Exchage Raes, Risk Premia, ad Iflaio Idexed Bod Yields by Richard Clarida Columbia Uiversiy, NBER, ad PIMCO ad Shaowe Luo Columbia Uiversiy Jue 14, 2014 I. Iroducio Drawig o ad exedig Clarida (2012; 2013) ad Luo (2013) his chaper derives ad empirically esimaes a srucural relaioship bewee he omial exchage rae, aioal price levels, ad observed yields o log mauriy iflaio - idexed bods. This relaioship ca be ierpreed as defiig he fair value of he exchage rae ha will prevail i ay model or real world ecoomy i which iflaio idexed bods are raded. Fair value is he level of he omial exchage rae ha equaes he kow real reur o holdig a log mauriy home currecy iflaio idexed bod o he expeced real reur o holdig a iflaio idexed bod payable i foreig currecy. We derive a ovel, empirically observable real ime measure of he risk premium ha ca ope up a wedge bewee he observed level of he omial exchage rae ad is fair value ad we relae our measure of he log horizo real risk premium o he Fama (1984) measure of he shor horizo omial risk premium. We ake our heory o a daily daa se spaig he period Jauary 2001 February 2011 ad sudy high frequecy, real ime decomposiios of poud, euro, ad ye exchage raes io heir fair value ad risk premium compoes. The relaive imporace of hese wo facors varies depedig o he sub sample sudied. However, sub samples i which we fid correlaios of 0.30 o 0.60 bewee daily exchage rae chages ad 1
daily chages i fair value are o ucommo. We also show ha o average i our daa se, for all hree exchage raes sudied, a 1 perceage poi rise i he foreig currecy risk premium is coemporaeously associaed wih a roughly 50 basis poi rise i he iflaio idexed bod reur differeial i favor of he foreig coury ad a 50 basis poi appreciaio of he dollar. Tha is, he dollar eds o sreghe o a day whe our measure of he foreig currecy risk premium o he poud, euro, or ye rises. This implies ha our measure of he level of he risk premium should help forecas subseque chages i he omial exchage rae ad prese empirical evidece of his forecasabiliy is prese i our sample. I paricular we show ha ideed, a rise i foreig currecy risk premium oday help o forecas subseque dollar depreciaio. II. A Equilibrium Implicaio Cosider a US ivesor who, amog he may asses he ca hold, ca hold iflaio idexed sovereig bods deomiaed i dollars ad pouds. Le ρ deoe he dollar price a zero coupo iflaio idexed bod ha pays off 1 dollar i years muliplied by cumulaive realized US iflaio over he ex years. The realized omial gross reur o his ivesme, if held o mauriy, will be 1) R hh, = (1/ ρ)p+/p where P is he CPI. Le ρ* deoe he poud price of a zero coupo iflaio idexed bod ha pays off 1 poud i years muliplied by cumulaive realized UK iflaio over he ex years. The realized omial gross reur o a US ivesor o his ivesme, if held o mauriy, will be 2) R h,f, = (1/Sρ*)S+ P*+/P* where S is he dollar price of a poud. The realized real gross reur o holdig a US iflaio idexed bod o mauriy is 3) RR hh, = R hh, (P/P+) = (1/ ρ) exp(r,) 2
Thus, of course, he US iflaio idexed bod offers a kow, osochasic realized real reur r, if held o mauriy. Bu wha is he realized real reur o a US ivesor of holdig a UK iflaio idexed bod? 4) RR hf,=r hf, (P/P+) = {(1/S ρ*) (S+ P*+/P*)} P/P+ = exp(r*,)q+/q where Q = S P*/P is he CPI real exchage rae. For a US ivesor he UK iflaio idexed bod is o riskless if held o mauriy: eve hough r*, is kow, he US ivesors bears real exchage rae risk. Noice ha his is o a assumpio: i follows from he aure of he sochasic omial cash flows o a US ivesor for holdig a UK liker o mauriy. Defie exp θ, as he raio of expeced real reur o a US ivesor of holdig a UK liker versus kow real reur o a US ivesor of holdig a TIP 5) exp θ, E RR hf,/rr hh, = (exp r*, / exp r,) E Q+/Q where E Q+ is he expeced level of he CPI real exchage rae i years. Whe θ, > (<) 0 he expeced real reur o holdig he UK liker exceeds (is less ha) he kow real reur o holdig he US TIP. Equaio (5) has a paricularly coveie ierpreaio whe he horizo is log eough so ha E Q+ = Q, he assumed cosa ucodiioal mea of he real exchage rae defied by log ru relaive PPP. I his case, muliplyig hrough by Q ad re arragig we see ha: 6) S = (P/P* ) exp (r*, r,) Q exp(-θ,) or ~ 7) S S exp, Thus period by period, he observed spo exchage rae is he produc of he fair value S ~ - he level of he spo exchage rae ha equaes he kow real reur o ivesig i he US TIP o he expeced real reur 3
o ivesig i he UK liker - ad he risk premium exp(-θ,), he raio of he kow real reur o ivesig i he US TIP o he expeced reur o ivesig i he UK liker (he reciprocal of he risk premium o he ucovered UK liker ivesme). Takig logs of boh sides ad leig q = l Q 8) s p p* ( r *, r, ) q, We see ha i firs differeces we mus have s ( p p* ) ( r *, r, ), A rise i he edogeous θ, is a icrease i he risk premium o a UK ivesme which is jus a icrease he expeced excess reur a US ivesor ears o a UK liker ivesme compared wih he kow real reur he ears o a US TIP. Thus from (6) a rise i expeced real reur o he UK liker ivesme mus be brough abou by some combiaio of a rise i UK US real ieres differeial ad a coemporaeous real appreciaio of he dollar relaive o he poud. A appreciaio of he dollar relaive o he poud oday coribues o a posiive expeced fuure excess reur o a iflaio hedged poud ivesme because i ses up he expecaio of a subseque (real) depreciaio of he dollar relaive o he poud as is evide from (6). Whe he kow real reur o a US ivesor i ivesig i he US TIP is equal o he expeced real reur o he US ivesor i he UK liker. I is his sese i which is he fair value of he omial exchage rae. Whe he exchage rae is equal o fair value by his defiiio, expeced real reurs ha ca be achieved by holdig iflaio idexed bods are equalized across couries. Thus our oio of fair value is relaed o he level of he exchage rae implied by he ex-ae real versio of ucovered ieres pariy over a shor holdig period for omial bods. However as we illusrae below, he wo are disic ways o assess currecy valuaio. A Asse Pricig Ierpreaio 4
We make a miimal umber of assumpios o provide a asse pricig ierpreaio of Equaio (7). We do o assume complee markes or a represeaive age. We do o assume ha we kow he model, le aloe he parameers, ha lik he prese value of macro fudameals o exchage rae valuaio. Uder our assumpios our framework is cosise wih almos ay uderlyig model i which iflaio idexed bods are raded. We assume ha, i a global fiacial equilibrium, here is a fucioal relaioship bewee he omial (US dollar) price oday of a asse ha delivers a radom dollar cash flow a some dae i fuure (for cocreeess, 10 years hece) ad o cash flow a ay dae oher ha +: F ( N;, ) where, is he codiioal probabiliy disribuio of he radom omial cash flow from he asse ha pays off i years. We specialize F so ha E ( m N;,, ) So oday s price of a asse wih radom omial cash flow i years is he codiioal expecaio of he produc of ha cash flow ad he radom variable m,. Assumpio: m, is homogeous i he price levels P ad P+ m, z, P P This is a sadard propery i may asse pricig models (Cochrae (2001). For example i Lucas (1982) we have m, U'( C U'( C ) ) P P 5
Agai, we do o require a represeaive age, complee markes, or really ay addiioal srucure o z,. This is a iuiive resricio o omial asse prices ha says ha he real price of he asse oday depeds upo he real value of he cash flow i delivers sae by sae a mauriy ad o he price level iself a + iself (afer, of course, corollig for facors oher ha he price level iself ha are icluded i z,.). Wih his backgroud, cosider how o price a zero coupo iflaio idexed bod ha pays off 1 dollar i years muliplied by cumulaive realized iflaio over he ex years. E P ( m 1 ) E ( z, P, 1) Or, dividig by ρ 1 {exp r } E ( z,, ) Where r, is he coiuous compouded kow real reur o he iflaio idexed bod. US ivesors ca also obai US dollar cash flows by ivesig i a UK iflaio idexed bod ad sellig he poud proceeds for dollars i years. Le S be he dollar price of a poud ad * represe a UK variable. Le Q = SP*/P defie he real exchage rae ad Q is ucodiioal mea Q = E Q. The we have S * E ( m, 1 S P* P* ) Or dividig hough: 6
1 exp r *, E ( z, Q 1 Q ) Wih hese buildig blocks we ow derive a srucural exchage rae equaio ha will hold i ay model ha seeks o describe a world i which log mauriy iflaio idexed bods rade. Sice such bods rade i may couries (US, UK, Frace, Caada, Japa) his should apply o a large umber of models. We see ha 9) S P P* exp r * exp r,, Q QE ( z, 1 Q E ( z ), ) Comparig wih equaio (6) we see ha θ, is give by 10) exp, Q E ( z, 1 Q E ( z ), ) Alhough o ecessary for wha follows, we gai addiioal isigh by lookig a he log ormal case i which we have equaio (11). 11 ) cov, (lz,, q q) var, ( q q) E ( q q) We oe ha he firs erm i he above expressio is he codiioal covariace bewee he sochasic discou facor ad real exchage rae ha prevails whe he zero coupo iflaio liked bods maure. This ca be ierpreed as a risk premium ha opes up a wedge bewee kow real reur (o a US ivesor) of holdig a log mauriy TIP ad he sochasic real reur o a US ivesor of holdig a UK liker. Whe his covariace is egaive, a uhedged posiio i a UK liker pays off less (because of realized real appreciaio of dollar relaive o he poud) whe he sochasic discou facor is high. Thus a posiive hea correspods o a posiive risk premium o he UK liker. Tha is, he kow 7
real reur o he US liker is less ha he expeced real reur o he US ivesor, iclusive of expeced appreciaio of he poud, of holdig a UK liker whe θ is posiive. A icrease i he expeced excess reur o he UK liker will require some combiaio of a icrease i r*, - r, ad a jump appreciaio of he dollar. I wha follows we shall assume for ease of exposiio ha expeced deviaios from PPP a a 10 year horizo are sufficiely close o zero so as o be igored. Imporaly, however, researchers who have a view o log horizo PPP deviaios ca iclude ha view direcly ad use i as a ipu o he accouig framework we develop below. Thus, i wha follows, we shall refer o θ, as he risk premium. I is worh oig ha he complee markes assumpio which we do require o derive (7) or (9) would pu a umber of addiioal resricios o he joi behavior of exchage raes ad bod yields, boh iflaio idexed ad omial. For example, uder complee markes, Backus e. al. (2001) show ha 12) S S 1 m m*,1,1 We see ha i our oaio his would also imply 13) Q Q 1 z,1 z *,1 These are elega, powerful implicaios bu we do o impose hem o he daa or use hem o ierpre real ime exchage rae flucuaios. III. Compariso wih he Lieraure Buildig o earlier work by Hase ad Hodrick (1980) ad Cumby ad Obsfeld (1981), Fama (1984) is he classic sudy of he risk premium o holdig a log posiio i a foreig currecy omial bod for oe period (bu see also Clarida, Davis, ad Pederse (2009) for a rece aalysis of 8
wha ca ad ca be leaed from a Fama regressio). I he lieraure (see Egel 2010 for a review ad exesio) his cocep of he risk premium is usually defied by he firs order log approximaio 14) rp, 1 E s 1 s i*,1 i,1 where lower case i deoes he shor erm omial ieres rae. There is of course a log ad proud radiio i he ieraioal fiace lieraure, begiig wih Frakel (1978), of empirically relaig real exchage raes o real ieres differeials (Shafer ad Loopesko (1983); Campbell ad Clarida (1987); Clarida Gali (1994) are early examples). For he mos par, his lieraure pre daes he widespread iroducio of log mauriy iflaio idexed bods ad of ecessiy solves forward he real versio of he deviaios from UIP equaio. 15) rp, 1 Eq 1 q er *,1 er,1 where er,1 = i,1 Eπ,1 is he ex ae shor erm real ieres rae o omial bods a home ad similarly abroad. Solvig forward ad assumig lq is saioary we obai (see Egle (2010) for a lucid discussio ad Bruermeier, Nagel, ad Pederse (2008) for a ierpreaio of he forward soluio for he omial exchage rae uder ucovered ieres pariy): 16) q ( er * i0 i 1 i, 1, er ) ( rp ) E l Q i0 i, 1 Noe ha covergece of hese o-discoued prese value equaio requires he ucodiioal mea of he ex-ae real rae differeial μ o equal he mea of he Fama risk premium λ. Comparig erms we see, i rp I r r i er i er i q { ( * ) ( *,1, )} 0,1,, 17) 1 El Q where for cocreeess we suppose ha i 18) ( er * i, 1 er i, 1 ) 0 9
ad similarly for he Fama premium afer periods. We see ha our measure of he risk premium θ, is he sum of hree erms: he period sum of he expeced Fama risk premiums; he differece (i brackes) bewee home ad foreig log mauriy iflaio idexed bod reurs ad he expeced real reurs o rollig oe period omial bods; ad a Jese s iequaliy erm. 10
IV. Daa Our daa se is comprised of daily observaios o spo exchage raes, iflaio idexed bod yields, ad mohly observaios o cosumer price idexes for he US, UK, ad Euro area for he period Jauary 2001 hrough Jauary 2011 ad for Japa sice December 2004 shorly afer iflaio idexed bods were iroduced. We cover mohly CPI levels o daily observaios via ierpolaio. Give he low ad relaive sable rae of iflaio for hese couries over his period, he approximaio of he uobserved daily iflaio differeial wih he observed per day mohly average iflaio differeial iroduces measureme error, bu his error is small relaive o he observed daily volailiy of exchage raes ad iflaio idexed bod yields. Our heoreical model is derived i erms of he yields o iflaio idexed zero coupo bods. Iflaio idexed bods are ypically issued i coupo form. However, i he US here is a marke i which iflaio idexed coupo Tips are sripped of heir coupos ad rade i zero coupo form. I our empirical aalysis we will use daily daa o cosa 10 years o mauriy yields o zero coupo Tips provided by Barclays. For he UK, we use he daa o zero coupo liker yields provided by he Bak of Eglad. For he Euro, we use esimaes of he zero coupo iflaio idexed yield curve for Frech ad Germa iflaio idexed bods provided by Morga Saley. For Japa, o daa o zero coupo iflaio idexed yields could be foud so we use he observed yield o coupo bearig iflaio idexed bods. Oe fial poi o discuss is how we calibrae he cosa erm i Equaio (6) for fair value. This cosa erm is o impora for much of wha we do sice we will ofe seek o accou for perce chages i observed omial exchage raes i erms of perce chages i fair value ad chages i he risk premium. For hese exercises, he cosa drops ou. However, i drawig he some of he graphs we will wish o preserve he levels iformaio, ad will selec he cosa erm equal o he average real exchage rae durig he sample period depiced. I Secio VI we relax his assumpio ha he expecaio a each dae of he real exchage rae 10 years forward from ha dae is cosa. The e effec of relaxig his assumpio is o icrease somewha he coribuio of fair value i accouig for omial exchage rae flucuaios. 11
V. Empirical Resuls We ow use he framework developed above o ierpre he behavior of he Euro, Poud, ad Ye exchage raes over he pas 10 years. There are o ecoomeric esimaes o prese because our framework (Equaio 7) provides day by day a real ime decomposiio he chage i he exchage io he chage i he fair value ad he chage i risk premium. Our framework allows ideed we expec o fid periods i which shocks o he risk premium are large ad die ou slowly while here may be oher periods i which exchage rae movemes, corary o he origial Meese-Rogoff (1983) fidig ha exchage rae chages are difficul o explai eve give ex pos realizaios of fudameals, are well accoued for by shifs i our measure of fair value derived above. We prese our mai fidigs i a series of chars. For each exchage rae, he chars will help us o ideify as well as quaify he imporace of shocks o fair value ad shocks o he risk premiums i accouig for exchage rae flucuaios over differe periods as well as over various horizos of ieres. As our sample icludes he global fiacial crisis ad is afermah (a leas hrough Jauary 2011!), we are paricularly ieresed o deermie ad quaify he shifs i risk premium ad fair value ha occurred over his period. 12
Euro Char 1 I Char 1, ad i all subseque chars, he dark blue lie depics he spo exchage rae, i his case he US dollar price of a Euro, he aqua blue lie is he fair value defied by Equaio 11. The amou by which he exchage rae EUR exceeds FV measures he risk premium i favor of he USD ha is refleced i he EUR spo exchage rae. This correspods o θ,. The amou by which he exchage rae EUR falls shor of FV measures he risk premium i favor of he EUR ha is refleced i he EUR spo exchage rae. This correspods o θ,. Our framework we believe provides a compellig qualiaive as well as a plausible quaiaive accou of he swigs i Euro exchage rae sice 2005. As ca be see from he char, he broad move i he Euro from 1.25 i he summer of 2005 o 1.45 i he sprig of 2008 is well 13
accoued for, boh i direcio ad i magiude, by he rise i he fair value durig ha period. Accordig o our model, he ex move i he Euro from 1.45 o he brual level of 1.60 reached i he summer of 2008 was due almos eirely a equal move i he risk premium, i favor of he dollar ad hus agais he Euro. Sice he ose he global fiacial crisis i Sepember 2008, movemes i he Euro have bee domiaed by flucuaios i risk premium wih fair value flucuaig i a raher arrow rage ceered a roughly 1.37. I Ocober 2008, our measure of he risk premium swigs i favor of he Euro (e.g. i appreciaed he dollar price of he Euro o such a exe i se up he expecaio of a deprecaio ad hus capial gai o a Euro ivesme). The risk premium swigs back i favor of he dollar i he secod half of 2009 as he dollar depreciaes i adem wih he Fed s quaiaive easig programs aouced i March of ha year. Sice 2010, our framework idicaes ha he foreig exchage marke has required a posiive risk premium o hold he Euro. This period of course coicides he crisis i he Euro periphery. Of course, i is impora o cofirm ha he visual impressio coveyed by he char is evide i he acual empirical correlaio bewee he Euro exchage rae ad our measure of fair value. Char 2: Correlaio i Daily Chages i Eur ad FV (60 day widow) 1 0.8 Correlaio bewee Daily Chages i Eur ad Daily Chages i RNFV - 60 Day Widow + ad - Oe Sadard Error Bads 11/23/2010 8/23/2010 5/23/2010 2/23/2010 11/23/2009 8/23/2009 5/23/2009 2/23/2009 11/23/2008 8/23/2008 5/23/2008 2/23/2008 11/23/2007 8/23/2007 5/23/2007 2/23/2007 11/23/2006 8/23/2006 5/23/2006 2/23/2006 11/23/2005 8/23/2005 5/23/2005 2/23/2005 11/23/2004 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 14
Char 2 depics he correlaio (over rollig 60 day widows) bewee daily chages i Euro exchage raes ad daily chages i our measure of fair value which of course is domiaed by daily chages i real ieres rae differeials bewee Europe ad US iflaio idexed bods. We see ha periods i which he correlaio is i he rage of 0.3 o 0.4 are o ucommo. We also see ha i periods i which shocks o he risk premium are see o domiae, he correlaio bewee he Euro ad fv falls o zero or is eve egaive. Oe is emped o ideify periods i which he exchage rae is well accoued for by movemes i fv (such as 2005 o 2008 i Char 1) as periods i which fudameals mosly maer for exchage rae deermiaio, i coras o periods sice Sepember 2008 i which fudameal are pushed aside ad risk aversio appears o ake over. Bu wihi he sric logic of our framework, his empaio would o be jusified. Fudameals may drive he risk premium as well, bu wihou imposig much more addiioal srucure o m, we ca really say more. However, ulike he radiioal approach (Fama (1984)) i which a uobserved currecy risk premium mus be iferred by exracig he forecasable compoe from realized reurs o currecy carry rades, our framework provides a ecoomeric free measure of he releva risk premium give observed yields o iflaio idexed bods ad he spo exchage rae. Char 3 15
Poud Char 4 Char 4 depics our decomposiio of he GBP exchage rae io is fair value ad risk premium compoes. From 2001 hrough summer of 2005, he appreciaio of he poud from 1.50 o 1.75 is almos fully accoued for by a equal rise i our esimae of fair value from he iflaio idexed bod marke. However, our framework accous for he subseque move up from 1.75 o 2.05 reached i Jauary 2008 almos eirely by he emergece of a subsaial risk premium i favor of he dollar (i.e. a risk premium ha se up expecaio of a higher reur o a US iflaio liked bods). This risk premium is elimiaed ad shifs i favor of he GBP i Sepember 2008 ad has remaied i place sice. Sice 2009, our esimae of fair value has sayed i a arrow rage ceered aroud 1.65. 16
Char 5: Correlaio i Daily Chages i GBP ad FV (60 day widow) 1 Correlaio bewee Daily Chage i Gbp ad Daily Chage i RNFV - 60 Day Widow 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 11/28/2010 7/28/2010 3/28/2010 11/28/2009 7/28/2009 3/28/2009 11/28/2008 7/28/2008 3/28/2008 11/28/2007 7/28/2007 3/28/2007 11/28/2006 7/28/2006 3/28/2006 11/28/2005 7/28/2005 3/28/2005 11/28/2004 7/28/2004 3/28/2004 11/28/2003 7/28/2003 3/28/2003 11/28/2002 7/28/2002 3/28/2002 11/28/2001 7/28/2001 3/28/2001 Char 5 depics he correlaio (over rollig 60 day widows) bewee daily chages i GBP exchage raes ad daily chages i our measure of fair value. Agai we see ha periods i which he correlaio bewee daily chages is i he rage of 0.3 o 0.5 are o ucommo. We also see ha i periods i which shocks o he risk premium are see o domiae, he correlaio bewee he GBP ad fv falls o zero or is eve egaive. This implies ha large shocks o he risk premium i favor of he poud (or i Char 1 he Euro) ed o require boh depreciaios of he exchage rae relaive o he dollar o se up he expecaio of fuure appreciaio - as well as a rise i he real ieres rae differeial i favor of he poud (or he Euro). 17
Ye Char 6 Char 6 depics our decomposiio of he JPY exchage rae io is fair value ad risk premium compoes. From 2005 hrough summer of 2010, he appreciaio of he ye from 120 o 88 is almos fully accoued for by a equal shif i our esimae of fair value. Durig mos of his period here was also a modes ad o very volaile risk premium i favor of he ye. This risk premium wideed i he fall of 2008 bu was almos eirely elimiaed by he summer of 2009. Sice ha ime, we esimae ha a risk premium i favor of he dollar opeed up as he ye coiued o appreciae owihsadig a shif i fair value i he direcio of a weaker ye. Our las daa poi is February 11, 2011. Fially Char 7 cofirms ha, if ayhig, chages i he ye ad our measure of fair value have bee more highly correlaed ha we foud for he Euro ad he poud. 18
Char 7: Correlaio i Daily Chages i JPY ad FV (60 day widow) 1 Correlaio of Daily Chages i JPY ad RNFV - 60 day Widow 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 12/2/2010 9/2/2010 6/2/2010 3/2/2010 12/2/2009 9/2/2009 6/2/2009 3/2/2009 12/2/2008 9/2/2008 6/2/2008 3/2/2008 12/2/2007 9/2/2007 6/2/2007 3/2/2007 12/2/2006 9/2/2006 6/2/2006 3/2/2006 12/2/2005 9/2/2005 6/2/2005 3/2/2005 VI. Risk Premium Esimaes Codiioal o he Half Life of PPP Deviaios Up uil ow, we have assumed ha expeced deviaios of he real exchage rae from relaive PPP a horizo of 10 years are equal o 0. Empirically some researchers (Rogoff (1996)) fid slower speeds of adjusme for example esimaig half lives of 3 o 5 years. I his secio, we esimae he ime varyig risk premium codiioal o assumpio of a half life of five years. Followig secio II, we re-arrage Equaio 5 as, 19) where is he expeced cosa ucodiioal mea of he real exchage rae defied by log ru relaive PPP i log form. To esimae he 19
risk premium, we assume ha auocorrelaio ad variace, follows a AR(1) process wih where equals o 0.87, if he half life is five years. Thus, he risk premium ca be expressed as, Muliplyig hrough by ad re-arragig, we obai, or where Thus, he spo exchage rae is he produc of he fair value ad he risk premium codiioal o he half life PPP deviaio. Takig aural logs ad i firs differeces, we mus have Hece, a rise i expeced real reur o he UK liker ivesme mus be brough abou by some combiaio of a rise i UK-US real ieres differeial, a coemporaeous real appreciaio of he dollar 20
relaive o he poud ad he real exchage rae deviaio from is log ru mea. Usig he daa se described i Secio V, we decompose he spo exchage rae io he risk premium ad fair value codiioal o he half life of PPP deviaio o be five years. I Char 8 Char 10, he dark blue lie depics he spo exchage rae; he aqua blue lie is he fair value defied by Equaio 11; he red lie is he fair value ( defied by Equaio 20 codiioal o he half life PPP deviaio. The risk premium (, ) is illusraed as he gap bewee he spo exchage rae ad he fair value. Whe he exchage rae EUR (Char 8) or GBR (Char 9) exceeds fair value, he risk premium is i favor of he USD. I Char 10, risk premium is i favor of he dollar whe spo exchage rae YEN is lower ha he fair value, sice ye is deoed as ye price per dollar. Euro Char 8 21
I is clear ha he risk premium codiioal o he half life PPP deviaio ( ) is smaller ha he risk premium wihou PPP adjusme ( ) i geeral. This pheomeo makes sese iuiively, because he assumpio of a cosa log ru real exchage rae would aribue he chage of he log ru real exchage rae o be par of he risk premium. However, he differece bewee ad, or he differece bewee ad, is o sigifica, especially i firs differece erms. Poud Char 9 22
Ye Char 10 VII. Regressios of (r*, r,) o θ, ad s o θ, 21) Recall i our framework, period by period, we have A posiive shock o θ, is a icrease i he risk premium o a UK ivesme which icreases he expeced excess reur a US ivesor ears o a UK ivesme. This mus be brough abou by some combiaio of a rise i UK US iflaio idexed yield differeial ad or a appreciaio of he dollar relaive o he poud. I daily daa, we ca recover θ, sice every oher erm i equaio (19) is observable (up o he egligible error i approximaig uobserved daily iflaio differeials wih he per day 23
mohly average iflaio differeial). We ca hus for each coury regress (r*, r,) o θ, ad s o θ, o quaify how much of a chage i he risk premium is o average refleced i idexed bod yields ad how much is refleced i he omial exchage rae. Whe we do so, we fid ha a srikig feaure i he daa is ha, for all hree exchage raes, roughly half of a give rise i he risk premium is refleced i a rise i he iflaio idexed bod reur differeials i favor of he foreig coury ad he remaiig half is refleced i a appreciaio of he dollar. These regressio resuls are preseed below i he followig chars. Char 11: Regressio of 10 (r* - r) o θ for Euro Slope = 0.4437 -sa = 41.3633 r-square = 0.3756 θ Char 11 preses he regressio of he chage of he Euro US iflaio idexed yield differeial (r*, r,) o he chage of euro risk premium θ,. Blue dos are he observaios, while he red lie preses liear regressio resul. The coefficie is 0.44, which meas aroud half of he rise i risk premium is refleced i a rise i he iflaio idexed bod reur differeials i favor of he Euro. Addiioally, Char 12 plos he regressio resul of he chage of he euro s o θ,. The regressio coefficie is -0.53, which meas he oher half of he rise i risk premium is refleced i a appreciaio of he dollar. 24
Char 12: Regressio of Eur o θ θ Slope = -0.5268 -sa = -50.8078 r-square = 0.4758 I Char 13 ad 14, we regress he chage of he UK-US iflaio idexed yield differeial (r*, r,) o he chage of poud risk premium θ, ad he chage of he poud s o θ,. Agai he resul cofirms ha roughly half of a give rise i he risk premium is refleced i a rise i he iflaio idexed bod reur differeial i favor of he UK (he regressio coefficie is 0.48) ad he remaiig half is refleced i a appreciaio of he dollar (he regressio coefficie is -0.50). 25
Slope = 0.4778 -sa = 42.1957 r-square = 0.3850 Char 13: Regressio of 10(r* - r) o θ for GBP 10(r* - r) Char 14: Regressio of GBP o θ GBP Slope = -0.5034 -sa = -44.221 r-square = 0.4074 Fially, Char 15 ad 16 furher cofirm his feaure usig iflaio idexed bod marke iformaio of Japa ad he US. 26
Char 15: Regressio of 10 (r* - r) o θ for Jpy 10 (r* - r) Slope = 0.5208 -sa = 44.154 r-square = 0.4091 θ Char 16: Regressio of Jpy o θ Jpy θ Slope = -0.4271 -sa = -34.96 r-square = 0.3028 27
We oe ha while cosa erms are icluded i each regressio, oe is saisically sigifica. We do o make ay iferece of cause ad effec. All hree variables are edogeous. Bu ay macro model of risk ad reur i he foreig exchage marke i a world wih iflaio idexed bods should edeavor o mach hese correlaios. VIII Does he Risk Premium Forecas Subseque Exchage Rae Chages? As we saw i Secio VII, oly abou half of a shock o he risk premium o he foreig currecy is refleced i liker yield differeials. This implies ha some of he ex ae risk premium is eared via a expeced fuure depreciaio of he dollar eve hough, as we docume i Secio VII o average he dollar appreciaes coemporaeously whe he risk premium rises. We ow explore wheher or o i our daase we i fac esimae ha a rise i he foreig currecy risk premium oday helps o forecas dollar depreciaio i he fuure. We would like o forecas he subseque period spo exchage rae i respose o a coemporaeous risk premium shock. Thus, we ru he followig regressio o mohly daa: 22) Our ieres is i esimaig he sig ad sigificace of he coefficie. Regressios of he form (22) suffer he complicaios overlappig daa ad as well small sample bias sice θ, while pre-deermied is o exogeous.. O he oe had, he explaaory variable, is persise i he daa. O he oher had, he shock o he regressor is correlaed wih he shock o he exchage rae. Therefore, we impleme he augmeed regressio mehod iroduced i Hjalmarsso (2008) o correc for smallsample bias. We compue robus sadard errors followig Hase ad Hodrick (1980) GMM. Usig he augmeed regressio mehod ad he auo-correlaio robus sadard errors, we plo he esimaed ad is 90% cofidece 28
ierval i Char 17. The solid black lie depics, while he dashed black lies represe he 90% cofidece ierval. The x-axis is he forecas horizo. Also, we esimae ad plo i i he char, o capure he coemporaeous effec of he risk premium shock. As is evide i Char 17, he bias adjused regressio coefficies are ideed of he expeced posiive sig ad are saisically sigifica. I a moh whe he risk premium o he foreig currecy rises, o average i he same moh he dollar appreciaes, bu i subseque mohs he dollar eds o depreciae which is a ex-ae source of expeced reur ideified by he risk premium. I he coex of he Fama regressio, researchers usually regress he subseque chage of spo exchage rae o he lagged shor erm ieres rae differeial bu o o he esimae of he lagged risk premium per se. Ad we also oe ha he exisece of a risk premium does o i ad of iself imply ha exchage rae chages are forecasable. Isead, if he exchage rae were uforecasable, flucuaios i he risk premium would be refleced eirely i iflaio idexed yield differeials. 29
Char 17 Noe: whe, we regress o, ad plo he coefficie. Whe, i Equaio 22 is ploed. 30
IX. Cocludig Remarks This chaper has derived a ovel srucural relaioship bewee he omial exchage rae, aioal price levels, ad observed yields o log mauriy iflaio - idexed bods. This relaioship ca be ierpreed as defiig he fair value of he exchage rae as well as a empirically observable measure of he risk premium ha ca ope up a wedge bewee he observed level of he omial exchage rae ad is fair value. We ake our heory o he daa o sudy high frequecy, real ime decomposiios of poud, euro, ad ye exchage raes io heir fair value ad risk premium compoes ad fid ha he relaive imporace of hese wo facors varies depedig o he sub sample sudied. However, sub samples i which, corary o he Meese-Rogoff (1983) puzzle, 30 o 60 perce of he flucuaios i daily exchage rae chages are explaied by coemporaeous chages i fair value are o ucommo. We also fid ha a srikig feaure i he daa is ha, for all hree major exchage raes i our sudy, o average roughly half of a give rise i he risk premium is refleced i a rise i he iflaio idexed bod reur differeials i favor he foreig coury ad he remaiig half is refleced i a appreciaio of he dollar. Moreover, we docume ha our measure of he risk premium, as prediced by heory, coais useful iformaio ha ca forecas subseque chages i he exchage rae. 31
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