Affine model of inflation-indexed derivatives and inflation risk premium

European Journal of Operational Research 235 (2014) 159–169

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Stochastics and Statistics

Affine model of inflation-indexed derivatives and inflation risk premium Hsiao-Wei Ho a, Henry H. Huang b, Yildiray Yildirim c,⇑ a

Department of Finance and Banking, Shih Chien University, No. 70, Ta-Chih Street, Chung-Shan District, Taipei, Taiwan, ROC Department of Finance, National Central University, No. 300, Jung-da Rd., Jung-Li 320, Taiwan, ROC c Whitman School of Management, Syracuse University, 721 University Ave Suite 600, Syracuse, NY 13244, United States b

a r t i c l e

i n f o

Article history: Received 22 November 2012 Accepted 5 December 2013 Available online 18 December 2013 Keywords: Inflation-indexed derivatives Inflation risk premium Affine models

a b s t r a c t This paper proposes an affine-based approach which jointly captures the nominal interest rate, the real interest rate, and the inflation risk premium to price inflation-indexed derivatives, including zerocoupon inflation-indexed swaps, year-on-year inflation-indexed swaps, inflation-indexed swaptions, and inflation-indexed caps and floors. We provide an example and explain how to use traded zerocoupon inflation-indexed swap rates to estimate inflation risk premiums. Crown Copyright Ó 2013 Published by Elsevier B.V. All rights reserved.

1. Introduction Inflation modeling is of great interest in financial research. The volatile oil and commodity prices before the financial crisis in 2008 led to a wave of inflation investing. After the crisis, expanding monetary policies have brought the global economy into another financial turmoil and are expected to push inflation uncertainties to a higher level. For example, Warren Buffet, the CEO of Berkshire Hathaway, worried on May 2, 2010, about the prospect for ’’significant inflation’’ in the United States and elsewhere. Many financial innovations that were created have been linked to inflation. For example, a futures contract linked to CPI index started trading at the Chicago Board of Trade (CBOT) in 2004. High borrowing needs combined with the desire to improve liquidity has resulted in a surge in the issuance of Treasury Inflation Protected Securities (TIPS) in 1997. In addition, the increasing volatility of inflation rates has introduced inflation as the newest class of assets.1 In the over-the-counter (OTC) market, there are a variety of inflation-linked derivatives issued. Among these products, the inflation-indexed swap is one of the most popular instruments for financial institutions to hedge the inflation risk in their portfolios. The benefits of the inflationindexed swap are the exemption of liquidity premium due to its

high trading volume and that it is only linked to inflation risk. TIPS, on the other hand, are related to both inflation risk and real interest rate risk.2 From these perspectives, the inflation-indexed swap is not only a good choice for hedging inflation risk, but also a good instrument for estimating real interest rate and inflation risk premium. The investor’s perception about inflation risk can be gauged from traded inflation-linked assets. To properly estimate the real interest rate and market-based inflation risk premium, a valuation model considering the inflation risk premium is necessary. Therefore, the main purpose of this paper is to propose an affine model which jointly captures the nominal interest rate, the real interest rate, the inflation expectation, and the inflation risk premium to price inflation-indexed derivatives, including zero-coupon inflation-indexed swaps, year-on-year inflation-indexed swaps, zero-coupon inflation-indexed swaptions, year-on-year inflation-indexed swaptions, and inflation-indexed caps and floors. We also take zero-coupon inflation-indexed swaps as an example that shows how to use the market prices of inflation derivatives to estimate both the real interest rate and inflation risk premiums. In this exercise, we perform two different estimations: one model-free and the other a model-based Kalman Filter, to retrieve the model-implied real interest rate and inflation risk premiums.

⇑ Corresponding author. Tel.: +1 315 443 4885. E-mail addresses: [email protected] (H.-W. Ho), [email protected] (H.H. Huang), [email protected] (Y. Yildirim). 1 About the detail introduction of inflation derivatives, please refer to Deacon, Derry, and Mirfendereski (2004).

2 Fleckenstein, Longstaff, and Lustig (in press) provides a replicating portfolio which is composed of inflation swaps and Nominal Treasury Bonds to replicate the payoffs of TIPS. They confirms that the illiquidity of TIPS market causes a significant extra issuing cost compared with the replicating portfolio.

0377-2217/$ - see front matter Crown Copyright Ó 2013 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2013.12.010

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To estimate inflation rates, the relations between the term structure of nominal interest rates, real interest rates, and inflation rates need to be specified correctly.3 However, due to the latent property of real interest rates, correctly estimating them is challenging for both macroeconomists and financial economists. In addition, conventional Fisher’s hypothesis assumes that the difference between the real and nominal interest rate is only the expected inflation rate, ignoring the uncertainties about future inflation rates. More and more empirical evidences from previous researches show that the Fisher’s hypothesis may not be valid, asserting the difference between the real and nominal interest rate is composed of the expected inflation rate and inflation risk premium. Furthermore, the independence of real interest rate and inflation rate is not supported by empirical evidences. Therefore, a correct specification coping with these two features is essential to construct pricing models for inflation-linked products. In literature, there are a few studies addressing pricing inflation-linked derivatives. Jarrow and Yildirim (2003) introduce the foreign-currency analogy and model nominal interest rates and real interest rates as domestic and foreign assets respectively, using a three-factor HJM model to price TIPS and inflation-indexed options. Mercurio (2005) applies the LIBOR market model to study the inflation-indexed swaps, inflation-indexed caps and floors pricing. In addition, Mercurio and Moreni (2006) further derive closedform formulas of inflation-indexed caplets and floorlets under the assumption of stochastic volatility. Recently, Hinnerich (2008) proposes an extended HJM framework, allowing for both jumps and stochastic volatility for a market consisting of a money market account, zero-coupon bonds and indexed zero-coupon bonds that are based on a non-traded index. However, none of these models consider the inflation risk premium into modeling; they ignore the inflation risk premium by assuming that Fisher’s equation holds.4 Therefore, to the best of our knowledge, this paper is the first one to provide a valuation framework which considers inflation risk premium in pricing inflation-linked derivatives. This paper also uses zero-coupon inflation swap rates and provides an example by using a model-free approach and a model-based Kalman-Filter estimation to estimate real interest rate and inflation risk premium. We organize the remainder of this paper as follows: In Section 2, we introduce the affine model proposed by D’Amico, Kim, and Wei (2009) and extend this model to Section 3 where we derive pricing formulas for inflation-indexed derivatives. In Section 4, we demonstrate how to implement our proposed model by taking a zerocoupon inflation-indexed swap as an example and use a simple model-free approach and a Kalman-Filter estimation approach to estimate the real interest rate and inflation risk premium. We conclude the paper in Section 5.

2. The model Affine models have been widely used in fixed income modeling due to their flexibility in capturing the time-varying risk premium (see Eraker (2008)). Previous researches also use an affine model to estimate the real bond yield and inflation risk premium, e.g.: Campbell and Viceira (2001), D’Amico et al. (2009). In this paper, we adopt D’Amico et al. (2009)’s model, which is based on the specification of Dai and Singleton (2000) and Duffie, Pan, and Singleton (2000), and assumes that the economic system is driven by a Gaussian vector of latent state variables such that the nominal 3 Boero and Torricelli (1996) provide a comparative evaluation of alternative models of the term structure of interest rates. Schmidt (2011) surveys approaches to modeling the term structure of interest rates. 4 The literature review about estimating inflation risk premium is given in the Internet Appendix. In this paper, we focus on pricing inflation derivatives and its applications.

bond yield, real bond yield, and expected inflation are all affine functions of the latent state vector. By specifying the dynamics of latent state vector, we can explicitly model the time-varying risk premium as well as the correlation between the real bond yield, nominal bond yield, and expected inflation. In this section we first state the affine model for the nominal interest rate, the inflation level, and the real interest rate respectively. We then use this model as the basis to derive the valuation formulas for inflation-linked derivatives. In the following subsections, we introduce the definition of inflation risk premium and its relation to Fisher’s Equation. Later, we introduce the affine model used in valuation and model estimation. 2.1. Fisher’s equation and inflation risk premium In Fisher’s hypothesis, the relation between the nominal yield and the real yield can be represented as e

yNt;T M ¼ yRt;T M þ it;T M

ð1Þ

where yNt;T M is the nominal yield during ½t; T M , and yRt;T M is the real e yield during ½t; T M . it;T M is the expected inflation between ½t; T M ,   N N equivalent to 1=ðT M  tÞEPt log IT M =It , and EPt ðÞ is the expectation under nominal physical measure PN given information at time t. Based on Fisher’s equation, the difference between yNt;T M and yRt;T M is known as breakeven inflation and is the benchmark for a Central Bank to control inflation. However, the breakeven inflation is not only composed of expected inflation, but also the inflation risk premium required by investors. We can derive the relation between yNt;T M and yRt;T M based on the stochastic discount factor approach by defining a nominal pricing kernel at time t; MNt , and a real pricing kernel at time t; MRt , where M Nt ¼ M Rt =It . In this framework, the price of a T M -maturity nominal zero-coupon bond and the real zero-coupon bond at time t can be expressed respectively as:

PN ðt; T M Þ ¼

N EPt

PR ðt; T M Þ ¼

N EPt

MNTM M Nt MRTM

! !

M Rt

With these assumptions, the relation between the nominal yield and the real yield can be expressed as5: e

yNt;T M ¼ yRt;T M þ it;T M þ /It;T M

ð2Þ

where /It;T M is the inflation risk premium which is composed of covariance effect, ct;T M , and Jensen’s effect, J t;T M . These two effects can be expressed as

 R  3 M cov t MTRM ; ITIt 6 7 1 t M 7 log 6 ct;T M   41 þ PN MRT  PN  5 TM  t It M Et I T Et M Rt M  

      N N 1 I It t log EPt  EPt log J t;T M   TM  t IT M IT M 



2

The covariance effect captures the covariance between the real pricing kernel and the inflation between ½t; T M  since the real pricing kernel and the inflation are all stochastic. Jensen’s effect reflects the concavity of logarithmic function. The difference between Eqs. (1) and (2) lies in the modeling of inflation risk premium. Fisher’s equation assumes that investors do not require a premium to compensate for inflation risk. However, inflation dynamics are stochastic and investors would require an inflation risk premium to compensate when the market is in equilibrium. 5

Similar argument was introduced and proved in D’Amico et al. (2009).

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2.2. The affine model

where

We adopt the model of D’Amico et al. (2009) as our vehicle, and assume that the nominal pricing kernel, M Nt , the price level, It , and the n-dimensional latent state vector processes, xt , are specified as follows:

qR0 ¼ qN0  qp0 

M Nt

0

N

N

¼ r ðxt Þdt  k ðxt Þ dWt 0 q dWt

d ln It ¼ pðxt Þdt þ r

þr

ð3Þ

? ?0 q dWt

ð4Þ

dxt ¼ Kðl  xt Þdt þ RdWt

ð5Þ

where xt ; rq ; r? q ; l are n  1 vectors, K; r are n  n constant matrices, dWt and dW? n-dimensional mutually orthogonal t are Brownian motions under nominal physical measure, and every element in Wt or W? t is independent. Thus, the variance–covariance N matrices for Wt or W? t are diagonal matrices. k ðxt Þ denotes the 6 nominal market price of risk, which is driven by the stochastic state vector process xt ; r N ðxt Þ is stochastic nominal interest rate and is also driven by xt . We follow Duffee (2002), assuming that xt follows vector Vasicek process, implying the latent state variable vector is mean-reverting with constant volatility. In Eq. (3), the nominal market price of risk, kN ðxt Þ, is a function of xt , implying that the nominal market price of risk is time-varying. Eq. (4) shows the dynamics of inflation. The inflation dynamics can be decomposed into three parts: a stochastic mean growth rate captured by pðxt Þ, a homogeneous common shock r0q dWt , and a exogenous homogeneous shock r?q dW?t . The homogenous common shock of inflation shares the same uncertainties as the nominal pricing kernel and the latent state vector, while volatility remains constant.7 Similarly, the exogenous homogeneous shock represents the homogeneous shock outside the affine economic system in order to capture the unexpected impact on inflation. This affine system assumes the stochastic interest rate, market price of risk, and mean growth rate of inflation. These features are essential in modeling inflation linked products and especially important in estimating inflation risk premium by using prices of these products. Following the specifications of Dai and Singleton (2000), Duffee (2002), D’Amico et al. (2009), the affine systems for r N ðxt Þ; pðxt Þ, and kN ðxt Þ can be assumed as follows

r N ðxt Þ ¼ qN0 þ qN0 1 xt

ð6Þ

pðxt Þ ¼ qp0 þ qp1 0 xt

ð7Þ

kN ðxt Þ ¼ kN0 þ KN xt N 0;

qp

ð8Þ N 1;

qp ; kN

N

where q 0 are scalers. q 1 0 are n  1 vectors. K is a n  n matrix. Similarly, the real pricing kernel, which is defined as M Nt ¼ MRt =It , can be derived as: R

dM t MRt

¼ rR ðxt Þdt  kR ðxt Þ0 dWt  r?q dW?t

and the real interest rate and real market price of risk can be shown as:

r R ðxt Þ ¼ qR0 þ qR0 1 xt

ð9Þ

kR ðxt Þ ¼ kR0 þ KR xt

ð10Þ

6

kR0 ¼ kN0  rq KR ¼ KN

N

dMt

1 0 r rq þ r?2 þ kN0 q 0 rq 2 q N0 R N p q1 ¼ q1  q1 þ K rq

The market price of risk represents the degree of aggregate risk aversion. 7 This assumption rules out the possibility of time-varying volatility. One may argue that values of swaps, swaptions, caps, and floors can critically depends on volatility dynamics. However, the incorporation of time-varying volatility in our model may drastically complicate the derivation of pricing formulae for inflationindexed derivatives. Alternatively, Campbell, Sunderam, and Viceira (2009) have relaxed this assumption and assume that the inflation’s conditional volatility is heteroskedastic following Engle (1982).

We assume that the nominal yield, real yield and inflation expectation at time t with time period s to maturity follow affine functions of latent state variables xt , which can be expressed as: N0

ð11Þ

R0

ð12Þ

yNt;s ¼ aNs þ bs xt yRt;s e

R

¼ as þ bs xt I0

I

is ¼ as þ bs xt

ð13Þ

yNt;s

e

yRt;s

where is the nominal yield, is the real yield, and is is the j inflation expectation. ajs ¼ Ajs =s; bs ¼ Wjs =s, for j=fN; Rg. Ajs ; Wjs can be solved by following equations j  1 dAs ¼ qj0 þ Wj0s Kl  Rkj0 þ Wj0s RR0s Wjs 2 ds  0 dWjs j ¼ q1  K þ RKj Wjs s ds

Aj0

with the initial conditions ¼ 0 and I tors aIs and bs can be shown as:

Wj0

ð14Þ ð15Þ

¼ 0, for j ¼ fN; Rg. The fac-

  Z s   1 p0 aIs ¼ qp0 þ q ds I  eKs l s 1 0  Z s 0 1 I bs ¼ eK s qp1 ds

s

0

In Eq. (14), Ajs is influenced by market price of risk through kj0 ; in (15), Wjs is influenced by market price of risk through Kj . Kj brings the stochastic feature of xt into the market price of risk process; hence, it also differentiates this model from other models with constant market price of risk. With the model implied real yield (12) and inflation expectation (13), the inflation risk premium can be obtained by (2). Based on Eqs. (11) and (12), the real zero coupon bond from t to T M and its nominal counterpart can be expressed as:

h i R0 PR ðt; T M Þ ¼ exp ðaRðT M tÞ þ bðT M tÞ xt ÞðT M  tÞ h  i N0 PN ðt; T M Þ ¼ exp  aNðT M tÞ þ bðT M tÞ xt ðT M  tÞ

ð16Þ ð17Þ

where PR ðt; T M Þ is the real zero-coupon bond from t to T M ; P N ðt; T M Þ is the nominal counterpart,8 and ðT M  t Þ is the time period from time t to time T M . Eqs. (16) and (17) together show that zero-coupon bond price is an exponential affine function of state variables. The expression of nominal and real zero-coupon bonds distinguish our paper from others in that the bond prices are exponential affine functions of latent factors. This setting is easier to formulate in empirical study as suggested by Duffie et al. (2000). 3. Valuing inflation-linked derivatives 3.1. Inflation-indexed swaps Inflation-linked swaps are swap contracts whose payoffs are linked to a specific inflation index.9 The purpose of these instruments is for an inflation risk exposed investor to hedge against or exchange the inflation risk undertaken. Inflation-indexed swap 8 Similar affine model also has been applied to price Treasury bonds, and TIPS in D’Amico et al. (2009). 9 HICPxT in European market; Retail Price Index (RPI) in UK; CPI-Urban in US.

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contracts widely used in practice include zero-coupon inflation-indexed swaps and year-on-year inflation-indexed swaps. Among these derivatives, zero-coupon inflation-indexed swaps are the most actively traded instruments and their quotations have been proven to provide additional information on inflation expectation (see Kerkhof (2005), Hurd & Rellen (2006)).10 Year-on-year inflation-indexed swaps provide the advantage of hedging the year-by-year inflation risk in one contract, which matches the cash-flow schedule of interest rate swaps. However, because year-on-year inflation-indexed swaps still have a debate on it convexity adjustment in long-term maturity contracts, the market of year-on-year inflation-indexed swaps is still in low liquidity.11 In this subsection, we introduce zero-coupon inflation-indexed swaps and year-on-year inflationindexed swaps and explain how to extract inflation risk premium directly from zero-coupon inflation-indexed swap rates. 3.1.1. Zero-coupon inflation-indexed swaps A zero-coupon inflation-indexed swap contract is an bilateral agreement that enables an investor or a hedger to secure an inflation-protected return with respect to an inflation index. The inflation receiver pays a predetermined fixed rate, and receives from the inflation seller inflation-linked payments. In the zero-coupon inflation-indexed swap starting at time T 0 with final time T M , and h nominal amountiN, the fixed-leg payer pays another party N ð1 þ KZCIIS ÞT M T 0  1 when the contract matures, whereas KZCIIS is the contract fixed rate and also the quotation in the swap market. On the other hand, the floating-leg payer pays another party N½IT M =IT 0  1, whereas I is the price index. In the US, the price index applied to the inflation-indexed swap contract is the non-seasonally adjusted US. City Average All Items Consumer Price Index for All Urban Consumers (CPI-U) lagged by two months. This indexation causes two errors which need to be corrected when pricing the contract. First, due to the 2 month lag, the CPI-U does not reflect the true price level at that moment. Second, the CPI-U index is non-seasonally adjusted such that the indexation will be affected by the seasonal variation.12 With the zero-coupon inflation-indexed swap, the inflation payer can exchange inflation sensitive cash flows for constant cash flows. For example, a real estate lessor may engage in a zero-coupon inflation-indexed swap contract to exchange his/ her inflation-linked cash flow (lease payments) for constant cash flows. In a pension scheme which is linked to inflation, the pension fund manager may engage in a zero-coupon inflation-indexed swap contract to hedge the inflation risk in pension liabilities. Let ZCIISðt; T M ; IT 0 Þ denote the time t value of a payer zero-coupon inflation-indexed swap contract that starts at time T 0 with payment date T M and has a swap rate KZCIIS . Without loss of generality, we normalize the nominal amount of the contract to be 1. Following Mercurio (2005), the time t value of the inflation leg of zero-coupon inflation-indexed swap contract can be expressed as:



 It PR ðt; T M Þ  PN ðt; T M Þ IT 0

ð18Þ

where PR ðt; T M Þ is the real zero-coupon bond from t to T M and PN ðt; T M Þ is the nominal counterpart. From this equation, we know that the term structure of real interest rates can be derived if we know the price index value and the term structure of nominal interest rates. From this perspective, we gauge that zero-coupon 10 The participants in inflation swap market include some big players. As reports in p39, Risk, Aug 2011, ‘‘Not only are central banks and real-money clients buying Tips, but they are also increasingly active in swaps, say dealers. Hedge funds continue to trade inflation swaps and Tips as well’’. 11 See the special report of Societe Generale in Risk, September 2007. 12 If the inflation derivatives pays on an annual basis, the seasonal effects do not take effect in initial pricing. However, the seasonal effects influence the mark-tomarket valuation of the inflation securities positions.

inflation-indexed swaps provide a handy instrument to carry out the information of real interest rates. When t ¼ T M ,

ZCIISðT M ; T M ; IT 0 Þ ¼ ¼

  h i IT M  1  ð1 þ KZCIIS ÞT M T 0  1 IT 0 IT M  ð1 þ KZCIIS ÞT M T 0 IT 0

ð19Þ

When t ¼ T 0 , the reference date of starting price index equals to initiation date of the inflation swap contract,

ZCIISðT 0 ; T M ; IT 0 Þ ¼ ½PR ðT 0 ; T M Þ  PN ðT 0 ; T M Þ h i  PN ðT 0 ; T M Þ ð1 þ KZCIIS ÞT M T 0  1 ¼ PR ðT 0 ; T M Þ  PN ðT 0 ; T M Þð1 þ KZCIIS ÞT M T 0

ð20Þ

As suggested by Mercurio (2005), Eqs. (18)–(20) yield modelfree values of real zero-coupon bonds at different time points since they are not based on any specific assumption of the interest rate model, but simply follow the no-arbitrage condition. Let KZCIIS ðT M Þ denote the swap rate at T 0 of a zero-coupon inflation-indexed swap contract with maturity T M . By no-arbitrage condition, the fair value of a zero-coupon inflation-indexed swap contract at time T 0 must be zero, implying

h i PR ðT 0 ; T M Þ  PN ðT 0 ; T M Þ ¼ PN ðT 0 ; T M Þ ð1 þ KZCIIS ðT M ÞÞT M T 0  1 ; and the real zero-coupon bond price in terms of the fixed-leg quotation can be shown as:

PR ðT 0 ; T M Þ ¼ PN ðT 0 ; T M Þð1 þ KZCIIS ðT M ÞÞT M T 0

ð21Þ

Given this affine model, we can rewrite Eq. (21) and obtain the fair rate of a zero-coupon inflation-indexed swap at time T 0 under the no arbitrage condition:



 1 PR ðT 0 ; T M Þ T M T 0 1 PN ðT 0 ; T M Þ

½aN aR þ½bN0 bR0 x ¼ e ðT M T 0 Þ ðT M T 0 Þ ðT M T 0 Þ ðT M T 0 Þ T 0  1:

KZCIIS ðT M Þ ¼

ð22Þ

From Eq. (22), we find that zero-coupon inflation-indexed swap rates are determined by nominal zero-coupon bond prices and real zero-coupon bond prices. The real or nominal zero-coupon bond prices are determined by the affine specification of state variables N R system. With different affined specification structure, aNs ; aRs ; bs ; bs for determining zero-coupon bond prices change accordingly. Eq. (22) provides a formula for estimating the parameters in the specified affine model by using zero-coupon inflation swap rates. One example is Haubrich, Pennacchi, and Ritchken (2011), in which they use Treasury bond data and inflation-indexed swap rates to jointly estimate inflation risk premium. Our model provides an alternative structure to achieve a similar purpose. 3.1.2. Year-on-year inflation-indexed swaps A year-on-year inflation-indexed swap contract is a swap which exchanges the cash flows linked to a number of accrued periods for a series of constant cash flows. Suppose that a year-on-year inflation-indexed swap starts at time T m with payment dates T mþ1 ; T mþ2 ; . . . ; T M . For each period ½T i1 ; T i ; i ¼ m þ 1; . . . ; M, the fixed-leg payer pays another party N ui KYYIIS , whereas N is the nominal amount of the swap contract, ui is the contract fixed-leg year fraction for the interval ½T i1 ; T i , and KYYIIS is the contract fixed rate, while the floating-leg payer pays another party   N ui IT i =IT i1  1 at the end of that specific year. Note that the payments of a zero-coupon inflation-indexed swap are based on the cumulative inflation from the contract starting date to the coupon date. In contrast, the payments of a year-on-year inflation-indexed

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swap are based on the inflation over a number of accrual periods. The benefit of year-on-year inflation-indexed swap contract is its match with the interest rate swap contract in the payment schedule. Combining nominal interest rate swaps and year-on-year nominal inflation-indexed swaps, the contract of real interest rate swap is readily available. The year-on-year inflation-indexed swap rate can also be derived based on our affine model. In this contract, the fixed-leg payer pays another party N ui KYYIIS , whereas ui is the contract fixed-leg year fraction for the interval ½T i1 ; T i , and t < T i1 , while float-leg payer pays another party N ui ½IT i =IT i1  1 at the end of the specific year. Without loss of generality, we normalize the nominal amount of the contract, N, to be 1. Following Mercurio (2005), the value at time t of this year-on-year inflation-indexed swap contract of the inflation leg during ½T i1 ; T i  can be expressed as:

RT  i1 N N  r ðxs Þds t YYIISðt; ½T i1 ; T i ; ui Þ ¼ ui EQ e P ð T ; T Þ R i1 i t  ui ð1 þ KYYIIS ÞP N ðt; T i Þ

ð23Þ

N EQ t ½

where is the conditional expectation under nominal risk-neutral measure QN . r N ðxs Þ is the nominal interest rate. In Eq. (23), the only term needing to be solved is the conditional expectation, and it can be further reduced by the technique of forward measure. Thus, we have following proposition13: Proposition 1. Under the affine system defined by (6)–(8), the value of a year-on-year inflation-indexed swap contract during ½T i1 ; T i  can be expressed as YYIISðt; ½T i1 ; T i ; ui Þ ¼ ui P N ðt; T i1 ÞCðT i1 ;T i ÞDðT i1 ; T i Þ  ui ð1 þ KÞP N ðt; T i Þ

where

CðT i1 ; T i Þ ¼ e

aRðT

i T i1 Þ

ðT i T i1 Þ

i bR0 x ðT T Þ DðT i1 ; T i Þ ¼ EQ e ðT i T i1 Þ T i1 i i1 t  R0  N;T N;T i1 Q b lQ i1 ðxT i1 ÞðT i T i1 Þþ12bR0 ðxT ÞbRðT T Þ ðT i T i1 Þ2 ðT i T i1 Þ Rt i1 i i1 ¼ e ðT i T i1 Þ t N;T i1

h

In Proposition 1, C ðT i1 ; T i ÞDðT i1 ; T i Þ is equal to N;T i1 ½ ð Þ , EQ which is the conditional expectation of forward P T ; T R i1 i t real zero-coupon bond under nominal forward measure. This formula provides a basis for the following analyses. Suppose there is a year-on-year inflation swap contract which swaps a series of inflation-linked cash flows. Let C ¼ fT m ; T mþ1 ; . . . ; T M g denote the set of the starting date and the payments dates of the year-on-year  inflation-indexed swap contract and W ¼ umþ1 ; umþ2 ; . . . ; uM denote the set of the corresponding year fractions. Let YYIISðt; C; WÞ denotes the value at time t of a payer year-on-year inflation-indexed swap contract. Then,

YYIISðt; C; WÞ ¼

M X

YYIISðt; ½T i1 ; T i ; ui Þ

i¼mþ1

¼

M X

ui PN ðt; T i1 ÞCðT i1 ; T i ÞDðT i1 ; T i Þ

i¼mþ1



M X

ui ð1 þ KYYIIS ÞPN ðt; T i Þ

ð24Þ

i¼mþ1

Following Hinnerich (2008), we define the forward swap rate of a year-on-year inflation-indexed swap contract to be the one at which the value of the swap is zero. We denote the forward swap 13

All proofs are provided in the Internet Appendix of this paper.

rate of this year-on-year inflation-indexed swap at time t as Rðt; m; MÞ. From Eq. (24), we can obtain that PM Rðt;m; MÞ ¼

i¼mþ1

ui PN ðt; T i1 ÞCðT i1 ; T i ÞDðT i1 ;T i Þ  PM i¼mþ1 ui P N ðt; T i Þ

PM

i¼mþ1

ui PN ðt; T i Þ ð25Þ

Particularly, when t ¼ T m , i.e. the swap contract initiation date equals to the beginning of first swap period, the forward swap rate RðT m ; m; MÞ will equal to the fair swap rate. In Eq. (25), the year-toyear inflation-indexed swap rate is composed of functions of xT i1 . Through these components, different affine specifications about xT i1 will lead to different inflation-indexed swap rates. Based on our affine model and the closed-form solution of inflation-indexed swap rate, we can estimate the parameters by Kalman-Filter Method, which is widely used in estimating affine model.14 Similar results are provided in Hinnerich (2008) (Eqs. (23) and (24), P.2301). We note that Hinnerich (2008) adopts a different approach from ours in estimating real interest rate parameters. As suggested by Hinnerich (2008), she achieves the estimation of real zero-coupon bonds by using the fact that the price of a real zerocoupon bond at time t is equal to the ratio of the price of a inflation-protected zero-coupon bond to the inflation level at time t. Since a real zero-coupon bond price can be expressed in terms of an inflation-protected zero-coupon bond price and the inflation level at the same time point t, the value of a year-on-year inflationindexed swap contract and the forward swap rate can be expressed as functions of nominal bond prices, inflation-protected bond prices, and inflation levels, which do not contain any real interest rate parameters. In contrast, from Eqs. (24) and (25) we observe that the real interest rate parameters appear in CðT i1 ; T i Þ and DðT i1 ; T i Þ. If the affine models for the nominal interest rate, the inflation level, and the real rate are specified, we can estimate real interest rate parameters by using year-on-year inflation-indexed swap rates and compute the inflation risk premium with those estimated parameters, which cannot be achieved using Hinnerich (2008)’s framework. The inclusion of inflation risk premium into our model makes YYIIS to be another source of estimating inflation risk premium. 3.2. Inflation-indexed swaptions A swaption is an option on a swap with the fixed rate on the swap given by the strike. A payer swaption gives the holder of the swaption the right to enter into a swap in which the holder pays the fixed leg and receives the floating leg, while a receiver swaption gives the holder of the swaption the right to enter into a swap where the holder receives the fixed leg, and pays the floating leg. Accordingly, an inflation swaption is an option to enter into a inflation-indexed swap (e.g.: zero-coupon inflation-indexed swap, year-on-year inflation-indexed swap, etc.), at a pre-specified date and a pre-determined rate in the future. As vanilla options on equities, the holder of an inflation swaption will decide to enter into the underlying inflation-indexed swap if the option is in the money at maturity. The two main types of inflation swaptions are call and put inflation swaptions, which give the holder the right to receive or pay inflation in exchange for a certain fixed nominal rate. Inflation swaptions are important financial instruments for pension fund managers due to the vital roles they play when implementing inflation hedging strategies (see Kerkhof (2005)). In addition, the implied volatilities derived from inflation swaptions are real-time estimates for the volatilities of forward inflation 14 For example, Chen, Liu, and Cheng (2005), Jacobs and Li (2008), Bikov and Chernov (2009), D’Amico et al. (2009), Date and Wang (2009), and Haubrich et al. (2011).

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par swap rates. The derived volatilities provide another dimension in estimating affine term-structure models, especially in fitting the second moment of the underlying asset. 3.2.1. Zero-coupon inflation-indexed swaptions Recall that we denote ZCIISðt; T M ; IT 0 Þ as the value at time t of a payer zero-coupon inflation-indexed swap contract that starts at time T 0 with payment date T M and has a swap rate KZCIIS . We have that

ZCIISðT 0 ; T M ; IT 0 Þ ¼ PR ðT 0 ; T M Þ  PN ðT 0 ; T M Þð1 þ KZCIIS ÞT M T 0 As defined in Hinnerich (2008), a payer zero-coupon inflation-indexed swaption is an option to enter into a payer zero-coupon inflation-indexed swap contract for a given pre-specified swap rate, KZCIIS , at a pre-specified date. Let ZCIISOðt; T M ; KZCIIS Þ denotes the value at time t of the corresponding swaption. Then, ZCIISOðT 0 ; T M ; KZCIIS Þ ¼ maxfZCIISðT 0 ; T M ; IT 0 Þ; 0g nh i o ¼ max P R ðT 0 ;T M Þ  PN ðT 0 ; T M Þð1 þ KZCIIS ÞT M T 0 ; 0

P R ðT 0 ; T M Þ  ð1 þ KZCIIS ÞT M T 0 ; 0 ¼ P N ðT 0 ; T M Þ max P N ðT 0 ; T M Þ ¼ P N ðT 0 ; T M Þ maxfUðT 0 ; T M Þ  G; 0g

YYIISðt; ½T i1 ; T i ; ui Þ ¼

M X

ui PN ðt; T i1 ÞCðT i1 ; T i Þ

i¼mþ1

 DðT i1 ; T i Þ 

M X

ui ð1 þ KYCIIS ÞPN ðt; T i Þ

i¼mþ1

Proposition 2. Under the affine system defined by (6)–(8), the value of the payer zero-coupon inflation-indexed swaption at time t can be expressed as 1 2

M X

i¼mþ1

ð26Þ

ZCIISOðt; T M ; KZCIIS Þ ¼ P N ðt; T M Þ elln U þ2rln U N½ðy0  rln U Þ  GN½y0 

3.2.2. Year-on-year inflation-indexed swaptions Recall that YYIISðt; C; WÞ denotes the value at time t of a payer year-on-year inflation-indexed swap contract, and KYCIIS is the year-on-year inflation-indexed swap rate. Then, YYIISðt; C; WÞ ¼

where UðT 0 ; T M Þ ¼ PR ðT 0 ; T M Þ=PN ðT 0 ; T M Þ and G ¼ ð1 þ KZCIIS ÞT M T 0 . Thus, we have following proposition:

n

we can obtain the mean and the variance–covariance matrix of xT 0 under the measure QN;T M and subsequently the distribution of ln UðT 0 ; T M Þ. We do not have to derive the dynamic of ln UðT 0 ; T M Þ to obtain its distribution under the measure QN;T M . In Hinnerich (2008)’s formula, when pricing a zero-coupon inflation-indexed swaption, one has to estimate real zero-coupon bonds by the relation between real bond prices, inflation-protected bond prices and inflation levels. The availability of the data for comparable inflation-protected zero-coupon bonds may raise additional difficulties.

o

ð27Þ

where h i CðT i1 ; T i Þ ¼ exp aRðT i T i1 Þ ðT i  T i1 Þ h  i N;T i1 R0 DðT i1 ; T i Þ ¼ EQ exp bðT i T i1 Þ xT i1 ðT i  T i1 Þ t bR0 ðT

¼ e

lQ i T i1 Þ t

N;T i1

ðxT

i1

ÞðT i T i1 Þþ12bR0 ðT

i T i1 Þ

RtQ

N;T i1

ðxT

i1

G ¼ ð1 þ KZCIIS ÞT M T 0 h i N;T lln U ¼ ðT M  T 0 Þ aðT 0 ; T M Þ  b0 ðT 0 ; T M ÞlQt M ðxT 0 Þ h i N;T r2ln U ¼ ðT M  T 0 Þ2 b0 ðT 0 ; T M ÞrQt M ðxT 0 ÞbðT 0 ; T M Þ

aðT 0 ;T M Þ ¼ aNðT M T 0 Þ  aRðT M T 0 Þ 0

b ðT 0 ;T M Þ ¼

N0 bðT M T 0 Þ

R0  bðT M T 0 Þ

y0 ¼ ðln G  lln U Þ=rln U

l

QN;T M t

ðxT 0 Þ ¼ eðKþRK

N

ÞðT 0 tÞ

xt 

 N N In  eðKþRK ÞðT 0 tÞ Kl  RkN0  RR0 bT M t Z T 0  N;T N N N 0 N 0 rQt M ðxT 0 Þ ¼ eðKþRK ÞðT 0 tÞ r e½ðKþRK ÞþðKþRK Þ s ds r0 eðKþRK Þ ðT 0 tÞ þ ðK þ RKN Þ

1

i T i1 Þ

ðT i T i1 Þ2

Furthermore, we denote the forward swap rate at time t by Rðt; m; MÞ,

PM Rðt; m; MÞ ¼

i¼mþ1

ui PN ðt; T i1 ÞCðT i1 ; T i ÞDðT i1 ; T i Þ  Sðt; m; MÞ Sðt; m; MÞ

where N½ is the p.d.f. of a standard normal distribution, and UðT 0 ;T M Þ ¼ P R ðT 0 ; T M Þ=P N ðT 0 ; T M Þ

ÞbRðT

ð28Þ PM

where Sðt; m; MÞ ¼ i¼mþ1 ui P N ðt; T i Þ, which is the so-called present value of basis point (PVBP). Hence, the value of a payer year-on-year inflation-index swap contract at time t is:

YYIISðt; C; WÞ ¼ Sðt; m; MÞ½Rðt; m; MÞ  KYCIIS : As defined in Hinnerich (2008), a payer year-on-year inflationindexed swaption is an option to enter into a payer year-on-year inflation-indexed swap contract for a given pre-determined swap rate at a pre-determined date. Let YYIISOðt; m; M; KÞ denote the value at time t of the year-on-year inflation-indexed swaption. Then,

YYIISOðT m ; m; M; KYCIIS Þ ¼ max fYYIISðt; C; WÞ; 0g ¼ Sðt; m; MÞ maxfRðt; m; MÞ  KYCIIS ; 0g

t

It appears that Eq. (27) is similar to, but more complicated than, the formula in Eq. (34) of Hinnerich (2008). The difference, however, lies in the derivations and intuitions of our results for the value of a payer zero-coupon inflation-indexed swaption. Hinnerich (2008) assumes standard HJM models without jumps for nominal interest rates and real interest rates while suggesting that UðT 0 ; T M Þ (corresponding to our notations) is lognormally distributed. Specifically, Hinnerich (2008) derives first the dynamic of ln UðT 0 ; T M Þ, given the specifications of nominal interest rates and real interest rates, and shows that ln UðT 0 ; T M Þ is normally distributed under the measure QN;T M . Using straight forward calculations, she finds a pricing formula for the value of a zero-coupon inflationindexed swaption, which is a analogy with the Black–Scholes formula. In contrast, in our model UðT 0 ; T M Þ can be expressed as an exponential function of xT 0 . Once the dynamic of xT 0 is specified,

Based on this formula, it requires the assumption for the distribution of Rðt; m; MÞ to derive the closed form solution. Since Rðt; m; M Þ is defined as a ratio with Sðt; m; MÞ, to be its numenaire, Eq. (28) is a martingale under QM m measure which is a measure with Sðt; m; MÞ to be the numenaire. In previous research, strong assumptions are imposed on Sðt; m; MÞ and assume Sðt; m; M Þ either to be lognormal distribution or normal distribution under QM m . Following these assumptions, the closed-form solution of YYIISO under our affine structure reduce to the one in Hinnerich (2008). 3.3. Inflation-indexed caplets and floorlets An inflation-indexed caplet is a call option on the inflation rate implied by the CPI index. Similarly, an inflation-indexed floorlet is a put option on the same inflation rate. Inflation-indexed caplets

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and floorlets provide a good instrument for hedging inflation risk, especially when the direction of future inflation is not clear.15 In another perspective, the implied volatilities derived from inflationindexed caplets and floorlets provide another sources to estimate the volatilities of inflation rates. This information helps for identifying the second moment of inflation rate dynamics in model estimation. The payoff of an inflation-indexed caplet/floorlet at time T i is equal to



ui x

IT i

IT i1

þ 1j ;

N;T i

f½xðni  hÞþ g

¼ ui P N ðt; T i Þxflni N½xni0   h½xðni0  rlnni Þg; ð30Þ

where IT i IT i1 ln

ni0 ¼

lni K

þ 12 r2lnni

rlnni

P ðt; T Þ lni ¼ N i1 CðT i1 ; T i ÞDðT i1 ; T i Þ P N ðt;T i Þ   N;T i QN;T i ? r2lnni ¼ T i  T i1 r0q rq þ r?0 ðVI2 Þ þ Cov Q ðVI2 ; VI3 Þ t q rq þ Var t CðT i1 ; T i Þ ¼ e

aRðT

i T i1 Þ

ðT i T i1 Þ

H ¼ K þ RKN N;T i

Cov Q t

N;T i

ðVI2 Þ ¼

0

g0 eHðT i tÞ RR2;1 r0 ½eHðT i tÞ  g

n  o ðVI2 ;VI3 Þ ¼ g0 eHðT i tÞ RH1 eHT i ðT i  T i1 Þ  H1 eHT i  eHT i1 rq

3.4. Inflation-indexed caps and floors An inflation-indexed cap (floor) is a stream of inflation-indexed caplets (floorlets). The total payoff of an inflation-indexed cap (floor) is equal to the sum of the payoffs of inflation-indexed caplets (floorlets) over all payment dates. Specifically, L et C ¼ fT m ; T mþ1 ; . . . ; T M g denote the set of payments dates of an inflation-indexed cap (floor) that starts at time T m , and W ¼ fumþ1 ; umþ2 ; . . . ; uM g denote the set of the corresponding year fractions. For i ¼ m þ 1; m þ 2; . . . ; M, the payoff of inflation-indexed caplets (floorlets) is equal to



ui x

IT i

IT i1

M X

N;T i

ui PN ðT m ; T i ÞEQT m



½xðni  hÞ

þ

i¼mþ1

¼x

M X

n

ui PN ðT m ; T i Þ lni N½xni0   h½xðni0  rln ni Þ

o

3.5. Discussion In Eqs. (8) and (10), the nominal and real market prices of risk are specified as linear functions of latent variables, which are stochastic, leading to time-varying market prices of risk. The linear coefficients for Eqs. (8) and (10) are incorporated into the prices of nominal and real zero-coupon bonds, and the prices of inflation-linked derivatives are essentially functions of nominal and real zero-coupon bonds; therefore, the inflation risk premium can be estimated by market prices of inflation-linked products. The market prices of inflation derivatives provide another dimension of information in addition to TIPS. For example, the inflation caplets and floorlets can be used to estimate the implied volatilities of inflation rates, which provide a forward-looking measure of future price changes. In contrast, the CPI index used in economic analysis is a historical measure and is not sufficient for prediction. When estimating quadratic terms of the affine model, prices of inflation caplets and floorlets are more efficient than inflation swaps because swaps are contracts with linear payoff. Similarly, inflation swaptions provide information about the volatilities of inflation swap rates and the informational disadvantage of inflation swaps can be improved by including the information of inflation swaptions. 4. An example for model estimation

h R0 i N;T i1 b x ðT T Þ DðT i1 ;T i Þ ¼ EQ e ðT i T i1 Þ T i1 i i1 t h  i KN g0 ¼ qp1 0  r0q þ r?0 q Var Q t

¼

i¼mþ1

Proposition 3. The time t value (t 6 T i1 ) of the payoff of an inflation-indexed caplet/floorlet at time T i can be expressed as

ni ¼

i¼mþ1

ð29Þ

where ui is the year fraction of time period ½T i1 ; T i ; j is the strike price of the inflation-indexed caplet/floorlet, and x ¼ 1 for an inflation-indexed caplet and x ¼ 1 for an inflation-indexed floorlet. ½Xþ ¼ maxðX; 0Þ. For brevity, we assume ni ¼ IT i =IT i1 and h ¼ 1 þ j. Let IICFletðt; ½T i1 ; T i ; ui ; h; xÞ denote the time t value (t 6 T i1 ) of the payoff of an inflation-indexed caplet/floorlet at time T i . Then we have following proposition:

IICFletðt; ½T i1 ; T i ; ui ; h; xÞ ¼ ui P N ðt; T i ÞEQ t

Then the value of an inflation-indexed cap (floor) at time t ¼ T m can be expressed as:

R Ti  M X N  rN ðx Þds IICFðT m ; C; W; E; xÞ ¼ ui EQT m e T m s ½xðni  hÞþ

þ 1j ¼ ui ½xðni  hÞþ :

In this section, we take zero-coupon inflation swaps as an example to estimate the real interest rate and inflation risk premium. In existing literature, inflation risk premium is estimated either directly from the nominal bond yield (see Ang, Bekaert, & Wei (2008), Buraschi & Jiltsov (2005), Campbell & Viceira (2001)) or the joint information of nominal bond yield and TIPS bond yield (see Chen et al. (2005), Risa (2001), Evans (1998)). With the appearance of zero-coupon inflation-indexed swaps and their enormous growing trading volume in this market, we can directly use their trading information and derive the inflation risk premium implied by the market variation of price index. In this section, we propose two approaches: The first approach, based on the no-arbitrage argument and free of model selection, decomposes the valuation formula for zero-coupon inflation swaps to infer the inflation risk premium while assuming the TIPS yields as the true real interest rates, and the survey for inflation expectation efficiently reflects the true inflation expectation. The second approach is based on Kalman-Filter estimation by using nominal yields, TIPS yields, survey data for inflation expectation, and zero-coupon inflation swap rates to infer the real interest rate, expected inflation, and inflation risk premium. 4.1. Data

15 As reported in p38, Risk, August 2008, ‘‘In 2010, market participants fretted about two distinct but equally worrying scenarios – deflation on one hand, and runaway inflation on the other. Among other things, those fears sparked massive interest in zero-coupon inflation caps and floors, including several large trades between California-based asset manager Pimco and Toronto-based insurer Fairfax Financial’’.

In our estimation, we use weekly data of CPI level, US Treasury yields, TIPS yields, survey forecasts of inflation, and inflation swap rates. The sample period is from July 28, 2004 to September 15, 2010, using weekly data which is recorded on Wednesdays. The

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reasons for using this sample period are due to (1) As suggested by D’Amico et al. (2009) and Haubrich et al. (2011), TIPS market becomes more liquid after 2004; therefore, using samples in this period can avoid the contamination of liquidity premium; (2) we will not have large amount of missing values in Kalman-Filter estimation. Although Kalman-Filter estimation can automatically update the information even if there are missing values, large amounts of missing values makes the estimation greatly dependent on the choice of initial values. While the CPI inflation is available monthly, we assumed that the data series are observed on the last Wednesday in each month during this sample period. Furthermore, we use the seasonallyadjusted CPI level in the estimation. Both the nominal and the TIPS yields are based on zero-coupon yield curves fitted at the Federal Reserve Board,16 and CPI inflation is obtained from the website of the Federal Reserve Bank. The US 1- and 10-year zero-coupon inflation swap rates are downloaded from Bloomberg.17 In economic sense, zero-coupon inflation swap rates are a composite reflection of future inflation expectation and inflation risk premium for its tenure. The SPF economists’ inflation expectations are obtained from the website of the Federal Reserve Bank and the SPF survey is an aggregate inflation prediction of professional economists. Following Haubrich et al. (2011), we obtain nominal Treasury yields from two sources. First, we observe daily secondary market yields for 3- and 6-month Treasury bills from the Federal Reserve System’s H.15 Release. Second, we obtain zero-coupon yields of 1, 3, 5, 7, 10, and 15 years to maturity from daily Treasure yield curves constructed by Gurkaynak et al. (2007). All of the nominal yields are observed on Wednesdays in each week. As for TIPS yields, Gurkaynak et al. (2008) provide data of zerocoupon TIPS yields on a daily frequency, and we use 10- and 15year zero-coupon TIPS yields for estimation. Similarly, we only take the weekly data series observed on Wednesdays in our estimation. As argued by Ang, Bekaert, and Wei (2007), survey data provide more accurate information about expected inflation than time-series models, production-based models, and no-arbitrage termstructure models. Therefore, we estimate expected inflation by survey data. We use the seasonal-adjusted forecasts of CPI inflation over next 1 and 10 years obtained from the Survey of Professional Forecasters (SPF). While the 1- and 10-years SPF inflation forecasts are made quarterly, we assume the inflation forecasts are identical during each week in a quarter. Moreover, in the estimation we use 10-, 12-, 15-, 20-, 25-, and 30-year zero-coupon inflation-indexed swap rates, which are observed daily. Similarly, we only choose the data series of inflation swap rates on every Wednesday to match the weekly frequency. With the inclusion of inflation-linked derivatives data, the estimates are more efficient than the ones estimated by nominal treasury yield data or TIPs data. 4.2. Model-free estimation and Kalman-Filter estimation In the existence of risk premium, we can link Eq. (2) with the zero-coupon inflation swap pricing formula. Let ZCIIS t; T M ; IT 0 denotes the time t value of a payer zero-coupon inflation-indexed swap contract that starts at time T 0 with payment date T M and has a swap rate KZCIIS . Without loss of generality, we normalize the nominal amount of the contract to be 1. Following the argument in Section 2, the real zero-coupon bond price in terms of the fixed-leg quotation can be shown as:

16 See Gurkaynak, Sack, and Wright (2007), Gurkaynak, Sack, and Wright (2008) for details. 17 The tickers of US 1-year and 10-year zero-coupon inflation swap rates are USSWIT1, and USSWIT10, respectively.

PR ðT 0 ; T M Þ ¼ PN ðT 0 ; T M Þð1 þ KZCIIS ðT M ÞÞT M T 0

ð31Þ

In detail, the relation between nominal yield and real yield can be expressed as:

ðT M  T 0 Þ logð1 þ KZCIIS ðT M ÞÞ ¼ yRT0 ;T M ðT M  T 0 Þ

  N 1 IT M þ EPT 0 log ðT M  T 0 Þ IT 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

yNT0 ;T M ¼ yRT0 ;T M þ

ieT

0 ;T M

  N 1 IT M þ log ð1 þ KZCIIS ðT M ÞÞ  EPT 0 log ðT M  T 0 Þ IT 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} /IT

ð32Þ

0 ;T M

e iT 0 ;T M

where is the annualized expected inflation, and /IT 0 ;T M is the inflation risk premium. Therefore, if we know the zero-coupon inflation-indexed from T 0 to T M ; KZCIIS , and the expected  swap rates  N inflation EPT 0 log IT M =IT 0 , we can reasonably estimate the inflation risk premium implied by the market from T 0 to T M . As it results, the real yields and inflation risk premium can be obtained when expected inflation and zero-coupon inflation rates are ready. In this estimation, we discretize the continuous-time process of state variables (latent variables) and formulate it as a transition equation. Then, we discretize the equations which link the market prices and latent variables as the observation equation. By applying the Kalman-Filter algorithm and maximal-likelihood estimation, we can obtain the estimated parameters for the affine system; hence the model-implied real interest rate, inflation expectation, and inflation risk premium can be obtained.18 4.3. Estimation results 4.3.1. Results of model-free estimation Fig. 1 shows the evolution of the 10-year nominal yields, real yields, expected inflation, and inflation risk premiums based on the model-free estimation. We observe that both the 10-year nominal and real yields increased slightly from September 2004 to July 2007, and then decreased rapidly in mid-2007 due to the occurrence of financial crisis. Furthermore, both the 10-year nominal and real yields fluctuated dramatically during the period from the third quarter of 2008 to mid 2009, which is the most severe period of the financial crisis. The 10-year nominal yields dropped from 4.70% on October 29, 2008 to 2.76% on December 24, 2008; the 10-year real yields also fell by 50% from October 29, 2008 to December 24, 2008 (from 3.61% to 1.81%). We find that these two yields decreased sharply in the fourth quarter of 2008, reflecting the investors’ worries about contagion effect due to the bankruptcy of Lehman Brothers. However, after the 3rd quarter of 2009, the 10-year nominal and real yields returned to relatively stable before declining again due to the launch of quantitative easing. With the exception of the period of financial crisis, the 10-year expected inflation and inflation risk premiums based on the model-free estimation presented a stable pattern in our sample period from September 2004 to September 2010. The 10-year expected inflation based on the model-free estimation stayed only in the range from 0.0815% to 0.0916%. The inflation risk premiums based on the model-free estimation reached the maximum value (53 bp) on December 30, 2009, reflecting the market expectation for higher inflation uncertainty due to quantitative easing. On the other hand, inflation risk premium fell to the minimum value on November 26, 2008, reflecting the recession expectation due to the financial crisis. 18

The Kalman-Filter formulation is given in the Internet Appendix.

H.-W. Ho et al. / European Journal of Operational Research 235 (2014) 159–169

167

Fig. 1. Model-free decomposition of 10-year nominal yields. In this figure, the decomposition is based on Eq. (7) as follows,

yNT0 ;T M |fflffl{zfflffl}

Nominal Yield

 

  N N 1 IT M 1 IT M ¼ yRT0 ;T M þ EPT 0 log þ logð1 þ K ZCIIS ðT M ÞÞ  EPT 0 log I0 IT 0 ðT M  T M Þ |fflffl{zfflffl} ðT M  T 0 Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ð1Þ Real Yield Inflation Expectation ð2Þ

Inflation Risk Premium ð3Þ

In this estimation, the US 10 year zero-coupon inflation swap rates are from July 28, 2004 to September 15, 2010, SPF economists’ inflation expectation are for the same period are used to estimate the inflation risk premium. The nominal Treasury Yields are based on zero-coupon yield curves fitted at the Federal Reserve Board for the same period. The Real Yields is defined as the difference between nominal yields and inflation expectation and inflation risk premium, i.e.: (1)–(2)–(3).

4.3.2. Results of affine-model based estimation Fig. 2 shows the evolution of the 10-year nominal yields, real yields, expected inflation, and inflation risk premiums based on the affine-model estimation. Different from the case in the modelfree estimation, the 10-year real yields based on the affine-model estimation presented a relatively stable pattern in the sample period, ranging from 1.36% to 2.29%. Even in the most severe period of the financial crisis, the model-implied 10-year real yields remained stable. Similarly, the expected inflation based on the affine-model estimation also shows a steady pattern in the sample period, ranging from 1.60% to 2.97%. Note that the minimum value (1.60%) of the expected inflation implied by the affine-model estimation took place on 25 March, 2009, which is in the most severe time of financial crisis. As we observe in Fig. 2, the 10-year inflation risk premiums implied by the affine-model estimation proceeded in a similar pattern

to that of the 10-year nominal yields. The inflation risk premium shifted up slightly from the fourth quarter of 2004 to July 2007, and then fell down sharply from mid 2007 due to the occurrence of financial crises. Also, the model-implied inflation risk premiums fluctuated significantly during the period from the third quarter of 2008 to mid 2009. The inflation risk premiums reached the maximum value (86 bp) on March 25, 2009, and the minimum level is on July 28, 2004, reflecting investors’ worries about long-run deflation. 4.3.3. Comparison Fig. 3 shows the evolution of the 10-year TIPS yields, the 10year real yields based on the affine-model estimation, and the 10-year real yields based on the model-free estimation. We observe that the 10-year TIPS yields and the real yields based on the model-free estimation fluctuated in a similar pattern, but the

Fig. 2. Affine-model based decomposition of 10-year nominal yields. In this figure, we decompose the nominal yields implied by TIPS yields, model-free estimation, and Kalman–Filter estimation. In this estimation, the US 10 year zero-coupon inflation swap rates are from July 28, 2004 to September 15, 2010, SPF economists’ inflation expectation are for the same period are used to estimate the inflation risk premium. The nominal Treasury Yields are based on zero-coupon yield curves fitted at the Federal Reserve Board for the same period.

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Fig. 3. Comparison of 10-year real yields. In this figure, we compare the real yield implied by TIPS yields, model-free estimation, and Kalman-Filter estimation. In this estimation, the US 10 year zero-coupon inflation swap rates are from July 28, 2004 to September 15, 2010, SPF economists’ inflation expectation are for the same period are used to estimate the inflation risk premium. The nominal Treasury Yields are based on zero-coupon yield curves fitted at the Federal Reserve Board for the same period.

Fig. 4. Comparison of 10-Year Inflation Forecast and Inflation Risk Premium. In this figure, we present model-free inflation forecast and inflation risk premium based on the decomposition defined in Eq. (32), and the model-implied inflation forecast and inflation risk premium defined in Section 2.2. In this estimation, the US 10 year zero-coupon inflation swap rates are from July 28, 2004 to September 15, 2010, SPF economists’ inflation expectation are for the same period are used to estimate the inflation risk premium. The nominal Treasury Yields are based on zero-coupon yield curves fitted at the Federal Reserve Board for the same period.

model-free estimation is higher than the TIPS yield. The 10-year real yields based on the affine-model estimation presented a relatively stable level in the sample period and its values are closer to TIPS yields. During the period of financial crisis, both the 10-year TIPS yields and the model-free real yields varied dramatically, but the 10-year real yields based on the affine-model estimation stayed in a small range. Notice that the amplitude of which the 10-year TIPS yields shifted up is much larger than that of the 10year real yields based on the model-free estimation right after the bankruptcy of Lehman Brothers occurred. On the other hand, both the 10-year TIPS yields and the 10-year real yields based on the affine-model estimation showed similar levels in our sample period except for the period from the third quarter of 2008 to mid 2009. As noted by Christensen, Lopez, and Rudebusch (2010), they use TIPS yields data to estimate nominal and real term structures only from January 2003 to March 2008 since they argue that illiquidity was high before and after this period (see Haubrich et al. (2011)). Overall, the 10-year real yields based on the

model-free estimation exhibits the highest values while the other two present similar levels in the sample period. Fig. 4 shows the evolution of 10-year expected inflation and inflation risk premiums based on the affine-model estimation as well as on the model-free estimation respectively. We see that the 10-year expected inflation based on the affine-model estimation is more volatile than that based on the model-free estimation and the discrepancy is more significant during the period of the financial crisis. The 10-year expected inflation based on the affine-model estimation decreased after the occurrence of the bankruptcy of Lehman Brothers while the model-free 10-year expected inflation did not. Moreover, the 10-year expected inflation based on the affine-model estimation is much higher (ranging from 1.60% to 2.97%) than that based on the model-free estimation (ranging from 0.08% to 0.09%). Comparing the inflation risk premiums based on the affinemodel estimation with those based on the model-free estimation in Fig. 4, we find that the model-implied inflation risk premiums

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fluctuate much more severely than the model-free inflation risk premiums over the sample period. Overall, the inflation risk premiums based on the affine-model estimation exhibit values of similar order to those based on the model-free estimation over the whole sample period. 5. Conclusion This research applies affine model to price inflation-indexed derivatives, including zero-coupon inflation-indexed swaps, yearon-year inflation-indexed swaps, inflation-indexed swaptions, and inflation-indexed caps and floors. The affine model introduced in this research can be used to analytically estimate real interest rates and inflation risk premiums which are required by the investors to compensate the inflation uncertainty. Under this framework, we can estimate the inflation risk premiums by using Kalman-Filter estimation. Taking zero-coupon inflation swaps as an example, this paper provides a theoretical background to estimate real interest rates and inflation risk premiums either by a model-free approach implied by the no-arbitrage assumption or by a model-based approach in which the proposed affine model can directly estimate every element in the model. This paper provides an affine framework to value a variety of inflation derivatives, but only inflation swap is given as an example. The natural extension of this paper is to conduct an empirical study to estimate real interest rates or inflation risk premiums using other liquid market prices of inflation derivatives. As we discussed in Section 3.5, other inflation derivatives prices can provide another dimension of information in estimation. Acknowledgment We thank seminar participants at Central Bank of the Republic of Turkey for their helpful comments and suggestions Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.ejor.2013.12.010. References Ang, A., Bekaert, G., & Wei, M. (2007). Do macro variables, asset markets or surveys forecast inflation better? Journal of Monetary Economics, 54, 1163–1212. Ang, A., Bekaert, G., & Wei, M. (2008). The term structure of real rates and expected inflation. Journal of Finance, 63, 797–849. Bikov, R., & Chernov, M. (2009). Unspanned stochastic volatility in affine models: Evidence from eurodollar futures and options. Management Science, 55, 1292–1305.

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