D , is the contribution from directional yield movements. One should fix p=O if onedoes not have a directional view on interest rates. The expected TRR isthe annualized instantaneous holding period return. o and C are respectively the horizon normalized yield volatility and horizon convexity.The yield and the roll contributions to the TRR can be seen as deterministic contributions, while the convexity/volatility contribution stochastic. Because it includes the convexity/volatility contribution, theexpected TRR is also called the convexity adjusted TRR.Without a view on interest rates, the objective of making an investment is usually to maximize the expected TRR subject to certain riskconstraints.
13.4
Measurement of Risk Premium
The static yield curve that effectively produces the convexity adjustedTRR is said to be the convexity adjusted yield curve,134and is given byy c = y + o 2c .(1 3.10)The incremental yield pickup for taking an extra unit of risk can bemeasured from the convexity adjusted yield curve.12h
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133 It is also called the “slide” by some practitioners.134 If one performs a static instantaneous holding period return analysis on this curve, onewill recover the convexity adjusted TRR.
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Sec. 13.5 Expectation Curve
445
Figure 13.2 illustrates the convexity adjusted yield curve against thevolatility adjusted duration for August 26th, 1997. The shape of the frontpart of the curve, the 0 to 5 year region, is influenced by both market expectations and risk premiums. The long end of the curve, the 25 to 30year region, is usually affected by the supply-demand effect. We assume that the cumulative risk premium increases linearly with VAD, andmeasure the incremental risk premium for taking an extra unit of VADby computing the slope of the convexity adjusted yield curve in the 2 23 year region. The plot in Figure 13.2 and empirical studies of the yieldcurves on many other days all reveal such a linear relationship.The risk premium /z is defined as the slope of the convexity adjustedyield curve between 2 - 23 year region, and the cumulative risk premiumA(z) for the z year point is defined as:A(z) =AT.(13.11)We call A(z) the risk premium curve.135Obviously, the cumulativerisk premium for the zero year point is zero, corresponding to the riskfree investment.
13.5
Expectation Curve
In order to see the pure market expectation effect, we define the expectation curve as the convexity adjusted yield curve minus the cumulativerisk premium:(13.12)E = + C T ~ CA- .Similar to the curves in Figure 13.2, we can also draw expectationcurve against the volatility adjusted duration.The expectation curve is an effective static yield curve that has builtin the convexityholatility contribution but is stripped the cumulative riskpremium. It reveals the market’s view on the risk-free yield structure andis determined by the market expectation of future interest rates, supply demand - liquidity relationships, and market segmentation effects.We believe that, the front part (0 to 5 year sector) of the expectationcurve is mainly driven by market expectation of future interest ratesJJ
135
We must emphasize that the cumulative risk premium, A ( r ) ,is the risk premiumcontribution in yield terms. It is not the risk premium contribution to the TRR. The riskpremium contribution to the TRR has two terms, the cumulative risk premium A(r)itself plus the “roll down” along the A(z) curve.
446
Yield Decomposition Model
Chapter 13
rather than by liquidity or supply - demand concerns. The short termTreasury market is an infinite reservoir. It can absorb any ordinarytransactions. The expectation curve is the yield curve without risk premium. Freed from risk concerns, for making a one year investment, onecan either buy the one year point on the expectation curve, or buy the 9month and roll the investment into the 3 month point 9 months in thefuture, all depending on one's view of future interest rates. At anytimethe massive participants of the market tend to drive the rates to equilibrium, so that the expectation curve reflects the market's view on futureinterest rates.On the other hand, the forward rate implied by the spot curve is always biased high if one uses it as a measure of the market expected future interest rate. This can be explained by the following analysis. Letus assume that two investors, A and B, are trying to make a 3-month investment in 9 months in the future. Investor A does not take any positiontoday and just waits until 9 months later to make the investment.Whereas investor B goes short the 9 month T-bill and long the one yearT-bill to lock in a 9 month forward 3 month rate; he executes the trade sothat it is proceeds neutral today. Investor B has taken a position and assumed the risk associated with it. Nevertheless the outcome is uncertain.In only one realization, either investor A or B will achieve a better returnfrom the trade. But on average, investor B should be compensated fortaking the risk and achieve a better return. In other words, the forwardrate investor B locks in will be higher than the average realized forwardrates.We argue that the middle part (10 to 25 years) of the expectationcurve should be flat. First of all, the market cannot expect detailed variations of interest rates 10 years into the future; the market will agree onsome long term expected forward interest rate that extends out to thelong end. Secondly, there should be no strong reasons for why the demand for the 21 year should be much higher than that for the 22 year.The shape of the expectation curve in the middle part is mainly determined by the risk premium that has already been taken out, so that thecurve should not show significant ~ a r i a t i 0 n s . l ~ ~The long end (25 to 30 years) of the expectation curve is usually13' The expectation curve cannot be used to support this view, because otherwise we arerunning into a circular reasoning. By construction, we take out the slope of the convexityadjusted TRR curve to derive the expectation curve. However, the flatness of the expectation curve in the middle part does indicate the consistency of our assumptions.
Sec. 13.7 Yield Decomposition Analysis
447
driven by the high demand for very long term instruments. Any investorwho wants to go beyond 30 years will have no choice but buy the 30 yearor its neighbors.
13.6
Expected FED Funds Rate
The most important application of the expectation curve is to computethe market expected forward overnight rate and identify it with the market expected forward FED funds rate. The long dashed line in Figure 3is the expected forward FED funds rate. The expected forward FEDfunds rates associated with Figure 13.2 are presented in Table 13.1.Table 13.1: Expected FED Funds Rate.HorizonFwd Fed RateBuilt-inTightening
Today 3 M 6 M 9 M 1 Y 2 Y 3 Y 5 Y 7 Y 1OY5.50 5.54 5.66 5.73 5.76 5.73 5.62 5.41 5.25 5.250.00 0.04 0.16 0.23 0.26 0.23 0.12 -0.09 -0.25 -0.25
The real world FED funds rates are rounded off to the nearest quarterpoint. In the table we also provide the rounded off expected FED fundsrates. However, because it is biased by market segmentation and supply- demand effects, the expected forward curve at the long end( z > 20 years) is not a good indicator of the expected future FED fundsrate.The expected FED funds rate is not a market forecast. The realworld FED funds rate is driven by economic conditions, political influences, the mass psychology, and individual human decisions. They arebasically unpredictable. However, if today's economic/political situations and today's view of the future all stay the same, the expected FEDfunds rate computed by the model will provide the forecast with thehighest probability of being correct. Investors who do not have a viewon future interest rates can take the market's view by using the expectedFED funds rate.
13.7
Yield Decomposition Analysis
From a methodological point of view, a natural, social or financial phenomenon can be better understood through a decomposition analysis thatseparates a seemingly complicated problem into a number of more basicelements that can be more conveniently investigated.
YiefdDecomposifionModel
448
Chapter 13
The movement of the yield curve is difficult to understand, and evenmore difficult to predict. The yield decomposition model breaks theyield curve (spot curve) down to three more understandable constituents.Let us re-write Eq. (13.12) as
~ = E + A - o ~ c = E ( ~ ) +1 A2 ~2 -. - ~
2
(13.13)
In this representation, the spot curve is equal to the sum of threeterms: (1) the expectation curve, plus (2) the risk premium curve, minus(3) the convexity/volatility c ~ n t r i b u t i o n .The' ~ ~ expectation curve is basically a flat risk-free yield level for all maturities; on top of it the marketexpectation of future interest rates, supply - demand relationships, andmarket segmentation effects add some variations, such as the sharp riseor fall in the front, some kinks in the 5 to 7 year region, and the long enddip. The risk premium term compensates the investor for taking on market risks. Finally, in order to be fair, the market only provides a yieldlevel net the convexityholatility contribution. By making the investmentin the stochastic interest rate environment, on average, the investor willrecover the convexity value.13*
13.8
Discussion
In order to anticipate market movements, we need to understand themarket. In order to understand the market, we have to know how to describe it. The YDM and its products-the VAD curve, the risk premiumcurve, the expectation curve, and the expected forward FED funds ratecurve-is a thermometer of the market which tells us about today's market environment.The yield decomposition analysis makes it easier to understand thebehavior of the yield curve. In order to understand the whole, we canfirst analyze the constituents. Economists can more conveniently explainor predict the structure and the time evolution of the expectation curve.The risk premium curve can be separately modeled. For example, wecan study the risk premium curve and find out how it is correlated withthe risk - return relationships of the equity market and foreign markets.instead, we use the BEY convention, the equation becomes(1 f E / 2)(1+ A / 2)(l+y/2)=1+ 02c/213' The convexityholatility term explains why, for the US Treasury market, the yieldcurve levels off after 10 years, or even drops after 25 years.13'
If,
Sec. 13.8 Discussion
449
The convexityholatility term mainly depends on the volatility structurethat usually shows strong mean reversion tendencies. Without a view,one can do reasonably well by using the long term average historicalvolatility as a prophecy for future ones. By recombining the studies ofall three components, it is possible to achieve a better understanding ofthe yield curve.The YDM can help us generate investment or trading ideas. Onemay not have a view on the yield curve as a whole, but may have viewson the constituents. Suppose one has a view that the risk premium willincrease, but is neutral on expectations and volatilities, one can increasethe risk premium it in Eq. (13.13), and conclude that the yield curve willsteepen. Then one may execute a steepening curve trade. In another example, suppose one has a strong view that in the near future the volatilitywill increase significantly, but is neutral on the other components. Byapplying Eq. (13.13), one will expect a drop in longer term yields and aflattening movement because the convexity impact is larger on the longend. One may long the old bond and short the 5 year to hedge the uncertain movements of interest rate level (to hedge the uncertainty in E ). Inyet another example, an economist may have a view that the interest ratelevel will go up sharply. This view also implies that volatilities will goup and the curve will flatten. The YDM would recommend a bear flattener trade.The YDM is a 3-factor interest rate term structure model. The expectation curve drives the level of interest rates. The risk premium parameter it and the volatility determine the slope and curvature of theyield curve. The model itself cannot predict the movements of the expectation curve, the risk premium or the volatility. It is phenomenological. After all, interest rates are driven by human opinions. Another wayto formulate the YDM is to start with Eq. (13.13), taking it as the basicpremise. Then all other results can be given as consequences.
Chapter 14
Inflation LinkedInstrumentsModeling
This chapter is based on one of our proposals in which we designed andproposed a new interest rate product - inflation linked swap. The predictions about this new product may not exactly happen, and the productmay not be as successfhl as expected, however, we still want to publishour original proposal and share our original design with you, so that youcan have a rough idea about how to propose, design, and analyze a newinterest rate product.As the U.S. Treasury Inflation Protected Securities (TIPS) start totrade in the market, there opens up the opportunity for the InflationSwap'39business. The rationales are:Some financial firms may need it for managing inflation linked assetsand liabilities;Investors seek higher real returns (than offered by the TIPS) by swappingfixed assets into inflation linked cashflows;It provides a flexible hedging and trading vehicle;It has the potential to grow into a multi-billion dollar business.In this chapter we analyze the feasibility of promoting the inflationswap business and provide the analytic framework for structuring, valuing and hedging inflation swaps. A variety of possible applications arediscussed.
13' Throughout this book, we also interchangeably call it the Inflation Linked Swap, orthe CPI-U Swap.
451
Sec. 14.1 Inflation Swaps
14.1
Inflation Swaps
In this section, we define the inflation swap and propose a general methodology - the zero-coupon real rate swap curve method - for structuring,pricing and hedging inflation swaps.
Definition:
The inflation swap uses the CPI-U index as reference forinflation, just like the TIPS does. The generic inflation swap contractspecifies a maturity term T in whole years (such as 2, 3, 5, 7, and 10years to maturity). As illustrated in Figure 14.1, the long side of theswap, Party A, pays the short side, Party B, inflation adjusted floatingpayments F in exchange for fixed payments C at the rear of every halfyear interval. The contract specifies a notional principal amount P onwhich interest payments and principal accretions are calculated.In this book, we assume the usual 30/360 bond day count conventionand semi-annual exchanges of payments for an inflation linked swap.
CFigure 14.1. Flowchart for an inflation linked swap.
In order to analyze the inflation swap, we introduce the followingsymbols:A The long side party in the inflation swap.B The short side party in the inflation swap.P Notional par amount,C Fixed cash payment.F Inflation linked floating payment.S Fixed interest rate.X Inflation swap rate (real swap rate).T Maturity term of inflation swap.i The i* period in the N payments, i = 0,1,2,...,N .ti The "i time period in the N payments, i = O,l,2,... , N .Ii The CPI-U index vaIue at time ti, i = 0,1,2,...,N .d The delay in a payment, I k = I(tk - d ) .
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Inflation Linked Instruments Modeling
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Then the fixed payment at time t k for the time period ( t k - l , t k ) isgiven by(14.1)The inflation linked floating payment for the same period is
(14.2)According to the above two equations, the fixed payment calculationis straightforward, whereas the floating payment has two terms: one isthe inflation adjusted interest payment given by the first term on the righthand side of Eq. (14.2), and the second is the principal accretion given bythe second term. The payment of principal accretion at the end of eachperiod is necessary for the operation of the inflation swap business, for itinsures that the default risk faced by the two parties is limited to a oneperiod payment. Otherwise, each of the market participants would haveto acquire its own inflation swap level according to its credit rating sothat there cannot be a single swap market. Another argument againstpostponing the payment of principal accretion until maturity is that, ifinflation rates have remained high during the course of the maturity term,the long side would have to pay a large sum of cash to the short side atmaturity, resulting in a serious “duration mismatch” throughout the lifeof the swap contract.
14.2
Functions and Applications
Over the past ten years, the interest rate swap and currency swap markethas grown from nil to a three trillion dollar business. This clearly tells usthat there have been tremendous demands for such products.Similar to an interest rate swap that exchanges floating and fixedcashflows, an inflation linked swap exchanges inflation adjusted interestpayments for fixed payments. In this section, we analyze the potentialdemand for inflation linked swap products from the point of view of bothbuyers and sellers.
Sec. 14.2
Functions and Applications
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14.2.7 AsseVLiability ManagementThe concept of inflation linked asset/liability management can be bestillustrated by a real world example. In 1988, the World Financial Centerconstruction project was completed by Olympia York Development Ltdon borrowed money. Olympia York’s asset, which mainly included theproperty and future net cashflows from rents and maintenance, was inflation linked, while its liability was debt to be paid off in terms of fixedpayments. The construction company could have hedged its position bysigning an inflation linked swap contract with a dealer to exchange inflation linked incomes for fixed cashflows. At that time, interest rates wereat about 9%, and inflation was about 5%. The swap position should haveproven advantageous to Olympia York as inflation rates declined afterwards.In general, real-estate and construction companies, utility and energyindustries, and agricultural farms have inflation linked assets and theycan take long positions in inflation linked swaps to match their assets andliabilities. Financial institutions can provide their clients with asset/liability management ideas that are made possible by the inflationswap business.74.2.2 Inflation Swaps as Hedging and Trading InstrumentsAs TIPS start to trade in the U.S. market, inflation swaps can provide thebridge between inflation linked instruments and nominal yield instruments. Traders can trade on inflation linked swap spreads according totheir market views. Financial service firms can use them as an effectivetool for hedging their exposure to real interest rates or inflation rates.Inflation swaps can make the financial markets more complete and efficient.14.2.3 lnvestmen t Alternatives
For performance considerations, a pension fund portfolio manager whoseeks inflation protection can achieve higher real returns than those offered by the TIPS market by swapping fixed cashflows for inflationlinked ones through inflation linked swaps. From a different perspective,since the real yield levels of TIPS reflect the default-free credit ratings ofthe US. government, it will result in a credit mismatch if a corporatebond portfolio manager diversifies a portion of his portfolio directly intothe TIPS market, therefore it is necessary to use inflation swaps to obtain
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credit-matched real returns. The main concept would be buying spreadproducts and swapping the cashflows into inflation linked incomes.14.2.4 Inflation Linked Debt IssuanceWhat is the prospect for inflation linked swaps? The impacts of inflationswaps can be twofold:It will make possible new debt issuance;It may take a share of the existing callable debt market.Potential corporate debt issuers may find inflation linked debt moresuitable for their business needs. With the help of inflation swaps, theycan issue fixed rate debt and swap it into inflation linked obligations,thereby reducing their financing costs and better matching their assetsand liabilities.In a high interest rate environment, U.S. corporations and Government agencies issue callable bonds to raise capital and use the embeddedcall protection to reduce their risks if interest rates drop. Similarly in ahigh inflation environment, which is usually also a high interest rate environment, they can issue non-callable debt and use inflation swaps toconvert their fixed liabilities into inflation linked obligations. As inflation rates decline, which is usually accompanied by a decline in nominalinterest rates, their debt does not increase unless real interest rates alsodecline ~ignificant1y.l~'Callable bonds provide the issuer protectionagainst nominal interest rate risk at a cost to the issuer in terms of an embedded option premium. On the other hand, inflation linked debts provide protection against disinflationary risks without incurring costs.Some callable debt issuers may consider this alternative and may find itmore attractive than issuing callable instruments.Another advantage of issuing inflation linked debt is that it eliminates the cost of paying an inflationary risk premium and this is one ofthe major motivations behind the issuance of TIPS. Due to the massivedemands for the inflation linked debt business, the inflation linked debtissuance becomes a fast growing business,74.2.5
Complementary to lnterest Rate Swaps
From empirical studies, we discover that real interest rates and inflation
I4O From historical observations, real rates are generally uncorrelated with inflation rates.
Sec. 14.3 Inflation Swap Level
455
rates are uncorrelated or rather weakly ~orre1ated.l~'According to theyield decomposition model, the nominal interest rate can be decomposedinto two parts: a real interest rate and an inflation rate. Interest rateswaps establish the connection between floating nominal interest ratesand fixed nominal interest rate expectations. On the other hand, inflationswaps provide the connection between floating real interest rates andfixed nominal interest rate expectations.The role of the inflation linked swaps is a complementary to that ofinterest rate swaps. Without inflation swaps, the interest rate market isincomplete in the sense that one cannot hedge real rate exposure or inflation rate exposure without resorting to the triviality of a tautology: A position of TIPS and Treasuries is hedged only if the TIPS exactly hedgethemselves and the Treasuries also exactly hedge themselves. However,one can achieve a hedged position with both the real rate and the inflation hedged by using TIPS, Treasuries, and inflation swaps. We conclude this section by proclaiming that, with the introduction of TIPS intothe U.S. market, inflation swaps are indispensable.
14.3
Inflation Swap Level
The inflation swap business is possible because all market participantsbelieve that, given today's market conditions, there is a fair value relationship between the fixed rate C and the inflation swap rate X for anyinflation swap contract as specified in Section 14.1. We also understandthat the pair of fair value swap rates C and X is not unique; for any chosen value of C, there will be a corresponding fair value swap rate X. Forconsistency of market practice, there is only one reasonable choice forthe generic inflation swap rate:Inflation Swap Fixed Rate S = Interest Rate Swap Rate Sint. (14.3)By making this choice, we will need only one fixed side swap termstructure in the market; it is shared by both interest rate swaps and inflation swaps. This choice is most natural and logical because the interestrate swap levels represent the swap market's credit worthiness and it provides a common platform for comparing investments via fixed coupon,14' For different historical periods of time, they appear to be weakly positively or negatively correlated. Academicians do not agree with an economic reason why they shouldbe correlated one way or another. For practical applications, it is better to assume a neutral view that they are not correlated in the unknown future.
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Inflation Linked Instruments Modeling
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floating and real rate instruments.Now the question is: what the inflation swap rate X should be? Wehave to bear in mind that the answer to this question can only be found inthe market, just as the interest rate swap levels can only be determined bythe regular swap market. However, for a specified maturity term T,given the fixed swap rate S(=Sint),the TIPS real yield R, and theTreasury par coupon yield Y, the following relationship provides an estimate for the inflation swap rate X
s-YX=R++E,l+i
(14.4)
where i is the average inflation rate, and E a small term (a few basispoints) to be decided by the market. In order to understand Eq. (14.4),let us assume that the swap market is as creditable as the U.S. Treasurymarket so that the fixed swap rate is equal to the Treasury yield, i.e.,S - Y = 0 . Then the inflation swap rate should be approximately equalto the TIPS real yield, i.e.,
X=R+S,
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where 6 is a small correction due to the fact that they are two differentmarkets and the principal accretion payment from TIPS is delayed to mat ~ r i t 7 . NowI ~ ~ we can add back the swap spread S - Y to the real swaprate with an inflation adjustment factor l + i to obtain Eq. (14.4). Thecorrection term S and other market influences are absorbed into E .On January 24th, 1997, the pre-settlement trading of the 10-yearTIPS indicates that R = 3.48%. Inflation rate is about i = 3.5%. Fromthe Treasury and interest rate swap market, we obtain Y =6.62% andS = 6.87%. By substituting these numbers into Eq. (14.4), we obtainX=3.72%+~.
(14.6)
This provides a rough estimation for the inflation swap level.
14.4
Real Rate Swap Curve
The TIPS have become available for stripping since they started to trade'42 The delayed payment of principal accretion increases the duration risk faced by theTIPS holders, so that 6 should be negative based on this consideration.
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Sec. 14.5 Zero-coupon Inflation Swap Curve Valuation Methods
on January 29, 1997. Then a full spectrum of TIPS C-STRIP real yieldswill be available for us to construct a real yield curve R ( t ) . Once theinflation swaps of all terms start to trade in the market, we will be able toconstruct a real rate swap curve X ( t ) . Figure 14.2 plots the schematicchart of the Treasury par coupon yield curve Y ( t ), the TIPS real yieldcurve R ( t ) , the interest rate swap curve S ( t ) , and the inflation swapcurve X ( t ) .8
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By definition, the price of the generic inflation swap must be zero at settlement because the real swap rate X is defined such that the cashflowsin the inflation linked leg is equal to cashflows in the fixed leg. In otherwords, on any trading day, the real swap rate level X obtained from themarket establishes an equality between the price of all future fixed cashflows of an inflation swap and the price of all future inflation linked unknown cashflows. The real rate swap curve X ( t ) establishes the implied(or market expected) relationship among the inflation linked and fixedcashflows for all maturity terms. From arbitrage argument, we obtain thefollowing primary principle for valuing inflation swap products:The fair price of a future unknown inflation linked cashflow is equalto the fixed cashflow projected by the forward rate implied by the real
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Inflation Linked Instruments Modeling
Chapter 14
rate swap curve, and then discounted by the real rate swap curve.For the task of valuing a structured product that involves many fixedand unknown inflation linked cashflows, we need to calculate the forward rates implied by the real rate swap curve for different maturitiesand various forward time periods. The procedure can be simplified if wefirst derive the zero-coupon curve Z ( t ) from the real rate swap curveX ( t ) . The Exponential Spline Model can be applied to achieve thisgoal.The general pricing formula for inflation linked structured productscan be formulated as:(14.7)
where D S ( t ) is the discount function derived by using the interest rateswap curve S ( t ) , D x( t ) the discount function derived from the inflationswap curve X ( t ) , &fixed the known cashflow expected at future time ti,and <i,nf the unknown inflation linked cashflow expected at future time
t i . The unknown inflation linked cashflow is given by
(14.8)which is the inflation linked payment on a notional principal P for thefuture time period (t - z , t ) , and f&r,f) is the implied forward rate fromthe zero-coupon real rate swap curve Z(t) .
14.6
Risk Measures and Hedging
The present value of a basis point move (PVBP) method can be appliedto the inflation swap practice. The risk measure of an inflation swapportfolio is the PVBP vector corresponding to a few standard input rates(key rates) for creating the zero-coupon real rate swap curve. The set ofinput rates, rj, j = 1,2,...,n, should include a few short term real ratesand the generic par inflation swap rates. For example, the first short termreal rate can be the current one month LIBOR minus the current CPI-Uinflation rate. The rest of the inputs can be the 1, 2, 3, 5 , 7, and 10 year
Sec. 14.6 Risk Measures and Hedging
459
inflation swap rate quotes from the market. We can construct the inflation swap curve and the corresponding zero-coupon curve, and thencompute the present value of the portfolio by using Eq. (14.