Three Essays in Forward Rate Unbiasedness Hypothesis

Transcription

1 Uah Sae Universiy All Graduae Theses and Disseraions Graduae Sudies Three Essays in Forward Rae Unbiasedness Hypohesis Devalina Chaerjee Uah Sae Universiy Follow his and addiional works a: hps://digialcommons.usu.edu/ed Par of he Finance Commons Recommended Ciaion Chaerjee, Devalina, "Three Essays in Forward Rae Unbiasedness Hypohesis" (2010). All Graduae Theses and Disseraions hps://digialcommons.usu.edu/ed/644 This Disseraion is brough o you for free and open access by he Graduae Sudies a DigialCommons@USU. I has been acceped for inclusion in All Graduae Theses and Disseraions by an auhorized adminisraor of DigialCommons@USU. For more informaion, please conac dylan.burns@usu.edu.

2 THREE ESSAYS IN FORWARD RATE UNBIASEDNESS HYPOTHESIS by Devalina Chaerjee A disseraion submied in parial fulfillmen of he requiremens for he degree of DOCTOR OF PHILOSOPHY in Economics Approved: Dr. Basudeb Biswas Major Professor Dr. Dee Von Bailey Commiee Member Dr. Dwigh Israelsen Commiee Member Dr. Drew Dahl Commiee Member Dr. Frank Caliendo Commiee Member Dr. Byron R. Burnham Dean of Graduae Sudies UTAH STATE UNIVERSITY Logan, Uah 2010

3 ii ABSTRACT Three Essays in Forward Rae Unbiasedness Hypohesis by Devalina Chaerjee, Docor of Philosophy Uah Sae Universiy, 2010 Major Professor: Dr. Basudeb Biswas Deparmen: Economics and Finance The objecive of his disseraion is o verify and explain he forward exchange rae unbiasedness hypohesis in he foreign exchange marke. Since in mos of he cases he unbiasedness hypohesis fails o hold, we ry o provide hree differen explanaions of his puzzling behavior in he hree essays. The firs essay ries o resolve he forward premium puzzle by addressing he model misspecificaion issue and hereby adding a ime-varying risk premium erm in he percenage change specificaion. The risk premium erm is modeled using he GARCH-M represenaion and he model is esimaed by applying a GARCH (1, 1) specificaion. The second essay aribues he failure of he unbiasedness hypohesis o hold o he nonsaionariy of he spo and forward exchange rae. I verifies he exisence of a coinegraing relaionship beween he spo and he forward exchange raes and hus specifies an Error Correcion Model o beer capure he relaion beween he spo and he forward raes. Furher, a coinegraing or he exisence of a long run relaionship beween he spo and forward exchange raes and he domesic

4 iii and foreign ineres raes is esed. I can be viewed as a robusness check where we ensure wheher he coinegraed exchange raes are sill relaed in he long run wih he inclusion of he ineres raes. The objecive of he hird essay is o apply he generalized mehod of momens (GMM) o es he unbiasedness hypohesis in he foreign exchange marke. Empirical evidence suggess ha he spo and forward raes are nonsaionary wih uni roos and are coinegraed. Coinegraion furher suggess ha he changes in he spo rae can be modeled by an Error Correcion Model. The hird essay explicily derives an ECM from he levels specificaion and uses he GMM esimaion echnique o es he unbiasedness hypohesis. (116 pages)

5 iv DEDICATION This disseraion is dedicaed o my faher (lae) Mr. Hiranmoy Biswas.

6 ACKNOWLEDGMENTS v I is my pleasure o hank all hose who made his disseraion possible. I would like o hank Dr. Dee Von Bailey, Dr. Drew Dahl, Dr. Dwigh Israelsen, and Dr. Frank Caliendo for being a par of my disseraion commiee. They helped me improve he qualiy of his disseraion by useful suggesions. I am graeful o Dr. Basudeb Biswas, my major professor, for his guidance and inspiraion hroughou my PhD program. He provided encouragemen, sound advice, good eaching, and los of good ideas. Wih his enhusiasm and effors o explain hings clearly and simply, he made economics fun for me. He augh me how o ask quesions and express my ideas wih clariy. He showed me differen ways o approach a research problem and he need o be persisen o accomplish any goal. I is my pleasure o hank he faculy members of he Applied Economics and he Economics and Finance Deparmen of Uah Sae Universiy, especially Dr. Kenneh Lyon and Dr. Paul Jakus. I would like o hank he adminisraive saff of he Applied Economics and he Economics and Finance Deparmen, especially Trina. I would also like o hank Mrs. Renuka Biswas and all my friends in Logan for heir companionship and he wonderful imes I have spen wih hem. They have provided me encouragemen and suppor in imes of need. Special hanks goes o Dr. Kaushik Ghosh for providing me wih he forward exchange rae daa, wihou which his disseraion would no have been possible. Las, bu no leas, I hank my family: my grandfaher, Mani G. Biswas, for being a grea grandfaher and a consan source of inspiraion ill dae; my wonderful parens,

7 vi Hiranmoy and Sukla Biswas, for educaing me, for heir uncondiional love, suppor, and encouragemen o pursue my ineress. Mos imporanly I would like o hank my husband, Anjan Chaerjee, for his belief in my capabiliies, and for his love and suppor. Devalina Chaerjee

8 CONTENTS vii Page ABSTRACT... ii DEDICATION... iv ACKNOWLEDGMENTS...v LIST OF TABLES...x INTRODUCTION...1 References...4 ESSAY 1. EXPLAINING FORWARD RATE UNBIASEDNESS HYPOTHESIS, THE RISK PREMIUM APPROACH Inroducion Role of Risk Premium in Forward Rae Unbiasedness Hypohesis Lieraure Review Exchange Rae Deerminaion Purchasing power pariy Moneary model of exchange rae deerminaion Foreign exchange marke equilibrium Covered ineres rae pariy Economic Model of Foreign Exchange Risk Premium Economeric Model of Foreign Exchange Risk Premium ARCH M Model Modeling Risk Premium Daa Model Esimaion Empirical Resuls Tes of he unbiasedness hypohesis Tess for heeroskedasic OLS residuals ARCH-M model GARCH-M model...39

9 viii 10. Conclusions...41 References...42 ESSAY 2. EXPLAINING FORWARD RATE UNBIASEDNESS HYPOTHESIS, STATIONARITY, COINTEGRATION AND ERROR CORRECTION Inroducion Forward Premium Puzzle Objecive Lieraure Review Unbiasedness and Coinegraion Forward Rae Unbiasedness Hypohesis and Coinegraion Daa Saionariy es Tes for Coinegraion The Engle-Granger mehod The Johansen procedure Tes resuls Implicaions of he es resuls Error Correcion Model Are Exchange raes and Ineres raes coinegraed? Saionariy es for ineres raes Coinegraion es Implicaion of he es resuls Conclusions...84 References...85 ESSAY 3: EXPLAINING FORWARD RATE UNBIASEDNESS HYPOTHESIS, ERROR CORRECTION MODEL AND GENERIALIZED METHOD OF MOMENTS Inroducion The Generalized Mehod of Momens The Error Correcion Model Daa Empirical Resuls The Engle-Granger mehod The Johansen procedure...97

10 ix 6. Conclusions References CONCLUSIONS CURRICULUM VITAE...104

11 LIST OF TABLES x Table Page 1.1 Tes of he Unbiasedness Hypohesis Q and LM Tess for ARCH Disurbances ARCH-M Model GARCH-M Model Saionariy Tes for Spo and Forward Raes Saionariy Tes for Change in Spo Rae Saionariy Tes for he Forward Premium Saionariy Tes for he Residual Series Saionariy Tes for he Residual Series (Comemporaneous Raes) Trace Tes Maximum Eigenvalue Tes Trace Tes (India & U.K.) Maximum Eigenvalue Tes (India & U.K.) Trace Tes (Comemporaneous Raes) Maximum Eigenvalue Tes (Comemporaneous Raes) Trace Tes (India & U.K., Comemporaneous Raes) Maximum Eigenvalue Tes (India & U.K., Comemporaneous Raes) Parameer Esimaes of he ECM Saionariy Tes for Ineres Raes Trace Tes (Exchange Raes and Ineres Raes)...83

12 xi 2.17 Maximum Eigenvalue Tes (Exchange Raes and Ineres Raes) GMM Esimaion...99

13 INTRODUCTION Wih an increase in rade among counries, he significance of foreign exchange marke has been growing. Any from of currency rading is associaed wih risk due o changes in he currency exchange rae. Such risks can be lowered by he use of derivaives. The forward rae which is a derivaive emerged as a resul of marke s reacion o he risk associaed wih floaing exchange. The forward exchange rae f observed for an exchange a ime +1 is he marke deermined cerainy equivalen of he fuure spo exchange rae s +1 (Fama, 1984, p. 320). The forward rae is an unbiased predicor of he fuure spo rae since i fully reflecs available informaion abou exchange rae expecaions (Chiang, 1988). In inernaional asse marke he relaion beween he forward and spo prices of foreign exchange is of concern for invesors, porfolio managers, and policy makers. Any inernaional ransacion involving foreign exchange is risky due o unexpeced change in exchange raes. Hence, i is imporan ha he forward raes are efficien and raional forecass of fuure spo raes. The sudy of he unbiasedness hypohesis, he forward rae is an unbiased predicor of he fuure spo rae, is imporan because i answers quesions such as wheher a marke paricipan can do beer han acceping he forward rae as an opimal predicor of he fuure spo rae and wheher cenrl banks can conrol he exchange rae by inervenion in he foreign exchange marke. Anoher imporan quesion is ha wheher he gap beween he forward rae and he expeced fuure spo rae can be aribued o an exchange risk premium.

14 2 In finding a soluion o he puzzle ha he hypohesis has no been suppored by empirical evidence, exising research has aken wo main direcions - he risk-premium approach and he non-raionaliy approach. Anoher explanaion for his puzzling behavior lies in he use of saisical and economeric esimaion echniques. The objecive of his disseraion is o verify and explain he forward exchange rae unbiasedness hypohesis. Since in mos of he cases he hypohesis fails o hold, we ry o provide hree differen explanaion of his puzzling behavior in he hree essays. The rejecion of he forward rae unbiasedness hypohesis can be aribued o a misspecified heoreical model. In he firs essay, we consider he misspecificaion o be in he form of exclusion of an explanaory variable, he risk premium. This essay ries o resolve he puzzle by addressing he model misspecificaion issue and hereby adding a ime-varying risk premium erm in he percenage change specificaion of he es of he unbiasedness hypohesis. The risk premium erm was modeled by Domowiz and Hakkio (1985) using he ARCH represenaion. For he dae used in his disseraion, in mos of he cases, he saisical ess confirm ha he error erm is heeroskedasic and follows a GARCH (1, 1) process. The ess sugges he use of he GARCH model insead of he ARCH model o model he risk premium more accuraely. Nieuwland, Verschoor, and Wolff (2000) esimaed he risk premium by a GARCH (1, 1) process using survey daa for a se of EMS currencies from 1986 o The unique conribuion of his essay lies in he use of he GARCH model applied o he exchange raes for he period January 1991 o February 2008 for U.K., Canada,

15 3 Ausralia, and Japan, he four advanced economies and for India, he emerging marke economy, he daa ranges from January 1999 o February The second essay aribues he failure of he unbiasedness hypohesis o hold o he nonsaionariy of he spo and forward exchange raes. I verifies he exisence of a coinegraing relaionship beween he spo and he forward exchange raes and hus specifies an Error Correcion Model (ECM) o beer capure he relaion beween he spo and he forward raes. Furher, a coinegraing or he exisence of a long run relaionship beween he spo and forward exchange raes and he domesic and foreign ineres raes is esed. I can be viewed as a robusness check where we ensure wheher he coinegraed exchange raes are sill relaed in he long run wih he inclusion of he ineres raes. The objecive of he hird essay is o apply he generalized mehod of momens (GMM) o es he unbiasedness hypohesis. Empirical evidence suggess ha he spo and forward raes are nonsaionary wih uni roos and are coinegraed. Coinegraion furher suggess ha he changes in he spo rae can be modeled by an ECM. Naka and Whiney (1995) and Bakshi and Naka (1997) explicily derives an ECM under he assumpion ha he spo and he forward raes are coinegraed, he firs difference of forward raes is saionary, and he firs order auocorrelaion in he forecas error is allowed. Their resuls show ha when ess of he unbiasedness hypohesis are conduced wih an ECM using GMM, he unbiasedness hypohesis canno be rejeced. The daa used in Bakshi and Naka (1997) range from January 1974 o April 1991 for Canada, U.K., Japan, France, Ialy, Germany, and Swizerland. The hird essay applies he above

16 4 menioned ECM and he GMM esimaion echnique o es wheher he same resuls hold for a differen (more recen) ime period for U.K., Canada, Ausralia, Japan, and India. References Bakshi, G.S., & Naka, A. (1997). Unbiasedness of he forward exchange raes. The Financial Review, 32(1), Chiang, T.C. (1988). The forward rae as a predicor of he fuure spo rae - a sochasic coefficien approach. Journal of Money, Credi and Banking, 20(2), Domowiz, I., & Hakkio, C.S. (1985). Condiional variance and he risk premium in he foreign exchange marke. Journal of Inernaional Economics, 19, Fama, E. F. (1984). Forward and spo exchange raes. Journal of Moneary Economics, 14, Naka, A., & Whiney, G. (1995). The unbiased forward rae hypohesis re-examined. Journal of Inernaional Money and Finance, 14, Nieuwland, F.M.C., Verschoor, W.F.C., & Wolff, C.C.P. (2000). Exchange risk premia in he European moneary sysem. Applied Financial Economics, 10,

17 5 ESSAY 1: EXPLAINING FORWARD RATE UNBIASEDNESS HYPOTHESIS, THE RISK PREMIUM APPROACH 1. Inroducion The origin of he modern foreign exchange marke daes back o early 1970s when he Breon Woods sysem collapsed and counries gradually swiched from fixed o floaing exchange raes. The foreign exchange marke helps inernaional rade and invesmen. I is he marke in which inernaional currency is raded. Is major paricipans are he commercial banks, corporaions engaging in inernaional rade, non bank financial insiuions (asse managemen firms and insurance companies) and he cenral banks as well as individuals. Considered o be he larges financial marke in he world, he significance of foreign exchange marke has been growing wih an increase in rade among counries. The average daily rade in he global forex and relaed markes currenly is over US $3 rillion (Triennial Cenral Bank Survey (April 2007), Bank for Inernaional Selemens). In a floaing exchange rae sysem a currency s value is allowed o flucuae. Any from of currency rading is associaed wih risk due o changes in he currency exchange rae. Hence, wih he inroducion of he floaing exchange rae sysem, uncerainy regarding he fuure value of he currency emerged as a concern in he area of inernaional rade and finance. However, he marke came up wih an insrumen, he forward rae, o deal wih he risk of uncerainy. The forward rae which is a derivaive emerged as a resul of marke s reacion o he risk associaed wih floaing exchange rae and is hus an endogenous innovaion. A derivaive is a financial conrac or insrumen,

18 6 whose value is derived from he value of an underlying asse. The main ypes of derivaives are fuures, forwards, opions, and swaps. The foreign exchange risk can be deal wih by engaging in a forward ransacion, in which, a buyer and seller agree on an exchange rae for any dae in he fuure, and he ransacion occurs on ha dae, a ha agreed exchange rae, regardless of wha he marke raes are hen. The duraion of he rade can be a few days, monhs or years. Money does no acually change hands unil he agreed upon fuure dae. On he oher hand, a spo ransacion represens a direc exchange beween wo currencies involving cash raher han a conrac, he amoun being paid wihin a couple of days. I has he shores ime frame, and ineres is no included in he agreed-upon ransacion. In he simple efficiency specificaion of forward exchange markes, i is ofen argued ha he forward rae fully reflecs available informaion abou he exchange rae expecaions; he forward rae, hus, is usually viewed as an unbiased predicor of he fuure spo rae. (Chiang, 1988, p. 212). Unbiased implies ha here is no obvious alernaive predicor ha performs beer on average. According o he efficien marke hypohesis (EMH), financial markes are informaionally efficien, ha is, he price on raded asses already reflec all known informaion or collecive beliefs of all invesors abou fuure prospecs and herefore are unbiased. The foreign exchange markes are efficien in he sense ha he expeced rae of reurn o speculaion in he forward exchange marke will be zero. EMH holds if economic agens are risk neural, he marke is compeiive and uses all available informaion raionally, and if axes, ransacion coss, or oher fricions are ruled ou. The implicaion of EMH is ha forward raes should be unbiased forecass of fuure spo raes

19 7 since forward exchange raes fully reflec available informaion abou invesor s expecaions of fuure spo raes. In esing marke efficiency, we joinly es wo null hypoheses: one is he marke efficiency hypohesis and he oher is he unbiasedness or raional expecaions hypohesis. If we fail o rejec he null hypohesis ha he forward rae is an unbiased predicor of he fuure spo rae hen we can use he forward rae available a ime as a proxy for he predicion of he spo rae a ime +1. A large number of sudies have been conduced in he pas o es wheher he forward rae is an unbiased predicor of he fuure spo rae. In he exising lieraure, he unbiasedness hypohesis (UH hereafer) is esed by an esimaion echnique which regress he log of he curren spo (s ) on he one-period lagged log of he forward rae (f - 1). The spo rae is defined as domesic unis per foreign unis. We hen proceed o es he join hypohesis ha he consan erm does no differ from zero, he coefficien on he one-period lagged forward rae does no significanly differ from one, and ha he error erm is free of serial correlaion. which is same as: s β + β f + = ε s = + + ε (1) + 1 β0 β1 f + 1 Due o he nonsaionariy properies of he spo and he forward raes, ess based on a level regression of he fuure spo rae on he forward rae resuled in spurious regression problems. This led researchers o adop a difference version of he log level regression in which he log curren spo rae is subraced from he one period fuure log

20 8 spo and he log forward rae. We hus consider he regression of he change in he log of he spo exchange rae on he forward discoun (expressed in log form): s = + ( f s ) + ε (2) + 1 s β0 β1 + 1 The change in he log of he spo exchange rae, (s +1 s ) is he expeced depreciaion and f s is he forward discoun. The null hypohesis o be esed in boh he level and percenage change specificaion is: H 0 : β 0 = 0 and β 1 = 1 and E [ ε + 1] = 0. In much of he research on his opic (Engel, 1996; Froo & Thaler,1990) he sound heoreical foundaions and heoreically elegan hypohesis have no been suppored by he empirical evidence which has demonsraed he puzzling resul ha he slope coefficien is significanly less han uniy and mos ofen negaive in sign. This implies ha one canno use he forward rae direcly as a measure for he fuure spo rae. Thus he empirical evidence suggess ha forward raes are neiher efficien nor raional forecass of fuure spo raes. This is he Forward Premium Puzzle, indicaing foreign exchange marke inefficiency which holds in spie of large rading volumes and low rading coss in currency markes. The heoreical foundaions and hypohesis have no been suppored by empirical evidence. This migh be due o he use of an inappropriae or misspecified model o es he hypohesis or some kind of empirical error (for example, no accouning for nonnormaliy or nonsaionariy in he daa or improper saisical echnique employed o es he hypohesis) or boh.

21 1.1 Objecive 9 The forward rae on many occasions has been found o be a biased predicor of he fuure spo rae and he negaive bias has been aribued o a ime-varying risk premium, irraional expecaions, or cenral bank inervenion. The objecive of his paper is o explain he Forward Premium Puzzle by invesigaing he exisence of a risk premia. The lieraure explores alernaive mehodologies o measure ime-varying premia. Firs, models were examined based on he ime series properies of spo and forward exchange raes wih he forecas errors being condiionally heeroskedasic. A second approach aemps o es specific heories of he risk premium. Anoher approach invesigaes he exisence of ime-varying risk premia by measuring expeced depreciaion direcly using informaion from surveys (Nieuwland, Verschoor, & Wolff, 2000). The rejecion of he forward rae unbiasedness hypohesis (FRUH hereafer) or he exisence of he forward premium puzzle can be aribued o a misspecified heoreical model. In his paper we consider he misspecificaion o be in he form of exclusion of an explanaory variable, he risk premium. The UH equaes he forward rae wih he expeced fuure spo rae. While he former is observed wih cerainy, he laer is an expeced value. Fuure spo rae is unknown (random or sochasic) and he only way we can characerize i is in erms of a probabiliy disribuion. Uncerainy regarding he expeced value of he fuure spo rae inroduces an elemen of risk which gives rise o an exra compensaion, a risk premium. Secion 2 presens he role of risk premium in explaining he FRUH followed by a lieraure review in secion 3. In esing he FRUH we need o know how he forward rae is deermined. Forward rae is a derivaive, is value being derived from he value of he spo rae. In

22 10 secion 4, we review he models of exchange rae deerminaion. Provided ha he spo rae and he ineres raes are aken as given, he forward exchange rae can be deermined by he Covered Ineres Arbirage. In order o invesigae he exisence of risk premium in he foreign exchange marke and is role in explaining he UH we need o incorporae he risk premium erm in he empirical model used o es he UH. Our nex ask will be o describe an economic model of risk premium. If he probabiliy disribuion changes over ime we do no have a consan risk premium bu a ime varying risk premium. Lucas (1982) described he naure of he risk premium in a complee dynamic general equilibrium model of ineres and exchange rae deerminaion. The spo and forward exchange raes and he risk premium are deermined in he Lucas model. Since he risk premium depends on he condiional covariance of he ineremporal marginal raes of subsiuion (hus implicily on he concaviy of he uiliy funcion and he probabiliy disribuion funcion of he exogenous processes), empirical ess of such models are complex. We hus need an economeric model of risk premium, a specificaion which capures some major aspecs of risk in a foreign exchange conrac. Following Domowiz and Hakkio (1985), he risk premium is a funcion of he condiional variance of he forecas error (in forecasing he spo rae using he forward rae) which are assumed o follow he ARCH process. The condiional variance of he error is defined as a funcion of pas informaion which includes pas forecas errors. This paricular model also capures several empirical regulariies noed in he esimaion of exchange rae models, including condiional heeroskedasiciy in he forecas errors. (Domowiz & Hakkio, 1985, p. 49). Secion 5 presens he economic model of foreign exchange risk premium followed by an

23 11 economerically esable model in secion 6. Secions 7 and 8 describe he daa and he model esimaion. The empirical resuls are presened in secion 9 wih he conclusions in secion Role of Risk Premium in Forward Rae Unbiasedness Hypohesis According o he UH, he forward rae represens he marke s expecaion of he fuure spo rae. UH holds under he assumpions ha markes are efficien, agens are risk neural and have raional expecaions and is saed as: E [ ] s + 1 = f.. (3) where f is he log forward rae a ime for he delivery of a currency a ime +1, s +1 is he corresponding log spo rae a ime +1 and E [.] is he condiional expecaion based on informaion available a ime. Given he assumpion of raional expecaion, s +1 = E ( ) s ε + 1 The forward rae is hen an unbiased predicor of he fuure spo rae, s +1 = f + ε (4) ε +1, a random variable wih E [ ε + 1 ] = 0 is he raional-expecaion forecas error. Rejecion of he null hypohesis leads o he conclusion ha he forward rae is no an unbiased predicor of he fuure spo rae or ha he forward rae does no represen he marke s expecaion of he fuure spo rae. The equilibrium relaionship beween he forward rae and he fuure spo rae is misspecified and specificaion errors in he model lead o biased esimaes. In order o remove he bias we need o incorporae a relevan explanaory variable in he model. Inclusion of a ime-varying risk premium is he mos

24 12 widely acceped reamen for he misspecificaion. The risk premium is he exra expeced reurn ha invesors demand in compensaion for holding a currency ha hey perceive as riskier han ohers. Thus he relevan equaion is obained by adding a risk premium (rp) erm in equaion (2). The risk premium is defined as he gap beween he forward discoun (f s ) and he expeced depreciaion (s +1 s ). The forward discoun is expressed as he sum of he unobserved expeced depreciaion in he spo rae and a risk premium 1. Hence, he forward discoun premium can be decomposed ino expeced depreciaion and he risk premium: fd + rp e = s +1.. (5) The UH consiss of wo hypoheses ha are esed a he same ime. 1. he raional expecaion hypohesis : s + 1 = + ε + 1 e s (Invesors predic he change in he exchange rae wih a purely random error erm). 2. he zero exchange risk premium hypohesis : fd - e s = rp = 0 The forward rae is he marke deermined cerainy equivalen of he fuure spo rae which Fama splis ino an expeced fuure spo rae and a (risk) premium, f = E [ s + 1] + rp.. (6) A number of empirical ess have lead o he conclusion ha β is significanly less han 1 and also negaive. According o Fama (1984), he negaive β is due o he presence of a ime varying risk premium (rp ), where 1 A forward discoun is he proporion by which a counry's forward exchange rae exceeds is spo rae. The forward discoun is deermined by he ineres rae differenial beween he counries.

25 rp = f - E [ s + 1].. (6)` 13 If we assume risk neuraliy, agens would equae f wih E ] so ha expeced profis [ s + 1 from forward marke speculaion would be zero. However, if f > E [ s + 1] hen he invesor incurs a premium from buying he foreign currency forward a ime relaive o is expeced price on he spo marke a ime +1. Thus he relevan equaion is obained by adding a risk premium erm in equaion (2). This is he firs inerpreaion of he forward premium puzzle. 3. Lieraure Review There exiss a large lieraure esing he join hypohesis of marke efficiency and ime-invarian risk premia in foreign exchange markes. The ess on he validiy of he marke efficiency may be classified ino wo groups, one esing UH and he oher he EMH. As observed by several auhors, he resuls of hese sudies are amazingly varied. Geweke and Feige (1979) aribued he inefficiency in foreign exchange markes o marke paricipans risk-averse behavior combined wih he exisence of ransacion coss. According o Frankel and Poonawala (2006), 30 years ago researchers found ha he forward rae is a biased predicor of he fuure spo rae. [I]n a regression of he fuure change in he spo rae agains he forward discoun, he exchange rae was found on average o move in precisely he opposie direcion from wha was prediced (p. 2). The firs ess included Hansen and Hodrick (1980) who rejeced he EMH from he 1970s and he 1920s. Longworh (1981) has rejeced he join null hypohesis of an efficien exchange marke and no risk premium for he period ending in Ocober The marke efficiency hypohesis have been rejeced by Fama (1984), Hakkio and Rush

26 14 (1989), and Sephon and Larsen (1991). This rejecion has been credied o facors such as presence of risk premiums in forward raes, he (negaive) correlaion beween he forward risk premiums and expeced fuure spo raes, empirical irregulariies in regression ess, and he use of inappropriae economeric echniques. The conflicing resuls found in he lieraure depend upon he paricular economeric specificaion and esimaion echnique, differences in he ime period of esimaion and currencies. According o a version of he UH, oher han a consan risk premium, he difference beween forward raes and realized spo raes is unpredicable. In he exising lieraure, we find exensive use of wo economeric specificaions o es he UH. The firs is a level specificaion in which he realized spo rae is regressed on he oneperiod forward rae. The oher is he percen change specificaion in which he percen change in he realized spo rae relaive o he curren spo rae is regressed agains he forward premium or he difference beween he forward and spo raes. Some of he ess of unbiasedness suffer from specificaion error (misspecificaion). The ime series properies of spo and forward exchange rae daa rule ou cerain economeric specificaions used o es for he UFRH (Barnhar & Szakmary, 1991, p. 246). Gregory and McCurdy (1984) have addressed he misspecificaion issue while Chiang (1988) has aken a sochasic coefficien approach. Since he foreign exchange markes are subjec o common exernal shocks, Chiang uses Zellner's (1962) seemingly unrelaed regression (SUR) echnique o esimae he relaed equaions. This is more efficien and consisen wih an efficien marke analysis han single equaion esimaion echniques.

27 The regression analysis shows ha he parameers α and β in he simple efficiency specificaion are sensiive o he newly available informaion and vary hroughou all he sub sample periods. Our research suggess ha he ime-series properies of he parameers should be exploied effecively and incorporaed ino he exchange rae predicions. (Chiang, 1988, p ). 15 Froo and Frankel (1989) relax he join assumpions of risk neuraliy and raional expecaion and aribue (empirically) he bias o he sysemaic forecas error, which is negaively correlaed wih he forward premium, raher han o a forward marke risk premium. Engel (1996) saes ha he UH represens he equilibrium condiion when markes are efficien, agens are risk-neural and have raional expecaions. I does no rely on assumpions abou he environmen of agens, he naure of preferences or of echnology. However, agens value reurns in real erms. The real reurn on a financial asse will depend on he environmen and preferences of he risk-neural agen (Engel, 1996, p. 132). The marke efficiency condiion (no real profi opporuniies) for a domesic agen is given by F S + 1 E =0.. (7) P + 1 where he variables are in levels and P +1 is he domesic price level. Assuming ha all variables are condiionally log-normally disribued, his expression is wrien as E s ] f 0.5Var ( s ) Cov ( s, p ).. (8) [ + 1 = where p+ 1 is he logarihm of he domesic price level. Here, he forward rae is no an unbiased predicor because of he presence of he wo addiional erms Var ( s ) Cov ( s, p ) commonly referred o as he Jensen s inequaliy erms

28 16 (JIT). Many auhors have esed he hypohesis ha he JIT significanly affec he esimaion resuls (Backus, Gregory, & Telmer, 1993; Engel, 1984; McCulloch, 1975). Their resuls show ha including he JIT do no change he bias of he slope coefficien. The empirical ess on he UH are inconclusive and conflicing. The UH is suppored by Cornell (1977) and Kohlhagen (1979), bu Levich (1979), Bilson (1981), Gregory and McCurdy (1984), Hodrick and Srivasava (1986), among ohers, have rejeced he UH. Similarly, oher sudies (e.g., Barnhard & Szakmary, 1991; Domowiz & Hakkio, 1985; Edwards, 1982; Lin & Chen, 1998) have also provided mixed resuls for he UH. Such uncerainy is aribued o he limiaions of he saisical procedures used. Cornell (1989) inerpres he bias due o wo measuremen errors in he daa used o es he UH. The firs is because he daa used are bid or ask rae or average of hem, hus neglecing ransacion cos informaion embodied in he bid-ask spread. The second is iming problem beween he forward rae delivery day and he corresponding spo rae. However, Bekaer and Hodrick (1993) show ha Cornell s inerpreaions do no significanly change he slope coefficien. Breuer and Wohar (1996) find ha he sampling problems accoun for some, bu no all of he bias in he coefficien on he forward premium. A large number of recen papers empirically esing he UH adop some advanced economeric echniques in order o overcome he problems faced in pas due o inadequae saisical echniques. Hsu and Kugler (1997) analyze he UH using a nonlinear impulse-response funcion approach. Esimaing an exponenial GARCH-inmean model, hey rejec he UH and find ha he forward premium has a nonlinear influence on he spo rae. According o Baillie and Bollerslev (2000), he forward

29 17 premium puzzle is due o small sample sizes and persisen auocorrelaion of he forward premium. They esimae a fracionally inegraed GARCH-in-mean (FIGARCH-M) model for he German mark / US dollar pariy. Roll and Yan (2000) argue ha he forward rae is an unbiased predicor of he fuure spo rae and sugges ha he puzzle arises because he forward rae, he spo rae and he forward premium follow nearly nonsaionary ime series processes. I has been empirically verified ha he slope coefficiens are significanly differen from uniy which can be viewed as a sign of he presence of a risk premium ( rp ). The equilibrium relaionship beween he forward rae and he k-period ahead spo rae is misspecified. Inclusion of a ime-varying risk premium is he mos widely acceped reamen for he misspecificaion (Breuer, 2000, p. 211). The exisence of a ime-varying risk premia in he foreign exchange marke has been documened in he lieraure by Hansen and Hodrick (1980), Hodrick (1981), Frankel (1982), Fama (1984), Hodrick and Srivasava (1984), Domowiz and Hakkio (1985), Canova and Io (1991), Breuer (2000), Backus e al. (2002), and Verdelhan (2006). Hodrick (1981) modifies he asse pricing model o allow ime-varying expeced reurns on asses, eliminaes he assumpion of a risk-free real asse and derives he characerisics of he risk premia in he forward marke as well as he equilibrium yield relaionships among he equiies and riskless nominal bonds of all counries. Hodrick and Srivasava (1984) use a saisical model ha examines he deerminaion of risk premiums in foreign exchange markes based on a heoreical model of asse pricing. In examining he robusness of hese ess o ime variaion in parameers and o he presence of heeroskedasiciy, hey find evidence for heeroskedasiciy and

30 18 ha he condiional expecaion of he risk premium is a nonlinear funcion of he forward premium. Accouning for his nonlineariy, he specificaion appears o be ime invarian. Canova and Io confirmed he presence of a risk premium using VAR model. The condiional variance of he risk premium changed over ime, bu is uncondiional disribuion seems sable across sub samples. Despie hese feaures, he volailiy of he series was subsanial and varied considerably hroughou he sample (Canova and Io, 1991, p. 140). Breuer (2000) explains how [T]he lengh of he forward conrac, may, in par, conribue o he ime-varying risk premium and is effec on he bias in he coefficien on he forward premium (p. 218) Verdelhan (2006) alks abou habi based explanaion of he exchange rae risk premium. The model reproduces he uncovered ineres rae pariy puzzle, based on a ime-varying business-cycle relaed risk premium where agens have preferences wih exernal habis. Thus, we can sae ha in spie of a huge body of lieraure and empirical sudies in his area here is sill a scope of furher research in erms of coming up wih an improved economeric echnique ha is capable of esing he join null hypohesis of efficiency and unbiasedness for he foreign exchange marke. For example, some sudies while esing he UH have aken ino consideraion nonsaionariy bu have no correced he saisical echniques for non-normaliy in he daa. Wih he help of beer saisical procedures and longer ime periods we should be able o eliminae ambiguiy in our empirical resuls. Along wih a coninuous improvemen in he esimaion echniques, he null hypohesis ha he forward rae is an unbiased predicor of he fuure spo rae has been quesioned ime and again. One reason for he rejecion of he null hypohesis is due o

31 19 he misspecificaion of he model which can be correced by including a ime-varying risk premium. We aim a esing he UH by modeling he risk premia using he GARCH-M represenaion. We measure he exchange risk premia condiional on he assumpion of raional forecass of exchange raes using daa for Ausralia, Canada, India, Japan, and U.K. 4. Exchange Rae Deerminaion Foreign exchange is a financial asse and exchange rae is an asse price. Asse is a form of wealh, a way of ransferring purchasing power from presen o he fuure. Curren asse price is relaed o he purchasing power ha buyers expec i o yield in fuure. Curren dollar/pound exchange rae is relaed o peoples expecaions abou he fuure level of ha rae. The demand for a currency as a financial asse relaive o he demands of oher currencies deermines he price of a currency. This demand is based on he uiliy his currency provides in erms of a medium of exchange, sore of value and uni of accoun. The demand for an asse depends on is reurn and risk relaive o a marke index bu demand for foreign exchange is based on a broader range of argumens. The foreign exchange marke canno be explained by he capial marke heory alone (which is used o deermine asse price), hence i should be combined wih macroeconomic heory o explain he exchange rae movemens. Exchange raes are relaive prices beween wo currencies which are deermined by he desire of residens o hold domesic and foreign financial asses. Some of he models of exchange rae deerminaion discussed are discussed below.

32 4.1 Purchasing power pariy 20 Purchasing power pariy (PPP) saes ha in he long run, he exchange rae beween wo counries currencies equals he raio of he counries price level. E $ = P US P UK PPP follows from he law of one price, which saes ha in compeiive markes, idenical goods will sell for idenical prices when valued in he same currency. I relaes o an individual produc and is generalizaion is he absolue version of PPP. Relaive PPP relaes o changes in prices and exchange raes, raher han on absolue price levels. I saes ha change in exchange raes is proporional o he change in he raio of he wo naions price levels, srucural relaionships remaining unchanged. The assumpions for PPP o hold are ha goods are idenical, all goods are radable, here are no ransporaion coss, informaion gaps, axes, ariffs, or resricions of rade, and exchange raes are influenced only by relaive inflaion raes. Due o hese resricive assumpions and empirical violaion of he law of one price which is he building block of PPP, moneary models of exchange rae deerminaion was adoped. 4.2 Moneary model of exchange rae deerminaion Since currencies are considered asses, exchange raes are asse prices ha adjus o equilibrae inernaional rade in financial asses. Like oher asse prices, exchange raes are deermined by expecaions abou he fuure. Since currencies are reaed as asses his approach is called he asse approach. The exchange rae deerminaion is influenced only by money demand and money supply facors and hus i is known as he moneary approach. According o he moneary

33 21 approach o exchange rae deerminaion, exchange raes are deermined hrough he process of maching he oal supply of and he oal demand for he domesic money in each counry. I is a long run heory since i allows price o adjus insananeously o mainain full employmen and PPP. I is assumed ha in he long run he foreign exchange marke ses he rae such ha PPP holds or here are no marke rigidiies which preven he exchange rae and oher prices from immediaely adjusing o he levels consisen wih full employmen. I also assumes ha capial is fully mobile across naional borders, and ha domesic and foreign asses are perfec subsiues. In he simplified version, he moneary approach combines he PPP heory wih he quaniy heory of money - increases or decreases in he money supply lead o proporionae increases or decreases in he price level over ime, wihou any permanen effecs on oupu or ineres raes. The domesic money marke is in equilibrium when he demand for money equals he supply of money. Hence domesic price levels in US and UK can be expressed in erms of domesic money demands and supplies. S P = M L( R, Y ).. (9a) US US / $ US S P = M L( R, Y ).. (9b) UK UK / UK where M S is he money supply which can be conrolled by he naion s moneary auhoriies and L(R, Y) is he aggregae real money demand which is negaively relaed o he ineres rae, R, and posiively relaed o he real oupu, Y. The moneary approach saes ha he exchange rae which is he relaive price of he wo currencies, is fully deermined in he long run by he relaive money supply and

34 22 money demand. Any change in he ineres raes and oupu affec he exchange rae only hrough heir influence on money demand. Thus we can define he exchange rae as: S S E = [ M L( R, Y ) ] / [ M L( R, Y ) ].. (10) $ US / $ US UK / UK The moneary model can be exended o allow individuals or firms o hold a porfolio of wo or more currencies and subsiue among currencies. However demand for asses beyond currency is no considered. Modeling real disurbances wihin a moneary framework may no be efficien. These problems are resolved in he porfolio balance approach which specifies asse demand funcions and provides an explici role for he curren accoun. The porfolio balance model is an asse marke model of shor-run exchange rae deerminaion based on he relaive price of asses, specifically wih he relaionship beween he relaive price of domesic and foreign bonds and he exchange rae. Demand for currencies in he foreign exchange marke is derived from demand for financial asses. I is assumed ha supply and demand is affeced by changes in moneary and/or fiscal condiions. Empirical ess of he porfolio balance model have no me wih grea success (Levich, 1983). 4.3 Foreign exchange marke equilibrium The condiion ha deposis of all currencies offer he same expeced rae of reurn when measured in he same currency is called he ineres pariy condiion. The implicaion is ha holders of foreign currency deposis consider hem all o be equally desirable asses.

35 23 The foreign exchange marke is in equilibrium only when he ineres pariy condiion holds, ha is, here exiss no excess supply of some ype of deposi and no excess demand for anoher. The ineres pariy condiion holds when expeced raes of reurn are equal and only hen is he foreign exchange marke in equilibrium. R $ = R + ( e E $ - $ E ) / E $.. (11) where $ and E$ is he expeced and curren dollar pound exchange rae and R $ and e E R are he US and UK ineres raes, respecively. 4.4 Covered ineres rae pariy Under he assumpion ha he ineres pariy condiion always holds, a forward exchange rae equals he spo exchange rae expeced o prevail on he forward conrac s value dae. The covered ineres pariy condiion defines he foreign exchange marke equilibrium involving he forward exchange rae raher han he expeced fuure spo exchange rae as in he uncovered ineres pariy condiion. This condiion saes ha he raes of reurn on dollar deposi and covered foreign deposi mus be he same. The covered reurn on pound deposi is: which approximaely equals [ F $ (1+ R ) - E $ ] / E $ R + [ F$ - E $ ] / E $ when he produc R * [ F$ - E $ ] / E$ is a small number. The covered ineres pariy condiion is hen saed as: R $ = R + [ F$ - E $ ] / E $.. (12)

36 24 F - E $ ] / E$ is he forward premium on pound agains dollar or he forward discoun [ $ on dollar agains pound. The covered ineres pariy condiion is hus saed as: he ineres rae on dollar deposis equals he ineres rae on pound deposis plus he forward premium on pound agains dollar. The uncovered ineres pariy condiion which is: R $ = R + ( e E $ - $ E ) / E $ equals he covered ineres pariy condiion when he forward rae quoed oday equals he expeced fuure spo exchange rae, ha is, F $ = unbiased predicor of he fuure spo rae. e E $. Hence he forward rae is an 5. Economic Model of Foreign Exchange Risk Premium The uncovered ineres pariy (UIP) saes ha when domesic ineres rae is higher han he foreign ineres rae he domesic currency is expeced o depreciae by an amoun approximaely equal o he ineres rae differenial. e s + k s = R $ - R If foreign exchange marke paricipans are risk averse, hey would demand a higher rae of reurn han he ineres differenial in reurn for he risk of holding foreign currency. Then he uncovered ineres pariy condiion maybe disored by a risk premium which may be ime varying and correlaed wih ineres differenial. The foreign exchange risk migh arise due o he covariance of moneary shocks wih real oupu shocks.

37 25 The forward premium includes a risk-premium erm besides he raders predicion for he exchange rae depreciaion which arises because rade using forward rae hedges agains he risk of currency flucuaions. The risk premium is he exra expeced reurn ha invesors demand in compensaion for holding a currency ha hey perceive as riskier han ohers. The economic model for deermining foreign exchange risk premium is developed in Lucas (1982) and is exended by Hodrick and Srivasava (1984). Domowiz and Hakkio (1985) presen an example ha yields an exchange rae equaion wih a imevarying risk premium ha is a funcion of he condiional variance of domesic and foreign money. Also, he error erm is heeroskedasic. We presen a brief summery of he Lucas model and follow Hodrick and Srivasava o exend i o come up wih an expression of he risk premium. We consider he world consising of wo counries, A and B and wo goods x and y. A ime, consumers in counry A receive an endowmen ξ of good x and nohing of good y; consumers in counry B receive nohing of good x and an endowmen of η of good y. Agens have idenical preferences and each agen of counry i maximizes uiliy = 0 E { β U ( x, y )},0 < β < 1.. (13) i i where xi and y i are he consumpion of good x and y in counry i. The uiliy funcion is bounded, coninuously differeniable, increasing in boh argumens and sricly concave. The uiliy funcion and he discoun facor β are same for boh counries. The curren sae of he sysem is given by s = ξ, η ) and he ransiion funcion is ( F( s+ 1, s ) = F( ξ + 1, η+ 1, ξ, η ).

38 26 Knowledge of he equilibrium price and ransiion funcion implies knowledge of he probabiliy disribuion of all fuure prices or raional expecaions. In equilibrium each represenaive agen consumes half of he endowmen of each counry and he relaive price of y in erms of x is given by he raio of he marginal uiliy of y o he marginal uiliy of x. p s ( 1 1 ) / ( 1, 1 ) = U y ξ, ) 2 η 2 U x ξ 2 η.. (14) 2 y ( Lucas (1982) derives he equilibrium in a barer exchange economy wih no money and hen exends his model by inroducing money. He considers a single world currency and hen presens a flexible exchange rae version wih wo naional currencies. Endowmen of counry i can be purchases by currency of counry i only. A he ime of rade in securiies and goods, here exiss no uncerainy abou he sae of he economy and he finance consrain holds for all. The equilibrium nominal goods prices are p ( s, M ) = M / ξ.. (15a) x p ( s, N ) = N / η.. (15b) y where M and N are monies of counry A and B, respecively. The ransiion funcion for he wo monies follow a known, exogenous Markov process, K, ( w + 1 w s 1, s + ) where w = ( wa, wb ) is he vecor of he sochasic growh raes for he wo monies beween period and +1. The equilibrium exchange rae depends on he relaive currency supply and real endowmens and is given by he PPP heory as e s, M, N ) = [ p s, M ) / p s, N ) ] p s ) = [ M η N ξ ) ] p s ).. (16) ( x ( y ( y ( / y (

39 27 Money in his model is demanded only for ransacion moive; i is needed o purchase curren period goods. An individual s porfolio consiss of curren goods and nex period s (fuure) goods. The nominal prices of hese goods depend on he money supply and endowmens. There exiss no uncerainy regarding he curren period price or spo price of he goods bu price in he nex period is uncerain since he money supply and endowmens follow a Markov process. In order o form an expecaion regarding he fuure price of he goods, we need o consider boh he curren and fuure period s money supply and endowmens. In ineremporal asse pricing models, he equilibrium asse price is deermined a a poin where he marginal uiliy foregone by purchasing he asse is equal o he condiional expecaion of he marginal uiliy of he reurn from holding he asse. The deerminaion of he risk premium in he foreign exchange marke depends on he exchange rae which being he relaive price of wo currencies is influenced by he ineremporal raes of subsiuion of he currencies. As discussed in Hodrick and Srivasava (1984), le b s, w ) be he dollar price of ( an amoun invesed in period which gives a reurn of dollar one in period +1 which is N M equivalen o / p ( s, M ) = π unis of x in period +1. π is he purchasing power 1 x of dollar in period +1. The marginal uiliy los in paying for he invesmen in period mus equal he marginal uiliy gained from he reurn in period +1. The loss in marginal uiliy is given by he produc of he dollar price of an amoun invesed in period, he marginal uiliy of x in period and he purchasing power of dollar in period. The gain in marginal uiliy in period +1 is given by he produc of he marginal uiliy of x in + 1

40 28 period +1, he purchasing power of dollar in period +1 and he discoun facor β. Thus he dollar price of he amoun invesed in period is given by b M β U x( s+ 1) π + s, w ) = E[ ].. (17a) M U ( s ) π 1 x( x which is he condiional expecaion of he ineremporal marginal rae of subsiuion of dollar. Similarly for pound we have N β U y ( s+ 1) π + 1 by ( s, w ) = E[ ].. (17b) N U ( s ) π y The ineremporal raes of subsiuion of he wo currencies are defined as: and Q M M M + 1 = β U x( s + 1) π + 1 / U x ( s ) π.. (18a) Q = β / U ( s ) π.. (18b) N N + 1 U y ( s+ 1) π + 1 y N which are imporan deerminans of he risk premium in he forward marke since he exchange rae is he relaive price of he wo currencies. In order o find an expression for he risk premium, we need o derive he forward price of foreign exchange. In a risk free siuaion, an invesor will be indifferen beween invesing in he sure dollar denominaed asse which yields a reurn of 1/ b s, w ) per x( dollar invesed and he covered ineres arbirage sraegy in which dollar is convered ino pounds, he amoun is invesed in he sure pound denominaed asse and he proceeds is sold in he forward marke a a price of f s, w, M, N ) dollars per pound. This ( sraegy yields [1/ e s, M, N ) ][1/ b s, w ) ][ f s, w, M, N ) ] per dollar invesed. ( y ( (