugalFinland Ireland Luxembourg&Slovenia
-0.6%
-0.4%
-0.2%
0.0%
0.2%
0.4%
0.6%
J F M A M J J A S O N D
Basket GER, FRA, ITA, ESP Others
HICPxT MoM
Source: SG Quantitative Strategy Source: SG Quantitative Strategy
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
seasonal magnitude.
Evolution of January and March MoM seasonality
Evolution of the seasonal maximum magnitude
-0.80%
-0.60%
-0.40%
-0.20%
0.00%
0.20%
0.40%
0.60%
97 98 99 00 01 02 03 04 05 06
jan mar
MoM adjustments (%)
0.0%
0.2%
0.4%
0.6%
0.8%
1.0%
1.2%
97 98 99 00 01 02 03 04 05 06
Source: SG Quantitative Strategy Source: SG Quantitative Strategy
US seasonality In this section we analyse seasonal effects on US inflation. We ran analyses on the CPI-U excluding
tobacco and its main sectoral sub-indices.
Several points can be highlighted:
Increase in seasonality: The US market is naturally impacted by international competition, as US
prices are exhibiting bigger swing movements over time. For example, in 1996 the maximum difference
between two monthly inflation rates was 0.4%, while in 2006 this difference had increased to 0.82%.
Although the increase in price swing is less than in Europe, the internationalisation of the economy still
has a noticeable impact;
Importance of textiles, housing and transport: The US textiles sector follows the classical
seasonality pattern � the US sales periods are around the months of January, June and July. Housing
represents more than 40% of total US expenditure, and prices in the housing sector tend to be lower at
the end of the year and higher at the beginning of the year. Transport is more expensive in April and
cheaper in November.
US versus European inflation: The main differences between US and EU seasonality patterns occur
during the month of January and from July to December. This can be essentially explained by looking
at the composition of the two indices. On the one hand, clothing and footwear accounts for a very small
proportion of the US inflation index (3.8%), while in Europe it accounts for a larger portion in inflation
measurement (7%). And textiles are much more seasonal in Europe than in the US, with a maximum
spread of 12.4% in Europe compared with 7.6% in the US. On the other hand, the transportation sector
represents 17.4% in the US and 16% in Europe and there is a higher seasonality adjustment in the US
in October and November.
Seasonality Case study
Inflation Market Handbook – January 2008 39
Evolution of seasonal maximum magnitude in the US
Clothing and transport exhibit typical seasonal patterns
0.0%
0.1%
0.2%
0.3%
0.4%
0.5%
0.6%
0.7%
0.8%
0.9%
94 95 96 97 98 99 00 01 02 03 04 05 06
-4.00%
-3.00%
-2.00%
-1.00%
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
J F M A M J J A S O N D
Transport Clothing
Source: SG Quantitative Strategy Source: SG Quantitative Strategy
US versus EU seasonality Housing, clothing and transport sectors explain US seasonality
-0.80%
-0.60%
-0.40%
-0.20%
0.00%
0.20%
0.40%
0.60%
J F M A M J J A S O N D
HICP uscpi
-0.60%
-0.50%
-0.40%
-0.30%
-0.20%
-0.10%
0.00%
0.10%
0.20%
0.30%
0.40%
J F M A M J J A S O N D
USCPIHousing, Clothing and Transport
Source: SG Quantitative Strategy Source: SG Quantitative Strategy
Inflation Products
Inflation Market Handbook – January 2008
40
Inflation Products
Inflation Products Overview
Inflation Market Handbook – January 2008
41
Overview Before we look at inflation-linked products in detail, let us take a step back and quickly review the
different types of product and how they relate to each other. We will focus on the link between bonds
and swaps and that between swaps and options.
From inflation bonds to inflation swaps We can consider that financial products are distributed along two axes:
Nominal vs. real economy: As explained in the �inflation indices� section, the economy is �nominal�
or �real�, depending on whether market players look at the nominal value of financial investments or the
amount of goods and services they can buy. The inflation market aims to create and trade products
which have fixed features in the real economy - for example a fixed coupon - which in practice means
indexing cash flows on inflation indices.
Credit risk: sovereign vs. interbanking: Inflation derivatives are essentially used by sovereigns, via
bond issuance, or in the interbanking system7, with the recent development of inflation swaps. The
issuance of inflation-indexed products by other bodies (mainly long-term financials and corporate
issuers) is beyond the scope of this publication.
This gives us four kinds of product and relative value opportunity plus indicators for measuring relative
value.
These four categories are:
Government issuance in the real economy: As explained in greater detail in the �Inflation-linked
bonds� section (page 45), it is in sovereigns� interest to issue bonds which guarantee the notional at
maturity in real terms. This means that the bond holder will have the same purchasing power at maturity
as at inception. This kind of bond pays a real coupon, which also guarantees the bond holder�s
purchasing power. These products are commonly called inflation-linked bonds. As with any bond, a
real yield can be calculated to reflect the bond yield in real terms.
Government issuance in the nominal economy: This is traditional government bond issuance. It is
useful to mention this kind of bond here to provide an overall picture of the links between the nominal
and real economies. The difference between the usual nominal yield and the real yield is the bond breakeven, which is the main relative value indicator for inflation-linked versus nominal bond
strategies.
Interbank products in the nominal economy: All the traditional interest rate products fall into this
category. Standard vanilla swaps are particularly interesting as they are the equivalent of inflation
swaps. The difference between the nominal swap rate and the nominal bond yield is the swap spread. This is a relative value measure of sovereign and interbank risk: the higher the swap spread,
the more expensive is funding for banks compared to sovereigns and therefore the riskier the banks�
credit signature.
7 Used as a generic term covering banks and other institutional investors such as pension funds.
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
First Issuance 1981 1983 1994 1991 1997 1998 2001 2003 2003 2004 2006
Maturity 2006-2055 2010-2020 2008-2028 2021-2036 2007-2032 2009-2029 2012-2040 2025-2030 2008-2035 2014-2017 2013-2016
Amount
Outstanding (local
currency)
78 6 215 24 418 63 58 10.7 74 7917 15
Amount
Outstanding (USD)155 5.2 34 24 418 93 85 15.7 109 73 22
Reference Index RPI monthly CPI quarterly CPI monthly CPI monthly CPI-U monthlyCPI France
ex-tobacco
HICP EMU ex-
tobacco
HICP EMU ex-
tobacco
HICP EMU ex-
tobacco
CPI ex-fresh
food
HICP EMU ex-
tobacco
Bloomberg ticker UKRPI Index AUCPI Index SWCPI Index CACPI IndexCPURNSA
Index
FRCPXTOB
IndexCPTFEMU Index
CPTFEMU
Index
CPTFEMU
Index
JCPNJGBI
Index
CPTFEMU
Index
Bloomberg ILB
pageUKTI Govt SAFA Govt SGB Govt CAN Govt TII Govt FRTR Govt FRTR Govt GGB Govt BTPS Govt JGBI Govt
DBRI Govt
OBLI Govt
Coupon
Semi-annual
(pre-fixed for
8-month lag)
Quarterly (pre-
fixed)Annual Semi-annual Semi-annual Annual Annual Annual Semi-annual Semi-annual Annual
Principal
3M (after
2005) and 8M
lag
6M lag 3M lag 3M lag 3M lag 3M lag 3M lag 3M lag 3M lag 3M lag 3M lag
Repayment of
principalNo Floor
Coupon and
pricipal
protected
Floor at par No Floor Floor at par Floor at par Floor at par Floor at par Floor at par No Floor Floor at par
Source: SG inflation trading desk – SG Fixed Income Research
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
99
100
101
102
103
104
105
106
Nov-05 May-06 Nov-06 May-07 Nov-07
DIR
HICP ex tobacco
1 May 1 June 1 July
Payment in July
1 August 1 Sep 1 Oct
Coupon payment schedule
March CPI release
April CPI release
May CPI release
30 April 31 May 30 June 31 July
CPI release schedule
June CPI release
Accrued coupon in
August
1 May 1 June 1 July
Payment in July
1 August 1 Sep 1 Oct
Coupon payment schedule
March CPI release
April CPI release
May CPI release
30 April 31 May 30 June 31 July
CPI release schedule
June CPI release
Accrued coupon in
August
Source: SG Quantitative Strategy - Bloomberg Source: SG Quantitative Strategy - Bloomberg
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.
BEIR history on the OATei 2012
150
170
190
210
230
250
270
2010 2015 2020 2025 2030 2035 2040
OATei Curve @ Nov 07
ECB Target
OATei Curve @ Jan 04
OATei 3% 2012
OATei 1.6% 2015
OATei 2.25% 2020
OATei 3.15% 2032
OATei 1.8% 2040
150
160
170
180
190
200
210
220
230
240
250
Nov-02 Nov-03 Nov-04 Nov-05 Nov-06 Nov-07
BEIR OATei 2012 vs OAT Apr 2012
ECB Target
Source: SG Quantitative Strategy - Bloomberg Source: SG Quantitative Strategy - Bloomberg
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
-
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Jan-03 Jan-04 Jan-05 Jan-06 Jan-07
1 Year data (weekly return)
10 Weeks data (weekly return)
10 weeks data (daily return)
0%
20%
40%
60%
80%
100%
120%
140%
160%
05/02 05/03 05/04 05/05 05/06 05/07
OATei 2009 OATei 2012 OATei 2029
Source: SG Quantitative Strategy - Bloomberg Source: SG Quantitative Strategy - Bloomberg
What can Bloomberg tell us?
Bloomberg�s YA function provides a wide range of references. The screen is split into four boxes:
Yield Calculations: this shows the real yield of the bond, also called the street real yield on the
Bloomberg screen. It is the bond yield y corresponding to the market price using the usual formula:
( ) ( )∑=
−+
++
=N
iTT
quotedmd erestAccruedInt
yycP
Ni1
, 1100
1
Note that Bloomberg�s equivalent 2/yr compound is the US version of the equivalent semi-annual yield.
Sensitivity Analysis includes duration and convexity calculations. On the one hand, due to its lower
fixed coupon an inflation-linked bond has higher duration and convexity than a nominal bond with the
same maturity. On the other hand, real yields are less volatile than nominal yields. Standard calculations
applied to inflation-linked bonds can thus be misleading. In the sensitivity analysis box, the investor can
choose a beta between nominal and real yields to calculate effective duration and convexity. The
effective duration is the standard duration multiplied by the beta, and a linker�s convexity is standard
convexity multiplied by the square of the beta.
Inflation Products Inflation-linked bonds
Inflation Market Handbook – January 2008 54
Economic Factors provide information on the CPI fixings. It gives the base index value for the bond,
the last coupon value, the two latest CPI fixings and the current daily inflation reference.
Payment Invoice: this details the payment for a transaction on the secondary market. The quoted
price multiplied by the index ratio is the gross amount. The accrued interest is calculated and the total
is given by the net amount. This is the invoice payment that would be paid for the bond at that point in
time.
Carry and forward price Carry is a measure of how much an investor would gain or lose over a short horizon by holding an asset
rather than investing the corresponding amount in the money market. Generally defined for a bond
(nominal or inflation-linked), it can be measured by estimating the return on the following strategy:
1) Buy the bond and hold it over a given period (typically one to six months). If a coupon is paid,
reinvest it at the money market rate.
2) Finance the bond with a secured loan (using the bond as collateral) on the repo market. The
nominal repo amount will be equal to the invoice price when initiating the transaction.
3) At the end of the period, sell the bond and unwind the repo transaction. The amount to be
reimbursed is the initial repo nominal amount plus the accrued interest over the holding period.
This strategy breaks even for a given yield change between now and the end of the period. The
difference between this breakeven yield and the yield at inception gives the carry in yield terms. We will
look at two examples: one on a nominal bond, the OAT April 2012, and one on an inflation-linked bond,
the OATei 2012.
Inflation Products Inflation-linked bonds
Inflation Market Handbook – January 2008 55
Calculating carry for a nominal bond
Let�s apply the above strategy to the nominal OAT 5% April 2012:
� We buy the bond on 6 July 2007 (settlement date 11 July). The clean price quoted on the
market is �101.466 (yield 4.647%). The accrued interest is �1.0519. Buying the bond on this
date costs �102.5179 (101.466 + 1.0519).
� We finance the bond with a repo at a given rate (say 4.11%).
� After one month, we reimburse the repo and sell the bond. The loan has a total value of
�102.8758. The accrued interest on the bond is �1.4754 and the clean price of the bond
should be �101.40 (102.8758 � 1.4754) for the strategy to break even. The corresponding
yield is 4.656%.
� The carry in yield terms is 0.9bp (4.656% - 4.647%).
Calculating carry for an inflation-linked bond
We take the OATei 3% July 2012 as our example:
� On 5 November 2007 (settlement date 8 November), the market price is �105.706,
corresponding to a real yield of 1.727%. The index ratio is 1.12152 and the accrued
�0.86885, so that the invoice price is �119.5258 (1.12152 x (105.706 + 0.86885) ). The bond
is bought at this price.
� The bond is financed by a repo at 4.14%. After one month, the cash due to reimburse the
loan is �119.9325 and the index ratio is 1.12611. The accrued interest in real terms at this
date would be �1.11475. So the unadjusted clean bond price for the strategy to break even
is �105.3868 (119. 9325 / 1.12611 � 1.11475).
� The yield for a breakeven strategy is 1.776%. The carry in yield terms is 4.9bp (1.776% -
1.727%).
As explained above, the carry of an inflation-linked bond depends on the current index ratio and the
index ratio at the end of the period. This ratio is a function of the past values of the index, through the
lagging system (see �Lag and indexations� section on page 47 above). In most cases, the index ratio for
a one-month carry will be fully known. For a carry over a longer period, the index ratio will depend on
index forecasts, calculated either from market quotes or from economic forecasts. Dealers usually
prefer to use economic consensus for short-term forecasts. The methodology used to calculate inflation
forecasts from market prices will be explained in the section on calculating the CPI forward curve (page
65).
A last point to note is that the index ratio is not constant over time and can change significantly due to
seasonal effects. This has a large impact on linkers� carry, which is significantly more volatile than that
of nominal bonds. To illustrate this, we show the carry of the OATei 09 and the OAT July 09 historically
in the left-hand graph below. The nominal carry moves between 3 and -3 bp. The linker carry oscillates
between 26 and -30bp. The size of the oscillations increases as the maturity of the bond shortens,
meaning that the shorter the bond, the more important the seasonality effect on the bond carry (as
defined previously in yield terms). The seasonal impact on the carry defined in yield terms is therefore
less significant on long-dated issues.
Inflation Products Inflation-linked bonds
Inflation Market Handbook – January 2008 56
These seasonality effects also significantly impact the BEIR forward, especially for short-term bonds.
We illustrate this effect in the right-hand graph below. The table underneath provides some examples of
carry and forward BEIR for some inflation-linked euro zone bonds.
Carry of the OATei July 2009 versus the OAT April 2009 BEIR spot and forward on the OATei July 2009
-40
-30
-20
-10
0
10
20
30
Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06
OATei 3% 25-Jul-09
OAT 4% 25-Apr-09
98
118
138
158
178
198
218
238
Mar-01 Mar-02 Mar-03 Mar-04 Mar-05 Mar-06 Mar-07
beir beir fwd
Source: SG Quantitative Strategy Source: SG Quantitative Strategy
Example of carry measurement and BEIR forward for main euro zone inflation-linked bonds Bond type Description Real yield BEIR 1Mth Carry ILB 3Mth Carry ILB 6Mth Carry ILB 1Mth Fwd BEIR 3Mth Fwd BEIR 6Mth Fwd BEIROATei 3% Jul 2012 1.51 220.09 6.16 -9.49 6.82 212.94 226.38 206.51OATei 1.6% Jul 2015 1.68 222.37 3.86 -4.93 5.18 218.19 226.23 214.96OATei 2.25% Jul 2020 1.91 224.74 2.68 -2.61 4.36 222.08 227.32 220.29OATei 3.15% jul 2032 2.09 242.59 1.72 -1.33 3.14 241.10 244.53 240.65OATei 1.8% jul 2040 2.08 244.11 1.26 -0.99 2.27 243.04 245.61 242.85BTANei 1.25% Jul 2010 1.40 220.41 10.29 -18.23 10.18 207.86 231.06 193.16OATi 3% Jul 2009 1.54 218.59 17.55 0.15 22.41 197.92 207.03 164.77OATi 1.6% Jul 2011 1.55 210.79 7.39 0.24 7.91 201.96 205.82 192.71OATi 2.5% Jul 2013 1.61 216.14 4.92 0.38 5.43 210.53 213.53 206.03OATi 1% Jul 2017 1.87 218.36 3.05 0.96 4.51 215.22 217.01 213.05OATi 3.4% Jul 2029 2.17 232.30 1.89 1.03 3.48 230.64 231.90 230.06BUNDei 1.5% April 2016 1.75 215.99 3.63 -4.32 6.83 212.09 219.40 207.19BTPei 1.65% Sep 2008 1.71 213.61 49.19 -82.77 235.96 160.21 275.35 -137.70BTPei 0.95% Sep 2010 1.56 218.25 10.31 -15.45 13.02 206.64 229.34 194.96BTPei 1.85% Sep 2012 1.66 217.32 6.18 -8.06 8.19 210.41 223.04 203.98BTPei 2.15% Sep 2014 1.77 219.63 4.56 -5.23 6.68 214.82 224.03 211.15BTPei 2.1% Sep 2017 2.00 219.77 3.47 -2.98 6.03 216.46 223.16 214.53BTPei 2.6% Sep 2023 2.27 227.10 2.53 -1.47 5.07 224.91 229.52 223.94BTPei 2.35% Sep 2035 2.39 245.77 1.65 -0.79 3.48 244.52 247.73 244.65GGBei 2.9% Jul 2025 2.30 239.58 2.34 -1.34 4.85 237.60 241.92 236.75GGBei 2.3% Jul 2030 2.41 247.23 1.91 -0.91 4.13 245.71 249.24 245.31CADESi 3.4% Jul 2011 1.58 208.29 7.66 0.39 8.45 199.19 203.17 189.67CADESi 3.15% Jul 2013 1.66 211.45 5.08 0.63 6.04 205.68 208.58 200.73CADESi 1.85% Jul 2019 1.95 221.19 2.72 1.03 4.30 218.49 220.13 216.81 Source: SG Fixed income Research
Inflation Products Inflation-linked bonds
Inflation Market Handbook – January 2008 57
What can Bloomberg tell us?
Many investors use Bloomberg to analyse inflation-linked bonds. The FPA function calculates the
forward price and carry in terms of yield. We can input the settlement date (usually three working days
hence), the current market price, the repo or financing rate and the termination date or horizon of the
carry. Assumptions concerning the CPI fixing at termination can be specified and the index ratio at the
horizon (term index ratio) is calculated.
The bottom field summarises all the results: the forward price (unadjusted clean price), the full forward
price (adjusted dirty price or forward invoice price), the drop in price (gain or loss due to the passage of
time or carry in monetary amount), the YYIELD field (forward yield to maturity calculated to cancel the
P&L of the strategy) and yield drop (difference between the initial and the forward yield).
Inflation Products Inflation Swaps
Inflation 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 final expression:
( ) ( ) ( )[ ] ( )iNiiRiNNtii TtBTTBTtBETTtgYoYInflaLe ,,,,, 111 −= −−−
The first term is the price of a derivative paying the value of a real zero coupon bond at time Ti-1. If the real rates were
deterministic, this would be the present value of this real zero coupon bond paid in the nominal economy. Unfortunately, real
rates are stochastic, and the remaining expectation is not simple to calculate: it depends on the assumptions made on nominal
and real diffusion, and therefore on their volatility and correlation. This adjustment is known as the YoY convexity correction
and is fundamental for obtaining the correct price of YoY swaps and YoY options, as we will show in the section on Pricing
Models page 87).
Real swaps Real swaps are designed to synthetically replicate the flows of inflation-linked bonds. Two
counterparties sign up to the following kind of contract:
� The inflation seller will pay annually a fixed real rate X, applied to an inflated notional. As
with inflation-linked bonds, the notional is multiplied by the inflation ratio, whose reference is
the price index at inception date. At maturity the inflation seller pays back the total inflated
notional.
� In exchange, the inflation buyer pays a Libor rate, typically the 6M Euribor. At maturity, the
inflation buyer pays the non-inflated notional.
The fixed real rate X is calculated so that the transaction is zero-cost at inception. This product offers a
synthetic way of transforming a floating rate note into an inflation-linked one. Moreover, combined with
a standard vanilla swap, a fixed-rate bond can be synthetically changed into an inflation-linked one.
These swaps are increasingly popular. Dealers are now quoting real rates on screen and the number of
transactions is increasing substantially. They offer constant revenue in real terms and as such are an
attractive tool for asset and liability management.
Pricing details are given in the technical box below. Real swaps offer an alternative way to obtain the
real discount term structure, as they are expressed in pure real terms at inception. Over the life of the
transaction, a real swap receiver will essentially be exposed to real rates, as the sensitivity of the Libor
leg to the nominal curve is marginal.
Inflation Products Inflation Swaps
Inflation Market Handbook – January 2008 64
Other kinds of real swap could be envisaged. As it is possible to construct a real discount curve, one
could imagine defining a real Libor rate as the cost of borrowing money over a short period in real
terms. Swapping a fixed rate against this real Libor would be equivalent to a standard vanilla swap, but
expressed in real terms. Alternatively, one could envisage a transaction where there would be only one
real payment and one Libor payment. This would be the equivalent transaction to the zero coupon
inflation swap, but in the real economic space.
Flows in a real swap
InflationSeller
InflationBuyer
InflationSeller
X% x CPI(Ti)/CPIbase
InflationBuyer
Libor
Every year until maturity
Inception
InflationSeller
CPI(Ti)/CPI(Ti-1)
InflationBuyer
Par
At maturity
InflationSeller
InflationBuyer
InflationSeller
X% x CPI(Ti)/CPIbase
InflationBuyer
Libor
Every year until maturity
Inception
InflationSeller
CPI(Ti)/CPI(Ti-1)
InflationBuyer
Par
At maturity
Source: SG Quantitative Strategy
Real swap valuation
Valuing a real swap is very similar to zero coupon valuation and we can use an analogy to foreign currency exchange to
simplify the expression:
( ) ( ) ( ) ( ) ( ) ( ) ( )MR
M
iiRT
MNN
M
i
iiNT
Nt TtBTtBR
ITITtB
ITITtBREtaLegalSwapInfl ,,,,Re
101 0
+=⎥⎦
⎤⎢⎣
⎡+= ∑∑
==
In the previous expression, BR is the real zero coupon price and RT is the fixed real rate associated with the real swap of
maturity T. The Libor leg expression is:
( ) ( ) ( ) ( ) ( ) 1,,,,1
11 =+−=∑=
−− MN
M
iiiiNii TtBTTtLTtBtttLiborLeg
The real swap breakeven is calculated in such a way that the real swap is entered at zero cost:
( )( )∑
=
−= N
iiR
MRT
TtB
TtBR
1
,
,1
Moreover, quotes can be found in the market for the 1Y, 2Y � 30Y real swap breakeven. Using these quotes, the real zero
coupon prices can be calculated recursively:
Year 1 Year 2 (�) Year m
( )Y
R RYtB
1111,
+= ( ) ( )( )YtBR
RYtB RY
YR 1,1
112, 2
2
−+
= (�) ( ) ( )⎟⎠
⎞⎜⎝
⎛−
+= ∑
−
=
1
1,1
11,
m
iRmY
mYR iYtBR
RmYtB
Inflation Products Inflation Swaps
Inflation Market Handbook – January 2008 65
Building a CPI forward curve A CPI forward curve is calculated in two steps:
1. Using the most liquid swap instruments, we calculate the CPI forwards for the dates the market
quotes the transactions,
EU Inflation-linked ZC swap. The BEV term structure… … can be translated into a forward reference CPI curve
1.5
1.7
1.9
2.1
2.3
2008 2011 2014 2017 2020 2023 2026 2029 2032 2035
80
100
120
140
160
180
200
220
2008 2011 2014 2017 2020 2023 2026 2029 2032 2035
Source: SG Quantitative Strategy – SG Inflation Trading, November 2007 Source: SG Quantitative Strategy – SG Inflation Trading, November 2007
2. Once the CPI forwards are known for a certain date, we choose an interpolation method to
calculate intermediary points. The difficulty here lies in integrating seasonal adjustments.
Seasonally-adjusted EU interpolated swap breakeven Seasonally-adjusted EU interpolated annualised forward inflation rate
1.7
1.8
1.9
2
2.1
2.2
2.3
Aug-09 Aug-14 Aug-19 Aug-24 Aug-29 Aug-34 Aug-39
Unadjusted BEV Seasonality Adjusted BEV
-6.0%
-4.0%
-2.0%
0.0%
2.0%
4.0%
6.0%
8.0%
Aug-08 Aug-10 Aug-12 Aug-14 Aug-16 Aug-18
Adjusted Fwd Infla Yield Unadjusted Fwd Infla Yield
Source: SG Quantitative Strategy – SG Inflation Trading Source: SG Quantitative Strategy - Bloomberg
Let�s look at these two steps in more detail:
CPI forwards
Zero coupon swaps are the most liquid instruments on the inflation derivatives market. They are quoted
for annual maturity, with the maturity date corresponding to the reference month, which changes every
month.
In the technical box zero coupon swap valuation (page 61), we explained the link between zero
coupon breakevens and CPI forwards. Using these relationships, the forward price index can be simply
deduced from the swap breakevens:
( ) ( )TBEVITCPI += 10 , where T is the maturity of the swap and I0 the price index reference value.
Inflation Products Inflation Swaps
Inflation Market Handbook – January 2008 66
Let us take a numerical example. On 22 November, the mid breakeven for the 10Y zero coupon swap
on European inflation is 205.3bp, on an August 2007 fixed basis. In August, the unrevised HICP fixing is
104.19 and the 10Y nominal discount factor is 0.64. As explained in the technical box, the value of the
fixed leg is given by:
Fixed Leg = BN(21/11/07, 21/11/2017) x [ (1+ BEV(10Y) )10 � 1 ] = 0.64 x [ (1+0.02053)10 � 1 ] = 0.1443
However, the expected CPI value at maturity is also unknown. The inflation leg can be expressed as a
function of this number. Taking the indexation lags into account, this gives:
Inflation Leg = BN(21/11/07, 21/11/2017) [ CPI(31/08/2017) / CPI(31/08/2007) - 1 ] = 0.1443
The CPI projection for the month of August 2017 is therefore 127.6.
The zero coupon swap market is therefore the standard market way of obtaining the CPI projection
curve. However, it gives the CPI projection for one particular month (August in our example). In most
cases the CPI forwards are also needed for some intermediary dates, so it is vital to find an adequate
interpolation method. Such a method should incorporate some seasonal adjustment to account for
inflation variations over a year. Let us now define this interpolation method:
CPI interpolation
Once the CPI forwards have been calculated from zero coupon prices, intermediate values need to be
interpolated. A simple approach would involve the linear interpolation of CPI values estimated from the
zero coupon price. But this approach would completely ignore monthly seasonal variations and would
severely misprice some inflation-linked products. A better alternative is to consider that the CPI
reference numbers are the product of three components:
1) A reference level, which is the base level used to price current zero coupon breakevens in the
market;
2) An exponential inflation factor calculated from quoted zero coupon breakevens. The inflation
rate is assumed to be piecewise constant;
3) An exponential seasonal adjustment, which equals 100% on the fixing date of the base index.
The seasonality yield is also assumed to be piecewise constant.
To return to our example, the table below gives the summary of the zero coupon (fixed basis)
breakevens, as well as implied CPI projections at maturity for the reference fixing date of 21 November
2007. In November, the August fixings are completely known from market quotes.
Inflation Products Inflation Swaps
Inflation Market Handbook – January 2008 67
CPI projections as of the 21 November 2007
Break-even Maturity
Maturity
Reference
Fixing Base CPI
CPI
Projection
Inflation
forward
yield
1Y 2.119 22-Nov-08 Aug-08 104.19 106.40 2.10%
2Y 1.972 22-Nov-09 Aug-09 104.19 108.34 1.81%
3Y 1.925 22-Nov-10 Aug-10 104.19 110.32 1.81%
4Y 1.914 22-Nov-11 Aug-11 104.19 112.40 1.86%
5Y 1.917 22-Nov-12 Aug-12 104.19 114.57 1.91%
6Y 1.934 22-Nov-13 Aug-13 104.19 116.88 2.00%
7Y 1.957 22-Nov-14 Aug-14 104.19 119.33 2.07%
8Y 1.983 22-Nov-15 Aug-15 104.19 121.91 2.14%
9Y 2.016 22-Nov-16 Aug-16 104.19 124.69 2.25%
10Y 2.049 22-Nov-17 Aug-17 104.19 127.62 2.32%
Source: SG Quantitative Strategy
The projected fixing in August 2008 can be found the table above. Assuming that on the last day of
August the seasonal adjustment is 100%, we can deduce the constant inflation rate over the first year:
CPI(31Aug08) = CPI(31Aug07) x exp( i(0, 1Y) x 1 ) = 106.40, so that i(0,1Y) = 2.1%
Similarly, the August 2009 fixing is known and can be expressed in function of the August 2008 fixing.
The constant inflation rate for the 1Y to 2Y period can be calculated from this:
CPI(31Aug09) = CPI(31Aug08) x exp( i(1Y, 2Y) x 1 ) = 108.34, so that i(1Y,2Y) = 1.81%
We can calculate the whole term structure of the forward inflation rate recursively. The non-adjusted
CPI reference can then be calculated from this inflation rate curve.
The seasonal components are calculated using the seasonal factors, which are either found using
statistical software or provided by market consensus. Using the seasonality factors given in the
Seasonality section (page 29) as an example, we rebase the seasonal adjustments in the table on the
left below. The month of August is taken as a reference and its seasonal adjustment is therefore zero.
The unadjusted reference index for a given month is calculated as the product of the previous month�s
reference index and the monthly exponential yield. For example, CPI on 30 September 2009 is
calculated as follows:
CPIU(30Sep09) = CPIU(31Aug09) x exp( 1.81% / 12 ) = 108.5
The adjusted reference index is calculated as the unadjusted index multiplied by the corresponding
seasonal adjustment. With the previous example, this gives:
CPI(30Sep09) = CPIU(30Sep09) x exp(0.07%) = 108.58
The right-hand table shows calculations for a whole year. It is slightly more complicated to calculate a
date in the middle of the month. The exact formula is given in the technical box on the next page.
Inflation Products Inflation Swaps
Inflation Market Handbook – January 2008 68
CPI Projections as of 21 November 2007; adjusted reference and unadjusted reference over the year 2008.
Seasonality
Adjustment
MoM
Cumulated
Seasonality
August
Based
Unadjusted
Reference
Seasonal
Factor
Adjusted
Reference
31-Jan -0.35% -0.34% 31-Aug-09 108.34 100.00% 108.34
28-Feb 0.16% -0.18% 30-Sep-09 108.50 100.07% 108.58
31-Mar 0.23% 0.05% 31-Oct-09 108.67 100.03% 108.70
30-Apr 0.22% 0.27% 30-Nov-09 108.83 99.88% 108.70
31-May 0.06% 0.33% 31-Dec-09 109.00 100.01% 109.01
30-Jun -0.09% 0.25% 31-Jan-10 109.16 99.66% 108.79
31-Jul -0.19% 0.05% 28-Feb-10 109.33 99.82% 109.13
31-Aug -0.05% 0.00% 31-Mar-10 109.49 100.05% 109.55
30-Sep 0.07% 0.07% 30-Apr-10 109.66 100.27% 109.96
31-Oct -0.04% 0.03% 31-May-10 109.82 100.33% 110.19
30-Nov -0.15% -0.13% 30-Jun-10 109.99 100.25% 110.26
31-Dec 0.14% 0.01% 31-Jul-10 110.16 100.05% 110.21
Source: SG Quantitative Strategy
CPI interpolation
The projected CPI reference index is assumed to be the product of the initial CPI reference, a ‘discount’ function to
represent accreting inflation and a seasonality adjustment:
( ) ( ) ( ) ( )∫×∫×=TT
duusduuieeCPITCPI 000,0
In this expression, i is the inflation rate and s the seasonal adjustment.
Both the inflation rate and the seasonal component are generally assumed to be piecewise. The inflation rate is calibrated
using the available swap breakevens and seasonality can either be calculated using statistical analysis or market
consensus. The inflation rate is calibrated every year in the same month. This is the base month for the breakeven
quotations. During this month the seasonal adjustment is assumed to be null, so that:
( ) ( )( ) ( )
( ) ( )11
1
11
1,00,0 −−
−
=− −
−
−+−
×=∑
×= jjjjjj
j
kkkk TTi
j
TTiTTi
j eTCPIeCPITCPI
( )njjT ..0=
are the dates on which the breakevens are known from dealers in the market, assuming that they fall in the same
month of the year.
For any date between two market points, the CPI is calculated using the formula above and the calibrated inflation rates. For example, at a time t such as [ ]jj TTt ,1−∈ , where t is in the nth month after the base month and is the dth day of the
month, N is the number of days in the month and ( ) 12..1=kks is the MoM rebased vector of seasonality adjustment (monthly
seasonal adjustment):
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑××=
∫××=
−
=−−−×+
−−
−−
1
111111 ,0,0,0
n
kjj
t
jTjjNdnsks
Ttij
duusTtij eeTCPIeeTCPItCPI
This formula allows us to calculate the forward CPI for any date. In its construction, it is consistent with all market swap
breakevens.
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:
( ) ( )( ) ( ) ( )NN
N
iiNiiiIMPLIEDMKT TBTBsTTLibtPPLiborLegBondLegmentUpfrontPay ,0,0,1
11 −+∆−+−=++ ∑
=−
With ( ) ( )( )
( )( ) ( )∑
=
+=M
jMN
MijNIMPLIED TB
CPITCPI
CPITCPI
TBRP1
,000
,0
The spread is calculated so that this expression equals 0. If there is no accrued payment, i.e. the valuation is done on a fixing
date, the Libor leg is equivalent to a single upfront payment which is equal to 100%. The spread is then simply:
01PVPP
s MKTIMLIED −=
where PV01 is the value of a 1bp move on the Libor leg.
In a proceeds asset swap, the upfront payment is cancelled out and the notional applied on the Libor leg is the bond market
price at inception. The total cash flows can be represented as:
( ) ( )( ) ( ) ( )NNMKT
N
iiNiiiMKTIMPLIEDMKTMKT TBPTBsTTLibtPPPPLiborLegBondLegmentUpfrontPay ,0,0,
11 −+∆−+−=++ ∑
=−
Simplifying in the same way as above leads to the following spread value for the proceeds swap:
01PVPPP
sMKT
MKTIMLIED −=
The seasonal pattern is implicitly taken into account in the pricing above: the CPI estimates are derived from inflation swaps
(see Inflation-linked options below) and include seasonal effects.
Inflation Products Inflation-linked asset swaps
Inflation Market Handbook – January 2008 74
Below we show the cash flows of the OATi 2012 as at end-October 2007. The real annual coupon is 3%
and the underlying inflation index is HICPxT - based on 25 July 2001 - equal to 92.98. The current
market price is 104.8 and the current asset swap spread is -19.8bp. This is a par/par asset swap. The
notional paid on the Euribor leg is therefore 100 for the whole life of the transaction. The net cash flow
at inception favours the asset swap buyer as the dirty market price is generally above par.
Asset swap cash flows for the OATi July 2012: example of schedule and calculations.
Date
Swap break-evens
Index Ratio (1)
Nominal Discount (2)
Inflation Discounted Cash Flows (3)=(1)x(2) Libor Rate
Libor Discounted Cash Flows
25-Oct-07 104.19 1.12 186,040 25-Jan-08 0.989 4.24% 10,682 25-Jul-08 106.37 1.144 0.967 33,180 4.59% 22,141
25-Jan-09 0.946 4.30% 20,535 25-Jul-09 108.29 1.165 0.928 32,429 4.07% 18,740
25-Jan-10 0.908 4.26% 19,483 25-Jul-10 110.75 1.192 0.889 31,792 4.29% 18,904
25-Jan-11 0.870 4.29% 18,820 25-Jul-11 113.32 1.220 0.852 31,171 4.34% 18,352
25-Jan-12 0.834 4.38% 18,395 25-Jul-12 116.00 1.248 0.816 1,048,700 4.42% 833,550
Bond Value (A) 1,177,272 Upfront payment (B) 186,040 Libor Leg Value (C) 999,603 PV01 (D) 4.26 Spread (A-B-C)/D (19.66)
Source: SG Quantitative Strategy
Accreting asset swaps One of the characteristics of the par/par asset swap - and to a lesser extent the proceeds asset swap -
is the accreting notional on the inflation leg and the fixed notional on the nominal leg. This produces a
distortion of counterparty risk for the asset swap seller: at maturity the asset swap seller pays par or at
best the initial market dirty price, while he receives the inflated notional, potentially much higher than
par. The counterparty exposure of the asset swap seller increases over time.
Posting collateral can solve this issue, and the Credit Support Annex (CSA) of the standard ISDA swap
contract can be used. However, posting collateral is not necessarily convenient for all investors.
Another way to solve this issue would be to structure asset swaps with an accreting notional on the
nominal leg. Several possibilities are readily available, including:
� Fixing the accretion rate at a predefined ratio. Even if realised inflation cannot be calculated
exactly, this technique can significantly reduce counterparty risk. Once the accretion rate is
fixed, the asset swap valuation is very simple.
� Linking the accretion rate to the inflation fixings in the same way as in the inflation leg. This
is an ideal solution in terms of cash-flow matching, guaranteeing the same notional on the
inflation and nominal legs. However, the nominal leg also depends on the inflation index.
This makes pricing much more complicated, as the correlation between the inflation and the
Libor fixings is needed as an input. As this type of asset swap is unusual and its valuation is
more complicated, a risk premium is usually paid when entering this kind of transaction.
Inflation Products Inflation-linked asset swaps
Inflation Market Handbook – January 2008 75
Risk mitigation can be achieved most simply with a fixed accretion rate, which in most cases will
significantly reduce the counterparty risk. More complicated structures may introduce some other risks
which are not necessarily well understood. The asset swap spread calculation in the fixed accretion
case is fairly simple to calculate, as explained below.
Calculating accreting asset swap spread
The assumptions here are the same as in the par/par and proceeds asset swap case: the two counterparties are considered to
have an AA counterparty risk, so the usual nominal swap curve can be used for discounting. Accretion is assumed to be
constant: every 6 months, the notional on the Libor leg is multiplied by the accretion ratio, 1+r. The spread is calculated so that
the total flows cancel out.
( ) ( ) ( )( ) ( ) ( ) ( ) 0,01,0,111
1 =+−+∆+−+−=++ ∑=
− NNN
N
iiNiii
iIMPLIEDMKT TBrTBsTTLibtrPPLiborLegBondLegmentUpfrontPay
So the spread can easily be calculated from the relationship above:
( ) ( ) ( ) ( )( ) ( ) ( )
( ) ( )∑
∑
=
=−
∆+
+−−+−+−= N
iiNi
i
NNN
N
iiNiN
iIMPLIEDMKT
TBtr
TBrTBTBrPPs
1
11
,01
,01,0,011
Early redemption asset swaps Another counterparty risk mitigation technique uses an early redemption asset swap. The idea is to
prepay some of the total inflated notional prior to maturity. The total inflated notional is the initial
notional multiplied by the CPI return between the issue and maturity dates:
( ) ( )( ) Notional
CPITCPI
TtionalInflatedNo MM ⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛−= 1
0
The total inflated notional can also be viewed as the sum of the increments of the inflated notional at
two subsequent payment dates:
( ) ( ) ( ) ( )01
1 TNotionalTtionalInflatedNoTtionalInflatedNoTtionalInflatedNoM
iiiM +−= ∑
=−
In an early redemption asset swap, part of the notional is repaid at each coupon date. The amount of
notional repaid is proportional to the notional accretion and is multiplied by the nominal discount factor
until maturity:
( )( )
( )( ) ( )MiNii TTBNotional
CPITCPI
CPITCPI
,00
1 ⋅⋅⎟⎟⎠
⎞⎜⎜⎝
⎛− −
Another asset swap measure for bonds: Z-spread Z-spread can be used as an alternative relative value. Z-spread is the quantity by which the discount
curve needs to be shifted so that the market value of the bond and the present value of its discounted
cash flows are equal. The discount curve is in this case calculated from the vanilla swap prices.
Z-spread can be calculated for any type of bond and as such can be a useful indicator to compare one
bond with another. It shows the risk associated with the bond in terms of yield: if the Z-spread is
Inflation Products Inflation-linked asset swaps
Inflation Market Handbook – January 2008 76
positive and large, the bond is substantially riskier than the reference Libor curve, usually associated
with an AA counterparty risk. Conversely, if the Z-spread is negative, the bond is less risky than the
usual swap AA counterparty. This is the case for most government bonds from G8 countries, though
the long-term Italian bonds are an exception.
An inflation-linked bond and a similar nominal bond (same maturity, same issuer) do not have the same
cash flows, especially as the notional of an inflation-linked bond increases over time if inflation remains
positive. As inflation-linked coupons are smaller than nominal ones, the credit risk for linkers is
essentially concentrated at maturity. Intuitively, the Z-spread for inflation linked bonds should be higher
than that for their nominal counterparts, as the total long-term credit risk is more important with the
accreting notional.
Z-spread can also be a measure of how much the swap and cash market diverge from each other. To
understand this point, we can imagine that the discount curve is calculated from the reference
government bond. With such a reference, the nominal Z-spread would always be zero. Moreover, the
government credit risk would be directly priced in the discount curve. In this case, we can argue that
the inflation linked-bond Z-spread is also zero when measured with the government discount curve.
However in reality, the inflation-linked bond would have a positive Z-spread. This is due to the CPI
fixing used in pricing the bond. As we already explained, the standard market practice when calculating
inflation fixings is to use swap market breakevens. As inflation swaps are increasingly popular -
especially for asset liability management - swap breakevens are becoming more expensive than bond
breakevens and CPI fixings calculated from the swap market are slightly higher than those calculated
from the bond market. Pricing the inflation-linked bonds with CPI fixings from swap breakevens makes
the bond price higher than the market price. To compensate for this, the Z-spread calculated to match
the market price is positive.
Measuring Z-spread on inflation bonds and comparing it with that on nominal bonds gives a relative
measure of the bond market versus the swap market: the bigger the difference between nominal and
inflation Z-spread, the more expensive swap breakevens are compared with bond breakevens.
The table below gives some indicative levels of Z-spread, accreting, proceeds and par/par asset swap
for comparison and illustration purposes.
Inflation Products Inflation-linked asset swaps
Inflation Market Handbook – January 2008 77
Indicative levels of Z-spreads, accreting, proceeds and par/par asset swaps
Bond type Description ZSpread Accreting Proceeds Par/ParNominal ZSpread
Delta Zspread
Clean Price
Real Yield
BTPe 1.65% 15-Sep-2008 -9.00 -9.00 -8.60 -9.50 -19.10 10.00 99.54% 2.016BTPe 0.95% 15-Sep-2010 -6.30 -6.40 -6.40 -6.50 -15.60 9.30 96.25% 2.139BTPe 1.85% 15-Sep-2012 -3.90 -4.10 -4.10 -4.00 -13.00 9.10 98.34% 2.194BTPe 2.15% 15-Sep-2014 -2.20 -2.40 -2.30 -2.50 -11.40 9.20 99.53% 2.231BTPe 2.10% 15-Sep-2017 2.50 2.30 2.70 2.80 -7.00 9.50 98.07% 2.324GGBe 2.90% 25-Jul-2025 7.00 6.50 7.60 9.00 0.30 6.70 107.12% 2.411GGBe 2.30% 25-Jul-2030 11.50 11.00 14.00 14.00 11.50 0.00 97.21% 2.460BTPe 2.35% 15-Sep-2035 14.50 13.80 18.00 18.70 10.50 4.00 98.05% 2.461
BTANe 1.25% 25-Jul-2010 -10.70 -10.70 -10.80 -11.00 -22.70 12.00 97.63% 2.028OATe 3.00% 25-Jul-2012 -9.30 -9.40 -9.30 -11.30 -21.80 12.50 104.43% 2.087OATe 1.60% 25-Jul-2015 -10.20 -10.40 -11.10 -11.30 -22.00 11.80 96.03% 2.134BUNDe 1.50% 15-Apr-2016 -13.00 -13.20 -14.40 -13.80 -25.50 12.50 94.43% 2.196OATe 2.25% 25-Jul-2020 -11.60 -11.90 -12.80 -14.00 -22.80 11.20 100.61% 2.196OATe 3.15% 25-Jul-2032 -14.40 -15.10 -16.90 -21.80 -18.90 4.50 118.48% 2.187OATe 1.80% 25-Jul-2040 -12.50 -13.40 -17.10 -15.80 -17.80 5.30 91.71% 2.152
OATi 3.00% 25-Jul-2009 -10.00 -10.00 -9.50 -12.80 -23.10 13.10 101.69% 2.192OATi 1.60% 25-Jul-2011 -10.00 -10.10 -10.20 -10.80 -21.40 11.40 97.46% 2.244OATi 2.50% 25-Jul-2013 -9.50 -9.60 -9.70 -11.00 -20.60 11.10 101.48% 2.239OATi 1.00% 25-Jul-2017 -11.00 -11.20 -12.40 -11.30 -22.00 11.10 88.49% 2.280OATi 3.40% 25-Jul-2029 -15.20 -15.70 -17.20 -23.80 -19.70 4.50 119.63% 2.264
Source: SG Inflation Trading Desk
OATei 2012 versus OAT 2012 Z-spreads history Z-spread difference between the OAT and the OATei 2012
-50-45-40-35-30-25-20-15-10-50Feb-07 Apr-07 Jun-07 Aug-07 Oct-07
OAT 5% Apr-12 OATei 3% Jul 2012
-20
-18
-16
-14
-12
-10
-8
-6
-4Feb-07 Apr-07 Jun-07 Aug-07 Oct-07
OATei/OAT spread
Source: SG Fixed Income Strategy Research – Bloomberg Source: SG Fixed Income Strategy Research - Bloomberg
Inflation Products Inflation-linked options
Inflation 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.
Inflation YoY volatility surface Inflation YoY volatility smile
0.0%0.9%
1.8%2.7%
3.6%
1 6 11 16 21 26
0
0.5
1
1.5
2
2.5
3
3.5
4 bp vol /
Maturity Strike
1
1.5
2
2.5
3
3.5
4
4.5
0% 1% 2% 3% 4%1Y 2Y 5Y
30Y 10Y 20Y
bp vol/day
Source: SG Inflation Trading Desk Source: SG Inflation Trading Desk
The market quotes YoY option prices in terms of implied volatility, as illustrated in the graph above.
There is one volatility number per strike and per maturity. The strikes are usually quoted on an absolute
scale (1%, 2%, 3%...) and the maturity for a round number of years (1Y, 2Y, 10Y�). The one-year YoY
cap is special in that it has only one payment and the first fixing is already known. The one-year YoY
cap is therefore strictly equivalent to the 1Y zero coupon cap.
From the implied volatility number, we can calculate the option price. The market usually quotes
volatility in terms of Black volatility. This means that the option premium is calculated by inserting
market volatility and option characteristics into the Black formula (see Models section, page 98). The
main problem here lies in calculating the YoY forward value. In �Building a CPI forward curve� (page 65)
we explained how to calculate implied CPI forward values from the zero coupon breakeven. A simplistic
view would be to calculate the YoY forward ratio as the ratio of these CPI projections. By doing this we
implicitly assume that the forward value of a ratio is the ratio of the forward values. This is generally not
true and is not so in this particular case. The correct forward to use is the convexity adjusted one,
which is model-dependent. We show how the convexity adjustment can be calculated from some
models in the section on pricing inflation derivatives.
Inflation Products Inflation-linked options
Inflation Market Handbook – January 2008 80
Combined flows of a YoY swap and a YoY cap
InflationSeller
InflationBuyer
Libor – spreador fixed rate
Option Seller
max(CPI(Ti)/CPI(Ti-1)-(1+K),0) CPI(Ti)/CPI(Ti-1)-1
InflationSeller
InflationBuyer
Libor – spreador fixed rate
Option Seller
max(CPI(Ti)/CPI(Ti-1)-(1+K),0) CPI(Ti)/CPI(Ti-1)-1
Source: SG Quantitative Strategy
Real rate swaptions Like in the standard interest rate market, a real rate swaption provides the option of entering into a real
swap at expiry. As explained in the sections above, a real rate swap is a transaction in which two
counterparties exchange a floating nominal rate (6M Libor for example) for a real fixed rate multiplied by
the accrued inflation ratio over the period.
Like with the real rate swap, a real rate swaption can be used to hedge future payments linked to
inflation. However, the swaption also provides the flexibility to choose whether it is worth entering into
the swap at maturity or not. For example, an investor might want to hedge future real income in one
year�s time. He can choose to wait a year and then enter a into a real swap rate agreement. However, if
real rates are expected to decrease over the coming year, it is worth him entering into a real rate
swaption. This will lock in the current level as the future real rate to be received by the investor.
A real rate swaption can also be used to express a view on the evolution of real rates. For example, an
investor who expects real rates to increase can sell a real rate swap (i.e. pay the leg linked to inflation).
However, he can also express this view through real rate swaptions, selling a real rate receiver swaption
or buying a real rate payer swaption. Depending on the level of volatility, playing a view through
swaptions may be more attractive than plain swaps.
The whole range of speculative strategies using options can also be expressed through real rate
swaptions. For example, if an investor expects the real rate to move sharply, he may initiate a straddle
or strangle.
However, the problem with real swaptions is their lack of liquidity, which means that swaption
price/volatility has to be inferred from YoY caps and floors volatility.
An example of swaption smile
1
1.2
1.4
1.6
1.8
2
2.2
2.4
0.0% 1.0% 2.0% 3.0%
bp vol / day
Source: SG Inflation Trading Desk
Inflation Products Inflation-linked options
Inflation Market Handbook – January 2008 81
Strategies with caps and floors Inflation caps and floors can be used as building blocks to construct customised pay-off or to play out
strategic views on the inflation market. Here are some simple strategies which can be implemented by
buying or selling inflation options.
YoY caps and floors can enhance the inflation profile of YoY swaps. For example, the inflation seller in
a swap transaction can decide to limit his outflows by buying an inflation cap. If the YoY inflation ratio
hits the cap level, the amount paid by the inflation seller will be limited to the cap strike. In
compensation for the payment of the option premium up front, the inflation seller is hedged against a
high inflation scenario.
Another possible strategy is to enhance the yield of an inflation swap by selling a cap or a floor. For
example, the inflation buyer can cap his gains on the inflation leg by selling a cap. If inflation stays
below the cap level, the inflation buyer will have earned the option premium. The four different
possibilities are illustrated in the graph below.
Other classical strategies include the collar spread and the butterfly, both of which are linear
combinations of caps and floors. A collar is a position where the investor buys and finances the cap by
selling the floor. The transaction is made at zero cost. It is particularly appropriate for an investor
wanting to bet on an increase in inflation, and can also be advantageous in a situation where caps are
cheaper than floors.
A butterfly is a bet that inflation will remain within a given range. It can be implemented by buying two
options on the central strike and selling two options on either side of the strike.
Inflation Products Inflation-linked options
Inflation Market Handbook – January 2008 82
Application for caps and floors: hedging or investment strategy?
Hedge Strategy / Buy Option Investment Strategy / Sell Option
Yo
YC
ap
Yo
YF
loo
r
Inflation Buyer
Inflation Seller Inflation Buyer
Inflation Seller
-2%
0%
2%
4%
0% 1% 2% 3% 4% 5%
-2%
0%
2%
4%
6%
0% 1% 2% 3% 4% 5%
-6%
-4%
-2%
0%
2%
0% 1% 2% 3% 4% 5%
Buy the floor
Receive Inflation
Net Pay-off: floored inflation leg
Sell the cap
Receive InflationNet Pay-off: capped
inflation leg
Buy the cap
Pay InflationNet Pay-off: capped
inflation leg
-6%
-4%
-2%
0%
2%
0 0.01 0.02 0.03 0.04 0.05
Premium
Premium
Premium
Premium
Sell the floor
Net Pay-off: floored inflation leg
Pay Inflation
Hedge Strategy / Buy Option Investment Strategy / Sell Option
Yo
YC
ap
Yo
YF
loo
r
Inflation Buyer
Inflation Seller Inflation Buyer
Inflation Seller
-2%
0%
2%
4%
0% 1% 2% 3% 4% 5%
-2%
0%
2%
4%
6%
0% 1% 2% 3% 4% 5%
-6%
-4%
-2%
0%
2%
0% 1% 2% 3% 4% 5%
Buy the floor
Receive Inflation
Net Pay-off: floored inflation leg
Sell the cap
Receive InflationNet Pay-off: capped
inflation leg
Buy the cap
Pay InflationNet Pay-off: capped
inflation leg
-6%
-4%
-2%
0%
2%
0 0.01 0.02 0.03 0.04 0.05
Premium
Premium
Premium
Premium
Sell the floor
Net Pay-off: floored inflation leg
Pay Inflation
Source: SG Quantitative Strategy
Inflation Products Inflation-linked futures
Inflation Market Handbook – January 2008 83
Inflation-linked futures
CME future The Chicago Mercantile Exchange (CME) launched a US CPI in September 2004 and an HICPxT future
in September 2005. Prior to that, other US exchanges had made a few attempts to list exchange traded
futures.
Although it seems to have attracted some interest, the HICPxT future has only been modestly
successful. It was designed to offer the investor maximum flexibility. It tracks the annual changes in
HICPxT and represents the inflation on a �1,000,000 notional for 12 consecutive months. Twelve
contracts are quoted at any one time, maturing on the business day before the HICPxT announcement
is made and for 12 consecutive months. The future is quoted as 100 minus the inflation rate the market
expects when the contract expires. For example, if the market expects the annual inflation rate to be
2.22% as of end of November, the future quote is 97.78. The graph below gives the market expectation
for the YoY ratio, calculated from the future prices. The bid-ask spread is still wide (20 to 40bp),
denoting poor liquidity on this instrument.
However, there are several advantages in having an efficient market for inflation futures. First, it
provides a tool for short-term hedging and liability management. A strip of 12 futures is available at any
time, so that matching short-term exposure is very easy. Second, it allows counterparty risk mitigation:
with the system of daily margin calls, the counterparty risk associated with futures is almost zero,
compared with the AA counterparty risk associated with inflation swaps. Finally, as the futures are
quoted for 12 subsequent months, they can be used to hedge seasonal risk. However, investors can
only take advantage of all this if the market is sufficiently liquid, and the liquidity comes with investors
using the instruments. Liquidity is therefore the main issue for this instrument to succeed. This might
happen in the next few years, as the swap market continues to develop rapidly.
The US CPI future launched in 2004 has not been as successful as its European cousin. This is mainly
due to some of its features. It is very similar to the Eurodollar future in that it is based on CPI-U changes
over a three-month period. The contracts mature every three months (in March, June, September, and
December) as do the Eurodollar futures. This contradicts the way the inflation market is structured, as
YoY ratios are favoured and seasonal effects occur on a monthly basis. Having a quarterly contract
provides exposure to only four months for seasonality hedging. Moreover, seasonality runs over three
months so that interpolation of the CPI fixing is fairly complicated. In its current form, the US CPI future
does not appear sustainable and is less and less frequently exchanged on the market.
As with any listed future instrument, inflation futures are subject to daily margin calls. This process
guarantees final payment of the inflation rate. Unfortunately, it also makes the valuation task slightly
more complicated. If inflation increases, the margin calls are paid to the future holder daily and the
resulting cash can be invested in the money market. In addition, the future matures as soon as the CPI
fixing is known, while the zero coupon swap matures with a lag similar to that used for calculating the
bond fixings. This triggers a correction (usually called convexity) which depends on the volatility of the
inflation ratio and the correlation between inflation and nominal rates.
Developing a highly liquid inflation futures market would be extremely beneficial to inflation derivatives
in general, providing increased hedging capabilities in the short term and on bond fixings, transparent
consensus measuring seasonality and more tools for short term liability management.
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)
Trading Hours Pre-Trading 09:00-09:45 (CET) Opening Auction 09:45-10:00 (CET)Continuous Trading 10:00-16:45 (CET)Closing Auction 16:45-17:00 (CET)Post-Trading 17:00-17:30 (CET)
Source: EUREX
Pricing Inflation Derivatives
Inflation Market Handbook – January 2008 86
Pricing Inflation Derivatives
Pricing Inflation Derivatives Background to Pricing Models
Inflation Market Handbook – January 2008
87
Background to Pricing Models The inflation market is relatively new and fast-growing, with a promising future thanks to pressure from
some countries� regulators and especially with the development of pension funds. This has meant that
inflation pricing models have recently come under the spotlight. 2007 saw a big development in the
options market, driven by hedging flows from structured products and inflation range accruals in
particular. So option liquidity has greatly improved and it is getting easier to find market prices for a
wider range of options. This is where the modelling expertise developed on standard interest rate
products comes into play and can be applied to inflation derivatives. Most inflation models have so far
been derived from interest rate models. Here we will review these models and look at how far the
interest rate world has been applied to the inflation world.
The inflation market is the combination of three different elements: first, the nominal economy, which is
the world where we live and price financial products; second, the real economy, which is a hypothetical
inflation-free world; and finally, the CPI, which converts an asset in the real economy into an asset in
the nominal economy. The Fisher equation relates nominal economy yield to real economy and inflation
index yields. This equation provides an economics-based framework for pricing models (see more
detailed explanation below). In the light of the Fisher equation, the analogy between the inflation market
and the FX market is striking and provides the base for the first inflation model we look at.
One of the difficulties in inflation modelling is that a good framework needs to take at least two
elements into account - the CPI index and nominal interest rate dynamics. The real interest rate can
also be modelled, but calibration will be difficult as there is no market instrument trading real prices
directly. The challenge is to come up with pricing which is consistent with all the prices observed in the
market, either in the nominal economy or the inflation market. Inflation derivatives also have some
specific risks and are relatively unexplored in terms of theoretical pricing. The risks include correlation
risk (the correlation between inflation and the nominal market), fixing risk, seasonal effects, the inflation
options market-specific smile risk and the term structure of inflation volatility. The most common
market models tackle some but not all of these issues and there is still great potential for new research.
As far as we know, there are two models which market participants recognise and on which articles
have been published:
� The first is Jarrow and Yildirim�s model, which is based on the analogy between the inflation
market and the foreign exchange market.
� The second is inspired by nominal-world market models and has been proposed
independently by both Benhamou et al. and Brigo and Mercurio.
We will provide a general description of these two models, along with another model which is based on
a short-rate approach and is particularly well-adapted to year-on-year (YoY) pricing.
Pricing Inflation Derivatives Background to Pricing Models
Inflation Market Handbook – January 2008
88
The Fisher equation
Irving Fisher (1867-1947) was an American economist who developed a price level theory. He was the first economist to make
a clear distinction between the real and nominal economy. The Fisher equation is based on a modern version of the quantity
theory of money and the equation of exchange. This theory states that the product of the total amount of money in circulation
in an economy (M) and the speed of spending (V) is equal to the product of the price level (P) and a given index of the real
value of expenditure (Q): MV = PQ
The real value of expenditure is mostly measured by gross domestic product (GDP). Assuming that the spending speed is
stable in the long term, this relationship can be re-written using the yields:
( ) ( )( )πµ ++≈+ 111 i
Where µ is the yield of the monetary mass, i is the price-level yield and π GDP yield. This gives the basis of the Fisher
equation. GDP can be interpreted as a measure of the real economy, the amount of money in circulation reflects the nominal
economy and the price level is simply the inflation rate. Moreover, the yields are usually relatively small so the first order of the
previous relationship allows us to obtain the Fisher equation:
rin +=
where n is the nominal yield, r is the real yield and i the inflation rate.
Pricing Inflation Derivatives Foreign Currency Analogy
Inflation Market Handbook – January 2008
89
Foreign Currency Analogy An FX derivative is based on the exchange rate between two currencies. Its pricing depends on the two
economies (foreign and domestic) and on the exchange rate. The exchange rate makes the transition
between the foreign and domestic prices and the exchange rate yield is the difference between the
foreign and domestic yields.
Similarly, an inflation derivative depends on two economies, the nominal and the real economy, and on
the consumer price index. So it is very tempting to make an analogy. Jarrow and Yildirim (JY) proposed
a model along these lines in 200310 - one of the first inflation models that emerged in academic
literature. The paper was originally written to price TIPS and bond options but has been extended to
price other derivatives. In the original paper, the authors propose a model based on a short-rate
approach for both economies, similar to a three-factor model in the FX world:
� The nominal economy corresponds to the domestic economy. It has its own interest rate
and term structure. The nominal interest rate is based on an HJM (Heath Jarrow Merton)
diffusion model.
� The real economy corresponds to the foreign economy, with the real economy�s rate term
structure following a one-factor HJM diffusion model.
� The spot inflation index (CPI) corresponds to the exchange rate. Like the foreign exchange
framework, it is assumed to be lognormal (Black-Scholes type). The trend component of the
spot inflation process is the difference between the nominal and the real short rates,
consistent with the Fisher equation.
The HJM framework, first introduced by Heath, Jarrow and Merton in the late 1980s, has proved very
useful for the pricing of pure interest rate derivatives and is more or less universally used. The key point
in the model is the so-called HJM drift condition. This states that, assuming there is no arbitrage
opportunity, the dynamic of the underlying variables (forward rate, zero coupon bond prices) is
completely defined by their volatility. In other words, no drift estimation is required. And if the volatility
function is well-chosen, the model becomes Markovian (meaning that the state of the underlying
variables at a given time does not depend on their past values but only on the current one). The Markov
property makes the numerical implementations of the model particularly user-friendly. In addition, zero
coupon bond prices are lognormal martingales under a well-chosen probability. The analytical formula
for zero coupon prices can be derived easily, which makes the pricing of vanilla instruments (such as
caps, floors and swaptions) much easier. This tractability is particularly useful when calibrating the
nominal market to swap and swaption prices.
The model can be calibrated to market parameters in several steps:
� First, calibration of the initial term structures (initial zero coupon prices):
o Nominal term structure: In theory and as presented in the original JY paper, this
should be calculated from government bond prices. In practice it is more frequently
10 Pricing Treasury Inflation Protected Securities and Related Derivatives using an HJM Model � R.A. Jarrow, Y.
Yildirim � Journal of Financial and Quantitative Analysis � June 2003
Pricing Inflation Derivatives Foreign Currency Analogy
Inflation Market Handbook – January 2008
90
calculated from the money market (deposit and futures) and swap market instruments,
using traditional bootstrapping routines.
o Real term structure: Similarly, this should be calculated from government-issued
inflation-linked bond prices. However, as the inflation swap market is now fairly liquid,
one alternative is to use the initial nominal term structure and quoted zero coupon
swaps. In the previous (swap) section, we explained how to deduce the real zero
coupon term structure from zero coupon swap prices and the zero coupon nominal term
structure.
US nominal term structure from Treasury bills and TIPS (bid yield October 2007)
US real term structure from Treasury bills and TIPS (bid yield October 2007)
3.8
4
4.2
4.4
4.6
4.8
5
5.2
Oct-07 Oct-12 Oct-17 Oct-22 Oct-27 Oct-32 Oct-37
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Oct-07 Oct-12 Oct-17 Oct-22 Oct-27 Oct-32
Source: SG Quantitative Strategy - Bloomberg Source: SG Quantitative Strategy - Bloomberg
� Second, calculation of the volatility terms:
o Volatility term structure of nominal interest rate forwards: In their paper,
Jarrow and Yildirim calibrate the nominal term structure historically using
nominal bond prices. However, this volatility term structure can also be
calibrated to nominal market instruments, as it would be in a standard one-factor
model for the nominal interest rate curve. The instruments used for this purpose
would typically be standard swap options (swaptions) or the options on standard
Libor (caps and floors).
o Volatility term structure of real interest rate forwards: Calibration of this
parameter is much more difficult, as no market instruments trade the real curve
directly. It is usually done on a historical basis.
o Volatility of CPI spot process: This is usually a single number with no term
structure. So it could be calibrated either on ATM YoY options or historically,
using the CPI time series. Although several YoY option prices are known, the
model does not have enough parameters to reprice them all.
� Finally, calculation of the various correlations - between the real and nominal economies,
between the inflation index and the real economy and between the inflation index and the
nominal economy. These are usually calibrated historically using CPI time series and the
calibrated term structures of real and nominal zero coupon prices.
Pricing Inflation Derivatives Foreign Currency Analogy
Inflation Market Handbook – January 2008
91
After defining the model and parameterising all its coefficients, we can calculate the forward value of
the YoY ratio and an equivalent Black volatility. Pricing the YoY option is then only a question of
applying the Black formula.
The main problems of the Jarrow-Yildirim approach are its over-parameterisation and the number of a
priori assumptions, particularly with respect to the real economy. And as the real and nominal rates are
Gaussian, there is a higher than zero probability of rates becoming negative, which can be another
limiting factor. Also, smile effect is not taken into account.
The Jarrow-Yildirim (JY) Model
The JY assumes an HJM diffusion for real and nominal forward rates, under the risk-neutral measure
( ) ( ) ( )( ) ( ) ( ) rate forward real for the
rate forward nominal for the
,
,
RtRRR
NtNNN
dWTtdtTtTtdf
dWTtdtTtTtdf
,,,
,,,
σα
σα
+=
+=
The inflation CPI is lognormally distributed, and all Brownian motion is correlated:
,,,,,,
,
dtdWdWdtdWdWdtdWdW
rridWdtiIdI
RIRt
ItNI
It
NtRN
Rt
Nt
Rt
Ntt
ItIt
t
t
ρρρ
σ
===
−=+= CPI inflationfor ,
The arbitrage-free (HJM) condition gives the drift terms in function of volatility functions and under the risk-neutral measure
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )∫∫
−=
=T
t RIIRRRR
T
t NNN
tTtduutTtTt
duutTtTt
ρσσσσα
σσα
,,,,
,,,
And a volatility structure is chosen - for example, the classical Hull-White volatility function: ( ) ( ) ( )tTaXX
XetTt −−= σσ ,
Using this model, the bond prices and the value of the index can easily be calculated. This is the starting point for finding
analytically tractable expressions for normal vanilla products. For nominal zero coupon prices under the risk-neutral probability
measure, zero coupon diffusion and its price can be written as:
( )( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ⎟
⎠⎞
⎜⎝⎛ Γ−Γ−Γ−Γ−=
−=ΓΓ+=
∫∫
∫t N
uNN
t
NNN
NN
T
t XNNtN
Nt
N
N
dWtuTudutuTutBTB
TtB
duutTtdWTtdtrTtBTtdB
00
22 ,,,,21exp
,0,0
,
,,,,,,
σ
And the real zero coupon diffusion and prices are given by:
( )( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ⎟
⎠⎞
⎜⎝⎛ Γ−Γ−Γ−Γ+Γ−Γ−=
−=ΓΓ+Γ+=
∫∫∫
∫t R
uRR
t
RRIRI
t
RRR
RR
T
t RRRtRRIRI
Rt
R
R
dWtuTudutuTuudutuTutBTBTtB
duutTtdWTtdtTttrTtBTtdB
000
22 ,,,,,,21exp
,0,0,
,,,,,,,
σρ
σσρ
Pricing Inflation Derivatives Market Models
Inflation Market Handbook – January 2008
92
Market Models The models presented in this section derive from the so-called Libor nominal market models. When
Vasicek, Hull-White and others introduced short-rate models in the late 1980s and early 1990s, these
were efficient in terms of calibration. Unfortunately, because they had too few parameters and the
diversity of instruments on the market was growing, the models (at least the one-factor version with
deterministic volatility) quickly reached their limits. Brace et al. (1997), Mitersen et al. (1997) and
Jamshidian (1997) presented a new approach using observable market variables (forward Libor rates)
as underlying model variables.
The inflation market models are based on this approach. First, the assumptions about the real economy
are dropped. Second, instead of considering the CPI fixings as the same variable observed at different
times, the market models assume each fixing is a different stochastic variable observed at one point in
time.
For example, Benhamou et al. (2004)11 take a set of CPI fixings and assume that each follows a
lognormal diffusion process. This model takes two main types of uncertainty into account:
The nominal curve: nominal zero coupon bond prices are driven by one-dimensional Brownian
motion. This is usually an HJM type of diffusion.
A set of CPI forwards: Each CPI forward is lognormally distributed with its own uncertainty source.
Contrary to a Jarrow-Yildirim-type multi-currency model, the real curve is not used as an input. It is
enough to know all the CPI forwards in order to completely determine the value of any inflation-linked
derivatives. For example, real cash flow will always be valued in nominal terms and multiplied by an
inflation ratio. The model parameters are also more restricted:
The nominal zero coupon term structure: calibrated on nominal money market and swap prices,
as in the Jarrow-Yildirim model;
The volatility structure of the nominal curve: calculated using optional instruments from the
nominal market (swaptions, caps and floors);
The volatility structure of the CPI fixings.
In this model, the CPI volatility structure is particularly well-adapted to the available instruments.
Generally speaking, inflation options are written on a consumer price index ratio. This CPI ratio can
generically be defined by:
1. The first fixing date (denominator fixing date), T;
2. The time span between the two fixing dates, δ;
3. The option strike K.
These three elements form a volatility cube.
11 Reconciling year-on-year and zero-coupon inflation swap: a market model approach � N. Belgrade, E.
Benhamou � August 2004.
Pricing Inflation Derivatives Market Models
Inflation Market Handbook – January 2008
93
Here is an example: A 10Y zero coupon option is defined on the ratio between the CPI index fixing in
ten years and at inception date. This corresponds to a first fixing date at 0 and a time span of 10Y.
Similarly, a 10Y YoY option is defined on the ratio between the CPI index fixing in 10 years and in 11
years. This corresponds to a first fixing date in 10Y and a time span of 1Y. The most common options
correspond to two planes in the volatility cube (see graph below): zero coupon options are represented
by the plane T=0 and YoY options are represented by the plane δ =1. If the volatility cube is fully
defined in the future, this model will contain all the necessary information.
In addition, one of the market model�s main advantages is that it shows a natural relationship between
YoY and zero coupon market implied volatilities. YoY volatility depends on two zero coupon volatilities
and a covariance term, which in turn depends on the CPI local volatility function. Conversely, the
convexity adjustment - between the YoY forward and the CPI forward ratio - is a function of the nominal
and inflation volatilities and covariance terms (see technical box below).
To sum up, this model benefits from:
� its definition, directly compatible with market observable data (the consumer price index);
� a direct relationship between ZC and YoY volatilities.
However, as explained in the technical box below, the relationship between YoY and ZC volatilities
depends on:
� how CPI volatility is specified;
� the correlations chosen between the different CPI fixing dates. Estimating these correlations
is a fairly difficult task as the fixings are not known a priori.
The inflation volatility market is currently orientated towards YoY products, with the smile in particular
defined in term of YoY option prices. Although the CPI fixings are market observables, the price index
does not seem to be the natural underlying variable to use. A more natural state variable would be
either the inflation rate or the YoY CPI ratio, as explained in the following section.
Inflation volatility cube: most common options are ZC and YoY options
First fixing date, T
Time span between the two fixing dates, δ
Strike K
T=0
δ=1ZC V
ol, V
ol(0
,δ,K
)
YoY Vol, Vol(T,1,K)
TK
YoY
vol,
Vol(T
,1,K
)
δK
ZC V
ol, V
ol(0
,δ,K
)
First fixing date, T
Time span between the two fixing dates, δ
Strike K
T=0
δ=1ZC V
ol, V
ol(0
,δ,K
)
YoY Vol, Vol(T,1,K)
TK
YoY
vol,
Vol(T
,1,K
)
δK
ZC V
ol, V
ol(0
,δ,K
)
Source: SG Quantitative Strategy
Pricing Inflation Derivatives Market Models
Inflation Market Handbook – January 2008
94
A market model
In the model proposed by E. Benhamou et al, each CPI forward follows a lognormal diffusion, with its own driving Brownian
motion, drift and local volatility process:
( )( ) ( ) ( ) i
tiii
i dWTtdtTtTtCPITtdCPI
,,,,
σµ +=
The nominal zero coupon price is also lognormally distributed according to a standard HJM type of model:
( )( ) ( ) t
Nt dZTtdtr
TtBTtdB ,
,,
Γ+=
All the Brownian motions driving the diffusions are correlated. There are two types of correlation:
- nominal/inflation correlation: dtdZdW Nitit ,, ρ=
- correlation between the different CPI forwards: dtdWdW jijt
it ,, ρ=
Using this approach, the implied volatilities of the most common market instruments can be derived from CPI local volatility
and the various correlations. Generically, the terminal (market volatility) in the model and for any ratio is given by:
( ) ( ) ( ) ( ) ( ) ⎟⎠⎞⎜
⎝⎛ +−++= ∫∫∫
+ iii T
iiji
T
O i
T
O i dsTsTsdsTsdsTsTVol0 ,
222 ,,2,,1, δσσρδσσδ
δδ
This pricing formula contains the necessary information to interpolate any point in the volatility cube as defined in the text or
graph. Moreover, in the particular case of YoY and zero coupon options, this formula relates YoY and zero coupon volatilities:
( ) ( ) ( ) ( ) ( ) ⎟⎠⎞⎜
⎝⎛ −+
−=+= ∫
iT
jiijjZCjiZCiij
ijiYoY dsTsTsTVolTTVolTTT
TTTVol0
222 ,,211, σσρ
In this model, YoY convexity adjustment can be expressed as a function of zero coupon volatilities and a covariance term. The
YoY convexity adjustment is the difference between the ratio of the CPI forward and the ratio forward. It is crucial to get this
convexity adjustment right to correctly price options on YoY inflation rates.
YoY convexity adjustment
-25
-20
-15
-10
-5
0
5
0 5 10 15 20 25 30
bp
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.
French and Euro Inflation
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07
French HICP Euro HICP
France/Europe Spread vs 6M Euribor
-1.000%
-0.500%
0.000%
0.500%
1.000%
1.500%
2.000%
0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00%
Correl Infla Spread € -Fr & Euribor3M Linear (Correl Infla Spread € -Fr & Euribor3M)
Structured Products Catalogue
Inflation Market Handbook – January 2008 107
5Y EUR range accrual
Target Clients Risk Profile Currency Maturity Format
Investment EUR 5Y EMTN
Maturity: 5 years
Coupon: 30/360 annually
(Euribor 3M + X%) * n/N
Where:
n = number of observations during the interest period when YoY CPTFEMU1 is observed between 1.10% and 2.60%
N = total number of monthly observations during the interest period (= 12)
1 YoY CPTFEMU refers to the ratio of the CPI 3 months before the observation date/15 months before the observation
date, minus 1.
STRUCTURE DESCRIPTION
PRODUCT OVERVIEW
Market view: This note is aimed at investors who expect European inflation to
remain close to the ECB inflation target of �below, but close to, 2% over the
medium term�.
Mechanism: This 5-year structure pays a semi-annual coupon equivalent to
Euribor 6m + X% for each monthly observation of the YoY inflation rate
between 1.10% and 2.60%.
Advantages:
Historical: Since the creation of the euro, more than 73% of monthly fixings
have been within this range. Over the past five years, the spread has been
outside of this range on only one monthly fixing date.
X% carry & comfortable barriers: By receiving a floating rate (Euribor) in the
current increasing rate environment, the client benefits from improvements in
the note�s MTM value.
Risk: The most substantial risk is a sharp move in inflation, either up or down.
This seems quite unlikely considering the strong ECB commitment to keeping
inflation low.
Historical evolution of YoY relative
to limits
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07
Structured Products Catalogue
Inflation Market Handbook – January 2008 108
10Y EUR swap corridor
Target Clients Risk Profile Currency Maturity Format
Liability EUR 20Y Swap
Client Receives
Quarterly, Act/360
Y1 to Y20: Euribor 3M p.a.
Client pays
Annual Act / 360
Y1: X1%
Y2 �Y10: X1% + X2% * n / N
With
n= number of months where YoY European inflation1 is observed outside of
the range [1.00% to 2.60%]
N = total number of monthly observations during the interest period (= 12)
1 YoY CPTFEMU refers to the ratio of the CPI 3 months before the observation
date/15 months before the observation date, minus 1.
STRUCTURE DESCRIPTION
PRODUCT OVERVIEW
Market view: This structure is aimed at investors who consider that the ECB
will continue its hawkish monetary policy and continue to monitor inflation in
the coming years.
Mechanism: The client receives Euribor 3M every quarter. He pays a fixed
rate of X1% and an extra X2% p.a. for every month European inflation is
observed to be lower than 1.00% or higher than 2.60%.
Advantages:
Guaranteed unconditional Euribor - X1% carry for the first year.
After the first year, the client can continue to benefit from the same carry,
conditional on YoY European inflation fixing.
A comfortable range of YoY inflation evolution. The ECB has attained a
high level of credibility, especially by implementing a clear and efficient
monetary policy based on controlled inflation.
Risk: The most substantial risk is a sharp move up or down in inflation. The structure is capped (at X1% + X2%)
Historical & forward EUR inflation
0.000%
0.500%
1.000%
1.500%
2.000%
2.500%
3.000%
Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07 Jan-09 Jan-11 Jan-13 Jan-15 Jan-17
Histo Eur Inflation Fwd Eur Inflation Barrier
Structured Products Catalogue
Inflation Market Handbook – January 2008 109
20Y EUR Kheops
Target Clients Risk Profile Currency Maturity Format
Liability EUR 20Y Swap
Client receives
Quarterly, Act/360
Y1 to Y20: Euribor 3M p.a.
Client pays
Annual Act / 360
Y1 – Y 20 X% If inflation < 1.5%
X% + lev * (1.5% - inflation)
If inflation < 2.0% and inflation > 1.5%
X% + lev * (inflation – 2.30%)
If inflation < 2.5% and inflation > 1.5%
X% If inflation > 2.5%
With
Inflation = YoY Euro inflation, measured using the CPTFEMU index.
Each YoY CPTFEMU observation refers to the ratio of the CPI 3 months before
the observation date/15 months before the observation date, minus 1.
STRUCTURE DESCRIPTION
PRODUCT OVERVIEW
Market view: This structure is aimed at investors who consider that the ECB
will take action to control the inflation level.
Mechanism: This structure pays X% minus leverage multiplied by the value of
a butterfly. The butterfly is the result of a long position in two call options at
strike 1.5% and 2.5% and a short position in two call options struck at 2%.
Economic rationale: The ECB is committed to maintaining inflation levels
around its 2% target.
Advantages: As long as inflation remains below 2.3% and above 1.7%, the
structure benefits from a higher rate than the current 20Y swap rate (4.90%).
Risk: If inflation stays well above the 2% target, the client will pay a higher
rate than the current 20Y swap rate. However, it remains capped at X%.
Historical & forward EUR inflation
2%
3%
4%
5%
6%
1.0% 1.4% 1.8% 2.2% 2.6% 3.0%
KHEOPS Profile
Current 20Y
Structured Products Catalogue
Inflation Market Handbook – January 2008 110
10Y EUR HICP index-linked leverage slope
Target Clients Risk Profile Currency Maturity Format
Liability EUR 10Y EMTN
Maturity: 10 years
Coupon: 30/360 annually
Y1 X%
Y2 to 10 YoY European inflation + leverage * max( CMS10Y � CMS2Y, 0.00%)
Where:
Each YoY CPTFEMU observation refers to the ratio of the CPI 3 months before the observation date/15 months before
the observation date, minus 1.
CMS10Y and CMS2Y refer to the 10Y and 2Y swap rates on the fixing date, reference Reuters ISDAFIX2.
STRUCTURE DESCRIPTION
PRODUCT OVERVIEW
Market view: This note is aimed at investors who forecast a steepening of the
euro swap curve and want coupons also indexed on consumer price
evolution.
Mechanism: This 10-year structure pays an annual coupon equal to YoY
European inflation plus the EUR CMS10Y � CMS2Y spread, multiplied by a
predefined leverage.
Advantages:
This is excellent timing to enter strategies indexed on the swap curve
slope: the curve is flat on the 2Y-10Y segment and the proposed leverage is
relatively high.
The two underlyings are complementary: compared to the current
monetary cycle, the curve will steepen when the market forecasts the end of
the tightening cycle. Central banks stop raising rates when they consider
inflationary pressures are no longer threatening.
Risk: The most substantial risk is a flat 10Y-2Y curve over the next few years,
combined with low inflation. This seems quite unlikely if the ECB is forced to
cut rates in the coming years, and as inflationary pressure is increasing.
Historical evolution of YoY relative
to limits
0
50
100
150
200
Jan-00 Jan-02 Jan-04 Jan-06 Jan-08
spread (bp)
Structured Products Catalogue
Inflation Market Handbook – January 2008 111
Hybrid inflation/rate performance swap (HIRPS)
Target Investors Risk Profile Currency Maturity Format
LDI/AM EUR 5Y to 20Y EMTN
Fear of rate increase
Capping French inflation
French Inflation Performance Swap
Receive Euribor 3M
Pay
Years 1-2 X1% without condition
Years 3-20 Lev * French inflation if Euribor 12M < X2%Euribor 12M if Euribor 12M > X2%Inflation capped at 2.50%
French Inflation Bear Performance Swap
Receive Euribor 3M
Pay
Years 1-2 Fixed Rate without condition
Years 3-20 Market Rate if Euribor 12M < X1%Leverage * French inflation if X1% < Euribor 12M < X2%Market Rate if X2% < Euribor 12M < X3%Market Rate + Fixed Rate if Euribor 12M > X3%
French Inflation Carry Swap
Receive
Product Independent of Euribor condition
Pay Lev * French inflationInflation capped at 2.30%
Capped French Inflation Bear Performance Swap
Receive Euribor 3M
Pay
Years 1-2 Fixed Rate without condition
Years 3-20 Market Rate if Euribor 12M < X1%Leverage * French inflation if X1% < Euribor 12M < X2%Market Rate if X2% < Euribor 12M < X3%Market Rate + Fixed Rate if Euribor 12M > X3%Inflation capped at 2.50%
Euribor 3MFear of rate
increase
Capping French inflation
French Inflation Performance Swap
Receive Euribor 3M
Pay
Years 1-2 X1% without condition
Years 3-20 Lev * French inflation if Euribor 12M < X2%Euribor 12M if Euribor 12M > X2%Inflation capped at 2.50%
French Inflation Bear Performance Swap
Receive Euribor 3M
Pay
Years 1-2 Fixed Rate without condition
Years 3-20 Market Rate if Euribor 12M < X1%Leverage * French inflation if X1% < Euribor 12M < X2%Market Rate if X2% < Euribor 12M < X3%Market Rate + Fixed Rate if Euribor 12M > X3%
French Inflation Carry Swap
Receive
Product Independent of Euribor condition
Pay Lev * French inflationInflation capped at 2.30%
Capped French Inflation Bear Performance Swap
Receive Euribor 3M
Pay
Years 1-2 Fixed Rate without condition
Years 3-20 Market Rate if Euribor 12M < X1%Leverage * French inflation if X1% < Euribor 12M < X2%Market Rate if X2% < Euribor 12M < X3%Market Rate + Fixed Rate if Euribor 12M > X3%Inflation capped at 2.50%
Euribor 3M
STRUCTURE DESCRIPTION
The range of products available has widened dramatically and our specialists advise clients on how best to protect them-
selves against inflation or maximise returns. The attached decision tree gives a good example of how clients can fine-tune
investment decisions according to their risk appetite and macro views.
Clients usually trade performance swaps on nominal interest rates. This hybrid rate/inflation product allows them to benefit
from the correlation smile structure. This is a very versatile structure that can be tailored according to clients� expectations,
leading to HIRPS variations such as the Bear Performance Swap.
STRUCTURE DESCRIPTION
Structured Products Catalogue
Inflation Market Handbook – January 2008 112
20Y EUR Hybrid performance swap
Target Clients Risk Profile Currency Maturity Format
Liability EUR 20Y Swap
Client receives
Quarterly, Act/360
Y1 to Y20: Euribor 3M p.a.
Client pays
Annual Act / 360
Y1 – Y 2 X1% unconditional
Y3 – Y 20 leverage * inflation if Euribor 12M < X2%
Euribor 12M – 0.02% if Euribor 12M > X2%
Inflation capped at 2.5%
With
Inflation = yoy French Inflation
STRUCTURE DESCRIPTION
PRODUCT OVERVIEW
Market view: This strategy on debt is based on the fact that the
French inflation rate is consistently found to be low, both in
relation to the rest of Europe and in absolute terms. It also takes
the ECB�s strict monetary policy into account.
Economic rationale: In its inflation control policy, the ECB
tends to increase the nominal rate in response to an increase in
inflation. In addition, the Euribor rate and French inflation tend
to be correlated: when inflation is high, the Euribor rate is high
and vice versa. The graph on the right hand side shows the
regression coefficient of French inflation versus the Euribor
12M. Note that French inflation and the Euribor 12M are
positively correlated.
The average French inflation rate since 1997 has been 1.414%.
The top left hand corner of the graph, in grey, corresponds to a
situation where inflation is high and Euribor is below the X2%
level. This risk remains historically remote.
Advantages:
Benefits from a guaranteed 100 bps carry gain for the first two
years.
The carry gain remains higher than 80 bps every month when
the ECB achieves its objectives.
Risk: highest when the inflation rate goes beyond the range.
Historical correlation between French
inflation and Euribor 12M
0%
1%
2%
3%
4%
0% 2% 4% 6% 8% 10% 12%
Euribor
French inflation
0%
1%
2%
3%
4%
0% 2% 4% 6% 8% 10% 12%
Euribor
French inflation
Inflation Market Handbook – January 2008 113
INDEX
A
Accreting asset swaps ................................................... 74
AR, MA, ARMA and ARIMA models ............................... 34
B
Beta ................................................................................. 51
Bloomberg ................................................................ 53, 57
Butterfly........................................................................... 81
C
Calculation of indices ..................................................... 22
Carry and forward price.................................................. 54
Chained index ................................................................. 20
CME future ...................................................................... 83
Collar ............................................................................... 81
Convexity ........................................................................ 51
Convexity adjustment..................................................... 63
CPI forward curve........................................................... 65
CPI interpolation ....................................................... 66, 68
D
Dummies method ........................................................... 32
Duration........................................................................... 51
E
Early redemption asset swaps ....................................... 75
Eurex future..................................................................... 85
Euro HICP ....................................................................... 24
Euro inflation derivatives .................................................. 9
European Inflation Convergence.................................... 25
F
Fisher equation ............................................................... 88
Fisher index..................................................................... 20
Foreign Currency Analogy.............................................. 89
French CPI (Indice des prix à la consommation, IPC)... 27
H
History ............................................................................... 9
Hybrid inflation/rate performance swap (HIRPS) ........ 111
I
Index rebasing ................................................................ 19
inflation breakeven ..........................................................49
Inflation forecasting.........................................................10
Inflation payers................................................................14
Inflation Products ............................................................40
Inflation receivers ............................................................14
Inflation Swaps................................................................58
Inflation year-on-year caps and floors ...........................79
Inflation zero coupon caps and floors............................78
Inflation-linked asset swaps ...........................................70
Inflation-linked bonds .....................................................45
Inflation-linked futures ....................................................83
Inflation-linked options ...................................................78
Invoice price and quotation ............................................48
J
Jarrow-Yildirim (JY) Model..............................................91
L
Lag and indexation..........................................................47
Laspeyres index price .....................................................19
M
Market Models.................................................................92
Market participants .........................................................14
Marshall-Edgeworth index..............................................20
Measuring Inflation..........................................................17
N
Nominal economy ...........................................................21
Nominal vs. real economy...............................................41
P
Paasche’s price index.....................................................20
Par/par and proceeds asset swap spread .....................73
Par/par and proceeds asset swaps................................70
Price index calculation....................................................19
Pricing Models.................................................................87
R
Range accruals................................................................12
Real economy..................................................................21
Real interest rates ...........................................................21
Real rate swaptions.........................................................80
Real swap valuation ........................................................64
Real swaps ......................................................................63
Inflation Market Handbook – January 2008 114
Risk premium.................................................................. 50
S
Seasonality...................................................................... 30
Seasonality in the euro zone .......................................... 36
Short-Rate Models ......................................................... 95
T
TRAMO/SEATS ............................................................... 32
U
UK RPI (Retail Price Index)............................................. 27
US CPI............................................................................. 22
US seasonality ................................................................ 38
X
X12-ARIMA...................................................................... 32
Y
Year-on-Year inflation swaps......................................... 62
Z
Zero coupon swap valuation.......................................... 61
Zero coupon swaps ........................................................59
Z-spread ..........................................................................75
Inflation Market Handbook – January 2008 115
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Emerging Gaëlle Blanchard (33) 1 42 13 44 96 General Industries Nadia Yoshiyama, CFA (44) 20 7676 6985 ECB Watcher James Nixon (44) 20 7676 7385 Great Britain Brian Hilliard (44) 20 7676 7165
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Asia Glenn Maguire (HK) (852) 21 66 54 38 Guy Stear, CFA (33) 1 42 13 40 26 Juan Esteban Valencia (44) 20 7676 7059 Commodities Global Head Frédéric Lasserre (33) 1 42 13 44 06
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Carbon & Coal Emmanuel Fages (33) 1 42 13 30 29 Head Julien Turc (33) 1 42 13 40 90 Base & Precious Stephen Briggs (44) 20 7762 5384 David Benhamou (33) 1 42 13 94 75
Plastics Sébastien Castelli (44) 20 7762 5275 Benjamin Herzog (33) 1 42 13 67 49 X Commodity Strategy Jesper Dannesboe (44) 20 7762 5603 Marc Teyssier (33) 1 42 13 55 96
US Natural Gas - Strategy Michaël S. Haigh (1) 212 278 57 45 Sandrine Ungari (33) 1 42 13 43 02 Paris Tour Société Générale 17 cours Valmy 92987 Paris La Défense Cedex France
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