Inflation Expectations and Risk Premiums in an Arbitrage-Free Model of Nominal and Real Bond Yields

Transcription

1 Inflaion Expecaions and Risk Premiums in an Arbirage-Free Model of Nominal and Real Bond Yields Jens H. E. Chrisensen Jose A. Lopez Glenn D. Rudebusch Federal Reserve Bank of San Francisco 101 Marke Sree San Francisco, CA Absrac Differences beween yields on comparable-mauriy U.S. Treasury nominal and real deb, he so-called breakeven inflaion (BEI) raes, are widely used indicaors of inflaion expecaions. However, beer measures of inflaion expecaions could be obained by subracing inflaion risk premiums from he BEI raes. We provide such decomposiions using an esimaed affine arbirage-free model of he erm srucure ha capures he pricing of boh nominal and real Treasury securiies. Our empirical resuls sugges ha long-erm inflaion expecaions have been well anchored over he pas few years, and inflaion risk premiums, alhough volaile, have been close o zero on average. The views in his paper are solely he responsibiliy of he auhors and should no be inerpreed as reflecing he views of he Federal Reserve Bank of San Francisco or he Board of Governors of he Federal Reserve Sysem. Draf dae: December 8, 2008.

2 From he perspecive of moneary policy, jus as imporan as he behavior of acual inflaion is wha households and businesses expec o happen o inflaion in he fuure, paricularly over he longer erm. If people expec an increase in inflaion o be emporary and do no build i ino heir longer-erm plans for seing wages and prices, hen he inflaion creaed by a shock o oil prices will end o fade relaively quickly. Some indicaors of longer-erm inflaion expecaions have risen in recen monhs, which is a significan concern for he Federal Reserve. Federal Reserve Chairman Ben S. Bernanke (2008) 1 Inroducion In fulfilling heir mandae for price sabiliy, cenral banks around he world are keenly ineresed in he expecaions of fuure inflaion held by households and businesses. Such expecaions are widely viewed as key deerminans of fuure inflaion, and long-run inflaion expecaions are considered a useful gauge of cenral bank credibiliy. There are wo main sources for daa on inflaion expecaions: surveys and financial markes. Policymakers consider he former source o be of somewha limied use because surveys are ypically conduced a a quarerly or, a bes, a monhly frequency, so heir informaion on expecaions can be sale by he ime of a policy meeing. Their infrequency also precludes using surveys o measure he immediae response of inflaion expecaions o discree evens, such as moneary policy acions or saemens. In addiion, surveys ypically focus on he expecaions of inflaion over he nex year or so, bu given policy lags, cenral banks are ineresed in longererm informaion as well, especially o help assess he credibiliy of heir long-run inflaion objecives. In conras o surveys, prices in financial markes can provide daily even inraday readings on inflaion expecaions a a wide range of horizons. The markes for nominal bonds, which have a fixed noional principal, and real bonds, which are direcly indexed o overall price inflaion, are he ones mos closely followed for his purpose. 1 For example, he principal and coupon paymens of U.S. Treasury inflaion-proeced securiies (TIPS) vary wih changes in he consumer price index (CPI). Differences beween comparable-mauriy nominal and real yields are known as breakeven inflaion (BEI) raes. Like oher cenral banks, he Federal Reserve closely moniors such BEI raes as high-frequency indicaors of inflaion expecaions. However, as is widely appreciaed, BEI raes are imperfec measures of 1 Oher financial insrumens, such as inflaion swaps, are also being inroduced ha may be useful for gauging inflaion expecaions. 1

3 inflaion expecaions because hey also include compensaion for inflaion risk. Tha is, a BEI rae could rise if fuure inflaion uncerainy rose or if invesors required greaer compensaion for ha uncerainy even if expecaions for he fuure level of inflaion remained unchanged. Obaining a imely decomposiion of BEI raes ino inflaion expecaions and inflaion risk premiums is imporan o cenral bankers, because hey may wish o respond o a change in each componen quie differenly. The decomposiion of a BEI rae ino inflaion expecaions and an inflaion risk premium depends on he correlaions beween inflaion and he unobserved sochasic discoun facors of invesors. This decomposiion requires a model, and in his paper, we use an affine arbirage-free (AF) model, which is he mos widely used finance represenaion of he erm srucure. 2 These models specify he risk-neural evoluion of he underlying yield-curve facors as well as he dynamics of risk premiums under he key heoreical resricion ha here are no residual opporuniies for riskless arbirage across mauriies and over ime. Following Duffie and Kan (1996), affine AF models have been paricularly popular because yields are convenien linear funcions of underlying laen facors (i.e., sae variables ha are unobserved by he economerician) wih facor loadings ha can be calculaed from a sysem of ordinary differenial equaions. Unforunaely, affine AF models can exhibi very poor empirical ime-series performance, especially when forecasing fuure yields (Duffee, 2002). In addiion, here are many echnical difficulies involved wih he esimaion of hese models, which end o be overparameerized and have numerous likelihood maxima ha have essenially idenical fi o he daa bu very differen implicaions for economic behavior (Kim and Orphanides, 2005, and Duffee, 2008). Researchers have employed a variey of echniques o faciliae esimaion including he imposiion of addiional model srucure. 3 Noably, Chrisensen, Diebold, and Rudebusch (2007), henceforh CDR, impose general level, slope, and curvaure facor loadings ha are derived from he popular Nelson and Siegel (1987) yield curve. In his paper, we show ha he resuling affine arbirage-free Nelson-Siegel (AFNS) model can be readily esimaed for a join represenaion of nominal and real yield curves. Our esimaed join AFNS model for nominal and real yields describes he dynamics of he nominal and real sochasic discoun facors, and wih his model, we can decompose BEI raes of any mauriy ino inflaion expecaions and inflaion risk premiums. A sizable research lieraure has analyzed he inflaion risk premium including, for ex- 2 Oher sudies conduc his decomposiion using hisorical inflaion daa; see Grishchenko and Huang (2008). 3 For example, many researchers (e.g., Dai and Singleon, 2002) simply resric parameers wih small - saisics in he firs round of esimaion o zero. 2

4 ample, Ang, Bekaer, and Wei (2008), who use a regime-swiching AF model esimaed on daa for nominal yields and inflaion. However, only a few papers have used U.S. daa on real yields o esimae an AF model and decompose he BEI rae. Chen, Liu, and Cheng (2005) esimae a wo-facor AF represenaion of nominal and real yields. Also, Hördahl and Trisani (2008) decompose BEI raes using an AF macro-finance model wih monhly daa on nominal and real yields, inflaion, and he oupu gap, and D Amico, Kim, and Wei (2008), henceforh DKW, esimae an affine AF model of nominal and real yields augmened wih daa on inflaion and survey daa on forecass of shor-erm nominal ineres raes. We compare our resuls o his earlier work. The paper is srucured as follows. In Secions 2 and 3, we esimae separae affine AF models for yields on nominal and real Treasury bonds, respecively. The resuling hree-facor nominal model and wo-facor real model are of some ineres on heir own and provide useful performance benchmarks for our join model of nominal and real yields in Secion 4. The separae models also provide an imporan inpu ino he consrucion of ha join model. Indeed, based on he correlaions among he facors of he separae models, we are able o idenify a redundan facor, so we find a join four-facor AF model fis boh he nominal and real yield curves quie well. Secion 5 hen analyzes ha esimaed model s decomposiion of BEI raes ino inflaion expecaions and inflaion risk premiums. Secion 6 concludes. 2 An esimaed AF model for nominal yields In his secion, we esimae an affine AF model on weekly daa for nominal zero-coupon U.S. Treasury bond yields. An imporan preliminary sep is o characerize he number and general form of he laen sae variables. Researchers have ypically found ha hree facors, ofen referred o as level, slope, and curvaure, are sufficien o accoun for he ime variaion in he cross secion of nominal Treasury yields (e.g., Lierman and Scheinkman, 1991). This characerizaion is suppored by a principal componen analysis of our weekly daa se, which consiss of Friday observaions from January 6, 1995, o March 28, 2008, for eigh mauriies: hree monhs, six monhs, one year, wo years, hree years, five years, seven years, and en years. 4 Indeed, as shown in Table 1, 99.9 percen of he oal variaion in his se of yields is accouned for by he firs hree principal componens. Furhermore, he loadings across he 4 Our sample of nominal yields is relaively shor because we are persuaded ha here have been significan regime shifs in erm srucure behavior during he poswar period, no leas of which sem from changes in he moneary policy rule linking shor-erm nominal ineres raes and inflaion. In addiion, our sample of real yields is even more limied, so earlier daa on nominal yields would be of limied value. Our daa are obained from hp:// and are described in Gürkaynak, Sack, and Wrigh (2007). 3

5 Mauriy Firs Second Third (in monhs) P.C. P.C. P.C Explain Table 1: Firs Three Principal Componens in Nominal Yields. The loadings of yields of various mauriies on he firs hree principal componens are shown. The final row shows he proporion of all bond yield variabiliy accouned for by each principal componen. The daa consis of weekly nominal zero-coupon U.S. Treasury bond yields from January 6, 1995, o March 28, eigh mauriies for he firs componen are quie uniform; hus, like a level facor, a shock o his componen will change all yields by a similar amoun. The second componen has negaive loadings for shor mauriies and posiive loadings for long ones; hus, like a slope facor, a shock o his componen will seepen or flaen he yield curve. Finally, he hird componen has U-shaped facor loadings as a funcion of mauriy and is naurally inerpreed as a curvaure facor. These resuls help moivae our nominal AFNS model, which assumes ha here are hree laen sae variables relevan for pricing nominal Treasury yields. The AFNS facors are idenified as level, slope, and curvaure by imposing he facor loadings from he Nelson and Siegel (1987) yield curve, which is commonly used by financial marke praciioners and cenral banks due o is excellen fi o various real-world yield curves across counries and ime; see Bank for Inernaional Selemens (2005), Diebold and Li (2006), Gürkaynak, Sack, and Wrigh (2007), and CDR (2008). CDR show ha an AFNS model can closely fi he erm srucure of ineres raes over ime and forecass well ou of sample. CDR also show ha he AFNS model can be esimaed in a sraighforward and robus fashion, unlike he canonical maximally flexible affine AF model, which, as noed in he inroducion, is plagued wih esimaion difficulies. 5 5 Duffee (2008) describes he difficulies ha require a fairly elaborae hands-on esimaion procedure. As an alernaive sraegy, DKW augmen an AF model of nominal and real yields wih daa on inflaion and survey daa on forecass of shor-erm nominal ineres raes. However, adding inflaion daa o he model raises he horny issue of reproducing he appropriae real-ime informaion se of invesors in ligh of lagged daa releases and ex pos daa revisions. An advanage of our sraegy is ha our model only uses daa from financial markes. In addiion, adding survey daa ino he esimaion raises quesions abou he congruency 4

6 The sae vecor of he hree nominal AFNS model facors level, slope, and curvaure is denoed as X N = (L N,SN,CN ). As discussed in CDR, he insananeous nominal risk-free rae is assumed o be he sum of he level and slope facors: r N = L N + S N. Also, he dynamics of he sae variables under he risk-neural (or Q) probabiliy measure are given by he following sysem of sochasic differenial equaions: dl N ds N dc N = λ N λ N 0 0 λ N L N S N C N d + ΣN dw Q,LN dw Q,SN dw Q,CN, λn > 0, where W Q is a sandard Brownian moion in R 3 and Σ N is he volailiy marix. 6 An imporan resricion in his dynamic sysem is ha he facor mean-reversion marix (under he Q-measure) is consrained o ake on a very simple form wih λ N as he only free parameer. CDR show ha his AFNS srucure implies ha nominal zero-coupon yields wih mauriy τ a ime, y N (τ), ake he form: y N (τ) = LN + ( 1 e λn τ λ N τ ) S N + ( 1 e λn τ λ N τ e λn τ ) C N + AN (τ). τ Tha is, he hree facors are given exacly he same level, slope, and curvaure facor loadings as in he Nelson-Siegel (1987) yield curve. A shock o L N uniformly; a shock o S N o C N affecs yields a all mauriies affecs yields a shor mauriies more han long ones; and a shock affecs midrange mauriies mos. Again, i is his idenificaion of he general role of each facor, even hough he facors hemselves remain unobserved and he precise facor loadings depend on he esimaed λ N, ha ensures he esimaion of he AFNS model is sraighforward and robus unlike he maximally flexible affine AF model. The yield funcion also conains a yield-adjusmen erm, AN (τ) τ, ha is ime-invarian and only depends on he mauriy of he bond. CDR provide an analyical formula for his erm, which under our idenificaion scheme is enirely deermined by he volailiy marix Σ N. CDR find ha allowing for a maximally flexible parameerizaion of he volailiy marix diminishes ou-of-sample forecas performance, so we resric Σ N o be diagonal. beween he informaion ses of survey respondens and financial marke paricipans. 6 For idenificaion, we fix he mean vecor under he Q-measure a zero, which CDR show is wihou loss of generaliy. 5

7 The final elemen required for empirical implemenaion of an affine AF model is a specificaion of he price of risk. For racable implemenaion, we employ he popular essenially affine risk premium specificaion inroduced in Duffee (2002), which implies ha he price of risk, Γ, depends on he sae variables: Γ = γ 0 + γ 1 X N, where γ 0 R 3 and γ 1 R 3 3 are unresriced. The relaionship beween real-world yield curve dynamics under he P-measure and risk-neural dynamics under he Q-measure is given by he measure change dw Q = dw P + Γ d. Therefore, we can wrie he P-dynamics of he sae variables as dx N = K P,N (θ P,N X N )d + Σ N dw P, where boh K P,N and θ P,N are allowed o vary freely. 7 We esimae his model using he Kalman filer, as deailed in he appendix. The Kalman filer provides consisen and efficien parameer esimaes and easily handles missing daa, which will be useful for combining nominal and real yields in he join esimaion. 8 Table 2 presens he esimaed parameers for his model. The level facor is very persisen (wih a rae of own mean reversion of only 0.100), while he slope and curvaure facors rever o mean more quickly. Only a few of he off-diagonal elemens in K P,N are significan, which is consisen wih earlier work. For example, CDR find beer ou-of-sample forecas performance from an esimaed model wih a diagonal raher han an unresriced K P,N. However, we are largely ineresed in he separae nominal and real models in order o calculae he correlaions beween he associaed nominal and real facors. These facors are essenially insensiive o any resricions ha migh be placed on he K P,N marix, so we simply employ he unresriced, flexible version in he separae nominal and real models. However, for he join model, where he esimaes of he facor dynamics will affec he resuling decomposiion of he BEI rae, we provide a horough analysis of alernaive dynamic specificaions. Summary saisics for he fied errors of yields a each mauriy of he esimaed nominal AFNS model are given in he second and hird columns of Table 3. Wih he excepion of he 7 The srucure under he Q-measure places no resricions on he dynamic drif componens under he empirical P-measure beyond he requiremen of consan volailiy. 8 Noe ha yields a each mauriy have heir own i.i.d. measuremen error wihin he esimaion process. 6

8 K P,N K P,N,1 K P,N,2 K P,N,3 θ P,N Σ N K P,N 1, Σ N 1, (0.220) (0.096) (0.076) (0.0071) ( ) K P,N 2, Σ N 2, (0.515) (0.187) (0.163) (0.0113) ( ) K P,N 3, Σ N 3, (1.270) (0.494) (0.485) (0.0112) ( ) Table 2: Parameer Esimaes for Nominal AFNS Model. The esimaed parameers of he K P,N marix, θ P,N vecor, and diagonal Σ N marix are shown for he AFNS model of nominal Treasury bond yields. The esimaed value of λ N is wih a sandard deviaion of The maximum log-likelihood value is 32, The numbers in parenheses are esimaed parameer sandard deviaions. Mauriy in monhs Nominal AFNS model Real AFNS model Join AFNS model Nom. yields Mean RMSE Mean RMSE Mean RMSE TIPS yields Mean RMSE Mean RMSE Mean RMSE Table 3: Summary Saisics for AFNS Models. The means and roo mean squared errors (RMSE) of he fied errors of he nominal, real, and join AFNS models are shown. All numbers are measured in basis poins. The nominal yields cover he period from January 6, 1995, o March 28, 2008, while he real TIPS yields cover he period from January 3, 2003, o March 28, hree-monh and en-year yields, he errors are quie low and indicae a reasonable overall fi o he cross-secion of yields. 9 9 The hree-monh mauriy is difficul o fi parly because he shor end of he Treasury yield curve is buffeed by shor-erm idiosyncraic forces (Duffee, 1996). 7

9 3 An empirical AF model for real yields In his secion, we esimae an affine AF model for real zero-coupon U.S. Treasury bond yields derived from TIPS yields. 10 In he empirical lieraure on erm srucure modeling, he focus has been on nominal raher han real bond yields in par because of he relaive scarciy of real deb. The U.S. Treasury firs issued TIPS in 1997, bu for several years afer ha iniial issuance, he liquidiy of he secondary TIPS marke was grealy impaired by he small amoun of securiies ousanding and uncerainy abou he Treasury s commimen o he program. Indeed, as described by Roush (2008), secondary TIPS marke rading was very low a leas ino 2002, and DKW esimae ha such illiquidiy boosed TIPS yields by 1 o 2 percenage poins. To avoid spurious quoes from he illiquid nascen years of his marke, we begin our sample of TIPS yields in 2003; herefore, our real yield daa cover he period from January 3, 2003, o March 28, 2008, and are measured a he end of business each Friday. In addiion, due o he limied mauriy range in he TIPS marke, we only consider mauriies of five, six, seven, eigh, nine, and en years. As a preliminary analysis, Table 4 repors he loadings by mauriy ha correspond o he firs hree principal componens for our sample of real yields. The firs wo componens accoun for essenially all of he variaion in he daa, and hese componens have loadings ha are consisen wih level and slope inerpreaions. Given he limied range of available mauriies for real yields, i is no surprising ha a curvaure facor is no needed. Therefore, we esimae an AFNS model for real yields wih a sae vecor of wo facors, denoed as X R = (L R,S R ). 11 The insananeous risk-free real rae is defined as he sum of he level and slope facors: r R = L R + S R. The dynamics of hese wo facors under he Q-measure are given by he sochasic differenial equaions: dlr ds R = λ R LR S R d + Σ R dw Q,LR dw Q,SR, λ R > 0, where W Q is a sandard Brownian moion in R 2 and Σ R is a diagonal volailiy marix. 12 By imposing his srucure on he general affine model, real zero-coupon yields wih 10 Our daa are obained from he Federal Reserve Board of Governors; see Gürkaynak, Sack, and Wrigh (2008) and he websie hp:// 11 As an alernaive, we also esimaed a hree-facor real AFNS model and found i o be overparameerized. 12 Again, for idenificaion, we fix he mean vecor under he Q-measure a zero. 8

10 Mauriy Firs Second Third Explain Table 4: Firs Three Principal Componens in Real Yields. The loadings of yields of various mauriies on he firs hree principal componens are shown. The final row shows he proporion of all bond yield variabiliy accouned for by each principal componen. The daa consis of weekly real zero-coupon bond yields from January 3, 2003, o March 28, mauriy τ a ime, y R (τ), are given by y R (τ) = LR + ( 1 e λr τ λ R τ ) S R + AR (τ), τ which has Nelson-Siegel facor loadings for he level and slope facors and a mauriydependen yield-adjusmen erm, AR (τ) τ, as described in CDR. As above, we only consider diagonal volailiy marices. As before, we employ he essenially affine risk premium specificaion: Γ = γ 0 + γ 1 X R, where γ 0 R 2 and γ 1 R 2 2 are unresriced. The same relaionship beween real-world and risk-neural dynamics applies; herefore, we can wrie he P-dynamics of he sae variables as dx R = K P,R (θ P,R X R )d + ΣR dw P, where boh K P,R and θ P,R are allowed o vary freely. Table 5 presens he esimaed parameers for he dynamics of he wo sae variables based on he Kalman filer o obain maximum likelihood esimaes. Boh facors rever o mean fairly quickly. The real level facor has an esimaed volailiy similar o ha of he nominal level facor, bu he real slope facor is esimaed o be wice as volaile as he nominal slope facor. The fied errors of his real AFNS model are repored in he fourh and fifh columns of Table 3. Their small size indicaes ha wo facors are sufficien o model he variaion in our TIPS yield sample, which is consisen wih he principal componen analysis. 9

11 K P,R K P,R,1 K P,R,2 θ P,R Σ R K P,R 1, Σ R 1, (0.696) (0.150) (0.0020) ( ) K P,R 2, Σ R 2, (2.720) (0.574) (0.0152) ( ) Table 5: Parameer Esimaes for Real AFNS Model. The esimaed parameers of he K P,R marix, θ P,R vecor, and diagonal Σ R marix are shown for he AFNS model of TIPS yields. The esimaed value of λ R is wih a sandard deviaion of The maximum log-likelihood value is 12, The numbers in parenheses are esimaed parameer sandard deviaions. 4 A join AF model for nominal and real yields An aracive feaure of he AFNS model is ha i can be exended o incorporae as many facors as required. For example, Chrisensen and Lopez (2008) esimae a join AFNS model ha accouns for he sandard hree Treasury yield facors and wo addiional facors accouning for corporae credi spread dynamics. In his secion, we esimae a join AFNS model ha combines he separae nominal and real yield models presened above. Figure 1 compares he esimaed pahs of he wo real yield curve facors from he real AFNS model o he pahs of he corresponding nominal yield curve facors from he nominal AFNS model. The correlaion beween he wo level facors is 0.90, while he slope facors have a correlaion of Given hese high correlaions, i is emping o use jus hree facors o model he variaion in boh ses of bond yields; however, as described in he appendix, we found ha a hree-facor join AFNS model was oo resricive o fi boh nominal and real yields. Insead, we only impose he assumpion of a common slope facor across he nominal and real yields. Therefore, our join model has four facors: a real level facor (L R ) ha is specific o TIPS yields only, a nominal level facor (L N ) for nominal yields, and common slope and curvaure facors. (The curvaure facor, of course, is only needed for fiing he nominal yields.) The sae vecor of join AFNS model facors is denoed as X J = (L N,S,C,L R ), and he insananeous nominal and real risk-free raes are defined by: r N = L N + S, r R = L R + α R S. The differenial scaling of real raes o he common slope facor is capured by he parameer α R. To preserve he Nelson-Siegel facor loading srucure in he nominal yield funcion, he 10

12 Value of L() Nominal level facor Real level facor Value of S() Nominal slope facor Real slope facor Time Time (a) Esimaed level facors. (b) Esimaed slope facors. Figure 1: Esimaed Nominal and Real Level and Slope Facors. The esimaed level and slope facors from he wo-facor real AFNS model are shown wih he level and slope facors from he hree-facor nominal AFNS model. Q-dynamics of he sae variables are given by he sochasic differenial equaions: dl N L N dw Q,LN ds 0 λ λ 0 S = d + Σ J dw Q,S dc 0 0 λ 0 C dw Q,C. (1) dl R L R dw Q,LR Based on hese dynamics, nominal Treasury zero-coupon yields are ( ) ( 1 e y N (τ) = L N λτ 1 e λτ + S + λτ λτ and real zero-coupon yields are e λτ ) C + AN (τ), τ ( ( 1 e y R (τ) = LR + α R λτ 1 e )S + α R λτ λτ λτ e λτ ) C + AR (τ). τ Again, deails of he yield-adjusmen erms are in CDR. Using he essenially affine risk premium specificaion, he implied measure change is given by dw Q = dw P + Γ d, where Γ = γ J,0 +γ J,1 X J, γj,0 R 4, and γ J,1 R 4 4. The resuling four-facor AFNS model 11

13 has P-dynamics given by dl N ds dc dl R = κ P,J 11 κ P,J 12 κ P,J 13 κ P,J 14 κ P,J 21 κ P,J 22 κ P,J 23 κ P,J 24 κ P,J 31 κ P,J 32 κ P,J 33 κ P,J 34 κ P,J 41 κ P,J 42 κ P,J 43 κ P,J 44 θ P,J 1 θ P,J 2 θ P,J 3 θ P,J 4 L N S C L R d+σ J dw P,LN dw P,S dw P,C dw P,LR, where Σ J is diagonal. As alluded o earlier, he specificaion of he P-dynamics is an imporan elemen in deermining he model s decomposiion of BEI raes ino inflaion expecaions and risk premiums. Therefore, we conduc a careful evaluaion of various model specificaions, as summarized in Table 6. The firs column of his able describes he 13 alernaive specificaions considered. Specificaion (1) a he op corresponds o an unresriced 4 4 mean-reversion marix K P,J, which provides maximum flexibiliy in fiing he daa. We hen pare down his marix using a general-o-specific sraegy ha resrics he leas significan parameer (as measured by raio of he parameer value o is sandard error) o zero and hen re-esimae he model. Therefore, specificaion (2) ses κ P,J 31 = 0, so i has one fewer esimaed parameers. Specificaion (3) ses his parameer and κ P,J 32 boh equal o zero. This sraegy of eliminaing he leas significan coefficiens coninues o he final specificaion (13), which has a diagonal K P,J marix. Each esimaed specificaion is lised wih is log likelihood (log L), is number of esimaed parameers (k), and he p-value from a likelihood raio es of he hypohesis ha i differs from he specificaion wih one more free parameer ha is, comparing specificaion (s) wih specificaion (s 1). We also repor wo informaion crieria commonly used for model selecion: he Akaike informaion crierion, which is defined as AIC= 2log L + 2k, and he Bayes informaion crierion, which is defined as BIC= 2log L + k log T, where T is he number of daa observaions (see e.g., Harvey, 1989). 13 These informaion crieria are minimized by specificaions (8) and (9) (he boldface enries in he righmos columns), which are hus our favored models. Noably, he unresriced specificaion (1) appears overparameerized, and he diagonal specificaion (13) appears oo parsimonious. The likelihood raio es also suggess ha (a he 10-percen level) specificaion (8) is a parsimonious model ha sill provides as good a fi o he daa as he maximally flexible unresriced specificaion. Therefore, we selec specificaion (8) as our preferred join AFNS model. 13 We have 691 nominal yield and 273 real yield weekly observaions. We inerpre T as referring o he longes daa series and fix i a

14 Alernaive Goodness of fi saisics specificaions log L k p-value AIC BIC (1) Unresriced K P,J n.a (2) κ P,J 31 = (3) κ P,J 31 = κ P,J 32 = (4) κ P,J 31 = κ P,J 32 = κ P,J 13 = (5) κ P,J 31 =... = κ P,J 34 = (6) κ P,J 31 =... = κ P,J 12 = (7) κ P,J 31 =... = κ P,J 24 = (8) κ P,J 31 =... = κ P,J 43 = (9) κ P,J 31 =... = κ P,J 41 = (10) κ P,J 31 =... = κp,j 42 = < (11) κ P,J 31 =... = κp,j 21 = < (12) κ P,J 31 =... = κp,j 14 = < (13) κ P,J 31 =... = κp,j 23 = < Table 6: Evaluaion of Alernaive Specificaions of Join AFNS Model. Thireen alernaive esimaed specificaions of he join AFNS model are evaluaed. Each specificaion is lised wih is log likelihood (log L), number of parameers (k), he p-value from a likelihood raio es of he hypohesis ha he specificaion differs from he one direcly above ha has one more free parameer. The informaion crieria (AIC and BIC) are also repored, and heir minimum values are given in boldface. Table 7 conains he esimaed parameers of his preferred specificaion (8). Noe ha he off-diagonal elemens in he esimaed K P,J marix (excluding he seven zero resricions) are highly saisically significan. The mean and volailiy parameers for he hree nominal facors and he esimaed value of λ are very similar o hose repored in Table 2 for he hreefacor nominal AFNS model. Based on hese resuls, we anicipae he fi of he nominal yields and he esimaed pahs of he hree nominal yield risk facors o be very similar across hese wo models. Indeed, as shown in Table 3, which conains summary saisics for he fied errors of he join model, here is no discernible difference in fi beween he join model and he nominal model for he eigh mauriies of nominal yields. Table 3 does repor a worse fi of he join model relaive o he wo-facor model for real yields. However, he difference in fi appears o be reasonable in ha he esimaed real facors behave similarly. Tha is, he correlaion beween he esimaed real level facors from he join and real AFNS models is The correlaion beween he esimaed real slope facor from he real AFNS model and he join AFNS model-implied real slope facor (α R S ) is The close connecions beween hese facors provide furher suppor for he join model. 13

15 K P,J K P,J K P,J,1 K P,J,2 K P,J,3 K P,J,4 θ P,J Σ J 1, Σ J 1, (0.277) (0.517) ( ) ( ) K P,J 2, Σ J 2, (0.504) (0.164) (0.141) ( ) ( ) K P,J 3, Σ J 3, (0.382) ( ) ( ) K P,J 4, Σ J 4, (0.643) (0.110) (0.724) ( ) ( ) Table 7: Parameer Esimaes for Join AFNS Model. The esimaed parameers of he K P,J marix, θ P,J vecor, and diagonal Σ J marix are shown for he AFNS model of nominal and real yields. The esimaed value of λ is wih a sandard deviaion of , while α R is esimaed o be wih a sandard deviaion of The numbers in parenheses are he esimaed parameer sandard deviaions. 5 Inflaion expecaions and inflaion risk premiums In his secion, we decompose he BEI raes ino inflaion expecaions and inflaion risk premiums. We sar wih a heoreical discussion of how an AF model of nominal and real yields can produce his decomposiion, and hen we presen he empirical decomposiion provided by our esimaed join AFNS model. Finally, we compare our resuls o ohers in he lieraure. 5.1 Theoreical discussion To describe he connecions among nominal and real yields and inflaion wihin our modeling framework, i is convenien o work in coninuous ime (see Cochrane 2001 for a primer). We firs define he nominal and real sochasic discoun facors, denoed M N and M R, respecively. The no-arbirage condiion enforces a consisency of pricing for any securiy over ime. Specifically, he price of a nominal bond ha pays one dollar a ime τ and he price of a real bond ha pays one uni of he consumpion baske a ime τ mus saisfy P N (τ) = E P [ M N ] +τ M N and P R (τ) = E P [ M R +τ Given heir paymen srucure, he no-arbirage condiion also requires a consisency beween he prices of real and nominal bonds such ha he price of he consumpion baske, denoed as he overall price level Q, is he raio of he nominal and real sochasic discoun facors: M R ]. Q = MR M N. 14

16 We assume ha he nominal and real sochasic discoun facors have he sandard dynamics given by dm N /M N dm R /MR = r N d Γ dw P, = r R d Γ dw P. Then, by Io s lemma, he dynamic evoluion of Q is given by dq = 1 M N dm R = (r N r R )Q d. MR (M N ) 2dMN + 1 ( 2 dm R dm N ) 0 1 (M N )2 1 (M N )2 2 MR (M N )3 dmr dm N Thus, wih he absence of arbirage, he insananeous growh rae of he price level is equal o he difference beween he insananeous nominal and real risk-free raes. (Tha is, here is no risk premium for he insananeous raes, and he Fisher equaion applies.) Furhermore, by Io s lemma, dln (Q ) = 1 Q dq Q 2 dq 2 = (rn r R )d. By inegraing boh sides and aking exponenials, we can express he price level a ime +τ as +τ Q +τ = Q e (rs N rs R)ds. The connecion beween nominal and real zero-coupon yields and expeced inflaion can be readily expressed. Namely, we decompose he price of he nominal zero-coupon bond as [ ] P N (τ) = E P M+τ N M N [ = E P M+τ/Q R +τ M R/Q = E P [ M R +τ M R = E P [ M R +τ M R ] E P [ ] E P [ ] = E P [ M R +τ Q Q +τ Q Q +τ ] ] M R + cov ( Q Q +τ [ 1 + ] M R +τ M R, Q Q +τ ] M R cov[ +τ, E P [ M R +τ M R M R ] Q Q +τ ] [ ] E P Q Q +τ ). 15

17 Convering his price ino a yield-o-mauriy, we obain y N (τ) = yr (τ) + πe (τ) + φ (τ), where he marke-implied rae of inflaion expeced a ime from he period o + τ is π e (τ) = 1 τ ln EP [ Q Q +τ and he corresponding inflaion risk premium is φ (τ) = 1 τ ln (1 + ] = 1 τ ln EP M R cov[ +τ, E P [ M R +τ M R M R [e +τ (r N s rr s )ds], ] Q Q +τ ] [ ] E P Q Q +τ This las equaion highlighs ha he inflaion risk premium can be posiive or negaive. I will be posiive if and only if [ M R cov +τ, M R Tha is, he riskiness of nominal bonds depends on he covariance beween he real sochasic discoun facor and inflaion. We observe posiive inflaion risk premiums if he real discoun facor ends o be high (i.e., in a srucural model, marginal uiliy is high) a he same ime Q Q +τ ] < 0. ha price inflaion is high (i.e., purchasing power is low). Finally, he BEI rae is defined as ). BEI (τ) y N (τ) y R (τ) = π e (τ) + φ (τ). Namely, he BEI rae is he difference beween nominal and real yields and can be decomposed ino he sum of expeced inflaion and he inflaion risk premium. 5.2 Empirical resuls We now urn o our esimaed model. Figure 2 displays he five- and en-year nominal and real zero-coupon yields and heir differences a each mauriy i.e., he associaed observed BEI raes. Boh five- and en-year BEI raes increased a bi during he firs wo years of our sample, bu since 2004, hey have changed lile on balance. Figure 2 also compares hese observed BEI raes o comparable-mauriy model-implied BEI raes, which are calculaed as he differences beween he fied nominal and real yields from he esimaed join AFNS model. The small differences beween he observed and model-implied BEI raes reflec he 16

18 Rae yr nominal yield 5 yr real yield 5 yr observed BEI rae 5 yr model implied BEI rae Rae yr nominal yield 10 yr real yield 10 yr observed BEI rae 10 yr model implied BEI rae Time Time (a) Five-year mauriy (b) Ten-year mauriy Figure 2: Nominal and Real Yields and BEI Raes. Daa on five- and en-year nominal and real zero-coupon Treasury yields are ploed wih he associaed BEI raes and he implied BEI raes from he join AFNS model. overall good fi of his model. The join AFNS model also allows us o decompose he BEI rae ino inflaion expecaions and he inflaion risk premium a various horizons. Given he esimaed parameers in Table 7 and he esimaed pahs of he four sae variables, he model-implied average five- and en-year expeced inflaion series are illusraed in Figure 3. The five-year measure varied from 1.93 percen o 2.57 percen, and he en-year measure from 2.16 percen o 2.42 percen. These ranges sugges ha long-run inflaion expecaions were fairly well-anchored during our sample period. The model s measures of inflaion expecaions are generaed using only nominal and real yields wihou any daa on inflaion or inflaion expecaions. To provide some independen indicaion of accuracy, Figure 3 also plos survey-based measures of expecaions of CPI inflaion, which are obained from he Blue Chip Consensus survey a he five-year horizon and from he Survey of Professional Forecasers a he en-year horizon. The relaively close mach beween he model-implied and he survey-based measures of inflaion expecaions provides furher suppor for he model s decomposiion of he BEI rae. Noe ha he larges differences in Figure 3 occur for he five-year horizon during he firs half of 2003 wih he model-implied measure well below he survey-based one. This paern is consisen wih some remaining residual liquidiy deficiencies in he TIPS marke, which would hold down bond prices, boos real yields, and lead o an undersaemen of model-implied inflaion 17

19 Rae Model implied BEI rae Model implied expeced inflaion Survey based inflaion forecas Rae Model implied BEI rae Model implied expeced inflaion Survey based inflaion forecas Time Time (a) Five-year horizon. (b) Ten-year horizon. Figure 3: BEI Raes and Expeced Inflaion. The five- and en-year BEI raes and average expeced inflaion raes ha are implied from he join AFNS model are ploed along wih survey-based measures of inflaion expecaions. expecaions. The K P,J marix, which governs facor dynamics, plays a key role in he decomposiion of BEI raes. The dependence of model-implied inflaion expecaions on he specificaion of he K P,J marix is illusraed in Figure 4, which shows five- and en-year expeced inflaion implied by hree differen specificaions of he join AFNS model. The solid line is he preferred specificaion described earlier, and he dashed and doed lines are based on unresriced and diagonal K P,J marices, respecively. The preferred specificaion, which was seleced based on in-sample fi o he daa, also provides abou he closes mach o he survey-based inflaion forecass. Finally, for our preferred specificaion, we subrac each model-implied expeced inflaion rae from he comparable-mauriy model-implied BEI rae and obain he associaed inflaion risk premium (IRP). A boh he five- and en-year horizons, hese premiums are fairly small, as shown in Figure Indeed, during our sample, hese inflaion premiums have varied in a range around zero of abou ±50 basis poins This resul provides some suppor for he argumen ha he gain o he U.S. Treasury from issuing TIPS bonds insead of nominal bonds may be quie limied, as argued in Sack and Elsasser (2004). 15 Again, in heory, he sign of he inflaion risk premium depends on he covariance beween he real sochasic discoun facor and inflaion, bu here are real-world consideraions as well. For example, a liquidiy premium for holding TIPS insead of nominal Treasury bonds would show up as a negaive inflaion risk premium. 18

20 Rae Preferred K marix Full K marix Diagonal K marix Survey based inflaion forecas Rae Preferred K marix Full K marix Diagonal K marix Survey based inflaion forecas Time Time (a) Five-year expeced inflaion (b) Ten-year expeced inflaion Figure 4: Expeced Inflaion Implied by Alernaive Specificaions. Model-implied inflaion expecaions a he five- and en-year horizons are shown for our join AFNS model wih he preferred specificaion of he K P,J marix, wih an unresriced full K P,J marix, and wih a diagonal K P,J marix. Rae year IRP 5 year IRP Time Figure 5: Model-Implied Inflaion Risk Premiums. The five- and en-year inflaion risk premiums (IRP) ha are implied from he join AFNS model are ploed. 5.3 Comparison o he lieraure Our resuls can be usefully compared o he findings of hree recen papers in he lieraure ha also decompose U.S. BEI raes using empirical affine AF models of nominal and real 19

21 yields. The earlies of hese sudies is by Chen, Liu, and Cheng (2005), who esimae a wo-facor AF model using he weekly nominal and TIPS yields daa from January 1998 o December Their esimaed inflaion risk premiums are quie sable. The 5-year premium averages abou zero, similar o ours, while heir 10-year premium averages around 130 basis poins, which is much higher han our esimae (even for only he wo years of overlap beween heir esimaion sample for real yields and ours). However, i seems likely ha heir model esimaes are inappropriaely influenced by he use of he TIPS yields daa from 1998 hrough 2002, when he marke exhibied lile volume and poor liquidiy. As in our analysis, he wo oher recen U.S. sudies also discard hese earlier readings on TIPS yields. Specifically, Hördahl and Trisani (2008) decompose BEI raes using an AF macro-finance model ha incorporaes monhly daa on nominal yields, real yields (since 2003), inflaion, he oupu gap, and survey forecass for inflaion and he hree-monh ineres rae. Similar o our resuls, hey find ha he en-year inflaion risk premium over he pas several years flucuaes wihin a band of ±50 basis poins around zero. Of course, one of he disinguishing feaures of our analysis is ha i uses a yields-only specificaion, which can provide a marke-based reading of inflaion expecaions ha is separae from survey readings or inflaion daa. Finally, DKW also decompose BEI raes wih an affine AF model. Their esimaion uses weekly daa from he nominal Treasury yield curve, weekly daa from he real TIPS yield curve (since 2005), monhly daa on inflaion, and monhly survey forecass of shorerm nominal ineres raes and inflaion. Figure 6(a) compares heir esimaed five- and en-year inflaion expecaions (based on heir preferred model ha includes TIPS yields o our AFNS resuls.) 16 The average values for heir five- and en-year inflaion expecaion measures over our sample period of January 2003 hrough March 2008 are 2.40 percen and 2.39 percen, respecively, which are similar o our values of 2.28 percen and 2.30 percen, respecively. Noe, however, ha heir five- and en-year inflaion expecaion measures have almos idenical dynamics in conras o our resuls. Their resuls likely reflec he use of a single facor o capure he levels of boh nominal and real Treasury yields, while our model uses separae level facors. In addiion, he dynamics of he wo ses of measures are quie differen; he correlaion coefficiens beween he DKW and AFNS measures are 0.14 for he five-year horizon and for he en-year horizon. However, he DKW inflaion measures do no mach he survey measures of inflaion expecaions very well, even hough heir models include boh inflaion 16 We hank Min Wei for sharing up-o-dae, high frequency resuls wih us. 20

22 Rae AFNS, 10 year expeced inflaion AFNS, 5 year expeced inflaion DKW, 10 year expeced inflaion DKW, 5 year expexec inflaion Rae AFNS, 10 year IRP AFNS, 5 year IRP DKW, 10 year IRP DKW, 5 year IRP Time Time (a) Model-implied inflaion expecaions. (b) Model-implied IRP. Figure 6: Model-Implied Inflaion Expecaions and Risk Premiums. Model-implied inflaion expecaions and inflaion risk premiums (IRP) a he five-and enyear horizons are shown for our join AFNS model and he DKW model. and survey daa. The correlaion beween he DKW five-year and survey measures of five-year inflaion expecaions is -0.08, while our he AFNS and survey measures have a correlaion of We can also compare our inflaion risk premiums o he DKW esimaes, as shown in Figure 6(b). The uncondiional means for he AFNS five- and en-year inflaion risk premiums measures are boh abou -5 basis poins, while for he DKW measures, hese means are 36 and 64 basis poins, respecively. In addiion o hese differences in uncondiional momens, he correlaion coefficiens beween he AFNS and DKW measures of inflaion risk premiums are relaively low a 0.25 and 0.38, a he five- and en-year horizons, respecively. 6 Conclusion This paper esimaes an arbirage-free model wih four laen facors ha can capure he dynamics of boh he nominal and real Treasury yield curves well and can decompose BEI raes ino inflaion expecaions and inflaion risk premiums. The model-implied measures of inflaion expecaions are correlaed closely wih survey measures, while he esimaed inflaion risk premiums flucuae in fairly close range around zero. The empirical resuls sugges ha long-erm inflaion expecaions have been well-anchored in he period from year-end 2002 hrough he firs quarer of Our proposed model has a disinc advanage in ha i can be easily esimaed because i 21

23 adops he dynamic arbirage-free Nelson-Siegel srucure developed by CDR. 17 Such easy and robus esimaion, implemened wih he Kalman filer, enables quick updaing of he model o incorporae new observaions and faciliaes he monioring and forecasing of Treasury yield curves on a real-ime basis. The resuling high-frequency measures should be quie desirable o policymakers, cenral bank saff, and financial marke praciioners. 17 Chrisensen, Lopez, and Rudebusch (2008) provide anoher applicaion of he AFNS srucure ha also demonsraes is favorable esimaion properies. 22

24 Appendix Model esimaion procedures We esimae all models using he Kalman filer; see Harvey (1989) for furher deails. The measuremen equaion for he bond yields is given by y = A + BX + ε, where ε represens measuremen errors ha are assumed o be independenly and idenically disribued (i.i.d.) for each mauriy included in he daa sample. For coninuous-ime Gaussian models, he condiional mean vecor and covariance marix are given by E P [X T F ] = (I exp( K P ))µ P + exp( K P )X, V P [X T F ] = 0 e KP s ΣΣ (KP ) s ds, where = T and exp( K P i ) is a marix exponenial. Saionariy of he sysem under he P-measure is ensured if he real componen of all he eigenvalues of K P is posiive, and his condiion is imposed in all esimaions. For his reason, we can sar he Kalman filer a he uncondiional mean and covariance marix, 18 denoed as X 0 = µ P and Σ 0 = The sae equaion in he Kalman filer is given by 0 e KP s ΣΣ (KP ) s ds. X i = Φ 0 i + Φ 1 i X i 1 + η i, where ( i Φ 0 i = (I exp( K P i ))µ P, Φ 1 i = exp( K P i ), and η i N 0, e KP s ΣΣ (KP ) s ) ds 0 wih i = i i 1. In he Kalman filer esimaions, all measuremen errors are assumed o be i.i.d. whie 18 In he esimaion, 0 e KP s ΣΣ (KP ) s ds is approximaed by 10 0 e KP s ΣΣ (KP ) s ds. 23

25 noise. Thus, he error srucure is in general given by η ε N 0, 0 Q 0 0 H. In he esimaion, each mauriy of he Treasury bond yields has is own measuremen error sandard deviaion, σ 2 (τ i ). The linear leas-squares opimaliy of he Kalman filer requires ha he whie noise ransiion and measuremen errors be orhogonal o he iniial sae; i.e., E[f 0 η ] = 0, E[f 0 ε ] = 0. Finally, he sandard deviaions of he esimaed parameers are calculaed as Σ( ψ) = 1 T [ 1 T T =1 log l ( ψ) ψ log l ( ψ) ] 1, ψ where ψ denoes he opimal parameer se. A hree-facor join model for nominal and real yields We considered an alernaive hree-facor model of nominal and real yields. The insananeous nominal risk-free rae was given by r N = L N + S N, and he usual AFNS dynamics of he hree sae variables under he pricing measure were imposed. Given ha only wo facors were needed o model he variaion in he real TIPS yields and ha boh of hese facors were correlaed wih he corresponding nominal yield risk facors, a reasonable specificaion of he insananeous real yield process was r R = α R L LN + α R S SN. Thus, his insananeous real yield was driven by he same wo facors ha drive he nominal shor rae process. Unforunaely, he esimaed hree-facor model (wih a full K P,N marix) performs relaively poorly. Table 8 repors he fied errors for he eigh nominal yield mauriies and he six real yield mauriies. This able shows ha he hree-facor model has a significanly 24

26 Mauriy in monhs Three-Facor Model Nom. yields Mean RMSE TIPS yields Mean RMSE Table 8: Summary Saisics for Three-Facor Join AFNS Models. The mean and roo mean squared error of he fied errors for he hree-facor join AFNS model. All numbers are measured in basis poins. The nominal yields cover he period from January 6, 1995, o March 28, 2008, while he real TIPS yields cover he period from January 3, 2003, o March 28, deerioraed fi for he hree-monh, six-monh, and one-year nominal yields, and for he fiveyear, six-year, and seven-year real TIPS yields. Therefore, we adoped he join four-facor model described in he ex. 25

27 References Ang, Andrew, Geer Bekaer, and Min Wei, 2008, The Term Srucure of Real Raes and Expeced Inflaion, Journal of Finance, 63, Bank for Inernaional Selemens, 2005, Zero-Coupon Yield Curves: Technical Documenaion, BIS papers, No. 25. Bernanke, Ben S., 2008, Remarks on Class Day, Cambridge, Massachuses June 4. Chen, Ren-Raw, Bo Liu, and Xiaolin Cheng, 2005, Inflaion, Fisher Equaion, and The Term Srucure of Inflaion Risk Premia: Theory and Evidence from TIPS, working paper. Chrisensen, Jens H. E., Francis X. Diebold, and Glenn D. Rudebusch, 2007, The Affine Arbirage-Free Class of Nelson-Siegel Term Srucure Models, Federal Reserve Bank of San Francisco Working Paper Chrisensen, Jens H. E., Francis X. Diebold, and Glenn D. Rudebusch, 2008, An Arbirage- Free Generalized Nelson-Siegel Term Srucure Model, forhcoming in The Economerics Journal. Chrisensen, Jens H. E., and Jose A. Lopez, 2008, Common Risk Facors in he U.S. Treasury and Corporae Bond Markes: An Arbirage-Free Dynamic Nelson-Siegel Modeling Approach, Manuscrip, Federal Reserve Bank of San Francisco. Chrisensen, Jens H. E., Jose A. Lopez, Glenn D. Rudebusch, 2008, Do Cenral Bank Liquidiy Faciliies Affec Inerbank Lending Raes?, Manuscrip, Federal Reserve Bank of San Francisco. Cochrane, John, 2001, Asse Pricing, Princeon: Princeon Universiy Press. Dai, Qiang, and Kenneh J. Singleon, 2002, Expecaions Puzzles, Time-Varying Risk Premia, and Affine Models of he Term Srucure, Journal of Financial Economics, Vol. 63, D Amico, Sefania, Don H. Kim, and Min Wei, 2008, Tips from TIPS: he informaional conen of Treasury Inflaion-Proeced Securiy prices, Finance and Economics Discussion Series No. 30, Federal Reserve Board. 26