The TIPS Liquidity Premium

Transcription

1 FEDERAL RESERVE BANK OF SAN FRANCISCO WORKING PAPER SERIES The TIPS Liquidiy Premium Marin M. Andreasen Aarhus Universiy Jens H. E. Chrisensen Federal Reserve Bank of San Francisco Simon Riddell Amazon Ocober 2017 Working Paper hp:// Suggesed ciaion: Andreasen, Marin M., Jens H. E. Chrisensen, Simon Riddell The TIPS Liquidiy Premium Federal Reserve Bank of San Francisco Working Paper hps://doi.org/ /wp 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.

2 The TIPS Liquidiy Premium Marin M. Andreasen Jens H. E. Chrisensen Simon Riddell Absrac We inroduce an arbirage-free erm srucure model of nominal and real yields ha accouns for liquidiy risk in Treasury inflaion-proeced securiies (TIPS). The novel feaure of our model is o idenify liquidiy risk from individual TIPS prices by accouning for he endency ha TIPS, like mos fixed-income securiies, go ino buy-and-hold invesors porfolios as ime passes. We find a sizable and counercyclical TIPS liquidiy premium, which grealy helps our model in maching TIPS prices. Accouning for liquidiy risk also improves he model s abiliy o forecas inflaion and mach surveys of inflaion expecaions, alhough none of hese series are included in he esimaion. JEL Classificaion: E43, E47, G12, G13 Keywords: erm srucure modeling, liquidiy risk, financial marke fricions We hank paricipans a he 9h Annual SoFiE Conference, he Financial Economerics and Empirical Asse Pricing Conference in Lancaser, he 2016 NBER Summer Insiue, he Vienna-Copenhagen Conference on Financial Economerics, and he 2017 IBEFA Summer Meeing, including our discussan Azama Abdymomunov, 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, he Office of Financial Research, CREATES a Aarhus Universiy, he Bank of Canada, he Federal Reserve Bank of San Francisco, he IMF, and he Swiss Naional Bank for helpful commens. Furhermore, we are graeful o Jean-Sébasien Fonaine, Jose Lopez, and Thomas Merens for helpful commens and suggesions on earlier drafs of he paper. Finally, Kevin Cook deserves a special acknowledgemen for ousanding research assisance during he iniial phase of he projec. 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 and Business Economics, Aarhus Universiy and CREATES, Denmark, phone: ; 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. Amazon; simonruw@gmail.com. This version: Ocober 10, 2017.

3 1 Inroducion In 1997, he U.S. Treasury sared o issue inflaion-indexed bonds, which are now commonly known as Treasury inflaion-proeced securiies (TIPS). The marke for TIPS has seadily expanded since hen and had a oal ousanding noional amoun of $973 billion, or 8.2 percen of all markeable deb issued by he Treasury, by he end of Despie he large size of he TIPS marke, an overwhelming amoun of research suggess ha TIPS are less liquid han Treasury securiies wihou inflaion indexaion commonly referred o simply as Treasuries. Fleming and Krishnan(2012) repor marke characerisics of TIPS ha indicae smaller rading volume, longer urnaround ime, and wider bid-ask spreads han observed in Treasuries(see also Sack and Elsasser (2004), Campbell e al. (2009), Dudley e al. (2009), and Gürkaynak e al. (2010), among many ohers). These facors are likely o raise he implied yields from TIPS because invesors generally require compensaion for carrying liquidiy risk. However, he size of his TIPS liquidiy premium remains a opic of debae because i canno be direcly observed. A leas hree idenificaion schemes have been considered in he lieraure o esimae he TIPS liquidiy premium so far. The work of Fleckensein e al. (2014) uses marke prices on TIPS and inflaion swaps o documen sysemaic mispricing of TIPS relaive o Treasuries, which may be inerpreed as a liquidiy premium in TIPS. Their approach relies on a liquid marke for inflaion swaps, bu his assumpion seems quesionable given ha U.S. inflaion swaps have low rading volumes and wide bid-ask spreads (see Fleming and Sporn (2013) and Chrisensen and Gillan (2012)). The second idenificaion scheme approximaes inflaion expecaions in he TIPS marke by hose from surveys (e.g., D Amico e al. (2014)). Bu inflaion expecaions from surveys are unavailable in real ime and may easily differ from he desired expecaions of he marginal invesor in he TIPS marke. The final idenificaion scheme relies on a se of observable characerisics for he TIPS marke (e.g., marke volume) as noisy proxies for liquidiy risk (see, e.g., Abrahams e al. (2016) and Pflueger and Viceira (2016)). The accuracy of his approach is clearly dependen on having good proxies for liquidiy risk, which in general is hard o ensure. The presen paper inroduces a new idenificaion scheme for he TIPS liquidiy premium wihin a dynamic affine erm srucure model (ATSM) for nominal and real yields. Our model idenifies liquidiy risk direcly from individual TIPS prices by accouning for he ypical marke phenomenon ha many TIPS go ino buy-and-hold invesors porfolios as ime passes. This in urn limis he number of securiies available for rading and hence increases he liquidiy risk. We formally accoun for his effec by pricing each TIPS using a sochasic discoun facor wih a unique bond-specific erm ha reflecs he added compensaion invesors demand for buying a bond wih low expeced fuure liquidiy. A key implicaion of he proposed model is ha liquidiy risk is idenified from he implied price differenial of oherwise idenical principal and coupon paymens. Individual TIPS prices are herefore suf- 1

4 ficien o esimae he TIPS liquidiy premium wihin our model, meaning ha we avoid he limiaions associaed wih he exising idenificaion schemes in he lieraure. The proposed idenificaion scheme is hus relaed o he approach aken in Fonaine and Garcia (2012), as hey also exploi he relaive price differences of very similar coupon bonds o esimae a liquidiy premium in Treasuries, alhough our model and is applicaion differ along oher dimensions from he analysis in Fonaine and Garcia (2012). The proposed model is esimaed based on TIPS prices and a sandard sample of nominal Treasury yields from Gürkaynak e al. (2007), boh covering he period from mid-1997 hrough he end of To ge a clean read of he liquidiy facor, we also accoun for he deflaion proecion opion embedded in TIPS during he esimaion using formulas provided in Chrisensen e al. (2012). Our main analysis uses TIPS prices and Treasury yields wihin he commonly considered en-year mauriy specrum, where our key findings are as follows. Firs, he average liquidiy premium for TIPS is sizable and fairly volaile, wih a mean of 38 basis poins and a sandard deviaion of 34 basis poins. To suppor he proposed idenificaion scheme, we also show ha he esimaed liquidiy premium is highly correlaed wih well-known observable proxies for liquidiy risk such as he VIX opions-implied volailiy index, he on-he-run spread on Treasuries, and he TIPS mean absolue fied errors from Gürkaynak e al. (2010). Second, we find a large improvemen in he abiliy of our ATSM o fi individual TIPS prices by accouning for liquidiy risk. The roo mean-squared error of he fied TIPS prices convered ino yields o mauriy falls from 14.6 basis poins o jus 4.9 basis poins when he liquidiy facor is included, meaning ha TIPS pricing errors are a he same low level as found for nominal yields. Third, by accouning for liquidiy risk, he proposed model avoids he well-known posiive bias in real yields, and hence he negaive bias in breakeven inflaion, i.e., he difference beween nominal and real yields of he same mauriy. This implies ha he proposed model does no predic spells of deflaion fears during our sample, conrary o he resuls obained when ignoring liquidiy risk. Fourh, he model-implied forecass of one-year CPI inflaion are grealy improved by correcing for liquidiy risk in TIPS, and so is he abiliy of he model o mach inflaion expecaions from surveys. We emphasize ha he improved abiliy of he proposed model o forecas inflaion and mach inflaion surveys is obained wihou including any of hese series in he esimaion. The remainder of he paper is srucured as follows. Secion 2 provides reduced-form evidence on he liquidiy risk of TIPS, while Secion 3 inroduces he general ATSM framework and he specific Gaussian model we use o accoun for he liquidiy disadvanage of TIPS. The model is esimaed in Secion 4, while Secion 5 sudies he esimaed liquidiy premium in he TIPS marke. Is robusness is explored in Secion 6, while Secion 7 sudies he liquidiyadjused real yield curve and he implied inflaion forecass from he proposed model. Secion 8 concludes and offers direcions for fuure research. Addiional echnical deails are provided in wo appendices a he end of he paper and a supplemenary online appendix. 2

5 2 The Dynamics of TIPS Liquidiy Risk Building on he work of Amihud and Mendelson (1986), we define liquidiy risk as he cos of immediae execuion. For insance, if a bond holder is forced o liquidae his posiion premaurely a a disadvanageous price compared wih he mid-marke quoe, hen his price differenial reflecs he liquidiy cos. Given his definiion, i is well-esablished ha liquidiy risk in Treasuries is decreasing wih (i) high marke volume (Garbade and Silber (1976)), (ii) high compeiion among marke makers (Tinic and Wes (1972)), and (iii) high marke deph (Goldreich e al. (2005)). A commonly used observable proxy for liquidiy risk is he implied yield spread from he bid and ask prices. Thesespreads are repored in Figure 1 for each of he four TIPScaegories issued in he U.S. The spreads from Bloomberg appear unreliable before he spring of 2011, and we herefore resric our analysis in his secion o a weekly sample from May 2011 o December The op row in Figure 1 repors he bid-ask spreads for he mos recenly issued (on-he-run) five- and en-year TIPS and for he corresponding mos seasoned TIPS wih a leas wo years o mauriy. We highligh wo resuls from hese chars. Firs, he bid-ask spreads for seasoned five- and en-year TIPS are sysemaically above hose of newly issued TIPS. Second, he bid-ask spreads on seasoned TIPS are around 4 basis poins and hence of economic significance. In comparison, he bid-ask spreads for he en-year Treasuries issued beween 2011 and 2016 have an average of only 0.4 basis poins, i.e., a facor 10 smaller han he corresponding spread in he TIPS marke. 1 The boom par of Figure 1 reveals ha we generally see he same paern for weny- and hiry-year TIPS, alhough he bidask spreads for newly issued securiies here are somewha noisy due o he few raded bonds in his par of he mauriy specrum. Figure 1 herefore reveals ha TIPS carry sizable liquidiy risk, which is higher for seasoned TIPS han for newly issued securiies. We nex es for he saisical significance of he posiive relaionship beween liquidiy risk and he age of a bond. The considered panel regression is given by Spread i = τ +α i +β 1 Noional i +β 2Age i +εi, (1) where he bid-ask spread for he ih TIPS in period is denoed Spread i. To conrol for unobserved heerogeneiy, we allow for boh ime fixed effecs τ and bond-specific fixed effecs α i in equaion (1). As argued by Garbade and Silber (1976), securiies wih large ousanding noional amouns ofen have greaer rading volumes, and we herefore also include he noional value of each securiy Noional i, which for TIPS grows over ime wih CPI inflaion. Finally, Age i measures he ime since issuance of he ih TIPS and ε i is a zero-mean error erm. Using all available securiies, we hen esimae he panel regression 1 The average bid-ask spreads for he individual en-year Treasury noes issued in January of 2011, 2012, 2013, 2014, 2015, and 2016 are 0.52, 0.42, 0.38, 0.28, 0.26, and 0.21 basis poins, respecively. 3

6 Rae in basis poins Mos seasoned Mos recenly issued Rae in basis poins Mos seasoned Mos recenly issued (a) Five-year TIPS (b) Ten-year TIPS Rae in basis poins Mos seasoned Mos recenly issued Rae in basis poins Mos seasoned Mos recenly issued (c) Tweny-year TIPS (d) Thiry-year TIPS Figure 1: TIPS Bid-Ask Spreads For he five- and en-year TIPS, he bid-ask spreads are compued using he mos recenly issued TIPS and he corresponding mos seasoned TIPS wih a leas wo years o mauriy. For he weny-year TIPS, he series are obained by racking he bid-ask spreads of he same wo weny-year TIPS over he period due o he few issuances a his mauriy. For he hiry-year TIPS, he bid-ask spread for he mos seasoned TIPS is ha of he firs hiry-year TIPS issued back in 1998, while he mos recenly issued series racks he bid-ask spread of he newes hiry-year TIPS. All series (measured in basis poins) are weekly covering he period from May 31, 2011, o December 30, 2016, and smoohed by a four-week moving average o faciliae he ploing. in equaion (1) by OLS separaely wihin each of he four TIPS caegories. Table 1 shows ha TIPS wih a larger ousanding noional value have significanly lower bid-ask spreads and, more imporanly, ha he age of a securiy has a significan posiive effec on he bidask spread, as also suggesed by Figure 1. The laer resul is obviously very similar o he 4

7 n TIPS N Noional i 1,000 Agei No. of parameers adj R 2 5-year TIPS 9 1, (0.0037) 10-year TIPS 28 5, (0.0025) 20-year TIPS 5 1, (0.053) 30-year TIPS 10 2, (0.0022) (0.259) (0.046) (0.264) (0.008) Table 1: Panel Regression: The Bid-Ask Spread in he TIPS Marke This able repors he resuls of separaely esimaing equaion (1) by OLS for each of he four caegories of TIPS. The esimaed loadings for he ime and bond-specific fixed effecs are no provided. The variable Noional i is measured in millions of dollars and Age i in years since issuance. Whie s heeroscedasic sandard errors are repored in parenheses. Aserisks * and ** indicae significance a he 5 percen and 1 percen levels, respecively. The adjused R 2 is compued based on he demeaned variaion in dependen variable. The daa used are weekly covering he period from May 31, 2011, o December 30, well-known finding in he Treasury marke, where newly issued securiies also are more liquid han exising bonds (see, for insance, Krishnamurhy (2002), Gurkaynak e al. (2007), and Fonaine and Garcia (2012), among many ohers). However, hese spreads are much wider for TIPS compared wih Treasuries. Therefore, unlike he Treasury marke, i is crucial o accoun for his dynamic paern in TIPS liquidiy o fully undersand he price dynamics in he TIPS marke. We draw wo conclusions from hese reduced-form regressions. Firs, curren liquidiy in he TIPS marke exhibi noable variaion over ime, and liquidiy herefore represens a risk facor o bond invesors in his marke, as also emphasized in he work of Gürkaynak e al. (2010), Fleming and Krishnan (2012), and Fleckensein e al. (2014) among ohers. Second, seasoned TIPS are less liquid han more recenly issued securiies wihin he same mauriy caegory. Alhough equaion (1) does no provide an explanaion for his dynamic paern in liquidiy, anecdoal evidence suggess ha i mos likely arises because increasing amouns of he securiies ge locked up in buy-and-hold invesors porfolios as ime passes and becomes unavailable for rading. 2 The objecive in he presen paper is no o provide a more deailed explanaion for his dynamic paern in liquidiy bu insead o examine is asse pricing implicaions. The effec we wan o explore is based on he assumpion ha raional and forward-looking invesors are aware of his dynamic paern in liquidiy and herefore demand compensaion for holding bonds wih low fuure liquidiy. The ATSM we propose in he nex secion formalizes his effec and quanifies how curren TIPS prices are affeced by expeced fuure TIPS liquidiy. 2 See, for insance, he evidence provided in Sack and Elsasser (2004), which indicaes ha he primary paricipans in he TIPS marke are large insiuional invesors (e.g., pension funds and insurance companies) wih long-erm real liabiliy risks ha hey wan o hedge. 5

8 3 An ATSM of Nominal and Real Yields wih Liquidiy Risk This secion inroduces a general class of ATSMs of nominal and real bond prices ha accouns for liquidiy risk. We formally presen he model framework in Secion 3.1 and describe a Gaussian version of i in Secion 3.2. The proposed idenificaion sraegy of liquidiy risk is hen compared wih he exising lieraure in Secion A Canonical ATSM wih Liquidiy Risk As commonly assumed, he insananeous nominal shor rae r N is given by r N = ρ N 0 + ( ρ N ) X x, where ρ N 0 is a scalar and ρn x is an N 1 vecor. The dynamics of he N pricing facors in X wih dimension N 1 evolve as dx = K Q x ( ) θx Q X d+σ x Sx, d W Q, (2) where W Q is a sandard Wiener process in R N under he risk-neural measure Q and S is an N-dimensional diagonal marix. Is elemens are given by [S x, ] k,k = [δ 0 ] k + δ x,k X for k = 1,2,...,N, where [δ 0 ] k denoes he kh enry of δ 0 wih dimension N 1. Hence, θ Q and δ x,k are N 1 vecors, whereas K Q and Σ x have dimensions N N. Absence of arbirage implies ha he price of a nominal zero-coupon bond mauring a ime +τ is given by P N (τ) = exp { A N (τ)+b N (τ) X }, (3) where he funcions A N (τ) and B N (τ) saisfy well-known ordinary differenial equaions (ODEs); see, for insance, Dai and Singleon (2000). The price of bonds wih paymens indexed o inflaion (i.e., real bonds) may in principle be obained in a similar manner by leing he insananeous real shor rae be affine in he pricing facors, as done in Adrian and Wu (2010) and Joyce e al. (2010) among ohers. An implici assumpion wihin his classic asse pricing framework is ha bonds are rading in a fricionless marke wihou any supply- or demand-relaed consrains. This is ofen a reasonable assumpion for Treasuries due o he large size of his marke and is low bid-ask spreads. 3 This assumpion is much more debaable for TIPS, as seen from he wide bid-ask spreads in Secion 2. The main innovaion of he presen paper is o relax he assumpion of a fricionless marke for real bonds in ATSMs and explicily accoun for he dynamic paern in TIPS 3 A minor excepion relaes o he small liquidiy spread beween newly issued Treasuries ha are onhe-run and somewha older off-he-run bonds; see, for insance, Krishnamurhy (2002), Gürkaynak e al. (2007), and Fonaine and Garcia (2012). 6

9 liquidiy documened in Secion 2. Inspired by he work of Amihud and Mendelson (1986), our conribuion is o price TIPS by a real rae ha accouns for liquidaion coss, which we specify for he ih TIPS as h( 0 ;i)x liq. The firs erm h( 0 ;i) is a deerminisic funcion of ime since issuance 0 of he ih TIPS and serves o capure he empirical regulariy from Secion 2 ha liquidaion coss (i.e. he bid-ask spread) increase as he bond approaches mauriy. Here, we iniially only assumeha h( 0 ;i) is bounded, nonnegaive, and increasing in 0. The second erm in our specificaion of liquidaion coss is a laen facor X liq, which is included o capure he cyclical variaion in hese coss, as is eviden from he bid-ask spreads in Figure 1. Hence, we sugges accouning for liquidiy risk by discouning fuure cash flows from he ih TIPS using a liquidiy-adjused insananeous real shor rae of he form r R,i = ρ R 0 + ( ρ R x) X }{{} fricionless real rae +h( 0 ;i)x liq }{{}, liquidiy adjusmen (4) where ρ R 0 is a scalar and ρr x is an N 1 vecor. The firs erm in rr,i is he radiional affine specificaion for he fricionless par of he real rae, which is common o all TIPS, whereas he liquidiy adjusmen varies across securiies. The laer implies ha we will price TIPS using a bond-specific insananeous real shor rae or, equivalenly, a bond-specific sochasic discoun facor when combining equaion (4) wih a disribuion for he marke prices of risk. [ ], Leing Z X X liq he dynamics of his exended sae vecor is assumed o be dz = K Q z (θ Q z Z ) d+σ z Sz, dw Q, (5) wherew Q isasandardwiener processinr N+1. Similarly, K Q z, θq z, S z,, andσ z areappropriae exensions of he corresponding marices relaed o equaion (2). Thus, our specificaion in equaion (5) accommodaes he case where he liquidiy facor is resriced o only aain nonnegaive values, as assumed in Abrahams e al. (2016), by leing X liq follow a squareroo process ha eners in S z, o deermine he condiional volailiy in Z. Anoher and less resricive specificaion is o omi X liq in S z, and allow he liquidiy facor o occasionally aain negaive values and hence accommodae episodes wih negaive liquidiy risk. 4 For his second specificaion, he esimaed ime series of X liq may serve as an indirec es of he model s abiliy o capure liquidiy risk, as we predominanly expec X liq o be posiive. From he Feynman-Kac heorem and equaions (4) and (5), i follows ha he price a ime of a real zero-coupon bond mauring a ime T is given by P R,i ( 0,,T) = exp { A R,i ( 0,,T)+B R,i ( 0,,T) Z }, (6) 4 This corresponds o periods when an invesor pays o hold liquidiy risk. This may happen when a bond helps an invesor (e.g., a pension fund) o hedge some of his liabiliies. 7

10 whendiscouningcashflowsrelaedoheihtips.thefuncionsa R,i ( 0,,T)andB R,i ( 0,,T) wih dimensions (N +1) 1 saisfy he ODEs A R,i B R,i ( 0,,T) = δ0 R ( Kz Q ) B θq z R,i ( 0,,T) 1 2 [ ] ( 0,,T) = δ R x h( 0 ;i) N+1 k=1 + ( Kz Q ) B R,i ( 0,,T) 1 2 [ Σ z B R,i ( 0,,T) ] 2 k δ 0,k, (7) N+1 k=1 [ Σ z B R,i ( 0,,T) ] 2 k δ z,k (8) wih he erminal condiions A R,i ( 0,T,T) = 0 and B R,i ( 0,T,T) = 0. Here, [a] 2 k denoes he [ ] squared kh elemen of vecor a and δ z δ x δ x liq wih dimensions (N +1) 1. Thus, he price of a real zero-coupon bond is exponenially affine in Z even when accouning for liquidiy risk by using he modified real shor rae in equaion (4). The implied breakeven inflaion rae from equaions (3) and (6) is given by y N (τ) y R,i ( 0,,τ) = AR,i ( 0,,+τ) A(τ) τ τ [ B(τ) X + BR,i ( 0,,+τ) X τ τ X liq where y N (τ) 1 τ logpn (τ) and y R,i ( 0,,τ) 1 τ logpr,i ( 0,,+τ) denoe he yield o mauriy from nominal and real bonds, respecively, wih T + τ. Hence, X liq can also be viewed as capuring he relaive liquidiy difference beween Treasuries and TIPS. In his respec, our model is similar o he work of D Amico e al. (2014) and Abrahams e al. (2016), who also use a single facor o capure he relaive liquidiy differenial of TIPS compared wih Treasuries. We also noe ha he bond prices in equaion (6) depend on he calender ime, which eners as a sae variable in our model o deermine he ime since issuance 0 of a given securiy and hence is liquidiy adjusmen. This propery of our model is similar o he class of calibraion-based erm srucure models daing back o Ho and Lee (1986) and Hull and Whie (1990), where he drif is a deerminisic funcion of calendar ime and repeaedly recalibraed o perfecly mach he curren yield curve. These calibraion-based models are known o be ime-inconsisen, as he fuure drif a +τ is repeaedly modified unil reaching ime +τ. Our model does no suffer from he same shorcoming because we only use calender ime o deermine he liquidiy adjusmen and no o change any dynamic model parameers. The model is closed by adoping he exended affine specificaion for he marke prices of risk Γ, as described by Cheridio e al. (2007). We summarize our model presenaion by exending he classificaion scheme of Dai and Singleon (2000) o our ATSM, which is referred o as A L m (N +1). Tha is, we consider N fricionless pricing facors and one facor for he liquidiy risk of TIPS, as indicaed by he superscrip L. Among he N + 1 pricing facors, we allow for m variance-influencing facors and impose he same resricions for he model o be admissible (i.e., well-defined) as in Dai ], 8

11 and Singleon (2000) A Gaussian ATSM wih Liquidiy Risk We nex analyze a paricular Gaussian version of our model wih closed-form expressions for liquidiy-adjused real bond prices. Beyond providing useful inuiion on he liquidiy adjusmen, we also noe ha his special case of our model should be paricularly ineresing given he well-known success of Gaussian models in maching yields and risk premia, as also exploied in D Amico e al. (2014) and Abrahams e al. (2016). To faciliae he inerpreaion of our Gaussian model, we consider he familiar case where facor loadings for nominal yields and he fricionless par of real yields represen level, slope, and curvaure componens. This is done by using he parameerizaion in Chrisensen e al. (2010), which represens an exension of he arbirage-free Nelson-Siegel specificaion derived in Chrisensen e al. (2011) o a join model for nominal and real yields. Saring wih he nominal shor rae, i is defined as r N = L N +S, (9) where L N is he level facor of nominal yields and S is he slope facor. The parameerizaion of he liquidiy-adjused real shor rae for he ih TIPS is given by r R,i = L R +α R S +β i (1 e λl,i ( 0 ) )X liq. (10) The firs par L R +α R S consiues he fricionless real rae using he specificaion adoped in Chrisensen e al. (2010). The variable L R represens he level facor of real yields and is absen in he expression for nominal yields. This specificaion is consisen wih nominal yields conaining a hidden facor ha is observable from real yields and inflaion expecaions (see Chernov and Mueller (2012)). Noe also ha, for simpliciy, we define he real slope facor as α R S wih α R R based on he empirical evidence in Chrisensen e al. (2010). The adoped funcional form for h( 0 ;i) conrolling he liquidiy adjusmen is given by he parsimonious specificaion β i (1 e λl,i ( 0 ) ), where β i 0 and λ L,i 0. To provide some inerpreaion of β i and λ i, i is useful o hink of he rading aciviy in he ih TIPS as aking place in wo phases. The firs phase may be characerized by a large supply of bonds jus afer bond issuance, bu also srong demand pressure from buy-and-hold invesors, who gradually purchase a large fracion of he ousanding securiies. The second phase hen sars when buy-and-hold invesors have acquired heir share of he ih TIPS and he number of securiies available for rading has become relaively scarce. Given his caegorizaion of he rading cycle, he value of λ L,i deermines he lengh of he firs phase, where exposure o 5 I is sraighforward o verify ha he proposed specificaion o accoun for liquidiy risk can be exended o nonlinear dynamic erm srucure models. Secion 6.5 provides one illusraion of such an exension by incorporaing he zero lower bound on nominal ineres raes ino he model. 9

12 X liq is fairly low. Tha is, a low value of λ L,i implies ha his firs phase of bond rading is fairly long, whereas a high value of λ L,i means ha his firs phase of bond rading is much shorer. 6 The value of β i deermines he maximal exposure of he ih TIPS o he liquidiy facor X liq in he second phase, which appears when e λl,i ( 0 ) 0. I is obvious ha more sophisicaed specificaions of h( 0 ;i) may be considered, as opposed o he one used in equaion (10), alhough such exensions are no explored in he curren paper. [ ], Leing Z L N S C L R X liq we consider Q dynamics of he form [ ] [ ] K Q dz = x θ Q x κ Q liq }{{} θ Q Z liq }{{} K Q z θ Q z +Σ z dw Q, (11) where θx Q = due o he adoped normalizaion scheme. Following Chrisensen e al. (2010), wele [ Kx Q ] 2,2 = [ Kx Q ] 3,3 = λ and[ Kx Q ] = λfor λ > 0, wih all remainingelemens 2,3 of he 4 4 marix Kx Q equal o zero. This ensures ha he facor loadings represen level, slope, and curvaure componens in he nominal and real yield curves provided [ Kz Q ] 5,i = 0 for i = {1,2,3,4}. The nex resricions [ Kz Q = 0 for i = {1,2,3,4} imply ha Xliq ]i,5 eiher operaes as a level or slope facor depending on he value of κ Q liq, alhough hese resricions could be relaxed wihou alering he inerpreaion of he four fricionless facors. Tha is, ourparameerizaion doesnoaccommodaeacurvauresrucureforx liq, whichisconsisen wih our reduced-form evidence in Secion 2 ha older TIPS are more affeced by liquidiy risk han are newly issued securiies. Using equaions (9) and (11), he yield a ime for a nominal zero-coupon bondmauring a +τ is easily shown o have he well-known srucure from he saic model of Nelson and Siegel (1987) ( ) ( 1 e y N λτ 1 e λτ (τ) = LN + S + λτ λτ e λτ ) C AN (τ), (12) τ where A N (τ) is an addiional convexiy adjusmen provided in Chrisensen e al. (2011). The price for he ih real zero-coupon bond mauring a ime T is given by equaion (6) wih he closed-form expression for A R,i ( 0,,T) and B R,i ( 0,,T) provided in Appendix A. To faciliae he inerpreaion of his soluion, consider he implied yield o mauriy on he ih 6 For insance, a shor iniial rading phase may coincide wih he bond ceasing o be he mos recenly issued TIPS wihin is mauriy range and hence is no longer on-he-run. 10

13 real zero-coupon bond, which we wrie as ( ( y R,i 1 e ( 0,,τ) = L R +α R λτ 1 e )S +α R λτ e )C λτ λτ λτ }{{} fricionless loadings +β i 1 e κq liq τ κ Q liq τ e λl,i ( 0 ) 1 e (κ Q liq +λl,i )τ ) X (κ liq Q liq +λl,i τ }{{} liquidiy adjusmen AR,i ( 0,,τ), }{{ τ } convexiy adjusmen (13) where τ = T. The firs hree erms in y R,i ( 0,,τ) capure he fricionless par of real yields, where he facor loadings have he same familiar inerpreaion as in equaion (12) due o he imposed srucure on r R,i and is dynamics under Q. The nex erm in equaion ( }) ) (13) represens an adjusmen for liquidiy risk. Is firs erm 1 exp { κ Q liq τ / (κ Q liq τ describes he maximal effec of liquidiy risk, which is obained when e λl,i ( 0 ) 0 and he ih bond has full exposure o variaion in X liq. This upper limi for liquidiy risk clearly operaes as a radiional slope facor for κ Q liq > 0, where he size of he liquidiy adjusmen is decreasing in τ and hence increasing in as he bond approaches mauriy. We have he opposie paern when κ Q liq < 0, whereas he upper limi for liquidiy risk is consan in wih κ Q liq 0, i.e., a level facor. In oher words, he Q dynamics of Xliq deermines he erm srucure for he maximal effec of liquidiy risk. The second erm in he liquidiy adjusmen in equaion(13) serves as a negaive correcion ( }) ) o 1 exp { κ Q liq τ / (κ Q liq τ during he iniial phase wih large rading volume, where buy-and-hold invesors have no acquired a large proporion of he ih bond. The erm β i e λl,i ( 0 ) is clearly decreasing in, whereas he remaining erm is similar o he one for he maximal effec of liquidiy risk (excep wih decay parameer κ Q liq + λl,i ), and hence ypically increasing in. As a resul, he liquidiy adjusmen in equaion (13) may eiher increase or decrease in, depending on he relaive values of κ Q liq and λl,i. Focusing on he mos plausible parameerizaion wih κ Q liq > 0, he combined loading on Xliq is clearly posiive, meaning ha liquidiy risk increases he real yield whenever X liq > 0. 7 Hence, our model capures he effec ha forward-looking invesors require compensaion for carrying he risk ha low fuure liquidiy reduces he price of he bond if sold before mauriy. This in urn reduces he curren bond price or, equivalenly, raises he curren yield. An effec which is documened empirically for Treasuries in Goldreich e al. (2005). On he oher hand, liquidiy risk is absen if β i = 0 and 7 I is also sraighforward o show ha he combined loading on X liq remains posiive even if κ Q liq < 0, provided 0 λ L,i < κ Q liq. 11

14 real yields in equaion (13) herefore simplify o hose in he fricionless model of Chrisensen e al. (2010). Finally, in Gaussian models he exended affine specificaion for he marke prices of risk reduces o he essenial affine parameerizaion of Duffee (2002). Hence, we have Γ = Σ 1 z (γ 0 +γ z Z ), (14) where γ 0 and γ z have dimensions (N +1) 1 and (N +1) (N +1), respecively. We denoe his Gaussian version of our model as he G L (5) model. I is he focus of he remaining par of he presen paper and will be compared exensively wih he model of Chrisensen e al. (2010), denoed he G(4) model, which has he same fricionless dynamic facor srucure, bu omis accouning for TIPS liquidiy risk. 3.3 Idenificaion of Liquidiy Risk and he Exising Lieraure As described above, he proposed model discouns coupon and principal paymens from TIPS using bond-specific real shor raes, which only differ in heir loadings on he common liquidiy facor X liq. This implies ha liquidiy risk in TIPS is idenified from he implied price differenial of oherwise idenical cash flow paymens or equivalenly, he degree of mispricing based on he fricionless par of our ATSM. This is a very direc measure of liquidiy risk ha only requires a panel of marke prices for TIPS, which is readily available. The proposed idenificaion scheme based on marke prices is herefore closely relaed o he one by Fleckensein e al. (2014), who use marke prices on TIPS and inflaion swaps o documen sysemaic mispricing of TIPS relaive o Treasuries, which may be inerpreed as reflecing liquidiy premiums in TIPS. The approach of Fleckensein e al. (2014) relies on a liquid marke for inflaion swaps, bu as argued by Abrahams e al. (2016) his assumpion is quesionable for he U.S. given he low rading volumes and wide bid-ask spreads in he U.S. inflaion swap marke (see Fleming and Sporn (2013) and Chrisensen and Gillan (2012)). The idenificaion sraegy we propose does no rely on a well-funcioning and liquid marke for inflaion swaps bu insead uses an ATSM o idenify liquidiy risk solely from TIPS marke prices and a sandard panel of Treasury yields. Anoher commonly adoped procedure o idenify liquidiy risk is o regress breakeven inflaion from TIPS on inflaion expecaions from surveys and various proxies for liquidiy risk (see Gürkaynak e al. (2010) and Pflueger and Viceira (2016) among ohers). In conras o our idenificaion sraegy, such reduced-form esimaes do no accoun for he inflaion risk premia in breakeven inflaion, which ofen is sizable and quie volaile (see for insance D Amico e al. (2014) and Abrahams e al. (2016)). Obviously, he idea of relaing liquidiy risk o a limied supply of cerain bonds is no unique o our paper. For insance, Amihud and Mendelson (1991) consider he case where an 12

15 increasing fracion of Treasury noes are locked away in invesors porfolios o explain he yield differenial in Treasury noes and bills of he same mauriy. Anoher example is provided by Keane (1996), who uses a similar explanaion for he repo specialness of Treasuries. From his perspecive, our main conribuions are o incorporae effecs of a limied bond supply in an arbirage-free ATSM and o show how his effec may explain he liquidiy disadvanage of TIPS. Our model is also relaed o he ATSM of D Amico e al. (2014), where real bonds are discouned wih a modified real rae common o all TIPS, conrary o our specificaion in equaion (4) where each TIPS is priced using is own unique real rae. We also noe ha he model of D Amico e al. (2014) beyond nominal and real zero-coupon yields requires ime series dynamics of CPI inflaion and especially is expeced fuure level from surveys o properly idenify liquidiy risk. Bu inflaion expecaions from surveys are unavailable in real ime and may differ from he expecaions of he marginal invesor in he TIPS marke. The idenificaion scheme we propose avoids hese well-known limiaions of inflaion surveys by solely idenifying he TIPS liquidiy premium from individual TIPS prices. Anoher closely relaed paper is he one by Abrahams e al. (2016), which also relies on an ATSM o esimae liquidiy risk in TIPS. They ake he liquidiy facor o be observed and consruced from (i) he TIPS mean absolue fied errors from Gürkaynak e al. (2010) and (ii) a measure of he relaive ransacion volume beween Treasuries and TIPS. This alernaive and somewha more indirec approach o idenify he TIPS liquidiy premium relies heavily on having good observable proxies for liquidiy risk, which in general is hard o ensure. The idenificaion scheme of Abrahams e al. (2016), however, is similar o ours by no relying on CPI inflaion or inflaion expecaions from surveys o esimae he TIPS liquidiy premium. An imporan similariy beween our approach and he ones considered in D Amico e al. (2014) and Abrahams e al. (2016) is o include he liquidiy facor in he se of pricing facors when deriving TIPS prices based on no-arbirage condiions. Fonaine and Garcia (2012) adop he opposie approach when sudying he off-he-run liquidiy spread in Treasuries, as hey omi he liquidiy facor from he se of pricing facors in heir ATSM. This implies ha he model of Fonaine and Garcia (2012) allows for arbirage wihin bond prices, where liquidiy risk is idenified from he relaive price differences of mauring coupon bonds. We acknowledge ha marke paricipans may no always be able o exploi all arbirage opporuniies due o financial fricions, as emphasized by Fonaine and Garcia (2012), bu we neverheless find i useful o discipline our model by imposing no-arbirage requiremens, as ypically done in he lieraure. Overall, our model offers a new and very direc way o idenify liquidiy risk in ATSMs wihou including addiional informaion from inflaion swaps, CPI inflaion, or inflaion 13

16 Time o mauriy in years Number of securiies All TIPS All five and en year TIPS Sample of five and en year TIPS (a) Disribuion of TIPS (b) Number of TIPS Figure 2: Overview of he TIPS Daa Panel (a) shows he mauriy disribuion of all TIPS issued since The solid grey recangle indicaes he sample used in our benchmark analysis, where he sample is resriced o sar on July 11, 1997, and limied o TIPS prices wih less han en years o mauriy a issuance and more han wo years o mauriy afer issuance. Panel (b) repors he number of ousanding TIPS a a given poin in ime for various samples. surveys. 8 4 Empirical Findings As menioned above, he proposed model is consruced for a sample of TIPS marke prices in addiion o a sandard panel of Treasury yields. Given ha ATSMs are rarely esimaed direcly on marke prices for coupon bonds, we firs describe our daa se and esimaion procedure in Secions 4.1 and 4.2, respecively, before presening he esimaion resuls in Secion Daa TIPS have been available in he five- o hiry-year mauriy range since 1997, alhough only en-year TIPS have been issued regularly. Panel (a) in Figure 2 shows he remaining ime o mauriy of all TIPS a a given dae for all 50 bonds in our sample ending in Our empirical applicaion is primarily devoed o he en-year mauriy specrum as in D Amico e al. (2014) and Abrahams e al. (2016), excep for he robusness analysis in Secion 6.3. This reduces he considered number of TIPS o n TIPS = 38. The evoluion in he number of 8 We sress for compleeness ha our model and he subsequen esimaion approach in Secion 4 is sufficienly general o include such addiional informaion if desired. 14

17 ousanding TIPS is shown in Panel (b) of Figure 2 for all mauriies (he red line) and for he en-year mauriy specrum (he grey line). Given ha TIPS prices near mauriy end o exhibi erraic behavior due o seasonal variaion in CPI inflaion, we exclude TIPS from our sample when hey have less han wo years o mauriy. 9 Using his cuoff reduces he number of TIPS in our sample furher, as shown by he grey recangle in Panel (a) and he solid black line in Panel (b) of Figure 2. We use he clean mid-marke TIPS prices as repored each Friday by Bloomberg. 10 Given ha our model has wo pricing facors specific o TIPS, reliable idenificaion of hese facors requires a leas wo TIPS prices. This dicaes he sar of our weekly sample on July 11, 1997, when he second ever TIPS (wih five years o mauriy) becomes available. Our sample ends on December 27, Finally, he considered panel of nominal zero-coupon yields are aken from Gürkaynak e al. (2007), where we include he following n y = 12 mauriies: 3-monh, 6-monh, 1-year, 2- year,..., 10-year. We use weekly daa and limi our sample o he same period as considered for TIPS. 4.2 Esimaion Mehodology We esimae he G L (5) model using he convenional likelihood-based approach, where we exrac he laen pricing facors from he observables, which in our case are nominal zerocoupon yields and TIPS marke prices. The funcional form for nominal yields is provided in equaion (12), whereas he expression for he price of he ih TIPS is more evolved and given by P R,i ( 0,,T) = C ( 1 ) { } exp ( 1 )y R,i ( 0,, 1 ) 2 1/2 n C { } + 2 exp ( k )y R,i ( 0,, k ) k=2 +exp { } (T )y R,i ( 0,,T) +DOV [ Z ;T, Π ], Π 0 where Π /Π 0 is he accrued CPI inflaion compensaion since issuance of he ih TIPS. Tha is, a ime we use he liquidiy-adjused real yields in equaion (13) o discoun he coupon paymens aached o he ih bond. 11 The las erm in equaion (15) accouns for he deflaion opion value (DOV) embedded in TIPS, meaning ha he principal a mauriy is only adjused for inflaion if accumulaed inflaion since issuance of he bond is posiive. We 9 A similar procedure is used in Gürkaynak e al. (2010), who omi TIPS wih 18 monhs o mauriy and linearly downweigh TIPS wih 18 o 24 monhs o mauriy. Secion 6.2 explores he sensiiviy of our resuls o gradually including more observaions for each TIPS as i approaches mauriy. 10 If prices are unavailable on a paricular Friday, we use he repored price on he las rading day before his Friday. 11 The implemenaion here is grealy simplified by he coninuous-ime formulaion of our model. For discree-ime models wih one period exceeding one day (say, a week or a monh), sandard inerpolaion schemes may be used o price he coupon paymens relaed o he ih bond a ime. (15) 15

18 compue he value of his opion as oulined in Chrisensen e al. (2012). 12 Following Joslin e al. (2011), all nominal yields in equaion (12) have independen Gaussian measuremen errors ε i y, wih zero mean and a common sandard deviaion σ y, denoed ε i y, NID ( 0,σy 2 ) for i = 1,2,...,n y. We also accoun for measuremen errors in he price of each TIPS hrough ε i TIPS,, where εi TIPS, NID( 0,σ 2 TIPS) for i = 1,2,...,nTIPS. To ensure ha he TIPS measuremen errors are comparable across mauriies and of he same magniude as he errors for nominal yields, we use he procedure in Gürkaynak e al. (2007) and scale boh empirical and model-implied TIPS prices by duraion o conver he relaed pricing errors ino he same unis as zero-coupon yields. Here, we use he sandard Macaulay duraion, as i allows us o obain a model-free measure of duraion from TIPS marke prices and heir implied yield o mauriy, which is also available from Bloomberg. 13 Combiningequaions(11) and(14), hesaeransiiondynamicsforz underhephysical measure P is easily shown o be dz = K P z ( ) θz P Z d+σ z dw P, where θ P z and K P z are free parameers wih dimensions 5 1 and 5 5, respecively. Due o he nonlineariies in equaion (15) wih respec o Z when pricing TIPS, we canno apply he sandard Kalman filer for he model esimaion. Insead, he exended Kalman filer (EKF) is used o obain an approximaed log-likelihood funcion L EKF, which serves as he basis for he well-known quasi-maximum likelihood (QML) approach, as also used in Duan and Simonao (1999) and Kim and Singleon (2012), among many ohers. 14 Andreasen e al. (2017) demonsrae ha his approach works well in a seing like ours wih a large panel of bonds wih varying imes o mauriy. To faciliae he esimaion process, Appendix B provides a ailored expecaion-maximizaion (EM) algorihm ha efficienly deals wih opimizing he quasi log-likelihood funcion across he relaively large number of bond-specific parameers in our model. Iisobviousfromequaion(10) hahelevel ofx liq andallheloadings { β i} n TIPS are no i=1 joinly idenified, alhough he level of r R,i and he relaed TIPS liquidiy premium (defined below in Secion 5.1) are idenified in he proposed model. The model of Fonaine and Garcia (2012) displays he same feaure and we herefore follow heir suggesion and normalize he 12 We do no accoun for he approximaely 2.5 monh lag in he CPI indexaion of TIPS, given ha Grishchenko and Huang (2013) and D Amico e al. (2014) find ha his adjusmen normally is wihin a few basis poins for he implied yield on TIPS and hence very small. 13 For robusness, we have also esimaed he G L (5) model using he yields o mauriy for each TIPS and go very similar resuls. However, his alernaive implemenaion is exremely ime consuming as he yield o mauriy is defined as an implici fix-poin problem ha mus be solved numerically for each observaion. 14 The deails for implemening he EKF are provided in our Online Appendix. Using he more accurae cenral difference Kalman filer of Norgaard e al. (2000) gives basically idenical values for he quasi loglikelihood funcion compared wih he values implied by he EKF. For insance, he difference is 0.25 log poins a he opimum for our benchmark model presened in Secion 4.3. This suggess ha he nonlineariy in equaion (15) wih respec o Z are very small and ha he efficiency loss from using a QML approach as opposed o he infeasible maximum likelihood approach is likely o be very small in our case. 16

19 Mauriy G(4) G L (5) in monhs Mean RMSE Mean RMSE All mauriies Table 2: Pricing Errors of Nominal Yields This able repors he mean pricing errors (Mean) and he roo mean-squared pricing errors (RMSE) of nominal yields in he G(4) and G L (5) models esimaed wih a diagonal specificaion of K P z and Σ z. All errors are compued using he poserior sae esimaes in he EKF and repored in basis poins. loading on a given bond. In our case, he loading on he firs bond in our sample is fixed o one (i.e. β 1 = 1), which is he en-year TIPS issued in January This implies ha all remaining loadings for liquidiy risk are expressed relaive o his paricular bond. Preliminary esimaion shows ha he value of λ L,i is badly idenified when i is close o zero or aains large values, and we herefore impose λ L,i [0.01,10] for i = 1,2,...,n TIPS, which are wihou any pracical consequences for our resuls. Finally, o ensure numerical sabiliy of our esimaion rouine, we also impose he resricions β i [0,80] for i = 2,3,...,n TIPS, alhough hey are no binding a he opimum. 4.3 Esimaion Resuls This secion presens our benchmark esimaion resuls. In he ineres of simpliciy, in his secion we focus on a version of he G L (5) model where Kz P and Σ z are diagonal marices. As shown in Secion 6.4, hese resricions have hardly any effecs on he esimaed liquidiy premium for each TIPS, because i is idenified from he model s Q dynamics, which are independen of Kz P and only display a weak link o Σ z hrough he small convexiy-adjusmen in yields. Given ha he G L (5) model includes Treasury yields, i seems naural o firs explore how well i fis nominal yields. Table 2 documens ha i provides a very saisfying fi o all nominal yields, where he overall roo mean-squared error (RMSE) is jus 4.59 basis poins. The corresponding version of his model wihou a liquidiy facor is denoed he G(4) model 17

20 and gives broadly he same fi o nominal yields wih an overall RMSE of 4.61 basis poins. 15 Thus, accouning for he liquidiy disadvanage of TIPS does no affec he abiliy of he G L (5) model o mach nominal yields. The impac of accouning for liquidiy risk is, however, much more apparen in he TIPS marke. The firs wo columns in Table 3 show ha he TIPS pricing errors produced by he G(4) model are fairly large, wih an overall RMSE of basis poins. The following wo columns reveal a subsanial improvemen in he pricing errors when correcing for liquidiy risk, as he G L (5) model has a very low overall RMSE of jus 4.87 basis poins. Hence, accouning for liquidiy risk leads o a significan improvemen in he abiliy of our model o explain TIPS marke prices. Accordingly, he pricing errors of hese securiies wihin he G L (5) model are a he same low level as found for nominal yields in Table 2. The final columns of Table 3 repor he esimaes of he specific parameers aached o each TIPS. Excep for bond number 37, all bonds in our sample are exposed o liquidiy risk, as β i are significanly differen from zero a he convenional 5 percen level. An inspecion of λ L,i in Table 3 reveals ha all five-year TIPS issued before he financial crisis in 2008 have very high values of λ L,i, meaning ha he firs phase wih acive buy-and-hold invesors is very shor for hese bonds. For he remaining TIPS, we generally find somewha lower values of λ L,i and hence somewha longer iniial rading phases, where hese bonds are no fully exposed o variaion in he liquidiy facor. As explained in Secion 3.2, he impac of liquidiy risk on real yields a various mauriies is ambiguous, and Figure 3 herefore plos he liquidiy adjusmen in equaion (13) as a funcion of ime for each of he 38 bonds in our sample. For he five-year TIPS in panel (a), his erm srucure of liquidiy risk displays noable variaion across securiies due o he bond-specific esimaes of λ L,i. The corresponding loadings for en-year TIPS are shown in panel (b), where we also find ha he liquidiy adjusmen is increasing in due o he srong mean-reversion in X liq under he Q measure (κ Q liq = 0.90 according o Table 4). Thus, liquidiy risk operaes as a radiional slope facor wihin he G L (5) model, alhough is seepness varies across he universe of TIPS. The remaining esimaed model parameers are provided in Table 4, which shows ha he dynamics of he four fricionless facors are very similar across he G(4) and G L (5) models, bohunderhepandheqmeasure. We drawhesameconclusion fromfigure4, which plos he esimaed facors in he wo models. The only noiceable difference appears for he real level facor L R, which in he G(4) model generally exceeds he real level facor in he GL (5) model. This difference is mos pronounced from 2001 o 2002 following he 9/11 aacks and around he financial crisis in Thefricionless insananeous real rae r R,FL = L R +α R S herefore has a higher level in he G(4) model, which in urn implies ha his model has a much lower level for he insananeous inflaion rae r r R,FL compared wih he G L (5) model (see panel (f) of Figure 4). Finally, panel (e) shows he esimaed liquidiy facor X liq, 15 Unrepored resuls furher show ha omiing TIPS prices in he esimaion gives basically he same saisfying fi of nominal yields wih an overall RMSE of 4.41 basis poins. 18