The TIPS Liquidity Premium

Transcription

1 The TIPS Liquidiy Premium Marin M. Andreasen Jens H. E. Chrisensen Kevin Cook Simon Riddell Absrac We inroduce an arbirage-free dynamic erm srucure model of nominal and real yields wih a liquidiy risk facor o accoun for he liquidiy disadvanage of Treasury inflaionproeced securiies(tips) relaive o Treasury securiies. The idenificaion of he liquidiy facor comes from is unique loading, which mimics he idea ha, over ime, increasing amouns of ousanding securiies ge locked up in buy-and-hold invesors porfolios. The model is esimaed using prices for individual TIPS combined wih a sandard sample of nominal Treasury yields and delivers liquidiy premium esimaes for each TIPS. We find ha TIPS liquidiy premiums have averaged 38 basis poins wih noable ime variaion. Furhermore, accouning for liquidiy risk improves he model s abiliy o forecas inflaion. JEL Classificaion: E43, E47, G12, G13 Keywords: erm srucure modeling, liquidiy risk, financial marke fricions We hank paricipans a he 9h Annual SoFiE Conference 2016 for helpful commens. We also hank seminar paricipans a he Naional Bank of Belgium, he Deb Managemen Office of he U.S. Treasury Deparmen, he Federal Reserve Board, and he Office of Financial Research for helpful commens. Furhermore, we are graeful o Jose Lopez for helpful commens and suggesions on an early draf of he paper. 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. Deparmen of Economics, Aarhus Universiy, Denmark; mandreasen@econ.au.dk. Corresponding auhor: Federal Reserve Bank of San Francisco, 101 Marke Sree MS 1130, San Francisco, CA 94105, USA; phone: ; jens.chrisensen@sf.frb.org. Federal Reserve Bank of San Francisco; kevin.cook@sf.frb.org. simonruw@gmail.com. This version: June 20, 2016.

2 1 Inroducion In 1997, he U.S. Treasury sared issuing inflaion-indexed bonds, which are now commonly known as Treasury inflaion-proeced securiies (TIPS). Since hen he U.S. Treasury has seadily expanded he marke for TIPS, which by he end of 2013 had a oal ousanding amoun of $973 billion or 8.2 percen of all markeable Treasury securiies. 1 Despie he apparen large size of he marke for TIPS, here is an overwhelming amoun of research suggesing ha TIPS are less liquid han regular Treasury securiies. Fleming and Krishnan (2012) repor marke characerisics of TIPS ha indicae smaller rading volume, longer urnaround ime, and wider bid-ask spreads han are normally observed in he nominal Treasury bond marke (see also Campbell e al. 2009, Dudley e al. 2009, Gürkaynak e al. 2010, and Sack and Elsasser 2004). Moreover, here is evidence ha TIPS yields are elevaed for hese reasons as invesors require a premium for assuming he associaed illiquidiy risk (see Fleckensein e al for a deailed discussion). However, he degree o which hese fricions bias TIPS yields remains a opic of debae as no consensus has emerged on esimaes of he TIPS liquidiy premium. 2 In his paper, we inroduce an arbirage-free dynamic erm srucure model of nominal and real yields wih a laen liquidiy risk facor o accoun for he poenial liquidiy disadvanage of TIPSrelaive otreasurysecuriies. Themodelis anexension of hemodelof nominal and real yields inroduced in Chrisensen e al. (2010, henceforh CLR), referred o hroughou as he CLR model. The idenificaion of he liquidiy facor comes from is unique loading for each TIPS ha is supposed o mimic he noion ha, over ime, an increasing amoun of is ousanding noional ges locked up in buy-and-hold invesors porfolios. Thanks o invesors forward-looking behavior, his affecs is sensiiviy over ime o variaion in he marke-wide liquidiy as capured by he liquidiy facor. By observing a cross secion of TIPS prices over ime, he liquidiy facor can be separaely idenified. We esimae he CLR model and is exension using price informaion for individual TIPS rading from mid-1997 hrough he end of 2013 combined wih a sandard sample of nominal Treasury yields from Gürkaynak e al. (2007). To ge a clean read of he TIPS liquidiy facor, we accoun explicily in he model esimaion for he ime-varying value of he deflaion proecion opion embedded in he TIPS conrac using pricing formulas provided in Chrisensen e al. (2012). In addiion o delivering unique liquidiy premium esimaes for each TIPS, we find ha he model ha accouns for he TIPS liquidiy premium ouperforms he model ha does no when i comes o forecasing CPI inflaion. To have more accurae and less biased esimaes of bond invesor s inflaion expecaions embedded in nominal and real yields is crucial for porfolio risk managemen and moneary policy analysis. 1 The daa is available a: hp:// 2 Pflueger and Viceira (2013), D Amico e al. (2014), and Abrahams e al. (2015) are among he sudies ha esimae TIPS liquidiy premiums. 1

3 In erms of he exising lieraure, Chrisensen e al. (2016) and Grishchenko e al. (2016) sudy he pricing of he TIPS deflaion proecion opion, while Pflueger and Viceira (2013), D Amico e al. (2014), and Abrahams e al. (2015) are among he sudies ha aemp o accoun for he TIPS liquidiy premium. We do boh. Thus, o he bes of our knowledge, his is he firs paper o accoun simulaneously for he liquidiy premiums and he embedded deflaion proecion opion values in TIPS prices wihin an arbirage-free model of nominal and real yields. We make a horough comparison o his exising lieraure. Since we obain improvemen in one-year CPI inflaion forecass from accouning for TIPS liquidiy premiums and achieve addiional, bu smaller improvemens from aking he deflaion proecion opion values ino accoun, we are also he firs o documen he relaive imporance of accouning for boh of hese aspecs in he pricing of TIPS. The main finding of he empirical analysis, hough, is ha TIPS liquidiy premiums over he enire sample average 38 basis poins, which is lower han he resuls repored in exising sudies of TIPS liquidiy premiums. Furhermore, we documen on-he-run TIPS liquidiy premiums ha are slighly lower, averaging 33 basis poins and 30 basis poins a he fiveand en-year mauriy, respecively, and we show ha i is an order of magniude larger han he off-he-run liquidiy premium in he Treasury bond marke. 3 We noe ha our resuls could have implicaions for he managemen of he U.S. Treasury deb. However, o evaluae he benefi o he U.S. Treasury of coninuing is TIPS issuance, requires a comprehensive assessmen of he sign and magniude of he inflaion risk premium ha represens he gain from issuing TIPS relaive o he liquidiy disadvanage we documen (see Chrisensen and Gillan 2012 for a discussion and analysis). Thus, we cauion agains drawing policy conclusions from our findings wihou furher analysis. Finally, we sress ha our model approach is amenable o numerous exensions and modificaions. Firs, we esimae he model using yield daa only, bu he model esimaion could include survey daa as in Kim and Orphanides (2012). Second, he nominal par of he model can be cas as a shadow-rae model o respec he zero lower bound for nominal yields as in, for example, Priebsch (2013). 4 Third, he model can be modified o allow for sochasic yield volailiy o improve is abiliy o price deflaion risk following Chrisensen e al. (2016). Lasly, he model s objecive dynamics can be adjused for finie-sample bias as described and discussed in Bauer e al. (2012). However, we leave he exploraion of hese avenues for fuure research. The remainder of he paper is srucured as follows. Secion 2 inroduces he general heoreical framework for inferring inflaion dynamics from nominal and real Treasury yields. Secion 3 describes he CLR model and is exension wih a liquidiy risk facor. I also 3 Fonaine and Garcia (2012) documen sysemaic and pervasive posiive differences beween he prices of recenly issued Treasury bonds and hose of more seasoned, bu oherwise comparable Treasury bonds. 4 In Appendix E, we do provide a brief comparison o a shadow-rae version of our model using formulas from Chrisensen and Rudebusch (2015). 2

4 deails our mehodology for deriving model-implied values of he deflaion proecion opions embedded in TIPS. Secion 4 conains he daa descripion, while Secion 5 presens he empirical resuls. Secion 6 conains an analysis of he esimaed TIPS liquidiy premium, while Secion 7 is dedicaed o an analysis of he risk of deflaion. Secion 8 analyzes he model-implied inflaion expecaions. Finally, Secion 9 concludes and provides direcions for fuure research. Appendices conain addiional echnical deails and resuls. 2 Decomposing Breakeven Inflaion In his secion, we demonsrae how an arbirage-free erm srucure model can be used o decompose he difference beween nominal and real Treasury yields, also known as he breakeven inflaion (BEI) rae, ino he sum of he expeced inflaion and he associaed inflaion risk premium. To begin, we follow Meron (1974) and assume a coninuum of nominal and real zerocoupon bonds exiss wih no fricions o heir coninuous rading. The economic implicaion of his assumpion is ha he markes for inflaion risk are complee in he limi and spanned by he coninuum of nominal and real bond prices. Given nominal and real sochasic discoun facors, denoed M N and M R, he no-arbirage condiion enforces a consisency of pricing for any securiy over ime. Specifically, he price of a nominal bond ha pays one dollar in τ years and he price of a real bond ha pays one uni of he defined consumpion baske in τ years mus saisfy he condiions ha P N (τ) = E P [ M N ] +τ M N and P R (τ) = EP [ M R +τ where P N (τ) and P R (τ) are he observed prices of he zero-coupon, nominal and real bonds for mauriy τ on day and E P [.] is he condiional expecaions operaor under he realworld (or P-) probabiliy measure. The 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 Π, is he raio of he nominal and real sochasic discoun facors: Π = MR M N We assume ha he nominal and real sochasic discoun facors have he sandard dynamics given by. M R ], dm N /M N dm R /MR = r N d Γ dw P, = r R d Γ dwp, wherer N andr R areheinsananeous, risk-freenominalandreal raesof reurn, respecively, 3

5 and Γ is a vecor of premiums on he risks represened by he Wiener process W P. By Io s lemma, he dynamic evoluion of Π is given by dπ = (r N r R )Π d. Thus, in 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. 5 Correspondingly, we can express he sochasic price level a ime +τ as +τ Π +τ = Π e (rs N rs R)ds. The relaionship beween he yields and inflaion expecaions can be obained by decomposing he price of he nominal bond as follows P N (τ) = E P = E P [ M N +τ M N [ M R +τ M R = P R (τ) E P ] [ ] = E P M+τ/Π R +τ M R/Π = E P ] [ ] E P [ Π Π +τ Π Π +τ Convering his price ino a yield-o-mauriy using ] ( +cov P 1+ E P [ M R +τ [ M R +τ M R, Π Π Π +τ ] M R Π +τ [ cov P M R ] +τ Π, M R Π +τ [ M R +τ M R ] E P ] [ ] Π Π +τ ). y N (τ) = 1 τ lnpn (τ) and y R (τ) = 1 τ lnpr (τ), we obain y N (τ) = yr (τ)+πe (τ)+φ (τ), where he marke-implied average rae of inflaion expeced a ime for he period from o +τ is π e (τ) = 1 τ lnep [ Π Π +τ ] = 1 τ lnep [e +τ (r s N rr)ds] s and he associaed inflaion risk premium for he same ime period is φ (τ) = 1 τ ln (1+ E P cov P [ M R +τ M R [ M R +τ, M ] R E P ] Π Π +τ [ ] Π Π +τ This las equaion highlighs ha he inflaion risk premium can be posiive or negaive. I 5 We emphasize ha he price level Π is a sochasic process as long as r N and r R are sochasic processes. ). 4

6 is posiive if and only if cov P [ M+τ R M R, Π Π +τ ] < 0. Tha is, he riskiness of nominal bonds relaive o real bonds depends on he covariance beween he real sochasic discoun facor and inflaion, and is ulimaely deermined by invesor preferences. Finally, he BEI rae is defined as BEI (τ) y N (τ) y R (τ) = π e (τ)+φ (τ), (1) ha is, he difference beween nominal and real yields of he same mauriy and can be decomposed ino he sum of he expeced inflaion and he associaed inflaion risk premium. Equaion (1) highlighs ha he decomposiion of BEI can be disored if nominal and real yields are biased by liquidiy effecs, and he magniude of he disorion equals he size of he bias. However, he equaion also makes clear ha i is only he relaive liquidiy beween nominal and real yields ha we need o correc BEI raes for any liquidiy bias. In he following secion, we inroduce a dynamic erm srucure model ha accouns for he liquidiy differenial of TIPS relaive o Treasuries and hence provides esimaes of he fricionless nominal and real yields ha feaure in he expecaions above. 3 An Arbirage-Free Model of Nominal and Real Yields wih Liquidiy Risk In his secion, we firs describe how we exend he general framework inroduced in he previous secion o accoun for he liquidiy risk of a se of real-valued securiies relaive o a benchmark se of nominal securiies. Second, we deail he CLR model ha we subjec o his exension and apply in he subsequen empirical analysis. 3.1 The General Model wih a Liquidiy Risk Facor Due o he lower liquidiy of real-valued securiies relaive o he benchmark nominal securiies, he yields of he former are sensiive o liquidiy pressures on a relaive basis. As a consequence, he discouning of fuure cash flows from he real-valued securiies is no performed wih he fricionless real discoun funcion described in Secion 2, bu raher wih a discoun funcion ha also accouns for liquidiy risk. Recen research by Hu e al. (2013, henceforh HPW) and ohers sugges ha liquidiy is indeed a priced risk facor. Thus, we choose o represen his by a single liquidiy risk facor denoed X liq. 6 Furhermore, since liquidiy risk is securiy-specific in naure, he discoun funcion used o discoun he cash 6 D Amico e al. (2014) and Abrahams e al. (2015) also only allow for a single TIPS liquidiy facor. 5

7 flow of a given real-valued securiy indexed i is assumed o be unique. The single innovaion of his paper is o le he individual TIPS discoun funcion ake he following form: r R,i = r R +β i (1 e λl,i ( i 0 ) )X liq, (2) where r R is he fricionless real insananeous rae as before, i 0 denoes he dae of issuance of he securiy, β i is is sensiiviy o he variaion in he liquidiy risk facor, and λ L,i is a decay parameer. While we could expec he sensiiviies o be idenical across securiies, he resuls from our subsequen empirical applicaion shows ha i is imporan o allow for he possibiliy ha he sensiiviies differ across securiies. Furhermore, we allow he decay parameer λ L,i o vary across securiies as well. Since β i and λ L,i have a nonlinear relaionship in he bond pricing formula, i is possible o idenify boh empirically. Finally, we sress ha equaion (2) can be included in any dynamic erm srucure model o accoun for securiy-specific liquidiy risks. The inclusion of he issuance dae i 0 in he pricing formula is a proxy for he phenomenon ha, as ime passes, i is ypically he case ha an increasing fracion of a given securiy is held by buy-and-hold invesors. This limis he amoun of he securiy available for rading and affecs is sensiiviy o he liquidiy facor. Raional, forward-looking invesors will ake his dynamic paern ino consideraion when hey deermine wha hey are willing o pay for he securiy a any given poin in ime beween he dae of issuance and he mauriy of he bond. This dynamic paern is buil ino he model srucure. 3.2 The CLR Model Building on he insighs from he general heoreical discussion in Secion 2, we need an accurae model of he insananeous nominal and real rae, r N and r R, in order o measure he marke-implied inflaion expecaions precisely. Wih ha goal in mind we choose o focus on he racable affine dynamic erm srucure model of nominal and real yields inroduced in CLR and briefly summarized below. We emphasize ha even hough he model is no formulaed using he canonical form of affine erm srucure models inroduced by Dai and Singleon (2000), i can be viewed as a resriced version of he canonical Gaussian model. 7 The CLR model of nominal and real yields is a direc exension of he hree-facor, arbirage-free Nelson-Siegel (AFNS) model developed by Chrisensen e al. (2011, henceforh CDR) for nominal yields. In he CLR model, he sae vecor is denoed by X = (L N,S,C,L R ), where L N is he level facor for nominal yields, S and C represen slope and curvaure facors common o boh nominal and real yields, and L R is he level facor for 7 These resricions can be derived explicily, and he calculaions are available upon reques. 6

8 real yields. 8 The insananeous nominal and real risk-free raes are defined as r N = L N +S, (3) r R = L R +α R S. (4) Noe ha he differenial scaling of he real raes o he common slope facor is capured by he parameer α R. To preserve he Nelson and Siegel (1987) facor loading srucure in he yield funcions, he risk-neural (or Q-) dynamics of he sae variables are given by he sochasic differenial equaions: ds 0 λ λ 0 = dc 0 0 λ dl N dl R L N S C L R d+σ dw LN,Q dw S,Q dw C,Q dw LR,Q, (5) where Σ is he consan covariance (or volailiy) marix. 10 Based on his specificaion of he Q-dynamics, nominal zero-coupon bond yields preserve he Nelson-Siegel facor loading srucure as ( ) ( 1 e y N λτ 1 e λτ (τ) = LN + S + λτ λτ where he nominal yield-adjusmen erm is given by e λτ ) C AN (τ), (6) τ A N (τ) τ = σ τ2 +σ σ 2 33 [ 1 2λ e λτ λ 3 τ e 2λτ ] 4λ 3 τ [ 1 2λ λ 2e λτ 1 4λ τe 2λτ 3 4λ 2e 2λτ e 2λτ 8λ 3 τ 2 λ 3 1 e λτ τ ]. Similarly, real zero-coupon bond yields have a Nelson-Siegel facor loading srucure expressed as ( ( ) 1 e y R (τ) = LR +α R λτ 1 e )S +α R λτ e λτ C AR (τ), (7) λτ λτ τ where he real yield-adjusmen erm is given by A R (τ) τ = σ τ2 +σ22(α 2 R S) 2[ 1 2λ e λτ λ 3 τ e 2λτ ] 4λ 3 τ +σ33(α 2 R S) 2[ 1 2λ λ 2e λτ 1 4λ τe 2λτ 3 4λ 2e 2λτ e 2λτ 8λ 3 τ 2 λ 3 1 e λτ τ 8 ChernovandMueller(2012) provideevidenceofahiddenfacorinhenominalyieldcurvehaisobservable from real yields and inflaion expecaions. The CLR model accommodaes his sylized fac via he L R facor. 9 As discussed in CDR, wih uni roos 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. 10 As per CDR, Σ is a diagonal marix, and θ Q is se o zero wihou loss of generaliy. ]. 7

9 3.3 The CLR-L Model In his secion, we augmen he CLR model wih a liquidiy risk facor o accoun for he liquidiy risk in he pricing of TIPS relaive o Treasuries, hroughou referred o as he CLR-L model. To begin, le X = (L N,S,C,L R,Xliq ) denoe he sae vecor of his five-facor model. As before, L N and L R denoe he level facor unique o he nominal and real yield curve, respecively, while S and C represen slope and curvaure facors common o boh yield curves. Finally, X liq represens he added liquidiy facor. As in he CLR model, we le he fricionless insananeous nominal and real risk-free raes be defined by equaions (3) and (4), respecively, while he risk-neural dynamics of he sae variables used for pricing are given by dl N ds dc dl R dx liq λ λ 0 0 = 0 0 λ κ Q liq where Σ coninues o be a diagonal marix θ Q liq L N S C L R X liq d+σ dw LN,Q dw S,Q dw C,Q dw LR,Q dw liq,q Based on he Q-dynamics above, nominal Treasury zero-coupon bond yields preserve he Nelson-Siegel facor loading srucure in equaion (6). On he oher hand, due o he lower liquidiy in he TIPS marke relaive o he marke for nominal Treasuries, TIPS yields are sensiive o liquidiy pressures. As deailed in Secion 3.1, pricing of TIPS is no performed wih he fricionless real discoun funcion, bu raher wih a discoun funcion ha accouns for he liquidiy risk:, r R,i = r R +β i (1 e λl,i ( i 0 ) )X liq = L R +α R S +β i (1 e λl,i ( i 0 ) )X liq, (8) where i 0 denoes he dae of issuance of he specific TIPS and βi is is sensiiviy o he variaion in he liquidiy facor. Furhermore, he decay parameer λ L,i is assumed o vary across securiies as well. In Appendix A, we show ha he ne presen value of one uni of he consumpion baske paid by TIPS i a ime +τ has he following exponenial-affine form r R,i (s, i 0 )ds] P ( i 0,τ) = EQ[ e +τ ( = exp B 1 (τ)l N +B 2 (τ)s +B 3 (τ)c +B 4 (τ)l R +B 5 (, i 0,τ)X Liq +A(, i 0,τ) ). This resul implies ha he model belongs o he class of Gaussian affine erm srucure models, bu unlike sandard Gaussian models, P ( i 0,τi ) is ime-inhomogeneous. Noe also 8

10 ha, by fixing β i = 0 for all i, we recover he CLR model. 3.4 Valuing he TIPS Deflaion Proecion Opion TIPS provide inflaion proecion since heir coupons and principal paymens are indexed o he headline Consumer Price Index (CPI) produced by he Bureau of Labor Saisics. 11 Imporanly, TIPS also provide some proecion agains price deflaion since heir principal paymens are no permied o decrease below heir original par value. This implies ha here is an inflaion floor opion for he principal embedded in he TIPS conrac. In his secion, we describe how o value his deflaion proecion opion assuming no fricions o rading. To begin, consider a TIPS issued a ime 0 wih mauriy a ime +τ. By ime is accrued inflaion compensaion, also known as is index raio, is given by he change in he price level since issuance, i.e., Π /Π 0. To value he embedded deflaion proecion opion, we need o explicily conrol for he accrued inflaion compensaion; ha is, he opion will only be in he money a mauriy provided he change in he price level beween and +τ saisfies he following inequaliy Π +τ Π 1 Π /Π 0. Thus, for he opion o be in he money, he deflaion experienced over he remaining life of he bond, Π +τ /Π, has o negae he accumulaed inflaion experienced since he bond s issuance. Now, he presen value of he principal paymen of he TIPS is given by E Q [ ] Π +τ e +τ r s Nds 1 Π Π { +τ +E Q > 1 } Π Π /Π 0 [ 1 e +τ r N s ds 1 { Π +τ Π 1 Π /Π 0 } The firs erm represens he presen value of he principal paymen condiional on he ne change in he price index over he bond s remaining ime o mauriy is no offse by he accrued inflaion compensaion as of ime ; ha is, Π +τ Π > 1 Π /Π 0. Under his condiion, full inflaion indexaion applies, and he price change adjusmen of he principal Π +τ Π is placed wihin he expecaions operaor. The second erm represens he presen value of he floored TIPS principal condiional on he ne change in he price level unil he bond s mauriy eroding he accrued inflaion compensaion as of ime ; ha is, he price level change Π +τ Π is replaced by a value of one o provide he promised deflaion proecion. Nex, we exploi he fac ha absence of arbirage implies ha he price level change is 11 The acual indexaion has a lag srucure since he Bureau of Labor Saisics publishes price index values wih a one-monh lag; ha is, 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. See Gürkaynak e al. (2010) for a deailed discussion. ]. 9

11 given by Π +τ +τ = e (rs N rs R)ds. Π This allows us o rewrie he presen value of he principal paymen as E Q = E Q [e +τ [e +τ rs Rds 1 Π { +τ +E Q > 1 } Π Π /Π 0 [e +τ r R s ds] +E Q ] [e +τ r N s ds 1 { Π +τ Π 1 Π /Π 0 } E Q ] rs Nds 1 Π { +τ 1 } Π Π /Π ] 0 [e +τ r R s ds 1 { Π +τ Π 1 Π /Π 0 } Here, he firs erm is he ne presen value of he TIPS principal paymen wihou any deflaion proecion, while he wo remaining erms equal he ne presen value of he deflaion proecion opion, which is denoed DOV and given by DOV ( τ, Π Π 0 ) E Q [e +τ r N s ds 1 { Π +τ Π 1 Π /Π 0 } ] E Q [e +τ r R s ds 1 { Π +τ Π 1 Π /Π 0 } ]. ]. (9) This opion value needs o be added o he model-implied TIPS price o mach he observed TIPS price. Now, consider he whole value of TIPS i issued a ime i 0 wih mauriy a +τi ha pays an annual coupon C semi-annually and has accrued inflaion compensaion equal o Π /Π 0. Is price is given by P ( i 0,τ i,c, Π ) = C ( 1 ) Π 0 2 1/2 EQ[ e 1 r R,i (s, i)ds] 0 + N j=2 +E Q[ e +τ i r R,i (s, i 0 )ds] +DOV (τ i, Π Π 0 ), C 2 EQ[ e j r R,i (s, i)ds] 0 where he las erm is he value of he deflaion proecion opion provided in equaion (9) and calculaed using he dynamics for he fricionless nominal and real insananeous shor raes, r N and r R, in combinaion wih formulas provided in Chrisensen e al. (2012). The only minor omission in he bondprice formula above is ha we do no accoun for he lag in he inflaion indexaion of he TIPS payoff, bu he poenial error should be modes in mos cases, see Grishchenko and Huang (2013) for evidence. 3.5 Marke Prices of Risk So far, he descripion of he CLR-L model has relied solely on he dynamics of he sae variables under he Q-measure used for pricing. However, o complee he descripion of he model and o implemen i empirically, we will need o specify he risk premiums ha connec he facor dynamics under he Q-measure o he dynamics under he real-world (or hisorical) P-measure. I is imporan o noe ha here are no resricions on he dynamic drif componens under he empirical P-measure beyond he requiremen of consan volailiy. To faciliae empirical implemenaion, we use he essenially affine risk premium specificaion 10

12 inroduced in Duffee (2002). In he Gaussian framework, his specificaion implies ha he risk premiums Γ depend on he sae variables; ha is, Γ = γ 0 +γ 1 X, where γ 0 R 5 and γ 1 R 5 5 conain unresriced parameers. Thus, he resuling unresriced five-facor CLR-L model has P-dynamics given by dl N ds dc dl R dx liq κ P 11 κ P 12 κ P 13 κ P 14 κ P 15 κ P 21 κ P 22 κ P 23 κ P 24 κ P 25 = κ P 31 κ P 32 κ P 33 κ P 34 κ P 35 κ P 41 κ P 42 κ P 43 κ P 44 κ P 45 κ P 51 κ P 52 κ P 53 κ P 54 κ P 55 θ P 1 θ P 2 θ P 3 θ P 4 θ P 5 L N S C L R X liq This is he ransiion equaion in he exended Kalman filer esimaion. d+σ dw LN,P dw S,P dw C,P dw LR,P dw liq,p. 3.6 Model Esimaion and Economeric Idenificaion Due o he nonlineariy of he TIPS pricing formula, he model canno be esimaed wih he sandard Kalman filer. Insead, we use he exended Kalman filer as in Kim and Singleon (2012), see Appendix B for deails. To make he fied errors comparable across TIPS of various mauriies, we scale each TIPS price by is duraion. 12 Thus, he measuremen equaion for he TIPS prices ake he following form: P ( i 0,τi ) D (τ i ) = P ( i 0,τi ) D (τ i ) +ε i, where P ( i 0,τi ) is he model-implied price of TIPS i and D (τ i ) is is duraion, which is fixed and calculaed before esimaion. Furhermore, o faciliae model esimaion when we adjus for he deflaion opion values, we firs esimae he CLR-L model wihou he opion values. Then we use he esimaed parameers and filered sae variables o calculae he ime series of opion values for each TIPS. These are used as a fixed inpu ino a new model esimaion a he end of which a new se of opion values are calculaed, and he process is repeaed. This algorihm is coninued unil convergence is achieved. From he five-facor model srucure above i follows ha we will be fiing TIPS yields wih wo separae facors, he real level facor, L R, and he TIPS liquidiy facor, X liq, in addiion o he common slope and curvaure facor ha can be idenified from he nominal yields. Thus, for reasons of idenificaion, we need o have a leas wo TIPS securiies rading 12 For robusness, we repeaed he esimaions using he mid-marke yield-o-mauriies for each TIPS downloaded from Bloomberg insead and go very similar resuls. However, we noe ha hose esimaions are exremely ime consuming since yield-o-mauriy is only defined implicily as a fix poin ha needs o be calculaed for each observaion. Hence, we advise agains ha approach. 11

13 a each observaion dae. This requiremen implies ha he earlies saring poin for he model esimaion coincides wih he issuance dae of he second TIPS in mid-july Since he liquidiy facor is a laen facor ha we do no observe, is level is no idenified wihou addiional resricions. As a consequence, we le he firs TIPS issued, ha is, he en-year TIPS wih 3.375% coupon issued in January 1997 wih mauriy on January 15, 2007, have a uni loading on he liquidiy facor, ha is, β i = 1 for his securiy. This choice implies ha he β i sensiiviy parameers measure liquidiy sensiiviy relaive o ha of he en-year 2007 TIPS. Furhermore, we noe ha he λ L,i parameers can be hard o idenify if heir values are oo large or oo small. As a consequence, we impose he resricion ha hey fall wihin he range from 0.01 o 10, which is wihou pracical consequences. Also, for numerical sabiliy during model opimizaion, we impose he resricion ha he β i parameers fall wihin he range from 0 o 80, which urns ou no o be a binding consrain a he opimum. Finally, we assume ha all fied nominal yields in equaion (6) have i.i.d. measuremen errors wih sandard deviaion σε N. Similarly, all TIPS measuremen errors are assumed o be i.i.d. wih sandard deviaion σε R. 4 Daa This secion briefly describes he daa we use in he model esimaion. 4.1 Nominal Treasury Yields The specific nominal Treasury yields we use are zero-coupon yields aken from he Gürkaynak e al. (2007) daabase wih he following mauriies: 3-monh, 6-monh, 1-year, 2-year, 3- year, 4-year, 5-year, 6-year, 7-year, 8-year, 9-year, and 10-year. We use weekly daa and limi our sample o he period from July 11, 1997, o December 27, The summary saisics are provided in Table 1. Researchers have ypically found ha hree facors are sufficien o model he imevariaion in he cross secion of nominal Treasury bond yields(e.g., Lierman and Scheinkman, 1991). Indeed, for our weekly nominal Treasury bond yield daa, 99.98% of he oal variaion is accouned for by hree facors. Table 2 repors he eigenvecors ha correspond o he firs hree principal componens of our daa. The firs principal componen accouns for 95.3% of he variaion in he nominal Treasury bond yields, and is loading across mauriies is uniformly negaive. Thus, like a level facor, a shock o his componen changes all yields in he same direcion irrespecive of mauriy. The second principal componen accouns for 4.5% of he variaion in hese daa and has sizable negaive loadings for he shorer mauriies and sizable posiive loadings for he long mauriies. Thus, like a slope facor, a shock o his componen seepens or flaens he yield curve. Finally, he hird componen, which accouns 12

14 Mauriy Mean S. dev. in monhs in % in % Skewness Kurosis Table 1: Summary Saisics for he Nominal Treasury Yields. Summary saisics for he sample of weekly nominal Treasury zero-coupon bond yields covering he period from July 11, 1997, o December 27, 2013, a oal of 860 observaions. Mauriy Loading on in monhs Firs P.C. Second P.C. Third P.C % explained Table 2: Eigenvecors of he Firs Three Principal Componens in Nominal Treasury 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 July 11, 1997, o December 27, for only 0.2% of he variaion, has a hump shaped facor loading as a funcion of mauriy, which is naurally inerpreed as a curvaure facor. This moivaes our use of he AFNS model wih is level, slope, and curvaure srucure for he nominal yields even hough we emphasize ha he esimaed sae variables are no idenical o he principal componen facors discussed here A number of recen papers use principal componens as sae variables. Joslin e al. (2011) is an example. 13

15 Time o mauriy in years Figure 1: Mauriy Disribuion of TIPS. Illusraion of he mauriy disribuion of all TIPS issued since he incepion of he TIPS program. The solid grey recangle indicaes he subsample used in he main analysis in he paper and characerized by hree sample choices: (1) for reasons of idenificaion he sample sars on July 11, 1997; (2) he sample is limied o bonds wih less han 10 years o mauriy a issuance; (3) he price of each TIPS is censored when i has less han wo years o mauriy o avoid erraic prices close o expiry. 4.2 TIPS Daa The U.S. Treasury sared issuing TIPS in The firs TIPS was issued on February 6, 1997, wih mauriy on January 15, 2007, and a coupon rae of 3.375%. 14 Since hen he U.S. Treasury has issued five-, en-, weny-, and hiry-year TIPS. However, only en-year TIPS have been regularly issued since he incepion of he TIPS program. As of he end of 2013, a oal of 50 TIPS had been issued and he disribuion of heir remaining ime o mauriy across ime is shown in Figure 1. The oal number of TIPS ousanding a any poin in ime since he sar of he TIPS program is shown wih a solid red line in Figure 2. A he end of our sample period here was a oal of 37 TIPS ousanding. To faciliae he empirical implemenaion and improve model fi, we limi our focus o five- and en-year TIPS. 15 This reduces he oal number of TIPS o 38, while he number of hose TIPS ousanding a any poin in ime is shown wih a solid grey line in Figure 2. A he end of 2013, his subse included 25 TIPS. Furhermore, as TIPS prices near mauriy 14 TIPS are issued wih a minimum coupon of 0.125%. Since April 2011 his has been a binding consrain for five-year TIPS and occasionally for en-year TIPS. 15 As a robusness check, we esimaed he model wih all available TIPS in combinaion wih nominal yields wih mauriies up o hiry years. This produced qualiaively similar, bu less accurae resuls. See Appendix F for deails. 14

16 Number of securiies All TIPS All five and en year TIPS Sample of five and en year TIPS Figure 2: Number of TIPS Ousanding. Illusraion of he number of TIPS ousanding. The sample covers he period from February 6, 1997, o December 31, end o exhibi erraic behavior due o seasonal variaion in CPI, we drop TIPS from our sample when hey have less han wo years o mauriy, see Gürkaynak e al. (2010). Thus, our analysis is cenered around he wo- o en-year mauriy range ha is he mos widely used for boh bond risk managemen and moneary policy analysis. Using his cuoff, he number of TIPS in he sample is furher reduced and shown wih a solid black line in Figure 2. As of he end of 2013, i included 19 securiies. Our sample of TIPS is also indicaed wih a solid grey recangle in Figure 1, while summary saisics for our sample of 38 TIPS are repored in Table 3. To esimae he CLR-L model, we use mid-marke clean TIPS prices downloaded from Bloomberg. Since he model has wo TIPS specific facors, we sar he model esimaion on Friday July 11, 1997, when prices become available for he second ever TIPS, he five-year TIPS wih mauriy on July 15, We end he sample on December 27, 2013, wih 19 TIPS rading. The number of weekly observaions for each of our 38 TIPS is also repored in Table 3. 5 Esimaion Resuls In his secion, we firs describe he resuls from he CLR-L model esimaed wih and wihou adjusmen for he he value of he deflaion proecion opion embedded in he TIPS 15

17 No. Issuance Firs reopen Second reopen TIPS securiy obs. Dae Amoun Dae Amoun Dae Amoun (1) 3.375% 1/15/2007 TIPS 393 2/6/97 7,353 4/15/97 8,403 n.a. n.a. (2) 3.625% 7/15/2002 TIPS 158 7/15/97 8,401 10/15/97 8,412 n.a. n.a. (3) 3.625% 1/15/2008 TIPS 419 1/15/98 8,409 10/15/98 8,401 n.a. n.a. (4) 3.875% 1/15/2009 TIPS 419 1/15/99 8,531 7/15/99 7,368 n.a. n.a. (5) 4.25% 1/15/2010 TIPS 418 1/18/00 6,317 7/17/00 5,002 n.a. n.a. (6) 3.5% 1/15/2011 TIPS 418 1/16/01 6,000 7/16/01 5,000 n.a. n.a. (7) 3.375% 1/15/2012 TIPS 419 1/15/02 6,000 n.a. n.a. n.a. n.a. (8) 3% 7/15/2012 TIPS 419 7/15/02 10,010 10/15/02 7,000 1/15/03 6,000 (9) 1.875% 7/15/2013 TIPS 419 7/15/03 11,000 10/15/03 9,000 n.a. n.a. (10) 2% 1/15/2014 TIPS 419 1/15/04 12,000 4/15/04 9,000 n.a. n.a. (11) 2% 7/15/2014 TIPS 419 7/15/04 10,000 10/15/04 9,000 n.a. n.a. (12) 0.875% 4/15/2010 TIPS /29/04 12,000 4/29/05 9,000 10/28/05 7,000 (13) 1.625% 1/15/2015 TIPS 418 1/18/05 10,000 4/15/05 9,000 n.a. n.a. (14) 1.875% 7/15/2015 TIPS 418 7/15/05 9,000 10/17/05 8,000 n.a. n.a. (15) 2% 1/15/2016 TIPS 416 1/17/06 9,000 4/17/06 8,000 n.a. n.a. (16) 2.375% 4/15/2011 TIPS 155 4/28/06 11,000 10/31/06 9,181 n.a. n.a. (17) 2.5% 7/15/2016 TIPS 390 7/17/06 10,588 10/16/06 9,412 n.a. n.a. (18) 2.375% 1/15/2017 TIPS 364 1/16/07 11,250 4/16/07 6,000 n.a. n.a. (19) 2% 4/15/2012 TIPS 156 4/30/07 10,123 10/31/07 7,158 n.a. n.a. (20) 2.625% 7/15/2017 TIPS 338 7/16/07 8,000 10/15/07 6,000 n.a. n.a. (21) 1.625% 1/15/2018 TIPS 312 1/15/08 10,412 4/15/08 6,000 n.a. n.a. (22) 0.625% 4/15/2013 TIPS 156 4/30/08 8,734 10/31/08 6,266 n.a. n.a. (23) 1.375% 7/15/2018 TIPS 286 7/15/08 8,000 10/15/08 6,974 n.a. n.a. (24) 2.125% 1/15/2019 TIPS 260 1/15/09 8,662 4/15/09 6,096 n.a. n.a. (25) 1.25% 4/15/2014 TIPS 156 4/30/09 8,277 10/30/09 7,000 n.a. n.a. (26) 1.875% 7/15/2019 TIPS 234 7/15/09 8,135 10/15/09 7,055 n.a. n.a. (27) 1.375% 1/15/2020 TIPS 207 1/15/10 10,388 4/15/10 8,586 n.a. n.a. (28) 0.5% 4/15/2015 TIPS 155 4/30/10 11,235 10/29/10 10,000 n.a. n.a. (29) 1.25% 7/15/2020 TIPS 182 7/15/10 12,003 9/15/10 10,108 11/15/10 10,268 (30) 1.125% 1/15/2021 TIPS 154 1/31/11 13,259 3/31/11 11,493 5/31/11 11,926 (31) 0.125% 4/15/2016 TIPS 140 4/29/11 14,000 8/31/11 12,367 12/30/11 12,000 (32) 0.625% 7/15/2021 TIPS 128 7/29/11 13,000 9/30/11 11,342 11/30/11 11,498 (33) 0.125% 1/15/2022 TIPS 102 1/31/12 15,282 3/30/12 13,000 5/31/12 13,000 (34) 0.125% 4/15/2017 TIPS 89 4/30/12 16,430 8/31/12 14,000 12/31/12 14,000 (35) 0.125% 7/15/2022 TIPS 76 7/31/12 15,000 9/28/12 13,000 11/30/12 13,000 (36) 0.125% 1/15/2023 TIPS 49 1/31/13 15,000 3/28/13 13,000 5/31/13 13,000 (37) 0.125% 4/15/2018 TIPS 37 4/30/13 18,000 8/30/13 16,000 12/31/13 16,000 (38) 0.375% 7/15/2023 TIPS 24 7/31/13 15,000 9/30/13 13,000 11/29/13 13,000 Table 3: Sample of TIPS. The able repors he characerisics, issuance daes, and issuance amouns in millions of dollars for he 38 TIPS used in he analysis. Also repored are he number of weekly observaion daes for each TIPS during he sample period from July 11, 1997, o December 27, Aserisk * indicaes five-year TIPS. conrac. Second, we compare he esimaed sae variables and model fi o hose obained from sandard AFNS and CLR models. Upfron we noe ha, for each model class, we limi he focus o he mos parsimonious independen-facor specificaion o make he resuls as comparable as possible. 16 Table 4 conains he esimaed dynamic parameers for he CLR-L model esimaed wih and wihou accouning for he deflaion opion values, while Table 5 repors heir respecive 16 Since he model fi and he esimaed facors are insensiive o he specificaion of he mean-reversion marix K P, his limiaion comes a pracically no loss of generaliy for he resuls presened in his secion. 16

18 CLR-L model, no adjusmen K P K P,1 K P,2 K P,3 K P,4 K P,5 θ P Σ K1, P σ (0.1904) (0.0082) (0.0001) K2, P σ (0.1458) (0.0276) (0.0002) K3, P σ (0.2934) (0.0145) (0.0005) K4, P σ (0.3169) (0.0099) (0.0002) K5, P σ (0.3953) (0.0076) (0.0007) CLR-L model, opion adjused K P K P,1 K P,2 K P,3 K P,4 K P,5 θ P Σ K1, P σ (0.1907) (0.0083) (0.0001) K2, P σ (0.1461) (0.0297) (0.0002) K3, P σ (0.2941) (0.0158) (0.0005) K4, P σ (0.2782) (0.0102) (0.0002) K5, P σ (0.3936) (0.0075) (0.0007) Table 4: Esimaed Dynamic Parameers. The op panel shows he esimaed parameers of he K P marix, θ P vecor, and diagonal Σ marix for he CLR-L model. The esimaed value of λ is (0.0019), while α R = (0.0077), κ Q liq = (0.0436), and θ Q liq = (0.0001). The boom panel shows he corresponding esimaes for he CLR-L model wih deflaion opion adjusmen. In his case, he esimaed value of λ is (0.0019), while α R = (0.0072), κ Q liq = (0.0598), and θq liq = (0.0001). The numbers in parenheses are he esimaed parameer sandard deviaions. esimaed β i and λ L,i parameers. Overall, he model parameers, he dynamic parameers in paricular, are relaively insensiive o adjusing for he deflaion opion values. As for he sae variables, Figure 3 shows he esimaed pahs for (L N,S,C ), which are primarily deermined from nominal yields. We noe ha he esimaed pahs of hese hree facors are pracically indisinguishable from he esimaed pahs obained wih a sand-alone AFNS model of nominal yields only. This is also refleced in he summary saisics for he fied errors of nominal yields repored in Table 6. Theresuls show ha he CLR and CLR-L models, wih and wihou deflaion opion adjusmen, provide a very close fi o he enire cross secion of nominal yields. Imporanly, he fi is as good as ha obained wih a sandalone AFNS model for he nominal yields. Thus, allowing for a join modeling of nominal and real yields based on he CLR model framework comes a effecively no cos in erms of fi o he nominal yields as also emphasized by CLR. As for he esimaed level facor for real yields, Figure 4(a) shows ha i is sensiive o 17

19 CLR-L model, no adjusmen CLR-L model, opion adjused TIPS securiy β i Sd λ L,i Sd β i Sd λ L,i Sd (1) 3.375% 1/15/2007 TIPS 1 n.a n.a (2) 3.625% 7/15/2002 TIPS (3) 3.625% 1/15/2008 TIPS (4) 3.875% 1/15/2009 TIPS (5) 4.25% 1/15/2010 TIPS (6) 3.5% 1/15/2011 TIPS (7) 3.375% 1/15/2012 TIPS (8) 3% 7/15/2012 TIPS (9) 1.875% 7/15/2013 TIPS (10) 2% 1/15/2014 TIPS (11) 2% 7/15/2014 TIPS (12) 0.875% 4/15/2010 TIPS (13) 1.625% 1/15/2015 TIPS (14) 1.875% 7/15/2015 TIPS (15) 2% 1/15/2016 TIPS (16) 2.375% 4/15/2011 TIPS (17) 2.5% 7/15/2016 TIPS (18) 2.375% 1/15/2017 TIPS (19) 2% 4/15/2012 TIPS (20) 2.625% 7/15/2017 TIPS (21) 1.625% 1/15/2018 TIPS (22) 0.625% 4/15/2013 TIPS (23) 1.375% 7/15/2018 TIPS (24) 2.125% 1/15/2019 TIPS (25) 1.25% 4/15/2014 TIPS (26) 1.875% 7/15/2019 TIPS (27) 1.375% 1/15/2020 TIPS (28) 0.5% 4/15/2015 TIPS (29) 1.25% 7/15/2020 TIPS (30) 1.125% 1/15/2021 TIPS (31) 0.125% 4/15/2016 TIPS (32) 0.625% 7/15/2021 TIPS (33) 0.125% 1/15/2022 TIPS (34) 0.125% 4/15/2017 TIPS (35) 0.125% 7/15/2022 TIPS (36) 0.125% 1/15/2023 TIPS (37) 0.125% 4/15/2018 TIPS (38) 0.375% 7/15/2023 TIPS Table 5: Esimaed Liquidiy Sensiiviy Parameers. The esimaed β i sensiiviy and λ L,i decay parameersfor each TIPS from he CLR-L model wih and wihou deflaion opion adjusmen are shown. Also repored are he esimaed parameer sandard deviaions. Aserisk * indicaes five-year TIPS. The sample used in each model esimaion is weekly covering he period from July 11, 1997, o December 27, wheher a liquidiy risk facor is included or no. However, i is only periodically ha he differences are sizable. Figure 4(b) shows he esimaed liquidiy risk facor ha is unique o he CLR-L model. We will sudy his facor and is implicaions for he TIPS liquidiy premiums furher below, bu for now we noe ha i has lile sensiiviy o adjusmen for he deflaion opion values. To evaluae he fi of he models o he TIPS price daa, we calculae he model-implied ime series of he yield-o-mauriy for each TIPS and compare ha o he mid-marke TIPS 18

20 Esimaed value AFNS model CLR model CLR model, opion adjused CLR L model CLR L model, opion adjused Esimaed value AFNS model CLR model CLR model, opion adjused CLR L model CLR L model, opion adjused Esimaed value AFNS model CLR model CLR model, opion adjused CLR L model CLR L model, opion adjused (a) L N. (b) S. (c) C. Figure 3: Esimaed Sae Variables. Illusraion of he esimaed sae variables ha affec nominal yields from he AFNS model, he CLR model, he CLR model wih deflaion opion adjusmen, he CLR-L model, and he CLR-L model wih deflaion opion adjusmen. The daa are weekly covering he period from July 11, 1997, o December 27, CLR model CLR-L model Mauriy AFNS model No adjusmen Opion adjused No adjusmen Opion adjused in monhs Mean RMSE Mean RMSE Mean RMSE Mean RMSE Mean RMSE All yields Table 6: Summary Saisics of Fied Errors of Nominal Yields. The mean fied errors and he roo mean squared fied errors (RMSE) of nominal U.S. Treasury yields from five model esimaions are shown. The full sample used in each model esimaion is weekly covering he period from July 11, 1997, o December 27, All numbers are measured in basis poins. yield-o-mauriy available from Bloomberg. The summary saisics of he fied yield errors calculaed his way are repored in Table 7. Firs, we noe ha he CLR model ends o provide less han ideal fi as measured by RMSEs and wih noable bias for some TIPS. Second, accouning for he value of he deflaion proecion opion improves he fi of he model, bu his modificaion is no sufficien for he model o deliver saisfacory fi (as measured by RMSEs) or o eliminae he bias for selec TIPS. For all TIPS yields combined, he RMSEs remain close o 15 basis poins. Third, incorporaing he liquidiy facor ino he CLR model leads o a significan improvemen in model fi for pracically all TIPS in 19

21 Esimaed value CLR model CLR model, opion adjused CLR L model CLR L model, opion adjused Esimaed value CLR L model CLR L model, opion adjused (a) L R. (b) X liq. Figure 4: Esimaed Sae Variables. Panel (a) shows he esimaed real yield level facor from he CLR model and he CLR-L model wih and wihou deflaion opion adjusmen. Panel (b) shows he esimaed liquidiy facor from he CLR-L model wih and wihou deflaion opion adjusmen. The daa are weekly covering he period from July 11, 1997, o December 27, he sample. Also, and equally imporan, here is no maerial bias for any of he TIPS. Wih he liquidiy exension, he fi o he TIPS daa is abou as good as he fi o he nominal Treasury yields. Finally, accouning for he deflaion opion values in addiion o incorporaing he liquidiy facor provides a furher modes improvemen in model fi. 6 The TIPS Liquidiy Premium In his secion, we firs analyze he esimaed TIPS liquidiy premiums in deail and compare hem o alernaive esimaes from he exising lieraure. We hen follow Chrisensen and Gillan (2015, henceforh CG) and sudy he effecs on TIPS liquidiy premiums from he Fed s TIPS purchases during is second large-scale asse purchases program frequenly referred o as QE2 ha operaed from November 2010 hrough June Figure 5 shows he ime series of he average yield difference beween he fied yieldo-mauriy of individual TIPS and he corresponding fricionless yield-o-mauriy wih he liquidiy risk facor urned off. This represens he average TIPS liquidiy premium for each observaion dae in our sample. According o he CLR-L model, he sample average of he weekly average liquidiy premium is 42.3 basis poins. Once we accoun for he value of he deflaion opion in he model esimaion, he sample average drops o 37.8 basis poins. Thus, he assessmen of he average TIPS liquidiy premium is sensiive o he inclusion of he deflaion proecion opion. Furhermore, we noe ha over he main sample period analyzed 20