EXTRACTION OF FINANCIAL MARKET EXPECTATIONS ABOUT INFLATION AND INTEREST RATES FROM A LIQUID MARKET (*)

Transcription

1 ETRCTION OF FINNCIL MRKET EPECTTIONS OUT INFLTION ND INTEREST RTES FROM LIQUID MRKET (*) Ricardo Gimno and José Manul Marqués (**) NCO DE ESPÑ (*) Th rsuls and opinions xprssd in his papr ar hos of h auhors. (**) W grafully acnowldg h commns by Robro lanco, Juan ngl García,Eva Orga, an anonymous rfr, h paricipans a h sminar of h anco d España, Foro d Finanzas and Simposium d nálisis Económico. W also han Rudbusch and Chrisnsn for providing us wih h daa rquird o compar our approach wih hir modl. Documnos d Trabajo. N.º NCO DE ESPÑ 9 DOCUMENTO DE TRJO N.º 0906

2 bsrac In his papr w propos an affin modl ha uss as obsrvd facors h Nlson and Sigl (NS) componns summarising h rm srucur of inrs ras. y doing so, w ar abl o rformula h Dibold and Li (2006) approach o forcas h yild curv in a way ha allows us o incorpora a non-arbirag opporuniis condiion and ris avrsion ino h modl. Ths condiions sm o improv h forcasing abiliy of h rm srucur componns and provid us wih an simaion of h ris prmia. Our approach is somwha quivaln o h rcn conribuion of Chrisinsn, Dibold and Rudbusch (2008). Howvr, no only dos i sm o b mor inuiiv and far asir o sima, i also improvs ha modl in rms of fiing and forcasing propris. Morovr, wih his framwor i is possibl o incorpora dircly h inflaion ra as an addiional facor wihou rducing h forcasing abiliy of h modl. Th augmnd modl producs an simaion of mar xpcaions abou inflaion fr of liquidiy, counrpary and rm prmia. W provid a comparison of h propris of his indicaor wih ohrs usually mployd o proxy h inflaion xpcaions, such as h bra-vn ra, inflaion swaps and profssional survys. Kywords: Inrs Ra Forcas, Inflaion Expcaions, ffin Modl, Dibold and Li. JEL Classificaions: G2, E43, E44, C53. NCO DE ESPÑ 0 DOCUMENTO DE TRJO N.º 0906

3 Inroducion Th forcasing propris of h yild curv hav bn h focus of rcn anion. In his rspc Dibold and Li (2006) propos using h shap of h yild curv, capurd by h paramrs of h Nlson and Sigl (987) rm srucur modl, as prdicors of h fuur valu of inrs ras. In his papr w dpar from h modl originally inroducd by Dibold and Li (2006) by rwriing i wih an arbirag-fr spcificaion following Vasic (977) and Cox al. (985). In hs modls i is possibl o improv h consisncy of h modl by adding h non-arbirag opporuniis condiion oghr wih ris avrsion in ordr o compu addiionally h ris prmia. Chrisnsn al. (2007) proposd anohr approach o inroduc non-arbirag opporuniis ino h Dibold and Li modl basd on an unobsrvd componn. In hir modl, h undrlying Nlson and Sigl (987) paramrs ar usd as a subjacn srucur of h lan facors in h affin modls along h lins of Duffi and Kan (996). On h conrary, our modl rlis on h Nlson and Sigl (987) facors as bing complly xognous o h affin spcificaion, rducing h complxiy of h Kalman filr ha is usually rquird in h sandard lan modls [ohr xampl of his lan-facor approach can b found in ng al. (2008)]. y using h Nlson and Sigl paramrs w can guaran a good fi of h whol rm srucur of nominal inrs ras 2 and, by aing hs facors as xognous, h rsuls ar much mor robus (i.. hy ar no dpndn on iniial condiions or on h slcion of h inrs ras obsrvd wihou rror). Morovr, h ris prmia could b asily obaind simply by comparing h valu obaind wih h simad pric of ris and hos basd on a pric of ris qual o zro. Our mpirical rsuls for h Unid Sas and h Euro ra suggs ha h inroducion of non-arbirag opporuniis and ris avrsion prsns svral advanags whn compard wih h original Dibold and Li modl. Firsly, w obaind a slighly br forcas and i is possibl o provid an simaion for h ris prmia, somhing ha was no possibl undr h original modl. Scondly, h affin formulaion mas i asir o includ ohr variabls and, hrfor, o obain simaions for hs variabls ha ar compaibl wih h rm srucur. This is spcially usful whn daling wih macroconomic variabls, as has bn highlighd in h liraur [i.. Carriro al. (2006), Dwachr al. (2006), Dwachr and Lyrio (2006) or Dibold al. (2004)]. Nvrhlss, prvious approachs ha poind ou h imporanc of macroconomic variabls wr hindrd by h lan facors usd o dscrib h rm srucur, which xhibi poor forcasing propris [Duf (2002) and svral opima [Kim and Orphanids (2005)]. Rgarding h comparison wih h approach of Chrisnsn al. (2007), our modl sms o b asir o sima. Th mpirical rsuls for h sam sampl priod suggs ha h fiing accuracy of our modl is suprior, whil in rms of forcasing w aain far br rsuls ovr h long-rm horizon.. Thr is an xnsiv liraur ha uss h lan facor in ordr o sima h ral inrs ra and inflaion xpcaions. S for xampl Dai and Singlon (2000) or Laubach and Williams (2003). 2. Mos of h modls basd on ndognous facors ypically focus only on a slcion of four inrs ras ha hav o b rplicad. y conras, h modl usd hr rlis on h paramrs ha rplica h whol rm srucur of inrs ras and, by dfiniion, rplica all h bonds usd o sima hs paramrs. NCO DE ESPÑ DOCUMENTO DE TRJO N.º 0906

4 nohr imporan faur of our modl is ha i is possibl o inroduc h inflaion ra ino h componns of h VR in h affin modl whil mainaining h forcasing accuracy of h modl wih rspc o h inrs ra. Ohr approachs o h simaion of inflaion xpcaions from h yild curv includ hos of ng al. (2008) or García and Wrnr (2008); howvr, hs xrciss ar usually basd on lan facors and hir simaions condiional upon svral ad-hoc assumpions, and hy normally prsn robusnss problms. Our mpirical rsul suggss ha h rm srucur of h inrs ra conains usful informaion for forcasing h inflaion ra compard wih a simpl R modl. In his papr w do no prform a comparison of our rsuls wih hos obaind wih affin modls basd on unobsrvd componns; his ind of xrcis can b found in Gimno and Marqués (2008). Insad, w focus on h comparison of our simaions for inflaion xpcaions wih ohr indicaors usually an by analyss as proxis for inflaion xpcaions. Thorically, our masur of xpcd inflaion should conain br propris han ohr alrnaivs givn ha his simaion ra was no affcd by ris prmia, counrpary ris or liquidiy prmia (givn ha h nominal public db mar could b considrd as liquid). Our rsuls for long-rm inflaion ras ar clarly in lin wih h inflaion xpcaions of analys survys. Morovr, h voluion of our simas of xpcd inflaion sms o b lss volail and mor plausibl boh for h uro ara and h Unid Sas han ohr mar masurs such as inflaion-lind bonds or inflaion swaps, which could b affcd by ris prmia, liquidiy prmia or counrpary ris. In paricular, during h las par of 2008, a priod characrisd by significan shifs in liquidiy prmia, h indicaors calculad as a comparison of diffrn mars, such as inflaion-lind and nominal bonds, could b paricularly mislading. Finally, wih his modl i is possibl o prform an impuls-rspons analysis in ordr o valua how an unxpcd movmn in on of hs variabls changs mar xpcaions abou h ohrs. In paricular, w will focus on h ffcs on inflaion xpcaions afr a prmann incras in shor-rm inrs ras. Our rsuls suggs ha monary policy sms o hav a biggr ffc in h Unid Sas han in h uro ara. Th rs of h papr is srucurd in fiv addiional scions. Th scond scion analyss h forcasing propris of h Dibold and Li (2006) approach if w incorpora h non-arbirag opporuniis condiion, ris avrsion and inflaion ras. In h hird scion, w compar h rsuls on inflaion xpcaions and ris prmia wih h survys of analyss and profssional forcasrs and h informaion conn in ohr mars, such as inflaion-lind bonds or inflaion swaps. In Scion four, w analys h racion of inflaion xpcaions afr a monary policy dcision. Finally, scion 5 concluds. NCO DE ESPÑ 2 DOCUMENTO DE TRJO N.º 0906

5 Forcasing Inrs Ras. Forcasing Inrs Ras: Th S-Up Dibold and Li (2006) addrssd h problm of inrs ra xpcaions indircly by forcasing h valus of h paramrs of h yild curv. For his purpos, hy usd h Nlson and Sigl (987) modl for h rm srucur, y L S C, = u, ( 0, I) 2 u N σ () whr y, is h inrs ra a im mauring in priods; L is h long-rm inrs ra (boh forward and spo); S is h sprad (diffrnc bwn long-rm and shor-rm inrs); C is a masur of h rm srucur curvaur; is a paramr drmining u h spd of ransiion bwn h shor and h long-rm inrs ras; whil, is an rror rm. Dibold and Li (2006) proposd fixing h valu of in h man obsrvd valu hroughou h original sampl 3, so inrs ras bcoms a linar modl (quaion 2), y L, = S u, C (2) In his way, paramrs L, S and C can b asily simad by an OLS rgrssion. Dibold and Li (2006) showd ha hn i is possibl o forcas hs paramrs by VR quaions (quaion 3), = µ Σε ε N( 0 I) (3) Φ,, µ is a vcor of h consan drifs, Σ is h = L S C whr ( ) varianc-covarianc marix of h nois rm, Φ is a marix of h auorgrssiv cofficins and h i.i.d. Gaussian nois vcor ε rprsns h uncrainy in h fuur valus of inrs ras. Onc w hav forcass of h paramrs, i is possibl o rcovr good projcions of h whol rm srucur of inrs ras. Nvrhlss, h Dibold and Li (2006) approach lacs som propris ha should b prsn in a dynamic modling of inrs ras: non-arbirag opporuniis and ris avrsion. Chrisnsn al. (2007) and Gimno and Marqus (2008) addrssd his problm by ransforming h Dibold and Li (2006) modl ino an affin modl. ffin rm srucur modls hav bn widly usd in h financial liraur o pric fixd-incom asss sinc h sminal wors of Vasic (977) and Cox al. (985). 3. Gimno and Nav (2009) showd ha rying o sima joinly wih L, S and C producs non-rivial problms of idnificaion. NCO DE ESPÑ 3 DOCUMENTO DE TRJO N.º 0906

6 n affin modl assums ha inrs ras can b xplaind as a linar funcion of crain facors, 2 u N( 0, I) ( ) u y, =, σ (4) whr and ar cofficins. Changs in inrs ras ovr im will b h oucom of changs in h facors, whras diffrncs in h rm srucur will b drivn by h cofficins and applid. In fac, his linar approach o inrs ras is similar o h linarizaion of h Nlson and Sigl (987) modl proposd by Dibold and Li (2006). ddiionally, in an affin modl, facors hav a dynamic bhavior usually modld as a VR [s Dibold al. (2004)] in h sam vin as Dibold and Li (2006) did, w considr h paramrs of h Nlson and Sigl (987) modl as h affin facors. In an affin modl, arbirag opporuniis ar avoidd by imposing quaion 5. Chrisnsn al. (2007) proposd aaining his qualiy by modifying h facors. y doing so, hy mainain h original of h original Nlson and Sigl modl and lavs o chang according o quaion 5. [ ] Q = E (5) Th lf-hand sid of quaion 5 rprsns h valuaion of a zro-coupon bond mauring in which, undr h non-arbirag condiion, should b quivaln o h xpcd valu on priod ahad of h sam bond wih mauriy discound by h shor-rm inrs ra. Chrisnsn al. (2007) imply ha in ordr o nsur ha quaion 5 holds, h valu of mus b rsricd. Thrfor, if valus ar an as lan facors, hy mus b simad by a Kalman filr, adding complxiy o h rahr difficul as of simaing h Nlson and Sigl modl 4. ddiionally, h dynamics of h variabls ar obscurd by h rsricion imposd. In h prsn papr w propos an alrnaiv approach o includ non-arbirag opporuniis in h Dibold and Li (2006) modl. I is possibl o mainain h obaind from h Nlson and Sigl simaion as xognous variabls, avoiding h nd for h Kalman filr. Thrfor, h non-arbirag condiion is rachd by rsricing h valus of paramrs and according o quaion 5. This could b don by solving forward quaion 5, which would imply a rcursiv form for h and cofficins. Th considraion of ris avrsion in his affin modl framwor implis som compnsaion for h uncrainy abou longr mauriis 5, in which h random shocs ε accumula. In his rspc, i is clar ha h highr h varianc of random shocs on VR quaion (3) [idnifid by marix Σ which w will impos o b orhogonal in lin wih 4. S Gimno and Nav (2009). 5. ar and Hodric (200) rviwd h vidnc which suggss ha xpcd rurns on long bonds ar, on avrag, highr han on shor bonds, rflcing h xisnc of a ris prmium and ha his prmium is im-varying. NCO DE ESPÑ 4 DOCUMENTO DE TRJO N.º 0906

7 ng al. (2008) among ohrs], h grar h uncrainy abou fuur valus of inrs ras. So, in ordr o compnsa invsors for lnding mony a longr rms, som ris prmium rlad o Σ should b mbddd in h nominal inrs ras. Cofficins ha ransla marix Σ ino h ris prmium ar calld prics of ris ( λ ) and, following h liraur, hs cofficins can b s o b affin o h sam facors, λ = λ 0 λ (6) whr λ 0 is a vcor and λ a marix of cofficins. If λ is s o b qual o zro, hn h ris prmium will b consan, whil if w lav i unrsricd, w will obain a im-varying ris prmium. Taing oghr non-arbirag opporuniis and ris avrsion, i is possibl, afr som algbra, o ransform quaion 5 ino a rcursiv sysm of quaions rprsnd by quaions 7 and 8. = µ Σλ0 ΣΣ (7) 2 = Φ Σλ (8) In quaions 7 and 8 h cofficins drmining inrs ras mauring in ( and ) ar h rsul of h aggrgaion of h drminans of h shor-rm inrs ra ( and ), h diffrnc bwn h acual shor-rm inrs ra and is forcas valu (rflcd by µ and Φ rms, rspcivly) a compnsaion for ris ( Σλ 0 and Σλ rms, rspcivly), and a quadraic rm consqunc of h Jnsn Inqualiy ( ΣΣ ). s can b sn, ris compnsaion dpnds on marix 6 Σ and 2 h pric of ris λ. Summarizing, h affin modl o b simad undr our framwor will consis of quaions 3 and 4, wih h cofficins of quaion 4 bing subjc o rsricions 7 and 8. In ordr o compar our modl wih ha proposd by Chrisnsn al. (2007), w hav usd h sam sampl hy us 7. In abl, w prsn in-sampl fiing saisics of Dibold and Li (2006) and h Chrisnsn al. (2007) rpord roo man squard rror (RMSE) for ach mauriy. Ths rsuls ar hn compard wih hos obaind from our own modl in h las columns. s can b sn, our modl is clarly suprior o hos prviously proposd. This is mainly a consqunc of br prformanc a h long-rm nd of h yild curv, whr h ohr modls obain poor rsuls. 6. lhough h Σ marix is im-invarian, i appars joinly wih h im-varian pric of ris. Hnc an incras in h volailiy of any facor would b capurd via λ. 7. Monhly daa on U.S. Trasury scuriy yilds from January 987 o Dcmbr Th daa ar nd-of-monh, unsmoohd Fama-liss (987) zro-coupon yilds a h following 6 mauriis: 3, 6, 9, 2, 8, 24, 36, 48, 60, 84, 96, 08, 20, 80, 240, and 360 monhs. NCO DE ESPÑ 5 DOCUMENTO DE TRJO N.º 0906

8 Tabl : Summary Saisics of In-Sampl Fi Mauriy in monhs DNS indp.-facor DNS corr.-facor CDR indp.-facor CDR corr.-facor GMS Man RMSE Man RMSE Man RMSE Man RMSE Man RMSE ll mauriis Sourc: Chrisnsn al. (2007) and auhors own daa. Th mans and h roo man squard rrors for 6 diffrn mauriis. ll numbrs ar masurd in basis poins. DNS rprsns h original Dibold and Li (2006) modl, wih boh assuming indpndn and corrlad ; CDR is for h Chrisnsn al. (2007) approach; finally GMS rprsns h modl proposd in his papr scond analysis prformd in Chrisnsn al. (2007) is o asss h ou-of-sampl forcas propris of ach modl. W hav rplicad his analysis 8, and h rsuls ar prsnd in abl 2. s can b sn, in rms of forcas accuracy, oo, our proposal ouprforms h ohr modls. This is mor vidn in h 2-monh forcas. This br oucom han ha of Chrisnsn al. (2007) may b a consqunc of h way nonarbirag condiions ar imposd on h modl and h rsricions rquird for h Kalman Filr simaion. y fring h variabls from h burdn of hs rsricions, h VR quaion is hn abl o produc br simaions in rms of forcas. nohr consqunc is ha his VR can b modifid for adding all h variabls ha you may considr ncssary wihou affcing h non-arbirag condiions ha would fall on h rcursiv form of h and cofficins. This possibiliy will b xplord in a lar scion, whr inflaion ras would b addd o h modl. n addiional advanag of a non-rsricd VR quaion is ha w ar abl o prform impuls-rspons xrciss on h modl, giving us inrsing informaion on mar racions o chang in som rlvan variabls such as monary policy shor-rm inrs ras or inflaion ras. 8. On-, six-, and wlv-monh-ahad forcass ar consrucd for all h modls and for six yilds wih mauriis of 3 monhs and, 3, 5, 0, and 30 yars, wih a rcursiv procdur. For h firs s of forcass, h modl is simad from January 987 o Dcmbr 996; hn, on monh of daa ar addd, h modls ar r-simad, and anohr s of forcass is consrucd. Th largs simaion sampl for h on-monh-ahad forcass nds in Novmbr 2002 (72 forcass in all). For h six- and 2-monh horizons, h largs sampls nd in Jun 2002 and Dcmbr 200 (67 and 6 forcass), rspcivly. NCO DE ESPÑ 6 DOCUMENTO DE TRJO N.º 0906

9 Tabl 2: Forcas RMSE for h Fiv Modls Forcas horizon Modl On monh Six monhs Twlv monhs 3-monh yild DNSindp DNScorr CDRindp CDRcorr GMS yar yild DNSindp DNScorr CDRindp CDRcorr GMS yar yild DNSindp DNScorr CDRindp CDRcorr GMS yar yild DNSindp DNScorr CDRindp CDRcorr GMS yar yild DNSindp DNScorr CDRindp CDRcorr GMS yar yild DNSindp DNScorr CDRindp CDRcorr GMS Sourc: Chrisnsn al. (2007) and auhors own compuaion..2 Forcasing Inrs Ras: Empirical Rsuls In ordr o sima h quaions w will focus on h Euro ra and h Unid Sas. For h Euro ra w a h paramrs for h yild curv providd by h EC in is wb pag whil for h Unid Sas w hav o sima h Nlson and Sigl paramrs basd on h obsrvaion of yilds of rasuris in h scondary mars wih rms from monh o 30 yars, an from Daasram. In boh cass w us monhly obsrvaions from January 999 o May Firs, w ar inrsd in comparing h forcasing propris of h modl agains wo bnchmars ha hav bn widly usd in h liraur: h implici forcas for inrs ras ha w can obain from h yild curv using h forward ras, and h forcas ha can b obaind from h Dibold and Li modl wihou imposing h non-arbirag opporuniis. NCO DE ESPÑ 7 DOCUMENTO DE TRJO N.º 0906

10 Tabl 3: Man Squard Error on -yar inrs ras forcas -yar ahad using forward inrs ras, Dibold and Li (986) VR modl, and h affin modl Forward VR modl ffin modl Euro ra Unid Sas s can b sn in Tabl 3, hr ar clar forcasing improvmns on h VR modl proposd by Dibold and Li (2006) ovr forward inrs ras 9. Nvrhlss, rsricions imposd by h affin modl prsnd in his papr do no rduc h forcas accuracy bu vn marginally improv i. This is a clar sign ha his rsricion dos no hav any cos n rm of h abiliy o forcas. Ohr imporan finding on abl is ha i sms asir o forcas yar inrs ra in h uro ara han in h Unid Sas. This faur, ha could b obsrvd also whn using forward inrss, could b du o h low dgr of smoohnss of monary policy in h Unid Sas during h sampl priod, Morovr sinc h EC has lss ambiguiy on is manda han h Fdral Rsrv i sms mor plausibl ha financial mars has bn mor succssful forcasing inrs ras in h uro ara han in h US. n addiional advanag of h affin modl ovr h unrsricd VR of Dibold and Li (2006) is ha a clan masur of h ris prmia can b xracd from h modl as was shown in nnx 2. Ris or rm prmia ar usually blamd for h forward inrs ras poor forcasing prformanc. This can b sn in Tabl 4, whr i is shown ha forward inrs ras ovrsima spo inrs ras (by 7bp in h E and 30bp in h US). This forcasing swnss is qui lowr in h cas of h VR modls. Nvrhlss, h assumpion ha h diffrnc bwn forward Inrs ras and forcas inrs ras wih h VR modl is h consqunc of h rm prmia dos no a ino accoun h masurmn rror in h simaion of h VR. y conras, in h affin modl, h rm prmia can b obaind as h diffrnc bwn h simad nominal inrs ra and a compud inrs ra whr h pric of ris is s o b qual o zro, ( λ ) γ ˆ ˆ (9), = y, y = 0, This rm prmium will b a mixur of h uncrainy abou h fuur cours of shor-rm inrs ras and h pric of ris. lhough h uncrainy is originad by his spcific mar, h pric of ris should b gnrally h sam across financial mars. So, a similar parn should b found if compard wih ohr ris masurs. This similariy can b sn in figur, whr h rm prmia xracd from quaion 9 ar compard wih crdi dfaul swaps. Obviously, hs wo masurs of ris prmia ar of a diffrn magniud sinc h yild curv has o rflc a prmium rlad o h uncrainy ovr h broad conomy (and hrfor o monary policy, h growh ra and inflaion shocs) and h indics of CDS wr basd mainly on firms dfaul probabiliy. Howvr, macroconomic uncrainy is an imporan drminan vn in h cas of individual CDS [s, for xampl, lonso al. (2006)]. Morovr, in h cas of a CDS indx, h imporanc of h individual facors of ach CDS is dilud whil common facors, such as inflaion or ral growh ra 9. For h modl simaions w hav usd h paramrs of h yild curv obaind hrough h Nlson and Sigl (987) mhodology from nominal inrs ras a diffrn mauriis compild from Daasram. NCO DE ESPÑ 8 DOCUMENTO DE TRJO N.º 0906

11 uncrainy, hav an imporan rol o play. In fac, Figur shows a significan rlaionship bwn our masur of ris prmia uncrainy from h yild curv and h prmia of h indics of CDS. Figur : Trm prmia for h 2-yar inrs ras (blu) and indics of CDS (pin) for boh h Euro ra and h Unid Sas Euro ra Unid Sas rm prmia (ls) iraxx-main (rs) rm prmia (ls) cdx-main (rs).50%.25%.00% 0.75% 0.50% 0.25% 0.00% -0.25% -0.50% % 2.0%.5%.0% 0.5% 0.0% -0.5% -.0% Th ris prmia w obain follow a similar parn o h inflaion ris prmia simad by Garcia and Wrnr (2008) for h uro ara basd on an unobsrvd componn approach and on h informaion conaind in h Survy of Profssional Forcasrs and h ra-evn Inflaion Ras. NCO DE ESPÑ 9 DOCUMENTO DE TRJO N.º 0906

12 2 Forcasing Inflaion 2. Forcasing Inflaion: Th S-Up Th modl proposd by Dibold and Li (2006) focusd only on forcasing inrs ras. Howvr, by rformulaing h affin vrsion of h modl i sms naural o incorpora h inflaion ra as an addiional facor in h vcor. On on hand, h inflaion ra could hlp o forcas h rm srucur and, hrfor, h Nlson and Sigl paramrs. Th main rason bhind his rsids on h rol ha his variabl plays in h racion funcion of monary policymars and, consqunly, on xpcd inrs ras. In fac, h EC and ohr cnral bans hav as hir primary arg h achivmn of pric sabiliy masurd hrough an appropria lvl of h inflaion ra. In ohr counris, li h Unid Sas, inflaion ras ar no an xplici arg alhough hs cnral bans ar clarly involvd in mainaining pric sabiliy. In his sns, Esrlla and Mishin (997) show ha rm srucur was clarly affcd no only by h monary policy ra bu also by cnral ban crdibiliy. On h ohr hand, h imporanc of h rm srucur in prdicing fuur inflaion changs has bn xnsivly documnd in h financ liraur. In paricular, Mishin (990) and Jorion and Mishin (99) usd rm sprads o prdic fuur inflaion changs in h Unid Sas and found ha, alhough rm sprads ar no usful in h vry shor rm, hir accuracy incrass as prdicions xnd byond a yar. In h sam vin, Esrlla and Mishin (997) found a similar rlaionship no only for h Unid Sas bu also for mos of h Europan counris. In his rspc, w analyz h rlaionship bwn h rm srucur and inflaion ra from a mor gnral prspciv. y including h inflaion ra in h vcor, w considr ha his variabl is no only corrlad wih h slop of h yild curv bu also, and simulanously, wih ohr characrisics such as h long-rm lvl or curvaur. This approach is in lin wih ng al. (2007) or ng and Piazzsi (2003), alhough hy us an affin modl wih lan componns insad of h Nlson and Sigl facors of Dibold and Li (2006). In a prvious papr [s Gimno and Marqués (2008)] w showd how, if w hav o dal wih an conomy wih a significan srucural shif, h affin modl wih h paramrs of h rm srucur sms o prform much br han radiional affin modls basd on unobsrvd componns in ordr o obain som simaion of x-an ral inrs ras. 2.2 Dos h rm srucur conain any informaion on h inflaion ra? To ascrain h rlaionship bwn h inrs ra and h inflaion ra w can compar h rsuls of h modl wih h inflaion ra wih hos prviously obaind wihou his variabl. In ordr o proprly compar boh aras, w hav usd h sam masur of inflaion ra basd on cor inflaion insad of h hadlin ovr h CPI. In h cas of h Fdral Rsrv, svral paprs highlighd h fac ha h cor PCE is h main variabl NCO DE ESPÑ 20 DOCUMENTO DE TRJO N.º 0906

13 for pric sabiliy in h sing of inrs ras 0. Howvr, givn ha in h uro ara hr is no quivaln o h PCE, w will us cor inflaion ra (rfrrd o h CPI) for boh aras, sinc is voluion dos no diffr significanly from h PCE and has a mor comparabl variabl for h uro ara. For h uro ara, h official objciv in h EC manda for pric sabiliy sablishd hadlin inflaion compud wih h HICP as h official rfrnc for monary policy. Nvrhlss, h rducd sampl availabl sinc h bginning of h hird phas in h uro ara has coincidd wih a priod whr xpcd and rpad shocs in boh oil and food prics has pushd up inflaion. So, if w ry o sima h modl wih h HICP, w would hav upward biasd simaions of inflaion xpcaions. Morovr, Gali al. (2004) show ha monary policy dcisions in h uro ara ar br xplaind by mans of a Taylor rul basd on a snimn indx and cor inflaion rahr han h HICP. Givn hs cavas, i sms naural o xpc h rm srucur of inrs ras o b basd mor on cor inflaion han on h hadlin inflaion ra. W compar in abl 4 h prdicion of h -yar inrs ra yar ahad h radiional Dibold and Li (2006) VR modl wih h forcas onc w incorpora h informaion on h inflaion ra. asd on h man squard rror h inclusion of h inflaion ra sms o clarly improv h forcas of h inrs ra boh in h Euro ara and h Unid Sas. This rsul confirms prvious findings in h liraur ha sablishd ha h inflaion ra conains rlvan informaion for forcasing inrs ras [s for xampl Esrlla and Mishin (997)]. Morovr, h inclusion of h inflaion ra dos no chang h diffrnc in h forcas accuracy of financial mars for inrs ras bwn boh aras. Tabl 4: Man Squard Error on -yar inrs ras forcas -yar ahad using VR modl and h join simaion wih h inflaion rm Forward VR modl Join Esimaion Euro ra Unid Sas In abl 5 w compar h forcass of h inflaion ra using an R() modl (simaing h inflaion ra only wih is laggd valus) and h VR modl for h inrs ras augmnd wih h inflaion ra (considring, hrfor, no only h laggd valus of h variabl bu also h dynamics of h Nlson and Sigl facors). Rsuls clarly suggs ha in boh aras h rm srucur provids som informaion ha improvs h forcas of h inflaion ra. s was h cas for h nominal inrs ra, in h uro ara was asir o prdic h cor inflaion ra han US, ihr wih a simpl R modl or wih h informaion conaind in h bond mar, alhough h vidnc was no so clar as for h nominal inrs ra. This diffrnc in h forcasing rrors bwn boh aras could b simply rflcing h fac ha prics in h uro ara hav a highr dgr of prsisnc. 0. good rviw of h imporanc of cor inflaion in h monary policy of h Fdral Rsrv and ohr cnral bans can b found a Wynn (2008.) NCO DE ESPÑ 2 DOCUMENTO DE TRJO N.º 0906

14 Tabl 5: Man Squard Error on -yar inflaion ra forcas -yar ahad using R modl and h Join simaion wih Nlson and Sigl facors Inflaion R modl Join Esimaion Euro ra Unid Sas Nvrhlss, rsuls in abl 4 and 5 ar no mar-consisn givn ha h non-arbirag opporuniy condiion and ris avrsion wr no imposd. W could inroduc hs condiions in a similar way as in scion 2. o h augmnd VR, givn ha boh condiions hav bn drivd indpndnly from h facors conaind in h vcor. Tabl 6 compars h forcas prformanc of his augmnd modl wih arbirag opporuniis (labld as affin modl) wih h modl ha considrs joinly h Nlson and Sigl facors and h inflaion ra (labld as Join simaion) and h modl ha considrs h Nlson and Sigl facors and h inflaion ra sparaly (labld as VR/R modl). Th man squard rrors show ha h inclusion of h non-arbirag opporuniy condiion dos no lssn h improvmn in forcasing abiliy for h inflaion ra and for h nominal inrs ras ha w had obaind by combining h rm srucur and h inflaion ra. Esimad paramrs for boh modls ar prsnd in abls 7 and 8. Ths rsuls ar in conras wih ng al. (2007) who, using an affin modl wih lan facors, show ha whil inflaion is a vry imporan drminan of yild curv movmns, h rm srucur appars o provid lil marginal forcasing abiliy for h dynamics of fuur inflaion ovr simpl im sris modls. This apparn conradicion could b rlad o h nd for affin modls wih lan facors o incorpora som rsricions on h VR ha ar no ncssary in our cas. Tabl 6: Man Squard Error on -yar inrs ras forcas -yar ahad using VR/R modl Join Esimaion and h affin modl wih h inflaion rm Inrs ras VR modl ffin Modl Euro ra Unid Sas Inflaion R modl Join Esimaion ffin Modl Euro ra Unid Sas Esimaion of xpcd inflaion ra Th rsuls in h prvious scion rval ha h rm srucur of inrs ras is closly rlad o h inflaion ra. Thus, his framwor could allow us o obain a forcas for h inflaion ra ha could b considrd as h mar paricipans xpcaions ha undrli NCO DE ESPÑ 22 DOCUMENTO DE TRJO N.º 0906

15 h bond mars. Thrfor, w can compar h rsuls obaind wih ohr masurs of inflaion xpcaions ha ar also availabl from h financial mars SURVEY EPECTTIONS Th mos sraighforward masur of inflaion xpcaions is o dircly as agns in h mar. This is prcisly wha h Consnsus Forcas and h Survy of Profssional Forcasrs (SPF) do. Howvr, som cavas hav o b considrd bfor maing his comparison. Firsly, w a h Consnsus Forcas and h SPF as indicaors of xpcaions for h hadlin inflaion ra, givn h shor sampl of h saisics rlaing o cor inflaion. lhough h diffrnc bwn h xpcaions for hadlin inflaion and cor inflaion in h long run has o b rducd in h shor run ( yar), hr could b significan diffrncs. Scondly, h survys rfr in gnral o avrag inflaion ovr h sampl priod, a masur ha coincids wih ha which w rpord from our simaion. Howvr, in h cas of h SPF for h uro ara, h xpcd inflaion for shor-rm priods (i.. 2 yars) rfrs o h poin simaion. Figur 2: Expcd cor inflaion for h US compard wih h Survy of Profssional Forcasrs for h -yar (lf) and 0-yar (righ) horizon Expcd Inflaion.- US yar SPF US Inflaion xpcaions Ralizd cor inflaion Expcd Inflaion.- US 0 yar SPF US Inflaion xpcaions In h cas of h Unid Sas, rpord in Figur 2, rsuls show ha h forcas did no diffr subsanially from h xpcaions implici in h nominal bond mars. In fac, h -yar-ahad forcas follows a cours ha is mor closly rlad o h SPF han wih h final inflaion ouurn. W considr ha his is clar vidnc ha our modl is capuring inflaion xpcaions and no simply an inflaion forcas.. Th comparison of hs sris has o b viwd wih cauion givn ha Consnsus Forcas and h SPF as abou h arihmic avrag for hs yars and hrfor do no considr h compound ffc of inflaion ovr h yars. Th plod obsrvd inflaion and xpcd inflaion rfr o gomric avrag mans ha considr his compound ffc. NCO DE ESPÑ 23 DOCUMENTO DE TRJO N.º 0906

16 Figur 3: Expcd Inflaion for h Euro ra compard wih h Survy of Profssional Forcasrs and h Consnsus Forcas for h 2-yar (lf) and 5-yar (righ) horizon SPF uro ara Inflaion xpcaions 2 Yars. Euro ra Obsrvd Inflaion 2 yars ahad Yars. Euro ra SPF Consnsus Inflaion xpcaions Obsrvd inflaion 5 yars ahad In h uro ara i is possibl o compar h inflaion xpcaions rpord by Consnsus Forcas and h SPF for h 2- and 5-yars horizon wih h xpcaions w hav simad from h rm srucur (Figur 3). s can b sn, modl-implid inflaion xpcaions ar in lin wih hos obaind from h survys. Garcia and Wrnr (2008) showd ha radiional affin modls ha ry o sima inflaion xpcaions from h rm srucur for h uro ara gnrally ovrsima hm. In hir modl hy lin inflaion xpcaions o h SPF in a modl similar o ha of ng al. (2008), obaining rsuls no oo diffrn from hos prsnd hr EPECTTIONS DRWN FROM OTHER FINNCIL SSETS nohr possibiliy for obaining inflaion xpcaions is o us h prics of ohr financial asss whos rurn is somhow lind o inflaion, i.. inflaion swaps or inflaion-lind bonds. n inflaion swap is a conrac bwn wo invsors in which on of hm agrs o rciv an amoun ha is lind o fuur inflaion in rurn for a fixd amoun from h ohr pary. This amoun would b a signal of h xpcd inflaion of boh invsors. Inflaion-lind bonds (IL) ar similar o nominal bonds, bu in h formr cas h principal is rgularly updad wih h voluion of h CPI. Thrfor, h diffrnc in h rurn on h nominal bond and ha on h IL is calld bra-vn inflaion and can b considrd a masur of xpcd inflaion, sinc his will b h inflaion rquird o clos h gap bwn boh bonds. Ths inflaion xpcaions, li hos obaind from h affin modl of nominal inrs ras, hav an advanag ovr survys in ha hy can b obaind daily and calculad for any horizon. In ordr o compar h rsuls of h affin modl wih hs mar masurs, w will focus on h forward yar inflaion ra 4 yars ahad ha is rpord in figur 4 for boh h uro ara and h Unid Sas. This indicaor has bn widly usd in his mar in ordr o obain an indicaor for h long-rm inflaion ra ha parially rmovs som of h problms rlad o liquidiy prmia, which w will discuss lar. Th inflaion xpcaions xracd from h modl ar subsanially mor sabl han hos obaind from inflaion swaps or IL. In h uro ara, mar masurs incorpora ris prmia and show valus abov hos rpord by h affin modl. For h Unid Sas, inflaion-lind asss ar highly volail and prsn such discrpancis ha hy ar difficul o considr as an accpabl racr of long-rm inflaion xpcaions. NCO DE ESPÑ 24 DOCUMENTO DE TRJO N.º 0906

17 Figur 4: Expcd Inflaion for h Euro ra (lf) and h Unid Sas (righ) compard wih ha drivd from inflaion-lind bonds and inflaion swaps for yar on a 4 yars-ahad horizon Euro ra ravn Inflaion (IL) Inflaion Swap ffin Expcd Inflaion 3.00% 2.75% 2.50% 2.25% 2.00%.75%.50%.25%.00% Unid Sas ravn Inflaion (IL) Inflaion Swap ffin xpcd inflaion 3.5% 3.3% 3.0% 2.8% 2.5% 2.3% 2.0%.8%.5%.3%.0% 0.8% 0.5% Sourcs: EC, Fdral Rsrv and arclays Capial Inc. In fac, nihr inflaion swaps nor inflaion-lind bonds giv clan masurs of inflaion xpcaions. Nihr ass is fr from rm prmia, which would man ha h longr h rm of h ass, h highr h compnsaion rquird by h invsor o buy h ass. ddiionally, hy ar no rmoly as liquid as nominal bonds, so invsors would also as for compnsaion for h difficulis in rducing hir posiion if ndd. In h cas of Inflaion swaps hr ohr problms, such as counrpary ris, sinc h sourc of paymn would no b any govrnmn bu jus anohr invsor. Furhrmor, swaps ar radd ovr h counr, so informaion abou prics is no as rliabl as ha obaind from a rgulad mar. Morovr, boh IL and Inflaion swaps rfr o h ovrall indx of consumr prics (CPI), whil in our modl w rfr o h cor inflaion ra. oh pric indics should hav similar xpcd bhavior for long horizons, hough in h shor run hy could prsn sizabl discrpancis. Givn all hs characrisics, i is usual o find som anomalis in h bhavior of hs indicaors. In Figur 5 w compar h voluion of h bra-vn inflaion ra, Inflaion Swaps and our masur of inflaion xpcaion during h whol of In ordr o parially limina h liquidiy problm w will compar hs indicaors by using h implici xpcd avrag inflaion ra for 5 yars bu 5 yars ahad 2. Howvr, as w can s in h Figur, vn if using his implici masur of h long-rm inflaion ra, h informaion conaind in IL and in swap inflaion ras could b paricularly mislading. During h hird quarr of his yar h inflaion ra rnd changd suddnly, promping som significan shifs in h dmand for IL and inflaion swaps. I is no h purpos of his papr o go ino dail on h chnicaliis surrounding h rading of his ind of produc, bu i should b poind ou ha insiuional invsors, such as pnsion funds and insuranc companis, usually s inflaion procion wih cusomizd inflaion swaps offrd by bans. Ths bans hdg hir posiion by buying porfolios of inflaion-lind bonds. Thrfor, hr is a clar arbirag bwn h IL mar and inflaion swaps, and, as a consqunc, h xpcd inflaion pah drivd from boh mars is usually vry similar. During h scond half of 2008 h currn inflaion ra 2. During h las quarr of 2008 h liquidiy prmia componn bcam so imporan in hs mars ha h bravn spo ra and h spo swap ra had ngaiv valus for crain priods. This misanly gav ris o som analyss xplaining ha financial mars wr discouning a dflaionary scnario. s can b sn in Figur 4, his scnario was no rflcd onc considraion was givn o h implici forward xpcd inflaion ra, which parially rmovd h liquidiy componn. NCO DE ESPÑ 25 DOCUMENTO DE TRJO N.º 0906

18 changd sharply. Mos of his dclin was rlad o oil pric dvlopmns and will no ncssarily affc h long-rm inflaion ra (which is mor rlad o h cor pric indx). Howvr, his rvrsal in h inflaion ra rnd mas a hyprinflaion scnario mor implausibl, and his producs a dramaic dclin in h dmand for inflaion hdgs. This vn brings abou a dcoupling bwn h bra-vn ra and h swap inflaion and a considrabl incras in h volailiy of his indicaor (spcially so in h Unid Sas). Th voluion of his indicaor during h arly monhs of 2009 and h dgr of volailiy suggs ha mos of his chang could no b aribud o a gnuin upda of inflaion xpcaions and should rahr b rlad o h liquidiy posiions of mar paricipans. In fac, our masur of inflaion xpcaions (for h cor inflaion ra) during his priod shows a modra dclin ha may b compaibl wih h nw scnario onc h prolongd incras in oil prics urns around. Figur 5: Expcd Inflaion for h Euro ra (lf) and h Unid Sas (righ) during 2008 compard wih ha drivd from inflaion-lind bonds and inflaion swaps for 5 yar on 5 yars-ahad horizon Euro ra ra vn 5Yo5Y Swap 5Yo5Y ffin Modl Unid Sas ra vn 5Yo5Y Swap 5Yo5Y ffin Modl /08 02/0803/08 04/08 05/08 06/08 07/08 08/08 09/080/08 /08 2/08 0/09 Sourcs: EC, Fdral Rsrv and arclays Capial Inc /08 02/08 03/08 04/08 05/08 06/08 07/08 08/08 09/08 0/08 /08 2/08 0/09 NCO DE ESPÑ 26 DOCUMENTO DE TRJO N.º 0906

19 3 How do inflaion xpcaions rac o changs in monary policy? Th simad affin modls giv us a VR quaion ha rlas h paramrs of h rm srucur o xpcd inflaion. This quaion allows us o prform an analysis of h racion of inflaion xpcaions o a shoc in shor-rm inrs ras by mans of an impuls-rspons xrcis. To approach h ffc of a 25 basis poins incras in h monary policy ra (h Fd Fund Ra in h US and h Main rfinancing opraions inrs ra of h EC) w hav considrd an quivaln incras a h shorr nd of h rm srucur. This ffc is achivd by rducing h slop facor ( S ) ha was dfind as h diffrnc bwn h inrs ra a boh nds of h yild curv. In h affin modl, h VR quaion has bn dfind o includ an orhogonal Σ marix, so an impuls-rspons xrcis can b prformd by simply adding a prmann shoc o h random variabl associad wih h S variabl. Th rsul of his shoc on inflaion is shown in Figur 6. s can b sn, such a movmn would rduc xpcd inflaion, alhough h spd of ransmission of such a shoc is qui slow, h full impac of such a masur bing rcivd mor han a yar afr h movmn. This ransiion is prolongd for somwha longr han h acual prcpion of mor han 6 monhs rquird for a chang in monary policy o a ffc on h ral sid of h conomy. Figur 6: Rspons of xpcd inflaion o a 25bp incras in shor-rm inrs ras for boh h Euro ra (lf) and h Unid Sas (righ). Euro ra US 0.00% 0.00% -0.0% -0.0% -0.02% -0.02% -0.03% -0.03% -0.04% % Whn comparing h ffc on h wo modls for h uro ara and h US, i is possibl o s ha h impac of h movmn in shor-rm inrs ras on inflaion xpcaions is highr in Europ han in h Unid Sas. ddiionally, h shoc is rvrsd arlir in h uro ara. This highr snsiiviy is consisn wih h fac ha h EC rquirs fwr movmns in is monary policy ra han h Fd [s djmian al. (2008)]. NCO DE ESPÑ 27 DOCUMENTO DE TRJO N.º 0906

20 4 Conclusions In his papr w rformula h Dibold and Li (2006) modl as a radiional affin modl in ordr o incorpora h non-arbirag opporuniis condiion and ris avrsion. y doing so, w improv h consisncy of h modl whil mainaining, and vn incrasing, h forcasing abiliy of h Dibold and Li (2006) mhodology. This framwor provids a ool for xracing ris prmia implici in h rm srucur. This variabl sms o b h main drivr of h variaion of nominal inrs ras in h Unid Sas and Europ ovr h las dcad. Morovr, wih h affin formulaion i is possibl o inroduc h inflaion ra as an addiional facor. asd on his augmnd modl w could obain mar xpcaions abou h inflaion ra ha in rms of forcas ouprform hos compud wih an auorgrssiv approach. Morovr, h inflaion ra sms o conain usful informaion for forcasing h inrs ra. u on of h main advanags of his mhodology for obaining inflaion ra xpcaions rsids on h fac ha i is basd on a liquid mar and ha w can isola inflaion xpcaions from h ris prmia componn, which was no possibl wih ohr masurs such as inflaion-lind bonds or inflaion swaps. This advanag is of paricular rlvanc during srssd priods as h nd par of Lasly, h VR quaion of h affin modl could b usd o prform an impuls rspons analysis o assss h conncion bwn shor-rm inrs ras and inflaion xpcaions. Our rsuls suggs ha h impac of a shor-rm inrs ra on inflaion xpcaions is highr in h 8-monh horizon. Morovr, w could obsrv ha undr a shoc in monary policy of h sam magniud, inflaion xpcaions in h uro ara sms o rac mor quicly and wih a highr magniud han in Unid Sas. This finding, ha could b rlad, among ohr facors, wih diffrncs in h monary policy insiuional framwor could hlp o xplain h smoohr racion funcion of h monary policy in h uro ara. NCO DE ESPÑ 28 DOCUMENTO DE TRJO N.º 0906

21 REFERENCES DJEMIN, S., M. D. PRIÈS and F. SMETS (2008). quaniaiv prspciv on opimal monary policy coopraion bwn h US and h uro ara, EC Woring Papr No LONSO, F., J. MRQUÉS and S. FORTE (2006). Implid dfaul barrir in crdi dfaul swap prmia, anco d España Woring Papr No NG,., G. EKERT and M. WEI (2007). "Do macro variabls, ass mars, or survys forcas inflaion br?, Journal of Monary Economics, Vol. 54, pp (2008). Th Trm Srucur of Ral Ras and Expcd Inflaion, Journal of Financ, Vol. 63, pp NG,., and M. PIZZESI (2003). no-arbirag vcor auorgrssion of rm srucur dynamics wih macroconomic and lan variabls, Journal of Monary Economics, Vol. 50, pp EKERT, G., and R. HODRICK (200). Expcaions Hypohss Tss, Journal of Financ, Vol. 56, pp CRRIERO,., C.. FVERO and I. KMINSK (2006). Financial Facors, Macroconomic Informaion and h Expcaions Thory of h Trm Srucur of Inrs Ras, Journal of Economrics, Vol. 3, pp CHRISTENSEN, J. H. E., F.. DIEOLD and G. D. RUDEUSCH (2007). Th affin rbirag-fr Class of Nlson-Sigl Trm Srucur Modls, NER Woring Papr No. W36. CO, J. C., J. E. INGERSOLL and S.. ROSS (985). hory of h rm srucur of inrs ras, Economrica, Vol. 53, pp DI, Q., and K. J. SINGLETON (2000). Spcificaion nalysis of ffin Trm Srucur Modls, Journal of Financ, Vol. 55, pp DEWCHTER, H., M. LYRIO and K. MES (2006). Join Modl for h Trm Srucur of Inrs Ras and h Macroconomy, Journal of pplid Economrics, Vol. 2, pp DEWCHTER, H., and M. LYRIO (2006). Macro Facors and h Trm Srucur of Inrs Ras, Journal of Mony, Crdi, and aning, Vol. 38, pp DIEOLD, F.., and C. LI (2006). Forcasing h Trm Srucur of Govrnmn ond Yilds, Journal of Economrics, Vol. 30 (2), pp DIEOLD, F.., G. D. RUDEUCH and S. ORGN (2004). Th Macroconomy and h Yild Curv: Dynamic Lan Facor pproach, NER Woring Papr No. W066. DUFFIE, D., and R. KN (996). Yild-Facor Modl of Inrs Ras, Mahmaical Financ, Vol. 6, pp ESTRELL,., and F. MISHKIN (997). Th prdiciv powr of h rm srucur of inrs ras in Europ and h Unid Sas: Implicaions for h Europan Cnral an, Europan Economic Rviw, Vol. 4, pp FM, E., and R. LISS (987). Th Informaion in Long-Mauriy Forward Ras, mrican Economic Rviw, Vol. 77, pp GLI J., S. GERLCH, J. ROTENERG, H. UHLIG and M. WOODFORD (2004). Th monary policy sragy of h EC rconsidrd, Monioring h EC Sris No. 5, CEPR. GRCÍ, J.., and T. WERNER (2008). Inflaion riss and inflaion ris prmia, EC Woring Papr, forhcoming. GIMENO, R., and J. M. MRQUÉS (2008). Uncrainy and h pric of ris undr a nominal convrgnc procss, anco d España Woring Papr No GIMENO, R., and J. M. NVE (2009). gnic algorihm simaion of h rm srucur of inrs ras, Compuaional Saisics & Daa nalysis, Vol. 53, pp JORION, P., and F. MISHKIN (99). muli-counry comparison of rm srucur forcas a long horizons, Journal of Financial Economics, Vol. 29, pp KIM, D., and. ORPHNIDES (2005). Trm Srucur Esimaion wih Survy Daa on Inrs Ra Forcass, FEDS Woring Papr No LUCH, T., and J. WILLIMS (2003). Masuring h Naural ra of Inrs, Th Rviw of Economics and Saisics, 85 (4). NELSON, C., and. SIEGEL (987). Parsimonious Modlling of Yild Curvs, Journal of usinss, Vol. 60 (4), pp VSICEK, O. (977). n quilibrium characrizaion of h rm srucur, Journal of Financial Economics, Vol. 5, pp WYNNE, M. (2008). "Cor inflaion: a rviw of som concpual issus", Rviw of h Fdral Rsrv of S Louis, May/Jun. NCO DE ESPÑ 29 DOCUMENTO DE TRJO N.º 0906

22 NCO DE ESPÑ 30 DOCUMENTO DE TRJO N.º 0906 Tabl 7: Esimad modl for h Euro ra = C S L C S L ε ε ε ε π π = C S L π λ ( ),, = u C S L y π ( ) I N u 0,.0000, ( ) I N, 0 ε Tabl 8: Esimad modl for h Unid Sas = C S L C S L ε ε ε ε π π = C S L π λ ( ),, = u C S L y π ( ) I N u 0,.0000, ( ) I N, 0 ε

23 nnx : Rcursiv xprssion of rm srucur paramrs RISK VERSION ND NON-RITRGE CONDITIONS non-arbirag condiion guarans h xisnc of a ris-nural masur (nod as Q) ha allows inrs ras o b xprssd in rms of fuur rm srucur oucoms, [ ] Q = E (.) Ris-nural masurs (Q) ar usually convrd ino naural probabiliis using h Radon-Niodym drivaiv, as in ng and Piazzsi (2003), dnod by ξ, ξ = E (.2) ξ Usually, ξ in quaion.2 is assumd o follow a log-normal procss, 2 λ λ λ ε ξ = ξ (.3) whr λ is a im-varying vcor ha incorporas h concp of ris avrsion ino h valuaion framwor. Th firs par of h xponn ( λ ) is h Jnsn Convxiy ξ ξ = componn ha nsurs ha [ E ] λ, whil in h scond, λ muliplis h prurbaion vcor ε, scaling h uncrainy in h random variabls. This scond rm is rsponsibl for h inroducion of h ris prmium in h valuaion framwor, whrby λ can b considrd as a pric of ris. Tim-varian ris prmia [ar and Hodric (200) will b h consqunc of changs in his pric of ris ha is modlld assuming i o b also affin o h sam facors, λ = λ 0 λ (.4.) Finally, subsiuing.3 ino.2, w arriv a a modifid non-arbirag condiion ha now as ino accoun invsors ris avrsion. E λ 2 λ λε = (.5) NCO DE ESPÑ 3 DOCUMENTO DE TRJO N.º 0906

24 NCO DE ESPÑ 32 DOCUMENTO DE TRJO N.º 0906 Only and ε of xprssion (.5) ar no alrady nown in priod, whil h ohr rms in h xponns can b xracd from h xpcaions opraor, [ ] 2 = E λ ε λ λ (.6) Nvrhlss, vcor can b forcas using VR quaion (2), ( ) [ ] 2 Σ Φ = E ε λ λ λ µ (.7) Th xponn lf in h xpcaions opraor of xprssion (.7) is solvd aing ino accoun h Jnsn inqualiy. ( ) µ Σλ Φ ΣΣ = 2 (.8) Finally, rplacing h pric of ris λ in.8 by is dfiniion (quaion.4), w arriv a xprssion (.9), ( ) ( ) 0 2 λ λ µ Σ Φ ΣΣ = (.9) This las xprssion allows us o rcovr h rcursiv xprssion of cofficins and in h affin rprsnaion as a funcion of h shorr rms, ΣΣ Σ = 2 0 λ µ (.0) λ Σ Φ = (.). Valuaion wihou ris compnsaion Th ris nuraliy valuaion framwor usd in (.) allowd us o incorpora h ris prmium ino h rm srucur. In ordr o rcovr ris-fr ras w should considr a framwor whr agns ar no concrnd abou ris, so xpcaions drivd from h non-arbirag condiion ar valuad undr a naural masur, [ ] = E (.)

25 NCO DE ESPÑ 33 DOCUMENTO DE TRJO N.º 0906 whr j and j ar h cofficins of quaion, ha m non-arbirag condiions. Using h sam rasoning as in annx., rplacing by is forcas and applying Jnsn inqualiy o solv h xpcaions opraor, w arriv a xprssion (.2), ( ) Φ ΣΣ = 2 µ (.2) s can b sn, xprssion (.2) is quivaln o (.8), h only diffrnc bing ha onc ris avrsion is avoidd, h rm λ Σ is no longr ndd. In fac, his was h rm ha addd a ris prmium for ach xra priod of invsmn. ris-nural individual would hav a null pric of ris, wih boh xprssions bcoming quivaln. Undr his assumpion, h rm srucur rcursiv xprssion would now b, 2 Σ Σ = µ (.3) Φ = (.4).2 Exognous modl simaion Prior o h simaion of h affin modl w hav o drmin h facors rlad o h rm srucur. Following Dibold and Li (2006), w us h Nlson and Sigl (987) formula of h rm srucur. = C S L y, (C.) Dibold and Li (2006) fixd h valu of o b h man hroughou h sampl. Onc is consan, quaion C. can b simad by OLS for ach priod, rgrssing inrs ras for diffrn rms () agains marix Z. = Z (C.2) Onc L, S and C ar simad as h paramrs of hs rgrssions for ach priod, hy can b usd as facors for h affin modl. s vcor is complly drmind, w no longr nd o fix any inrs ra as obsrvd wihou rror. In fac, i is now qui asy o rcovr iniial valus of h paramrs via OLS simaions in hr sps.

26 Sinc vcor is xognously drmind, w can sima h VR quaion via OLS, which allows iniial valus o b obaind of µ, Φ and Σ = µ Φ Σε ε N( 0 I ) (C.3), W can also us vcor o rgrss i agains nominal inrs ras for diffrn rms using h rm srucur quaion, in ordr o sima conscuiv valus of and, 2 u N ( 0, σ I ) (C.4) y, = u, Finally, in ordr o incorpora a non-arbirag condiion and ris avrsion w go furhr han Dibold and Li (2006) and us  and ˆ simaions o rgrss hm agains shorr rm valus, rarranging quaions 5 and 6. Onc w hav naiv valus from C.3 and C.4, hn quaions 5 and 6 bcom, ( ˆ ˆ ) ˆ ˆ ˆ ˆ ˆ µ ΣΣ = Σλ0 (C.5) 2 ( ˆ ˆ Φ) ˆ ˆ = Σλ (C.6) Equaions C.5 and C.6 ar linar wih rspc o λ 0 and λ, and hrfor, hs paramrs can also b simad by OLS. Onc w hav simad sparaly C.3, C.4, C.5 and C.6 quaions, w hav naiv iniial valus of h affin modl ha allow for h swif compuaion of h join maximum lilihood simaion of h affin modl givn by, 2 u N ( 0, σ I ) ( ) u y, =, = µ Φ Σε ε N( 0 I ), subjc o (C.7) = µ Σλ0 ΣΣ 2 = Φ Σλ NCO DE ESPÑ 34 DOCUMENTO DE TRJO N.º 0906