Florian Kajuth und Sebastian Watzka: Inflation expectations from index-linked bonds: Correcting for liquidity and inflation risk premia

Transcription

1 Florian Kajuh und Sebasian Wazka: Inflaion expecaions from index-linked bonds: Correcing for liquidiy and inflaion risk premia Munich Discussion Paper No Deparmen of Economics Universiy of Munich Volkswirschafliche Fakulä Ludwig-Maximilians-Universiä München Online a hp://epub.ub.uni-muenchen.de/4858/

2 Inflaion expecaions from index-linked bonds: Correcing for liquidiy and inflaion risk premia Florian Kajuh Sebasian Wazka Ludwig-Maximilians-Universiä Munich Deparmen of Economics July 2008 Absrac We provide a criical assessmen of he mehod used by he Cleveland Fed o correc expeced inflaion derived from index-linked bonds for liquidiy and inflaion risk premia and show how heir mehod can be adaped o accoun for ime-varying inflaion risk premia. Furhermore, we show how sensiive he Cleveland Fed approach is o differen measures of he liquidiy premium. In addiion we propose an alernaive approach o decompose he bias in inflaion expecaions derived from index-linked bonds using a sae-space esimaion. Our resuls show ha once one accouns for ime-varying liquidiy and inflaion risk premia curren 10-year U.S. inflaion expecaions are lower han esimaed by he Cleveland Fed. Keywords: Inflaion expecaions, liquidiy risk premium, inflaion risk premium, reasury inflaion-proeced securiies (TIPS), sae-space model JEL Classificaion: E31, E52, G12 florian.kajuh@lrz.uni-muenchen.de sebasian.wazka@lrz.uni-muenchen.de We would like o hank Gerhard Illing for moivaing us o sudy he opic and for very simulaing discussions. We would also like o hank Tara Sinclair and paricipans a he Macro Seminar a he LMU Deparmen of Economics for helpful commens and suggesions. All errors are of course our own responsibiliy. 1

3 1 Inroducion In 1997 he U.S. governmen sared o issue a en-year inflaion-linked bond, a reasury inflaionproeced securiy (TIPS) 1. Inflaion linked bonds make i possible o observe he real ineres rae and furhermore allow o infer he so-called break-even inflaion rae (BEIR), which is he difference beween he nominal and real yield of a securiy wih he same characerisics such as he same mauriy. The BEIR is a marke based measure of expeced inflaion and is in many ways preferable o survey based measures. However, he yield on a nominal bond conains a premium for he risk ha inflaion changes unexpecedly, which leads he BEIR o oversae inflaion expecaions ceeris paribus. Conversely, he yield on an inflaion linked bond probably conains a premium for liquidiy risk, which resuls in an undersaemen of inflaion expecaions when looking a he BEIR ceeris paribus. Therefore i is essenial ha one correcly adjuss he BEIR for boh premia. The Federal Reserve Bank of Cleveland publishes an adjused measure for expeced inflaion each monh. For May 2008 he Cleveland Fed pus expeced inflaion afer adjusmen for liquidiy and inflaion risk premia a 3.2 percen. In his paper we provide a criical assessmen of he mehod he Cleveland Fed uses o adjus for liquidiy and inflaion risk premia. We show how heir mehod can be adaped o accoun for ime-varying inflaion risk premia and provide esimaes of expeced inflaion ha correc for a variable inflaion risk premium. In addiion, we quesion heir measure of he liquidiy premium and show ha using an alernaive measure yields differen resuls for curren inflaion expecaions. Furhermore, we propose an alernaive mehod based on a sae-space approach o correc BEIRs for boh risk premia wihou recurring o survey based measures of expeced inflaion. Our resuls show ha boh modificaions of he Cleveland Fed mehod resul in considerably lower values for U.S. en-year expeced inflaion. The paper is srucured as follows. Secion 2 provides a criical assessmen of he Cleveland Fed approach. In secion 3 we adap he Fed-mehod o include a ime-varying inflaion risk premium and presen new esimaes of he adjused measure for expeced inflaion. Secion 4 looks in more deail a he liquidiy premium in he TIPS marke. Secion 5 ses up our proposed sae-space model of nominal yields, real yields and expeced inflaion and presens esimaion resuls for he adjused values for expeced inflaion. Finally secion 6 concludes. 1 TIPS are linked o he urban no-seasonally adjused U.S. CPI. For a comprehensive inroducion o indexlinked bonds in he Euro Area see Garcia and van Rixel (2007). 2

4 2 Criicism of he Cleveland Fed approach The mehod used by he Cleveland Fed aims a explaining he difference beween he unadjused measure of expeced average annual inflaion over he nex 10 years 2, which is he difference i T bill r TIPS and ofen called break-even inflaion rae (BEIR), and he unbiased expeced average annual inflaion, E π,+10. Noe ha he observed nominal T-bill yield is equal o he unobserved naural real rae r plus expeced inflaion E π,+10 and an inflaion risk premium ρ π. i T bill = r TIPS + E π,+10 + ρ π (1) and he real yield from TIPS is equal o he unobserved naural real rae plus a liquidiy risk premium ρ LP. r TIPS = r + ρ LP (2) The Fisher equaion saes ha i T bill = r TIPS + E π,+10 + ρ π ρ LP (3) where ρ π is an inflaion risk premium and ρ LP is a liquidiy risk premium. Define in (3) Spread i T bill r TIPS E π,+10 (4) = BEIR E π,+10 (5) = ρ π ρ LP (6) To ge a measure for he spread he Cleveland Fed akes he 10-year CPI-inflaion expecaions from he Survey of Professional Forecasers (SPF) as an unbiased esimaor for E π,+10. As shown in equaion (6), he spread conains boh a liquidiy premium and an inflaion risk premium. The inflaion risk premium is expeced o lead o an oversaemen of inflaion expecaions, while he liquidiy premium o an undersaemen. The Cleveland Fed assumes he inflaion risk premium consan, ρ π = ρ π, and assumes he liquidiy premium in he yield of inflaion-linked bonds o be correlaed wih he liquidiy premium for nominal bonds of he 2 The mehod is documened a hp:// [13 May 2008]. 3

5 same mauriy. To quanify he liquidiy premium in nominal bonds he Cleveland Fed uses he difference beween he yield on off-he-run and on-he-run nominal 10-year reasury bills: LP = i off i on (7) On-he-run securiies of a paricular mauriy are he mos recenly issued ones. Once a new se of securiies wih he same original mauriy are issued, he former ones become offhe-run. Since on-he-run securiies are considered o be more liquid han off-he-run ones, hey command a premium over off-he-run ones, which resuls in a lower yield 3. Regressing he spread on a consan and he linear and squared measure of he liquidiy premium in he nominal bond marke he Cleveland Fed arrives a he following equaion: Spread = LP LP 2 (8) As expeced he consan inflaion risk premium biases he BEIR away from acual expeced inflaion and he liquidiy premium narrows he spread, however a a decreasing rae. The squared erm is mean o capure he idea ha invesors don like uncerainy abou liquidiy condiions. However, an increase in uncerainy from a relaively low level weighs more han he same increase in uncerainy from a relaively high level. Unforunaely, here is no informaion on he sample period used. Using his resul he Fed hen calculaes an adjused measure for expeced inflaion by subracing he spread from he BEIR. E π adj,+10 = BEIR LP 20.9LP 2 (9) In our opinion here are hree major problems wih his mehod. The firs is ha he mehod uses survey daa for expeced inflaion as an unbiased esimaor for acual expeced inflaion. However, he aim should really be o ge away from survey based measures and use nominal and real yields as marke measures o ge an esimae of acual expeced inflaion. Moreover, a survey based measure migh no be unbiased eiher. Le s however assume ha he SPF expeced inflaion is ruly unbiased and ha one could accoun for all he bias in he BEIR. Then one should be able o compue a perfecly adjused measure for expeced inflaion a daily frequency, he quarerly average of which should - on average - yield he SPF expeced inflaion again. 4 A deailed analysis of his poin is provided in he appendix. Thus, he only advanage gained 3 For a deailed accoun of how primary marke dealers use on-he-run securiies in heir business see Fisher (2002). Vayanos and Weill (2006) propose a heory for why on-he-run securiies come o be more liquid han off-he-run ones. 4 The SPF inflaion forecas is available a quarerly frequency only. 4

6 would be an unbiased measure for expeced inflaion a daily frequency, which however would flucuuae around he SPF expeced inflaion. A a 10-year horizon one would hen give probably more weigh o he SPF expeced inflaion because of is lower frequency, rendering he adjused series redundan. Now, in conras, assume he SPF forecas is biased. Then he mehod is flawed because i is based on a fauly measure of he spread, which hen addiionally conains he survey bias. Therefore i would be desirable o carry ou he adjusmen for he biases wihou referring o survey based measures a all. We propose an alernaive mehod based on a simple sae-space approach in secion 5. Our second objecion is ha he relaionship beween he liquidiy premia in he TIPS marke and he liquidiy premia on he nominal bond marke migh no be as sable as assumed by he Fed. In paricular, i is widely argued (e.g. Shen, 2006; Sack and Elsasser, 2004) ha he TIPS marke has gained a reasonable degree of liquidiy only over he las couple of years. Thus, we argue he liquidiy premium in he TIPS yields relaive o he nominal reasuries yields is no free of any rending paerns, be hey deerminisic or sochasic. The problem wih sochasic rends and univariae regression analysis is of course he possibiliy of spurious resuls. Moreover, aside from economeric issues regarding he liquidiy premium here migh be a problem wih using he on-/off-he-run spread LP as a measure for ρ LP. Consider he period from Augus 2007 o oday. I is likely ha markes experienced he so-called fligh o qualiy, where invesors increase heir holdings of safe reasury papers and reduce heir holdings of risky papers. This would depress he nominal bonds yield. To he exen ha he off-he-run yield decreases by less han he on-he-run yield LP rises. However, he change in LP is obviously no relaed o a change in he liquidiy in he TIPS marke. On he conrary, TIPS liquidiy is even likely o increase as rading volume increases because demand for TIPS increases due o fears of inflaion and inflaion risk. Daa for he ransacions volume in he TIPS marke confirm his conjecure. As a consequence he TIPS liquidiy premium hasn increased by as much and adjused inflaion expecaions didn rise as much as in he Fed approach. Finally, we argue ha i is implausible o assume a consan inflaion risk premium. A priori i is no obvious why he inflaion risk bias should be consan over ime. Inflaion volailiy is paricularly high in imes of high inflaion. Because i is inuiive o relae he inflaion risk premium o inflaion volailiy, i follows ha we should allow for a variable inflaion risk premium. If wha we wan o model are he dynamic properies of inflaion expecaions - and if hese properies are no consan - hen one should allow for inflaion volailiy and hence le 5

7 inflaion risk premia change wih expecaions abou he level of inflaion iself 5. Furhermore, he oulook for fuure inflaion migh become more uncerain during imes of economic and financial urbulance, such as he recen episode of financial disress during he pas monhs. Even if one was o look a inflaion expecaions over he nex en years as a gauge for he credibiliy of moneary policy, hen his judgemen could become more uncerain as cenral banks are faced wih new problems for which no esablished response exiss. Moreover a number of sudies have found considerable variabiliy in an esimaed inflaion risk premium (see references in Amico, Kim and Wei, 2008). Therefore we correc for his shorcoming and argue ha o correcly model inflaion expecaions one needs o ake ino accoun a variable inflaion risk premium. The nex secion adaps he Cleveland Fed mehod by including a ime-varying inflaion risk premium. In secion4 we provide some empirical evidence on he relaion beween liquidiy premia in he TIPS marke and he marke for nominal Treasuries. 3 Correcing for a ime-varying inflaion risk premium In his secion we exend he analysis by he Cleveland Fed and allow for a ime-varying inflaion risk premium, which he Cleveland Fed assumes consan. In paricular we esimae he following equaion. Spread = β 0 + β 1 LP + β 2 LP 2 + β 3 IP + ε (10) where Spread is defined as in (5), LP defined in (7) and IP is a measure for he inflaion risk premium, and ε is assumed normally disribued whied noise. Daily daa for spread and LP are aken from he Cleveland Fed homepage and run from 3/2/1997 o 28/3/2008. There are wo measures for he inflaion risk premium. One is he sandard deviaion of individual forecass of inflaion from he SPF. The higher he dispersion of he individual forecass he more uncerain are he survey paricipans and he higher should be he inflaion risk premium. This measure however is only available quarerly and we have aken he quarerly value o be valid on each day of he monh. The second measure is he esimaed volailiy of acual inflaion from a GARCH(1,1) model. The higher he volailiy of acual inflaion he higher he uncerainy in esimaing expeced inflaion, and herefore he higher he inflaion risk premium. We esimaed hree differen versions of (10) on he whole sample. One wih β 3 = 0 as a 5 For a deailed analysis of inflaion risk premia in European bond yields see e.g. Hördahl and Trisani (2007). 6

8 Spread = β 0 + β 1 LP + β 2 LP 2 + β 3 IP + ε Sample period 3/2/1997 o 28/3/2008 Version β 0 β 1 β 2 β 3 I II III Table 1: Esimaion resuls for differen versions of he spread equaion. Three aserisks denoe significance on he 1%-level. comparison o wha he Cleveland Fed did (version I), one wih he volailiy of he SPF forecas as measure for IP (version II), and one wih he esimaed volailiy of acual inflaion as a measure for IP (version III). Subsequenly we adjused he raw BEIR series by subracing he spread. The resuls are summarized in able 1 and ploed in figure 1. Table 1 shows ha he hree versions yield plausible signs for he coefficiens of all variables excep he coefficien on he sandard deviaion of he individual forecas from he SPF. The inflaion risk premium is expeced o lead o an overesimaion of he spread, which seems no confirmed by version II of he regression. However, he coefficien on he condiional volailiy of inflaion as a measure for inflaion risk yields he expeced sign. All coefficiens are significan on he 1%-level. Figure 1 plos he differen resuls for expeced inflaion over he nex en years for he period 1/1/2007 o 28/3/2008 along wih he SPF forecas. Firs hing o noice is ha he Cleveland Fed series differs considerably from our esimaed version I, which is supposed o replicae he Fed resuls. Obviously, he Fed does no include all available daa poins in heir esimaion. Insead hey appear o have esimaed he spread equaion on a subsample. Our resuls, however, show ha including all daa up o he presen yields a lower curren value for expeced inflaion even wihou correcing for inflaion risk. Furhermore, replacing he consan wih a ime-varying measure for he inflaion risk premium leads o markedly differen values for expeced inflaion. Figure 2 shows ha in paricular from he hird quarer 2007 o he end of sample adjused inflaion expecaions are up o 23 basis poins lower when accouning for a ime-varying inflaion risk premium. 7

9 Q1 2007Q2 2007Q3 2007Q4 2008Q1 1.6 adjused by Cleveland Fed version I version II version III SPF forecas Figure 1: Inflaion expecaions adjused for liquidiy and inflaion risk premia using wo differen measures for inflaion risk. 8

10 Q3 2007Q4 2008Q adjused by Cleveland Fed version III SPF forecas Figure 2: Inflaion expecaions adjused for liquidiy and inflaion risk premia using he condiional volailiy of inflaion. 9

11 4 A closer look a he liquidiy premium The major problem when sudying he relaionship beween he liquidiy premium on TIPS and he premium on nominal reasuries is ha no daa on he former is direcly available. In conras, he liquidiy premium on nominal reasuries migh be measured by he difference in he yields on on-he-run and off-he-run nominal bonds. The Cleveland Fed assumes ha here is a posiive and sable correlaion beween he TIPS liquidiy premium and he on/off-he-run premium, and in paricular assumes his is given by equaions (8) and (6). 4.1 Liquidiy premium on TIPS versus he on/off-he-run premium on nominal bonds In order o invesigae he relaion beween he wo premia, we ake he Cleveland Fed serious and urn heir approach upside-down. In oher words, we assume he Fed is indeed able o esimae inflaion expecaions correcly, i.e. E π,+10 = CF, where CF is he Cleveland Fed s adjused measure of expeced inflaion (see equaion 9). In addiion, we follow he Fed in assuming he inflaion risk premium o be consan oo, ρ π = ρ π. These wo assumpions allow us o solve equaion (3) for he liquidiy risk premium of TIPS yields versus nominal Treasuries, i.e. ρ LP. We hen relae he series for ρ LP o LP, he on/off-he-run liquidiy premium in he nominal Treasuries marke for which daa is in fac available. Thus, having daa on hese wo variables allows us o sudy heir relaionship given he Cleveland Fed s assumpions were indeed o hold. Figure 3 shows he resuling relaionship beween he wo premia. The posiive and nonlinear relaionship is no surprising since i resuls from he way he Fed calculaes he spread. In oher words, if he Fed did no regress he spread on a linear and quadraic erm (see equaion 8), he relaionship beween he wo liquidiy premia would look linear. To make he exac nonlinear relaionship beween he wo liquidiy premia more explici, we solve equaion (3) for he liquidiy premium and assume E π,+10 = CF and ρ π = ρ π : ρ LP = ρ π BEIR + CF (11) Subsiuing in for CF from equaion (9) we obain: ρ LP = 12.71LP 20.9LP 2 (12) To ge a feeling for he rue empirical relaionship, we use as a second measure for E π,+10 he inflaion forecas from he SPF. 10

12 Liquidiy premium (TIPS) Liquidiy premium (on/off-he-run) Figure 3: TIPS liquidiy premium vs. on/off liquidiy premium: The exac nonlinear relaionship is by consrucion because he Cleveland Fed regresses he spread on linear and quadraic erms of LP. See equaion (8). 11

13 Liquidiy premium (TIPS) Liquidiy premium (on/off-he-run) Figure 4: TIPS liquidiy premium vs. on/off liquidiy premium: The empirical relaionship looks linear, bu isn srong and here is no evidence of a nonlinear relaionship. ρ LP = ρ π BEIR + SPF (13) We again solve for he liquidiy premium on TIPS (equaion 11) and plo he resuling series in figure 4. Though he figure does reveal some correlaion beween he wo premia, his relaion is no very srong and here is no evidence of a nonlinear relaionship a all. Raher, here seems o be evidence of heeroskedasiciy 6. However, he fac ha he linear relaionship is somewha disurbed migh well be caused by, firs, a ime-varying slope parameer in he spread-regression (equaion 8) or second, by a ime-varying inflaion risk premium. The nex subsecion discusses alernaive measures for he TIPS liquidiy premium. 6 We accoun for heeroskedasiciy in our esimaions by using heeroskedasiciy robus sandard errors. 12

14 amoun ousanding as % of nominal bonds ransacions as % of nominal bonds volume Figure 5: Ousanding TIPS noes of mauriy of 5 and 10 years as percenage of nominal reasury bonds wih comparable mauriy. TIPS primary dealer ransacions volume as percenage of ransacions volume of nominal reasury bonds wih comparable mauriy, hree monh moving average. A higher value indicaes a lower TIPS liquidiy premium. 4.2 Measures for he TIPS liquidiy premium There are poenially more direc measures for he TIPS liquidiy premium han LP, e.g. he bid-ask spread, he amoun ousanding or he ransacions volume in he TIPS marke. Daa for he laer wo are available and are ploed in figure 5 (each as he raio o he number for nominal reasuries of comparable mauriy). Boh series have been increasing over ime 7. A higher value of each series makes he marke for TIPS more liquid. Consequenly, higher values should be associaed wih a lower liquidiy 7 We disregard any issues arising from poenial non-saionariy o make our analysis comparable o he Fed s, who don accoun for non-saionariy in heir series eiher. 13

15 on-/off-he-run spread in nominal reasuries marke Figure 6: Spread beween off-he-run and on-he-run nominal reasuries. A higher value indicaes a higher nominal bonds liquidiy premium. premium. Noe ha in paricular owards he end of he sample saring from he second half of 2007 he ransacions volume rises and he amoun ousanding says roughly consan. This is in conras o wha he on-/off-he-run spread indicaes. Figure 6 shows he on-/off-he-run spread on nominal reasuries, which he Fed uses as a measure for he liquidiy premium in he TIPS marke. Saring from he second half of 2007 he liquidiy premium in nominal on-he-run reasuries has increased, which increased he on-/off-he-run spread. Taking his spread as a measure for he TIPS liquidiy premium one migh conclude ha liquidiy in he TIPS marke has fallen. However, he ransacions volume and he amoun ousanding of TIPS convey evidence ha TIPS liquidiy has no decreased bu mos likely acually increased. In he nex secion we explain he consequences of he wo opposing views concering he change in he TIPS liquidiy premium for inflaion expecaions. 14

16 Figure 7: Schemaic depicion of he Fisher equaion and he effec on expeced inflaion of an increase in he liquidiy premium under he Cleveland Fed assumpions. 4.3 How are inflaion expecaions relaed o he TIPS liquidiy premium? To illusrae he mechanism behind he Fed and o moivae our own approach consider figure 7. I depics he unobserved naural real rae r along wih he real TIPS yield r TIPS and he nominal T-bill rae i T bill. Under he simplifying assumpion of a zero inflaion risk premium he difference beween he nominal T-bill rae and he unobserved naural real rae equals expeced inflaion. Furhermore, he difference beween he nominal T-bill rae and he real TIPS yield is equal o he BEIR. Lasly, he difference beween he real TIPS yield and he unobserved naural rae is he liquidiy risk premium. Now suppose a = 0 he liquidiy risk premium rises by ρ LP > 0 and keeps consanly rising, as he Cleveland Fed argues happend from Augus 2007 on. This increases he real TIPS yield by he same amoun. Under he assumpion of a consan naural real rae he effec on expeced inflaion depends on he behaviour of he BEIR. In he special case where he BEIR says consan expeced inflaion rises exacly by he change in he liquidiy premium. In any oher case he effec on expeced inflaion depends on he change in he nominal -bill rae. Even if he naural real rae is no consan, he correlaion beween he liquidiy risk premium and 15

17 TIPS yield nominal yield Figure 8: Nominal reasury yield and real TIPS yield. The difference is equal o he BEIR. Towards he end of he sample he TIPS yield falls bu he BEIR says roughly consan. expeced inflaion depends on he change in he BEIR, as can be seen from he Fisher-equaion. To find ou which case he daa suppor we plo he TIPS yield ogeher wih he nominal yield in figure 8. The difference beween he wo lines is he BEIR. Observe ha he TIPS yield has fallen owards he end of he sample and he BEIR has sayed roughly consan. Taken ogeher wih he observaions from figure 5 he daa raher sugges ha he TIPS liquidiy premium has fallen as opposed o wha a measure based on he on-/off-he-run spread would ell. In ha case expeced inflaion would fall. The explanaion for why he nominal yield spread beween on-/off-he-run securiies has risen while he TIPS liquidiy premium has fallen migh lie in he fligh-o-qualiy-phenomenon. The problems in financial markes saring in Augus 2007 led o an increased demand for nominal U.S. reasury bonds which resuled in an increase in he on-/off-he-run spread. However, i is plausible ha rading in TIPS increased as well, which is suppored by he daa. Therefore TIPS are likely o have gained in liquidiy a he 16

18 Figure 9: Schemaic depicion of he Fisher equaion and he effec on expeced inflaion of a decrease in he liquidiy premium. same ime as he on-/off-he-run spread in nominal reasuries has risen. We illusrae he case of a falling TIPS liquidiy premium wih a resuling fall in expeced inflaion in figure 9. I is imporan o noe ha up o here a consan inflaion risk premium was assumed. Leing i vary over ime adds anoher variable o he Fisher equaion, which could parly offse he response of expeced inflaion o an increase/decrease in he liquidiy premium for a given change in he BEIR. Our resuls in secion 5 will show exacly his resul: Using he on/offhe-run liquidiy premium we find ha i is he inflaion risk premium which has increased since Augus 2007, hus almos enirely compensaing he observed increase in he Cleveland Fed adjused inflaion expecaions series. No accouning for his increase in he inflaion risk premium hus leads he Cleveland Fed now o oversae acual inflaion expecaions. To quanify he effecs on he adjused measure of expeced inflaion according o he Cleveland Fed approach we repea heir adjusmen mehod making use of he ransacions volume in he TIPS marke as measure for he TIPS liquidiy premium. We re-esimae equaion (10). However, we use he raio of ransacions volume of TIPS o he one of nominal reasuries, 17

19 volume, as a measure for he TIPS liquidiy premium. Noe ha he expeced sign on he coefficien β 1 in (10) is now posiive because a higher ransacions volume decreases liquidiy risk and he premium, which in urn is associaed wih a higher spread. The spread regression now yields Spread = volume 4.24volume 2 (14) The coefficiens are all significan on he 1%-level, excep for he squared erm, which is significan on he 5%-level. Wha is peculiar is he negaive consan. However, a plo of he resuling adjused series for expeced inflaion yields plausible resuls. Figure 10 plos our adjused measure for expeced inflaion using he TIPS ransacions volume as a measure for TIPS liquidiy risk. Clearly expeced inflaion is much lower a he end of he sample han esimaed by he Fed. in Adding he condiional volailiy of inflaion as a measure for an inflaion risk premium resuls Spread = volume 4.21volume IP (15) However, he coefficien on he inflaion risk premium is economically and saisically insignifican. This secion has demonsraed ha i is crucial which measure one uses o esimae he TIPS liquidiy premium. Especially during he imes of he recen financial urmoil differen measures convey differen sories. The spread beween on-/off-he-run nominal bonds has increased while he ransacions volume in he TIPS marke relaive o he nominal bonds marke has increased. Up o here we have followed he Cleveland Fed approach and have evaluaed heir measures for he differen risk premia. In he following secion we ackle he problem of how o ge away from survey based measures and presen an alernaive approach. 5 Using a sae-space approach o esimae inflaion expecaions As an alernaive approach o model, esimae, and predic inflaion expecaions using yield daa on TIPS we employ a sae-space framework. This has been shown o work well for he esimaion 18

20 Q1 2007Q2 2007Q3 2007Q4 2008Q1 1.6 adjused by Cleveland Fed adjused using TIPS ransacions volume SPF forecas Figure 10: Adjused inflaion expecaions using he TIPS ransacions volume as measure for TIPS liquidiy risk compared o Fed adjusmen and SPF forecas. 19

21 of ex ane real ineres raes as e.g. shown in Hamilon (1994). Our approach acknowledges ha inflaion expecaions, as well as he relevan risk premia are inherenly unobservable and models hem in a sandard sae-space framework, ha allows us o derive inflaion forecass ha are compleely free of any survey-based measure. In his secion we provide he deails of he economeric modelling of our proposed sae-space model: z +1 = Fz + v +1 y = Hz + w (16) wih following unobservable sae variables: E π +1 expeced inflaion, ρ π he inflaion risk premium, and ρ LP E π +10 z = ρ π ρ LP he liquidiy risk premium. and he following observable variables: y = i T Bill r TIPS σ π LP where i T Bill r TIPS is he unadjused BEIR ime series obained by subracing he real yield on TIPS from he corresponding nominal rae. σ π is our GARCH-based measure for inflaion volailiy which we argue is a good indicaor for he uncerainy surrounding inflaion expecaions. 8 I seems prey naural o hen relae he unobservable inflaion risk premium o our series of inflaion forecas uncerainy. Finally, LP is he on/off-he-run liquidiy premium from he nominal Treasuries marke defined in equaion (7). In he model (16) we hen impose some srucure on he coefficien marices by making use of he arbirage condiion beween nominal and real reurns: 9 i T Bill r TIPS = E π ρ π ρ LP (17) 8 We also used as a measure of uncerainy around inflaion expecaions he sandard deviaions of forecass of inflaion from he Survey of Professional Forecass. Whils boh series reveal he same message, we prefer our GARCH-based measure of uncerainy of inflaion expecaions because of is monhly, as opposed o quarerly frequency. 9 See equaion (3) above. 20

22 We add an error erm o equaion (17) o allow for ransiory noise in he arbirage relaionship resuling from possible fricions in financial markes. We impose furher srucure on model (16) by assuming auoregressive processes for he inflaion risk and liquidiy risk premia, and by assuming inverse funcional relaionships beween he observable measures of risk/volailiy and he corresponding risk premia. In oher words, whils i is sandard pracise o assume ha risk premia depend on measures of risk and volailiy, we urn his relaionship upside down and assume a unique inverse relaionship exiss, i.e. ha he observables are linear funcions of he saes. 10 The model is closed by assuming he error erms are normally disribued whie-noise. We hen use a Kalman-Filer approach o esimae he following F and H marices ogeher wih he variances of he error erms: f F = 0 f f 3 H = h h 2 Our approach offers a number of advanages over he Cleveland Fed approach. Insead of again relying on a survey-based measure of inflaion expecaions in calculaing he bias in he unadjused BEIR series, we do no resor o surveys. Insead, we rea inflaion expecaions, as well as he relevan risk premia as inherenly unobservable - which hey undoubedly are. We employ a simple sae-space model in which we linearly relae he observable measures of inflaion compensaion and uncerainy o he unobservable expeced inflaion which we are ulimaely ineresed in, as well as he risk premia for liquidiy and inflaion. The model (16) is esimaed hrough a sandard Kalman filer algorihm. Whils he sae-space model poenially allows for a large number of free parameers and hence, for very general specificaions, we resric ourselves o wha we believe o be a simple, parsimonious, and economically reasonable specificaion. The resuls of our sae-space model are shown in figure 11. We plo he smoohed saeseries: he inflaion forecas, he liquidiy risk and he inflaion risk premium ogeher wih he 10 More general nonlinear funcional specificaions are possible, bu need much more complicaed and compuaionally-inensive soluion algorihm. Whils his is an ineresing possibiliy o improve on our resuls, we believe he curren model specificaion is sufficien o highligh our main poins of deparure and improvemens over he Cleveland Fed approach. 21

23 Cleveland Fed adjused inflaion forecas Sae-space inflaion forecas Liquidiy premium Inflaion risk premium Figure 11: Smoohed ime series from he sae-space model and Cleveland Fed adjused inflaion forecas: Cleveland Fed and sae-space inflaion forecas rising owards he end of he sample. Sae-space forecas, however, a lower level. 22

24 adjused Cleveland Fed measure for inflaion expecaions. There is a srong rise in inflaion expecaions of he Cleveland Fed series a he end of he sample. In consras he sae-space inflaion forecas series - in line wih he argumens of secion 3 - picks up much less a he end of he sample. Our mehod also allows us o back ou he liquidiy and inflaion risk premia as smoohed esimaes from he Kalman filer procedure. Ineresingly, we see a gradual rise in he inflaion risk premium a he end of he sample. However, having sudied he univariae properies of he daa series, he resuls have o be reaed wih cauion given he srong evidence of non-saionary behaviour in he series. Finally, o compare our approach and resuls wih he Cleveland Fed s we highligh again he implied relaion beween inflaion expecaions and he liquidiy premium. Figure 12 plos a scaer of our measures of inflaion expecaions and he liquidiy premium. The correlaion is now negaive, wih a coefficien of This resuls from he uncondiional naure of he plo: In oher words, he oher variables in he Fisher equaion are also changing. Because our approach allows for a variable inflaion risk premium, he resuling uncondiional correlaion beween he expeced inflaion series and he liquidiy premium no longer has o be posiive, bu now urns negaive. The inflaion risk premium and he BEIR are moving in he background oo. Our resuls in fac sugges he liquidiy premium is negaively relaed o he inflaion risk premium. In oher words, our resuls srongly sugges ha i is he BEIR ha adjuss o bring he variables back ino Fisher-equilibrium. The wo premia alone would insead cause deviaions from he Fisher-equaion. 6 Conclusion In his paper we aimed a undersanding how one should opimally correc inflaion-expecaions derived from TIPS yields for ime-varying liquidiy and inflaion risk premia. Saring from an approach by he Cleveland Fed we have shown ha, firs, heir mehod yields on average he inflaion forecas of he Survey of Professional Forecasers, second he relaion beween he unobserved liquidiy risk premium and measures for he liquidiy risk premium in nominal reasuries is likely no consan over ime, and hird he assumpion of a consan inflaion risk premium is no innocuous wih respec o he esimaed adjused inflaion expecaions. In paricular, once we accoun for a ime-varying inflaion risk premium he adjused figures for expeced inflaion are considerably lower. In addiion we propose as an alernaive approach 23

25 inflaion expecaions liquidiy premium Figure 12: Our sae-space model predics a negaive relaion beween inflaion expecaions and liquidiy risk premia. The BEIR insead adjuss. 24

26 a sae-space esimaion of he liquidiy premium, he inflaion risk premium and expeced inflaion. This approach, which is enirely marke-based, also yields lower curren inflaion expecaions. References [1] Amico, S., D. Kim and M. Wei (2008): Tips from TIPS: he informaional conen of reasury inflaion-proeced securiy prices. BIS Working Paper No [2] Fisher, M. (2002): Special repo raes: an inroducion. Economic Review, second quarer, Federal Reserve Bank of Alana. [3] Garcia, J. and A. van Rixel (2007): Inflaion-linked bonds from a cenral bank perspecive. ECB Occasional Paper No. 62. [4] Hamilon, J. D. (1994): Time Series Analysis, Princeon. [5] Hördahl, P. and O. Trisani (2007): Inflaion risk premia in he erm srucure of ineres raes. ECB Working Paper No [6] Sack, B. and R. Elsasser (2004): Treasury inflaion-indexed deb: a review of he U.S. experience. Economic Policy Review, May, Federal Reserve Bank of New York. [7] Shen, P. (2006): Liquidiy risk premia and breakeven inflaion raes. Economic Review, second quarer, Federal Reserve Bank of Kansas Ciy. [8] Vayanos, D. and P. Weill (2006): A search-based heory of he on-he-run phenomenon. NBER Working Paper No Appendix This appendix shows why he adjusmen mehod used by he Cleveland Fed on average leads o he survey resul obained by he Survey of Professional Forecasers (SPF). In oher words, we show ha he essenial advanage of he Fed-measure of inflaion expecaions is no is perceived independence from survey resuls, bu only he higher frequency for which inflaion forecass are possible using his mehod. 25

27 We assume he SPF publishes every monh a forecas of average annual inflaion over he nex 10 years. We denoe his survey-based inflaion forecas wih SPF where he subscrip here and in wha follows always indicaes he ime a which he forecas is made. In oher words, SPF denoes he inflaion forecas of he SPF made a ime for he average annual inflaion rae over he following 10 years. From equaion (5) in he ex we know ha he Fed calculaes on each day + j he spread as difference beween he unadjused BEIR a ime + j and SPF. Formally, we have he following: spread +j = BEIR +j SPF j = 0,1,2,..,29 (18) As laid ou in secion 2, he Fed hen regresses his spread on a consan and a linear and squared measure of he liquidiy premium. The Fed hen akes he prediced values from his regression and subracs hem from he unadjused BEIR o derive is measure of adjused TIPS-derived inflaion expecaions CF +j. Formally, his is given as: CF +j = BEIR +j spread +j (19) To show ha he average value of he Cleveland Fed inflaion forecass CF +j equals he SPF-forecas, we average ou he daily Cleveland Fed forecass over he monh in which he SPF forecas was announced j=0 CF +j = 1 30 = 1 30 = j=0 29 j=0 29 j=0 = SPF = SPF (BEIR +j spread +j ) (20) (BEIR +j (spread +j u +j )) (21) (SPF + u +j ) (22) 29 j=0 29 j=0 u +j (23) u +j (24) 26