Extracting Deflation Probability Forecasts from Treasury Yields

Transcription

1 Exracing Deflaion Probabiliy Forecass from Treasury Yields Jens H.E. Chrisensen, Jose A. Lopez, and Glenn D. Rudebusch Federal Reserve Bank of San Francisco We consruc probabiliy forecass for episodes of price deflaion (i.e., a falling price level) using yields on nominal and real U.S. Treasury bonds. The deflaion probabiliy forecass idenify wo deflaion scares during he pas decade: a mild one following he 2001 recession and a more serious one saring in lae 2008 wih he deepening of he financial crisis. The esimaed deflaion probabiliies are generally consisen wih hose from macroeconomic models and surveys of professional forecasers, bu hey also provide high-frequency insigh ino he views of financial marke paricipans. The probabiliies can also be used o price he deflaion proecion opion embedded in real Treasury bonds. JEL Codes: E31, E43, G12, G Inroducion Throughou much of he pos-war period, as overall price levels rose fairly rapidly in many counries, cenral banks were concerned wih reducing price inflaion in order o achieve heir mandae for price sabiliy. However, in recen years, as inflaion raes around he world have fallen o much lower levels, he risks o price sabiliy have become more symmeric, and fears ha inflaion may fall oo low have emerged. In paricular, he risk of negaive inflaion price deflaion has become a recurring concern for several cenral banks. 1 Mos seriously, Japan has been mired in deflaion and economic sagnaion since he mid-1990s. 1 See Bordo and Filardo (2005) and Kumar e al. (2003) for hisorical and cross-counry surveys of deflaionary episodes. 21

2 22 Inernaional Journal of Cenral Banking December 2012 Among Federal Reserve policymakers, worries abou deflaion surfaced wice during he pas decade. The firs episode followed he 2001 recession. Noably, he Federal Open Marke Commiee s (FOMC) saemen of May 6, 2003 publicly expressed he possibiliy of an unwelcome subsanial fall in inflaion. Then Federal Reserve Governor Ben Bernanke (2003) highlighed he imporance of his saemen, noing ha he May 6 saemen broke new ground as he firs occasion in which he FOMC expressed he concern ha inflaion migh acually fall oo low. The second period of deflaionary worries began during he recen financial crisis and ensuing recession. In he wake of a deepening worldwide financial upheaval in lae 2008, projecions of a slowing economy and possible price deflaion were key drivers of moneary policy. Federal Reserve Chairman Bernanke (2010) described he moivaion for policy acions during he crisis in his way: The FOMC s policy response also refleced concerns abou a possible unwelcome decline in inflaion. Taking noe of he painful experience of Japan, policymakers worried ha he Unied Saes migh sink ino deflaion and ha, as one consequence, he FOMC s arge ineres rae migh hi is zero lower bound, limiing he scope for furher moneary accommodaion. The desire o avoid deflaion is ofen based on he view ha a deflaionary episode is paricularly reacherous. Early on, Bernanke (2003) sressed he pernicious effecs of deflaion: In any case, I hope we can agree ha a subsanial fall in inflaion a his sage has he poenial o inerfere wih he ongoing U.S. recovery, and ha in conceivable alhough remoe circumsances, a serious deflaion would do significan economic harm. Deflaionary episodes are considered especially worrying because hey may have heir own unique and painful dynamics. The coninuing deflaionary ravails of Japan illusrae he possibiliy ha he ineracion of price deflaion wih he zero lower bound on nominal ineres raes may produce an inracable regime ha is difficul o exi. Even worse, some heoreical models sugges ha a reinforcing and escalaing deflaionary spiral could arise a he zero bound, in which he fall in prices booss real ineres raes and vice versa. In addiion, deflaions also may carry paricularly severe social coss relaed o he increased real burden on borrowers wih fixed nominal debs. If deflaions are special regimes ha have heir own unique dynamics or social coss, i would be paricularly useful o go beyond poin forecass of

3 Vol. 8 No. 4 Exracing Deflaion Probabiliy Forecass 23 inflaion and consider probabiliy forecass for he occurrence of a deflaionary episode. 2 Given he special ineres in periods of price deflaion, we examine probabiliy forecass for such episodes. 3 Of course, here are a variey of ways in which deflaion probabiliy forecass could be consruced. In his paper, we focus on high-frequency probabiliy forecass ha are derived from yields on nominal and real Treasury bonds. While nominal Treasury bonds have fixed coupons and principal, real Treasury bonds or Treasury inflaion-proeced securiies (TIPS) have coupons and principal ha vary wih changes in he headline consumer price index (CPI). Differences beween comparable-mauriy nominal and real yields are widely used as readings on he inflaion expecaions of marke paricipans; however, such readings are obscured by flucuaions in he compensaion for inflaion risk. To disenangle inflaion expecaions and risk premiums, Chrisensen, Lopez, and Rudebusch (2010), henceforh CLR, use an affine, arbirage-free (AF) dynamic erm srucure model of nominal and real yields. 4 In his paper, we show how o use such a model o calculae he enire probabiliy disribuion of fuure inflaion oucomes paricularly, he implied deflaion probabiliy forecass a any forecas horizon. Our deflaion probabiliy forecass esimaed from yield curves align well wih he resuls from simple macroeconomic benchmarks, surveys of professional forecasers, and oher financial marke prices. These comparisons provide some assurance ha our yieldsonly approach provides useful deflaion probabiliies under he 2 Similar argumens have long been used o moivae he widespread focus on recession probabiliy forecass. Tha is, since recessions are excepional episodes or regimes worhy of special aenion, i is useful o consider even probabiliies insead of jus poin forecass of fuure growh (e.g., Diebold and Rudebusch 1989, Rudebusch and Williams 2009). 3 Noe ha his conrass wih risk measures of deflaion, which combine he probabiliy disribuion of inflaion wih agens preferences over deflaionary oucomes; see Kilian and Manganelli (2007) for a deailed discussion. 4 AF 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. Noe ha hese models provide pricing informaion under boh he risk-neural and observed (or real-world ) probabiliy measures.

4 24 Inernaional Journal of Cenral Banking December 2012 real-world probabiliy measure needed for macroeconomic policy analysis and risk managemen (as opposed o jus he risk-neural pricing measure). Furhermore, we demonsrae how he deflaion probabiliy forecass can be used o value he deflaion proecion opion embedded in TIPS bonds exploiing he AF propery of he model. 5 Our resuls show ha he model-implied value of he embedded deflaion proecion is highly correlaed wih he observed price difference beween TIPS bonds of similar remaining mauriies bu differen degrees of accumulaed inflaion exposure. Our discussion proceeds as follows. Secion 2 summarizes he AF model and describes how deflaion probabiliy forecass are obained from a full-sample model esimae (i.e., in-sample fied values) and from real-ime model esimaes based on expanding samples (i.e., ou-of-sample forecass). The resuling deflaion probabiliies sugges ha over he pas decade wo deflaion scares occurred in he Unied Saes: a mild one following he 2001 recession, and a more serious one ha sared in he auumn of 2008 afer he deepening of he worldwide financial crisis. In secion 3, we compare our real-ime deflaion probabiliy forecass based on Treasury yields wih a variey of alernaive forecass from macroeconomic models and surveys. These alernaive forecass are roughly in line wih our esimaes, bu heir low frequency makes hem less useful as indicaors of fuure deflaion in real ime. In secion 4, we compare our real-ime forecass wih alernaives in he finance lieraure ha are based on yield differenials beween seasoned and newly issued TIPS. These alernaive measures conain no correcion for risk premiums and are hus expressed in risk-neural pricing erms. Using an AF model, we can generae similar deflaion probabiliies under he risk-neural pricing measure; however, our focus is on deflaion probabiliies under he alernaive real-world probabiliy measure, which is more relevan for macroeconomic policy and risk managemen. Indeed, secion 5 provides a salien example of he model s asse pricing applicaions by valuing he deflaion proecion opion embedded in he principal paymens of TIPS bonds. 5 To he bes of our knowledge, Grishchenko, Vanden, and Zhang (2011) is he only oher paper o explicily address he valuaion of he deflaion proecion embedded in TIPS bonds wihin a dynamic erm srucure model.

5 Vol. 8 No. 4 Exracing Deflaion Probabiliy Forecass Deflaion Probabiliies from a Term Srucure Model A dynamic erm srucure model can be used o decompose differences beween nominal and real yield curves ino marke-implied inflaion expecaions and inflaion risk premiums a various mauriies. Such models can also provide informaion on he full disribuion of expeced inflaion oucomes. In his secion, we summarize he CLR version of such a model and explain how o obain he implied deflaion probabiliies. 2.1 CLR Model Specificaion Affine, arbirage-free erm srucure models, of which he CLR model is an example, link yield-curve dynamics and invesor risk premiums wihin a consisen framework ha can produce boh risk-neural and real-world represenaions of yield curves over ime. Denoe he nominal and real sochasic discoun facors as M N and M R, respecively. The 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 are wrien as [ M P N (τ) =E P N ] [ +τ M and P R (τ) =E P R ] +τ. M N The no-arbirage condiion requires a consisency beween he prices of nominal and real bonds such ha he price of he consumpion baske, denoed as he overall price level Π, is he raio of he sochasic discoun facors: Π = M R M N. As derived in CLR, he relaionship beween nominal and real zero-coupon yields wih mauriy τ a ime, denoed as y N (τ) and y R (τ), and expeced inflaion is y N (τ) =y R (τ)+π e (τ)+φ (τ), where he marke-implied rae of inflaion expeced a ime for he period from o + τ is π e (τ) = 1 [ ] τ ln Π EP = 1 τ ln EP [e +τ (r N s rr)ds] s, (1) Π +τ M R

6 26 Inernaional Journal of Cenral Banking December 2012 where r N and r R are he insananeous nominal and real risk-free raes. The corresponding inflaion risk premium is denoed as [ φ (τ) = 1 τ ln cov P M R ] +τ Π, M 1+ R Π +τ [ E P M R ] [. (2) +τ E P Π M R Π +τ ] The CLR model is a four-facor version of he arbirage-free Nelson-Siegel (AFNS) represenaion developed by Chrisensen, Diebold, and Rudebusch (CDR 2011). The firs hree facors correspond o he level, slope, and curvaure facors commonly observed for nominal yields and are denoed L N, S, and C, respecively. The fourh facor, L R, corresponds o he level facor for real yields. The sae vecor is hus defined as X =(L N,S,C,L R ). The insananeous nominal and real risk-free raes are se o be r N = L N + S, r R = L R + α R S, where he 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 yield funcions, he risk-neural (or Q-) dynamics of he sae variables are given by he sochasic differenial equaions: 6 dl N ds dc = 0 λ λ λ dl R L N S C L R d +Σ dw LN,Q dw S,Q dw C,Q dw LR,Q. Wihin his framework, nominal Treasury zero-coupon bond yields are denoed as ( ) ( ) 1 e y N (τ) =L N λτ 1 e λτ + S + e λτ C + AN (τ), λτ λτ τ 6 As discussed in CDR (2011), wih a uni roo in he wo level facors, he model is no arbirage free wih an unbounded horizon; herefore, as is ofen done in heoreical discussions, we impose an arbirary maximum horizon.

7 Vol. 8 No. 4 Exracing Deflaion Probabiliy Forecass 27 where A N (τ)/τ is a nominal yield-adjusmen erm. The real TIPS zero-coupon bond yields are ( ) ( ) 1 e y R (τ) =L R + α R λτ 1 e S + α R λτ e λτ C λτ λτ + AR (τ), τ where A R (τ)/τ is a real yield-adjusmen erm. These wo equaions, when combined in sae-space form, consiue he measuremen equaion wihin our Kalman filer esimaion. To complee he model, we define he price of risk, which deermines he connecion beween he risk-neural and real-world yield dynamics. We assume ha he nominal and real sochasic discoun facors have sandard dynamics given by dm N /M N = r N d Γ dw P and dm R /M R = r R d Γ dw P, where W P is a Brownian moion process. We use he essenially affine risk premium specificaion inroduced by Duffee (2002), so he risk premium Γ is defined by he measure change dw Q = dw P +Γ d, wih Γ = γ 0 + γ 1 X,γ 0 R 4, and γ 1 R 4 4. Therefore, he real-world dynamics of he sae variables can be expressed as dx = K P (θ P X )d +ΣdW P. (3) In he unresriced case, boh K P and θ P are allowed o vary freely, bu CLR provide a deailed empirical analysis o jusify various zero-value resricions on he K P marix. Imposing hese resricions resuls in he equaion dl N κ P κ P 14 ds dc = κ P 21 κ P 22 κ P κ P dl R 33 0 κ P 41 κ P 42 0 κ P 44 dw P,LN dw P,S +Σ dw P,C, dw P,LR θ1 P θ2 P θ3 P θ4 P L N S C L R d

8 28 Inernaional Journal of Cenral Banking December 2012 where he covariance marix Σ is assumed diagonal and consan. This is he ransiion equaion in our Kalman filer esimaion. A shorcoming of he CLR model and of all Gaussian erm srucure models is ha hey do no explicily impose a zero lower bound on nominal ineres raes. The lack of his consrain becomes more imporan during periods when shor-erm raes approach zero, as observed in he Japanese governmen bond marke since 2001 and in he U.S. Treasury marke more recenly. 7 For his sudy, we proceed wihou imposing he zero lower bound since mos of our sample daa are no close o zero, and he fied nominal yield curves implied by our esimaion never fell below zero. The laer resul provides suppor for he asserion ha he model s lack of a zero bound should no weaken our empirical resuls. 2.2 Calculaion of Deflaion Probabiliies Using he CLR model, we can examine wheher he change in he price index (i.e., he inflaion rae) from ime o +τ will fall below a cerain criical level q. This even is denoed as Π +τ Π = e +τ (r N s rr s )ds (1 + q). Taking logs, his expression is equivalen o ( ) +τ Π+τ Y,+τ =ln = (rs N rs R )ds ln(1 + q). Π As shown in he appendix, he condiional disribuion of his inegral erm is Y,+τ N ( m P Y (, τ),σ P Y (τ) 2), where m P Y (, τ) and σp Y (τ)2 are he disribuion s condiional mean and variance, respecively, under he real-world probabiliy measure. 8 The probabiliy of he change in he price index being below he criical level q is herefore equivalen o 7 See Kim and Singleon (2012) for a deailed analysis of dynamic erm srucure models ha respec he zero bound. 8 Risk-neural inflaion probabiliies are readily obained by replacing he realworld dynamics of he sae variables wih heir risk-neural dynamics.

9 Vol. 8 No. 4 Exracing Deflaion Probabiliy Forecass 29 Prob (Y,+τ ln(1 + q)) ( Y,+τ m P Y (, τ) = Prob σy P (τ) ln(1 + q) mp Y σy P ( ) (τ) ln(1 + q) m P =Φ Y (, τ) σy P (τ). To assess deflaionary oucomes, q = 0, and ( ) m P Prob (Y,+τ 0)=Φ Y (, τ) σy P (τ). ) (, τ) 2.3 Full-Sample Deflaion Probabiliy Esimaes We sar wih a full-sample examinaion of he model and is fied (or in-sample) deflaion probabiliies. For his esimaion, we use nominal Treasury zero-coupon bond yields wih mauriies of hree and six monhs, and one, wo, hree, five, seven, and en years from January 3, 1995 o December 7, 2010, for a oal of 3,978 daily observaions. We also use real TIPS bond yields wih mauriies of five, six, seven, eigh, nine, and en years from January 4, 1999, o December 7, 2010, for a oal of 2,980 daily observaions. 9 The U.S. Treasury firs issued TIPS in 1997, bu for several years aferward, he liquidiy of he secondary TIPS marke was impaired by he small amoun of securiies ousanding and uncerainy abou he Treasury s commimen o he program. 10 Indeed, o avoid he illiquid nascen years of his marke, CLR began heir esimaion sample of TIPS yields in Here, in order o provide insighs on deflaion concerns early in he 2000s, we sar our sample of TIPS daa a his earlier dae, which does no significanly affec our resuls in he laer period. 9 The daa are described in Gürkaynak, Sack, and Wrigh (2007, 2010) and are available from he Board of Governors of he Federal Reserve web sie. Nominal yields are consruced from off-he-run Treasury bonds, while real yields are based on boh on-he-run and off-he-run TIPS bonds. 10 See Roush (2008) as well as D Amico, Kim, and Wei (2008) for furher deails regarding TIPS liquidiy during his period.

10 30 Inernaional Journal of Cenral Banking December 2012 Table 1. Parameer Esimaes for he Preferred Specificaion of he CLR Model K P K P,1 K P,2 K P,3 K P,4 θ P Σ K1, P Σ 1, (0.2539) (0.2157) (0.0039) (0.0001) K2, P Σ 2, (0.3873) (0.1561) (0.1055) (0.0105) (0.0001) K3, P Σ 3, (0.2584) (0.0094) (0.0002) K4, P Σ 4, (0.4303) (0.1213) (0.3198) (0.0047) (0.0001) Noes: The esimaed parameers of he K P marix, θ P vecor, and diagonal Σ marix are shown for he preferred specificaion of he CLR model. The esimaed value of λ is (0.0015), while α R is esimaed o be (0.0017). The numbers in parenheses are he esimaed parameer sandard deviaions. The maximum log-likelihood value is 298, Table 1 presens he esimaed parameers of he preferred CLR model specificaion over he full sample. The esimaes are comparable o hose given in CLR for a shorer sample of weekly daa. In paricular, he off-diagonal elemens in he esimaed K P marix are highly saisically significan excep for κ P 14, which has seen is significance decline since he onse of he financial crisis. However, a robusness check indicaes ha his parameer has a negligible effec on he fied, in-sample deflaion probabiliies; herefore, we proceed wih he preferred CLR specificaion hroughou he paper. Figure 1 shows our full-sample esimaes of he one-year probabiliy of deflaion (i.e., q = 0 and τ equals one year). Using a rough benchmark of a probabiliy greaer han 5 percen, wo deflaion scare episodes sand ou. Ouside hese wo periods of elevaed deflaion risk, he esimaed deflaion probabiliy is effecively zero. Figure 2 pus he wo deflaion scares ino sharper focus along wih grey shading o indicae recessions. Boh episodes were preceded by recessions, bu in each case, he risk of deflaion persised long afer he recessions ended. The firs episode spans Sepember 2001 hrough December During his period, as noed in he inroducion, he FOMC expressed concern for he firs ime ha

11 Vol. 8 No. 4 Exracing Deflaion Probabiliy Forecass 31 Figure 1. In-Sample One-Year Deflaion Probabiliies Probabiliy Noe: Illusraion of he probabiliy of non-posiive ne inflaion (or deflaion) over he forhcoming year as esimaed by he CLR model over he full-sample period. inflaion migh fall oo low. The one-year marke-implied deflaion probabiliy averaged 3 percen over his period, reaching highs of over 25 percen in Augus 2002 and Augus During his wenyeigh-monh period, negaive monh-o-monh values for headline CPI inflaion were recorded six imes. The second deflaion scare episode begins shorly afer he Lehman Brohers bankrupcy on Sepember 15, 2008 and runs hrough April This episode is marked by a sharp spike in he fied, one-year deflaion probabiliies up o near cerainy in lae Ocober and early November In response o concerns of a very severe and rapid economic collapse, he Federal Reserve enaced a variey of convenional and unconvenional moneary and liquidiy policy acions, which likely helped reduce he probabiliies in he firs quarer of 2009 o an average of 4.3 percen, wih emporary spikes up o 15 percen. The model s one-year-ahead deflaion probabiliies averaged 5.3 percen over he course of The probabiliies averaged 2.6 percen for

12 32 Inernaional Journal of Cenral Banking December 2012 Figure 2. In-Sample Deflaion Probabiliies over he Defined Deflaion Scares A. Firs Deflaion Scare Probabiliy B. Second Deflaion Scare Probabiliy Lehman Brohers Bankrupcy Sep. 15, Noes: Illusraion of he fied, in-sample one-year deflaion probabiliies from he preferred specificaion of he CLR model over he defined deflaion scares (i.e., probabiliy >5 percen). Shown in grey shading are recessions as deermined by he NBER.

13 Vol. 8 No. 4 Exracing Deflaion Probabiliy Forecass 33 he daa se in 2010 and dipped below 5 percen in April. Over his weny-monh period, seven monhs regisered negaive monh-omonh values for headline CPI, wih annualized values of more han 10 percen in Ocober and November The very high deflaion probabiliies immediaely afer he Lehman bankrupcy cerainly reflec he widespread fear of a resuling global macroeconomic free fall, bu hey are also likely boosed by marke illiquidiy during he financial crisis. Several asse classes faced impaired liquidiy during he fall of 2008 wih widening bid-ask spreads, lower rading volumes, and concurren increases in yields. The jump in risk aversion also helped creae a heighened global demand for safe asses, and his fligh-o-qualiy (or safe haven ) demand favored highly liquid nominal Treasury securiies and led o a sharp decline in heir yields, while real yields declined by less. Liquidiy in he TIPS marke was especially hard hi (see Campbell, Shiller, and Viceira 2009 and CLR for deailed discussions), which led o higher real yields and a narrowing of spreads o nominal Treasuries. Chrisensen and Gillan (2012) found ha he TIPS liquidiy premia rose sharply in Sepember 2008 before reurning o heir prior levels by mid Thus, he CLR model s deflaion probabiliies were cerainly boosed due o hese liquidiy evens and he model assumpions of he absence of arbirage and normal marke funcioning. Furher research is needed o refine he model s forecass from his challenging period. 2.4 Real-Time Deflaion Probabiliy Forecass In order o generae deflaion forecas probabiliies ha would be relevan o marke paricipans and policymakers in real ime, we complemen our fied, in-sample esimaes wih deflaion probabiliies based on expanding-sample model esimaions. In paricular, we reesimae he preferred specificaion of he model using weekly daa wih end-of-sample daes from January 7, 2005 o December 3, 2010, a oal of 309 esimaions. 11 We sar his analysis in Tha is, saring wih he firs week of January 2005, each new weekly observaion is included in he sample, and he model is reesimaed. Noe ha he shif o weekly daa from daily daa had lile effec on our analysis bu speeded he reesimaion process significanly.

14 34 Inernaional Journal of Cenral Banking December 2012 Figure 3. Real-Time, One-Year Probabiliy Forecass of Deflaion Full sample esimae Real ime esimae Probabiliy Lehman Brohers Bankrupcy Noes: Illusraion of he real-ime, ou-of-sample probabiliies of non-posiive ne inflaion (i.e., deflaion) over he following year as esimaed by he CLR model. Included are he corresponding esimaes from he full-sample esimaion. in order o provide a minimum sample size for accurae esimaion and hus canno examine he earlier deflaion episode in his manner. Figure 3 shows ha inference based on he wo ses of defaul probabiliies does no change qualiaively. To provide furher conex for hese deflaion probabiliy forecass, figure 4 examines he abiliy of he model s poin esimaes of one-year-ahead expeced inflaion o rack acually observed inflaion raes. In figure 4A, we graph hese expecaions relaive o he corresponding year-overyear changes in headline CPI, which is he index used for TIPS bonds. Noe ha he model does well a predicing deflaion in 2009, as headline CPI in May 2009 was 2.1 percen below is level he year before. However, he model does no capure he volaile movemens in headline CPI known o be caused by flucuaions in food and energy prices, which are widely considered o be difficul

15 Vol. 8 No. 4 Exracing Deflaion Probabiliy Forecass 35 Figure 4. Real-Time, One-Year Inflaion Forecass A. Comparison wih Headline CPI Inflaion Rae in percen Full sample esimae Real ime esimae Change in headline CPI B. Comparison wih Core CPI Inflaion Rae in percen Full sample esimae Real ime esimae Change in core CPI Noes: Illusraion of he real-ime esimaes of he one-year inflaion expecaions from he CLR model. Included are he corresponding esimaes from he full-sample esimaion as well as he subsequen year-over-year change in headline and core CPI.

16 36 Inernaional Journal of Cenral Banking December 2012 Figure 5. Comparison of Year-over-Year Core CPI Inflaion and Corresponding Forecass Rae in percen CLR (2010) model SPF one year core CPI inflaion forecas Year over year change in core CPI Noe: See he main ex for he descripion of he various inflaion forecass. o forecas. 12 In figure 4B, we compare he model s one-year inflaion expecaions o he observed year-over-year changes in he less volaile core CPI series. The model racks hese changes remarkably well using boh esimaion samples. This resul suggess ha bond marke paricipans appear o focus heir longer-erm inflaionforecasing effors on his measure of underlying inflaion. Macroeconomic forecasers in he Survey of Professional Forecasers also seem o concenrae on he core inflaion series for heir longer-erm inflaion forecass, as shown in figure 5. Thus, he CLR model and is deflaion probabiliy forecass should be useful ools for real-ime macroeconomic policy or hisorical analysis a longer horizons. 12 Hamilon (2009) repors ha while a random-walk poin forecas for oil prices is reasonable a he one-year horizon, is sandard deviaion is quie large. In his example, he $115 oil price in 2008:Q1 corresponds o a range from $62 o $212 for he forecas confidence band of wo sandard deviaions one year ahead.

17 Vol. 8 No. 4 Exracing Deflaion Probabiliy Forecass 37 Table 2. Summary Saisics for Inflaion-Forecas Errors Full-Sample Analysis, One-Year Forecas Two-Year Forecas Headline CPI Mean RMSE Mean RMSE Random Walk CLR Model One-Year Forecas Two-Year Forecas Core CPI Mean RMSE Mean RMSE Random Walk CLR Model Real-Time Analysis, One-Year Forecas Two-Year Forecas Headline CPI Mean RMSE Mean RMSE Random Walk CLR Model One-Year Forecas Two-Year Forecas Core CPI Mean RMSE Mean RMSE Random Walk CLR Model Noes: The summary saisics are for he forecas errors from he random-walk and he CLR models in forecasing boh headline and core CPI inflaion one and wo years ahead. The op panel is based on he full-sample esimae generaing 131 and 119 monhly forecas errors from he end of January 1999 unil he end of November 2009 and November 2008, respecively. The boom panel is based on he real-ime analysis, which generaes 59 and 47 monhly forecas errors from he end of January 2005 unil he end of November 2009 and November 2008, respecively. All numbers are measured in basis poins. Table 2 repors summary saisics for he model s forecass of boh headline and core CPI one and wo years ahead from boh he full-sample and he real-ime analysis and compares is performance wih ha of he random walk. In general, under he roo-meansquared-error loss funcion, he model is able o bea he random

18 38 Inernaional Journal of Cenral Banking December 2012 walk a forecasing headline CPI and is no ha far behind when i comes o forecasing changes in he more sable core CPI. During his sample period, he model generally underesimaed realized inflaion. Bond invesors likely underesimaed he persisen increases in energy prices from 2003 o mid In addiion, a imes, TIPS yields were likely arificially elevaed due o liquidiy premiums, which would ranslae ino low break-even inflaion raes and, presumably, correspondingly low model inflaion forecass. 3. Forecass from Macroeconomic Models and Surveys In his secion, we compare he deflaion probabiliy forecass from he CLR yields-only model wih hose obained from simple macroeconomic models and from professional forecasers. One source for deflaion probabiliies is he quarerly Survey of Professional Forecasers (SPF), which provides poin forecass for CPI inflaion over he nex four quarers. These forecass can be used o generae implied SPF deflaion probabiliy forecass based on he disribuion of pas SPF forecas errors. Specifically, he SPF probabiliy forecas in a paricular quarer assumes a normal disribuion of oucomes around he poin forecas wih a variance equal o ha of SPF CPI forecas errors from 1981:Q1 he firs quarer of SPF CPI forecass up o ha paricular quarer. Rudebusch and Williams (2009) apply a similar mehodology o obain implied SPF recession probabiliy forecass and show ha hey correspond quie closely wih he subjecive recession probabiliies ha are repored in he SPF. 13 The SPF inflaion forecass and implied deflaion probabiliy forecass are shown in figures 6 and 7, respecively. I is also of ineres o compare our yields-only deflaion probabiliies wih hose generaed from models of inflaion. From he vas 13 The SPF has long asked paricipans o repor inflaion forecas probabiliy disribuions; unforunaely, his direc subjecive assessmen refers o gross domesic produc (GDP) price inflaion on a calendar-year average over calendaryear average basis raher han he one-year-ahead CPI percen change relevan for he TIPS-based probabiliies. Since 2007, he SPF has also repored subjecive probabiliies for core CPI inflaion on a calendar-year basis. To he exen hey are comparable across heir differing price indexes, he repored SPF deflaion probabiliies appear o be consisen wih our model-implied deflaion probabiliies.

19 Vol. 8 No. 4 Exracing Deflaion Probabiliy Forecass 39 Figure 6. CPI and Core CPI Inflaion Forecass Rae in percen SPF (CPI) Random walk (CPI) Phillips curve (core CPI) CLR model Noe: See he main ex for he descripion of he various inflaion forecass. macroeconomic lieraure, we consider wo simple real-ime, macroeconomic benchmarks using daa and analysis ha arguably were available o forecasers conemporaneously ha is, in he early o mid-2000s. The firs model employs a random-walk forecas of he kind recommended by Akeson and Ohanian (2001); ha is, a each poin in ime, inflaion over he nex year is projeced o be he same as i was over he pas year. The second model is a simple Phillips curve along he lines considered by Sock and Wason (1999), as well as Rudebusch and Svensson (1999) in which inflaion depends on lagged inflaion and inversely on he degree of slack. For his purpose, we simply regress he one-quarer-ahead core CPI inflaion on four lags of iself and one lag of he unemploymen rae over he sample 1984:Q1 o 1999:Q4. 14 Each quarer from 2000 hrough 2010, his equaion is ieraed ahead (using SPF real-ime forecass for 14 Given he shor available samples, simple Phillips curves do no yield saisfacory esimaes wih headline CPI. Williams (2009) esimaes a similar model using he core PCE price index and ges broadly similar probabiliies.

20 40 Inernaional Journal of Cenral Banking December 2012 Figure 7. Probabiliy Forecass of CPI and Core CPI Inflaion Probabiliy SPF (CPI) Random walk (CPI) Phillips curve (core CPI) CLR model Noe: See he main ex for he descripion of he various deflaion probabiliy forecass. unemploymen) o produce four-quarer-ahead inflaion forecass. Probabiliy forecass from hese models also require a disribuion of likely oucomes. Afer he mid-1980s, U.S. oupu growh and inflaion exhibied much less volailiy han before, as deailed by Sock and Wason (2007). Therefore, like our SPF implici forecas disribuion, we assume a normal disribuion of oucomes around he poin forecas wih a variance equal o ha of he Phillipscurve CPI forecas errors afer The poin inflaion forecass and resuling deflaion probabiliy forecass are shown in figures 6 and 7, respecively. Figures 6 and 7 also show he CLR model s inflaion forecass and deflaion probabiliies for comparison. During he second half of he sample, he one-year-ahead CLR inflaion forecass are generally consisen wih he macroeconomic models and survey resuls. The divergence seen in he firs few years of he sample can mos likely be aribued o he TIPS liquidiy issues discussed earlier, which raised TIPS yields and reduced inflaion forecass. The deflaion

21 Vol. 8 No. 4 Exracing Deflaion Probabiliy Forecass 41 probabiliies in figure 7 are also generally consisen wih each oher in ha wo deflaionary episodes are idenified following he 2001 recession and he bankrupcy of Lehman Brohers in Sepember During oher periods, all of he deflaion probabiliies are close o zero. During 2010, he various deflaion probabiliy forecass have he larges differences, which is consisen wih he recen heighened uncerainy abou he fuure direcion of inflaion (e.g., Leduc, Rudebusch, and Weidner 2009). In summary, he deflaionary episodes idenified wih he CLR model have rough parallels wih hose from macroeconomic models and survey forecass, supporing he suggesion ha he yields-only approach encompasses macroeconomic daa quie well, even during he recen crisis. 4. Alernaive Forecass from Financial Marke Daa Alernaive yields-only approaches for generaing deflaion probabiliies have been examined in a few recen sudies. Wih respec o U.S. daa, Sack (2000) showed how o generae a proxy for inflaion expecaions based on TIPS yields and a similar porfolio of Treasury STRIPS. The auhor assumed ha risk-neural invesors arbirage away differences beween he paymen srucure of a real TIPS securiy (i.e., boh coupons and principal) and a porfolio of nominal Treasury STRIPS (separae rading of regisered ineres and principal securiies) ha has he same paymen schedule. The marke-implied inflaion compensaion measure is he consan rae of inflaion over he mauriy of he porfolio ha equaes he prices of he TIPS securiy and he maching STRIPS porfolio. Higgins (2010) exended his analysis by assuming ha invesors believe ha he inflaion process has a Gaussian disribuion wih a consan mean and variance. Once he model s parameers are esimaed, he probabiliy of deflaion over he mauriy of he invesmen, which was five years in his analysis, can be generaed as he inegral over he appropriae inerval of he inflaion process. Wrigh (2009) proposed an alernaive, risk-neural yields-only approach ha is based on comparing he yields on a pair of TIPS securiies ha have comparable mauriy daes bu differen issuance daes and hus differen reference CPI raes; see secion 5 for furher discussion of specific TIPS bond pairs. The inuiion here is

22 42 Inernaional Journal of Cenral Banking December 2012 Figure 8. Five-Year-Ahead Deflaion Probabiliy Forecass Probabiliy Model based real world Model based risk neural Higgins (2010) esimae Wrigh (2009) lower bound Lehman Brohers Bankrupcy Noes: Illusraion of he real-ime probabiliy forecass of non-posiive ne inflaion (i.e., deflaion) under boh he real-world (or P) probabiliy measure and he risk-neural (or Q) pricing measure over he following five years as esimaed by he CLR model. Included are wo alernaive measures. The one following Higgins (2010) is only available since April 27, The oher is our applicaion of he lower-bound calculaion described in Wrigh (2009) o he pairs of comparable TIPS analyzed in secion 5. o ake advanage of he marke pricing of he deflaion proecion opions wih differen srike prices ha are embedded in he wo TIPS bonds; i.e., he conracual feaure ha insures ha he principal repaymen canno be less han he face value of he bond. Based on he proposed Wrigh (2009) approach, a lower bound on he implied, risk-neural deflaion probabiliy can be calculaed over he period up o he mauriy of he bonds. Figure 8 shows our full-sample and real-ime five-year deflaion probabiliies under he risk-neural pricing measure. These deflaion probabiliies follow he same qualiaive paern as he one-year, real-world probabiliies presened in figures 1 and 2; ha is, hey rise sharply afer he Lehman bankrupcy in Sepember 2008 and hen decline in ligh of he various policy acions aken by he Federal Reserve and oher cenral banks over he course of he financial crisis.

23 Vol. 8 No. 4 Exracing Deflaion Probabiliy Forecass 43 However, he risk-neural probabiliies are generally more persisen, and heir decline hrough 2009 and 2010 is slower han ha ypically exhibied by real-world probabiliies. The differences beween he wo ses of deflaion probabiliies, especially during he financial crisis, underscores he usefulness of a dynamic erm srucure model ha allows us o generae and analyze boh of hem. In comparing our esimaes wih he exising alernaives described previously, we noe ha, wih few excepions, our realime risk-neural esimaes are above he Wrigh lower bound, bu ypically close o i. 15 On he oher hand, our esimaes are well below hose repored by Higgins (2010). Based on hese resuls, we conclude ha he CLR model produces very reasonable esimaes of he deflaion probabiliy under he risk-neural pricing measure, which is in line wih he model s abiliy o fi he cross-secions of nominal and real yields very well. As discussed in secion 2.3, all yields-only approaches are vulnerable o changes in liquidiy premiums in he Treasury markes ha could disor he marke pricing mechanism. In his regard, he Wrigh procedure has he advanage of no depending on differenial liquidiy premiums across he nominal and real Treasury markes, alhough liquidiy issues may remain across differen TIPS bonds. Alhough hese disorions direcly affec he model-implied deflaion probabiliies, we are comforable wih he CLR model s abiliy o smooh hrough he recen excess volailiy and capure he underlying rends in he daa. The mos imporan shorcoming of he alernaive yields-only approaches is ha hey only provide deflaion probabiliies under he risk-neural pricing measure and no under he real-world pricing measure. Thus, hese deflaion probabiliies are no comparable o hose generaed from macroeconomic sources. In conras, he CLR modeling srucure can generae real-world probabiliies ha can be used for macroeconomic policy analysis Mauriy mismaches likely accoun for mos violaions since our esimaes have consan five-year mauriies, while he Wrigh measure varies wih he mauriy of he underlying TIPS bonds. 16 The alernaive approaches also generae only a single deflaion probabiliy a he forecas horizon deermined by he mauriy of he bonds under analysis. In conras, he CLR model s use of he enire yield curve allows us o generae deflaion probabiliies for all horizons of ineres.

24 44 Inernaional Journal of Cenral Banking December 2012 Aside from TIPS bonds, derivaive conracs could be a separae source of financial marke daa from which o generae deflaion probabiliies. Noably, since i provides is purchaser wih proecion when realized inflaion exceeds a specified hreshold, an inflaion swap could provide a reading on he marke-based deflaion probabiliy on he day he conrac was sruck. 17 Since conracs are creaed every day a various possible mauriies, a ime series of deflaion probabiliies a various mauriies could be generaed. However, rading volume on such swaps conracs remains limied in he Unied Saes Pricing Inflaion Floors wih Deflaion Probabiliies TIPS coupons and principal have an asymmerical indexaion o he observed headline CPI changes. The indexaion in ligh of accumulaed inflaion leads o increases in he coupon and principal paymens accordingly. 19 If a mauriy he indexed principal paymen is less han he par amoun a issuance due o accumulaed deflaion, he paymen is increased back o is par value. This embedded deflaion floor proecs he invesor from declines in he price level. Under normal inflaionary circumsances, he opion value of he TIPS deflaion floor is negligible since he probabiliy of having negaive accrued inflaion compensaion a mauriy is very small. 17 See Hinnerich (2008) for a discussion of he pricing of various forms of inflaion-indexed derivaives. Haubrich, Pennacchi, and Richken (2012) use monhly inflaion swap daa in heir join model of nominal and real Treasury raes. Chrisensen and Gillan (2012) use his daa o gauge TIPS marke liquidiy. 18 In addiion, oher inflaion derivaives, such as inflaion caps or floors ha provide binary paymens when he observed inflaion rae is above or below he conraced inflaion rae, are said o be available and acively raded. Given exising opion pricing models, inflaion-indexed derivaives could be used o exrac marke-based deflaion probabiliies. However, aside from marke liquidiy issues, an imporan shorcoming of such an approach is ha he probabiliies would again be solely based on he risk-neural pricing measures. 19 As described in Gürkaynak, Sack, and Wrigh (2010), he reference CPI values used in he adjusmen have an indexaion lag since he Bureau of Labor Saisics publishes price index values wih a one-monh lag; i.e., he index for a given monh is released in he middle of he subsequen monh. The reference CPI is hus se o be a weighed average of he CPI for he second and hird monhs prior o he monh of mauriy.

25 Vol. 8 No. 4 Exracing Deflaion Probabiliy Forecass 45 However, a he peak of he financial crisis in lae 2008, neiher he perceived nor he priced probabiliies of deflaion were negligible. Under hese circumsances, a wedge developed beween he prices of seasoned TIPS bonds wih a significan amoun of accrued inflaion compensaion and recenly issued TIPS bonds ha had no accumulaed inflaion compensaion and herefore were a he deflaion floor upon issuance. Grishchenko, Vanden, and Zhang (2011) used a monhly wo-facor AF model o compue he value of he deflaion proecion opion during he recen financial crisis and showed ha is value peaked in mid In his secion, we use he coningen claim pricing derived by Duffie, Pan, and Singleon (2000) wihin he AF modeling framework described above o value his deflaion proecion opion. This exercise provides a relevan applicaion of our deflaion probabiliies and, as we shall see, an independen check on he overall fi of he model. We calculae he deflaion proecion opion value by comparing under he risk-neural pricing measure he prices of a newly issued TIPS bond wihou any accrued inflaion compensaion and a seasoned TIPS bond wih sufficien accrued inflaion compensaion. Firs, consider a hypoheical seasoned TIPS bond wih T years remaining o mauriy ha pays an annual coupon C semi-annually. Assume his bond has accrued sufficien inflaion compensaion so i is impossible o reach he deflaion floor before mauriy. The parcoupon bond saisfying hese crieria has a coupon rae deermined by he equaion 2T i=1 C 2 EQ [e i r R s ds ]+E Q [e T rr s ds ]=1. (4) The firs erm is he sum of he presen value of he 2T coupon paymens using he model s fied real yield curve a day. The second erm is he discouned value of he principal paymen. The coupon paymen for his seasonal bond ha solves his equaion is denoed as C S. Nex, consider a new TIPS bond wih no accrued inflaion compensaion wih T years o mauriy. Since he coupon paymens are no proeced agains deflaion, he difference is in accouning for he deflaion proecion on he principal paymen:

26 46 Inernaional Journal of Cenral Banking December T i=1 C 2 EQ [e i r R s ds ]+E Q + E Q [ 1 e T rn s ds 1 Π { T Π 1} [ ] ΠT e T rn s ds 1 Π Π { T Π >1} ] =1. The firs erm is he same as before. The second erm represens he presen value of he principal paymen condiional on a posiive ne change in he price index over he bond s mauriy; i.e., Π T Π > 1. Under his condiion, full inflaion indexaion applies, and he price change Π T Π is placed wihin he expecaions operaor and weighed by he probabiliy of accumulaed inflaion a ime T. The hird erm represens he presen value of he floored TIPS principal condiional on accumulaed ne deflaion; i.e., when he price level change is below one, Π T Π is replaced by a value of one o provide he promised deflaion proecion. Since Π T Π = e T (rn s rr s )ds, he equaion can be rewrien as 2T i=1 C 2 EQ [e [ E Q + i r R s ds ]+E Q [e T [ ] e T rn s ds 1 Π { T E Q Π 1} rr s ds] [ ]] e T rr s ds 1 Π { T =1, Π 1} where he las erm on he lef-hand side represens he ne presen value of he deflaion proecion of he principal in he TIPS conrac. 20 The par-coupon yield of a new hypoheical TIPS bond ha solves his equaion is denoed as C 0. The difference beween C S and C 0 is a measure of he advanage of being a he inflaion adjusmen floor for a newly issued TIPS bond. The black line in figure 9 shows he difference beween he C S 20 The appendix explains how hese coningen condiional expecaions are calculaed wihin he CLR model using he coningen claim pricing resuls of Duffie, Pan, and Singleon (2000).

27 Vol. 8 No. 4 Exracing Deflaion Probabiliy Forecass 47 Figure 9. Value of he Deflaion Proecion Embedded in TIPS Yield spread in basis poins Real ime model based yield spread Yield spread beween seasoned and new TIPS Bear Searns Collapse Lehman Brohers Bankrupcy Noes: Illusraion of he model-implied five-year par-coupon yield spread of a seasoned TIPS over a comparable newly issued TIPS. Included are he spread in he yield o mauriy as repored by Bloomberg beween he on-he-run pairs of seasoned and newly issued TIPS. and C 0 values ha solve he pricing equaions a he five-year mauriy using our real-ime model esimaes. Prior o he Lehman bankrupcy, he differences beween he wo synheic TIPS bond yields were quie near zero, bu slighly negaive. However, he yield differences jumped during he financial crisis, which is consisen wih our model-implied deflaion probabiliies shown in figure 3. Even by he end of our sample in December 2010, he spreads remained above 5 basis poins, suggesing ha deflaion proecion was sill of some value despie he fac ha he model-implied deflaion probabiliies had reached insignifican levels monhs before This spread suggess ha seasoned and newly issued TIPS bonds should no be pooled o consruc real yield curves, unless he prices of he recenly issued TIPS are correced for he value of he deflaion proecion.

28 48 Inernaional Journal of Cenral Banking December 2012 Figure 9 also compares our model-based esimaes wih he observed yield differences beween pairs of comparable seasoned and recenly issued TIPS bonds. The solid grey line represens he yield difference beween a seasoned en-year TIPS bond wih approximaely five years remaining o mauriy and he mos recenly issued five-year TIPS bond. 22 Since hese pairs of TIPS bonds have similar remaining paymen schedules and liquidiy, heir yield difference should be primarily due o he value of he embedded deflaion proecion opion. Our real-ime model esimaes rack he observed TIPS yield spread remarkably well, especially since he individual TIPS spreads are no used direcly in he model esimaion. These resuls provide furher suppor for he model s underlying deflaion probabiliy forecass Conclusion The possibiliy of deflaion has been an imporan risk facor for he Federal Reserve over he pas decade and, in ligh of he curren low inflaion environmen, is likely o coninue o be so going forward. In his paper, we use a yields-only dynamic erm srucure model developed by Chrisensen, Lopez, and Rudebusch (2010) o generae inflaion expecaions and corresponding deflaion probabiliy forecass ha could be used direcly for macroeconomic policy analysis and asse pricing purposes. A key advanage of he model is ha i can be updaed daily wih jus Treasury bond yields and used for real-ime analysis, unlike macroeconomic analysis based 22 From January 3, 2005 o April 24, 2006, we use he five-year TIPS ha maured in April 2010 and he en-year TIPS ha maured in January From April 25, 2006 o April 23, 2007, we use he five-year TIPS wih mauriy in April 2011 and he en-year TIPS wih mauriy in January From April 24, 2007 o April 22, 2008, we use he five-year TIPS wih mauriy in April 2012 and he en-year TIPS wih mauriy in July From April 23, 2008 o April 22, 2009, we use he five-year TIPS wih mauriy in April 2013 and he en-year TIPS wih mauriy in July From April 23, 2009 o April 23, 2010, we use he five-year TIPS wih mauriy in April 2014 and he en-year TIPS wih mauriy in July Since April 26, 2010, we use he five-year TIPS wih mauriy in April 2015 and he en-year TIPS wih mauriy in July The model-based yield spreads are lower han he observed spreads. Incorporaing sochasic volailiy ino he model, as per Chrisensen, Lopez, and Rudebusch (2012), appears o improve he pricing of he deflaion opion.

29 Vol. 8 No. 4 Exracing Deflaion Probabiliy Forecass 49 on lower-frequency daa. The model s deflaion probabiliies, boh under he risk-neural and real-world pricing measures, are shown o correspond well wih forecass from macroeconomic models, survey daa, and oher yields-only approaches in he lieraure. Finally, he model s abiliy o capure flucuaions of he value of he TIPS embedded deflaion proecion opion provides furher model validaion and an example of is usefulness. Appendix The Probabiliy of Deflaion in he CLR Model The probabiliy ha he change in he price index is below a cerain criical level q equals he probabiliy of he saes of he world where Π +τ 1+q. Π This is equivalen o requiring +τ (r N s r R s )ds ln(1 + q). Therefore, we are ineresed in he disribuional properies of he following process: Y 0, = 0 (r N s r R s )ds = dy 0, =(L N 0 (L N s + S s L R s α R S s )ds +(1 α R )S L R )d. In general, he real-world P -dynamics of he sae variables X are given by dx = K P (θ P X )d +ΣdW P. Adding he Y 0, process o his sysem leaves us wih a five-facor sochasic differenial equaion (SDE), dl N κ P 11 κ P 12 κ P 13 κ P 14 0 θ ds dc dl R = κ P 21 κ P 22 κ P 23 κ P 1 P 24 0 θ P κ P 31 κ P 32 κ P 33 κ P θ P κ P 41 κ P 42 κ P 43 κ P θ4 P d dy 0,