2 TÜSİAD-KOÇ UNIVESITY ECONOMIC ESEACH FOUM WOKING PAPE SEIES THE INTETEMPOAL ELATION BETWEEN EXPECTED ETUN AND ISK ON CUENCY Turan Bali Kamil Yılmaz Working Paper 99 evised - November 9 Sepember 9 hp://www.ku.edu.r/ku/images/eaf/erf_wp_99.pdf TÜSİAD-KOÇ UNIVESITY ECONOMIC ESEACH FOUM umeli Feneri Yolu 3445 Sarıyer/Isanbul
3 The ineremporal relaion beween expeced reurn and risk on currency Turan G. Bali a and Kamil Yilmaz b ABSTACT The lieraure has so far focused on he risk-reurn radeoff in equiy markes and ignored alernaive risky asses. This paper examines he presence and significance of an ineremporal relaion beween expeced reurn and risk in he foreign exchange marke. The paper provides new evidence on he ineremporal capial asse pricing model by using high-frequency inraday daa on currency and by presening significan ime-variaion in he risk aversion parameer. Five-minue reurns on he spo exchange raes of he U.S. dollar vis-à-vis six major currencies (he Euro, Japanese Yen, Briish Pound Serling, Swiss Franc, Ausralian Dollar, and Canadian Dollar) are used o es he exisence and significance of a daily risk-reurn radeoff in he FX marke based on he GACH, realized, and range volailiy esimaors. The resuls indicae a posiive, bu saisically weak relaion beween risk and reurn on currency. Key words: foreign exchange marke, ICAPM, high-frequency daa, ime-varying risk aversion, daily realized volailiy. JEL classificaion: G, C3, C November 9 a Turan G. Bali is he David Krell Chair Professor of Finance a he Deparmen of Economics and Finance, Zicklin School of Business, Baruch College, One Bernard Baruch Way, Box -5, New York, NY. Phone: (646) 3-356, Fax: (646) 3-345, b Kamil Yilmaz is Associae Professor of Economics a he College of Adminisraive Sciences and Economics, Koc Universiy, umeli Feneri Yolu, Isanbul, Turkey, Phone: , Fax: ,
4 I. Inroducion Meron s (973) ineremporal capial asse pricing model (ICAPM) indicaes ha he condiional expeced excess reurn on a risky marke porfolio is a linear funcion of is condiional variance plus a hedging componen ha capures he invesor s moive o hedge for fuure invesmen opporuniies. Meron (98) shows ha he hedging demand componen becomes negligible under cerain condiions and he equilibrium relaion beween risk and reurn is defined as: E ( + ) = β E ( σ +), () where E ) and E ( σ ) are, respecively, he condiional mean and variance of excess reurns on a ( + + risky marke porfolio, and β > is he risk aversion parameer of marke invesors. Equaion () esablishes he dynamic relaion ha invesors require a larger risk premium a imes when he marke is riskier. Many sudies invesigae he significance of an ineremporal relaion beween expeced reurn and risk in he aggregae sock marke. However, he exising lieraure has no ye reached an agreemen on he exisence of a posiive risk-reurn radeoff for sock marke indices. Due o he fac ha he condiional mean and volailiy of he marke porfolio are no observable, differen approaches, differen daa ses and sample periods used by previous sudies in esimaing he condiional mean and variance are largely responsible for he conradicory empirical evidence. The predicion of Meron (973, 98) ha expeced reurns should be relaed o condiional risk applies no only o he sock marke porfolio bu also o any risky porfolio. However, earlier sudies have so far focused on he risk-reurn radeoff in equiy markes and ignored oher risky financial asses. Alhough here are a few sudies esing he significance of a ime-series relaion beween risk and reurn in inernaional equiy markes, he focus is generally on he U.S. sock marke. I is also imporan o noe ha earlier sudies assume a consan risk-reurn radeoff and ignore ime-variaion in he risk aversion parameer β. This paper examines he ineremporal relaion beween expeced reurn and risk in currency markes. The paper no only invesigaes ICAPM in he foreign exchange marke, bu examines he significance of ime-varying risk aversion as well. The foreign exchange marke includes he rading of one currency agains anoher beween large banks, cenral banks, currency speculaors, mulinaional corporaions, governmens, and oher financial markes and insiuions. The FX marke is an iner-bank or iner-dealer nework firs esablished in 97 when many of he world s major currencies moved owards floaing exchange raes. I is considered an See French, Schwer, and Sambaugh (987), Campbell (987), Nelson (99), Campbell and Henschel (99), Chan, Karolyi, and Sulz (99), Glosen, Jagannahan and unkle (993), Scruggs (998), Harvey (), Goyal and Sana-Clara (3), Brand and Kang (4), Ghysels, Sana-Clara and Valkanov (5), Bali and Peng (6), Chrisoffersen and Diebold (6), Guo and Whielaw (6), Lundblad (7), and Bali (8). A few excepions are Chou, Engle and Kane (99), Harvey (), and Leau and Ludvigson (4).
5 over-he-couner (OTC) marke, meaning ha ransacions are conduced beween wo couner paries ha agree o rade via elephone or elecronic nework. Because foreign exchange is an OTC marke where brokers/dealers negoiae direcly wih one anoher, here is no cenral exchange or clearing house. 3 The FX marke has grown rapidly since he early 99s. According o he riennial cenral bank surveys conduced by he Bank for Inernaional Selemens (BIS), he April 7 daa show an unprecedened rise in aciviy in radiional foreign exchange markes compared o 4. As shown in Table, average daily urnover rose o US $3. rillion in April 7, an increase of 69% (compared o April 4) a curren exchange raes and 63% a consan exchange raes. 4 Since April, average daily urnover in foreign exchange markes worldwide (adjused for cross-border and local doublecouning and evaluaed a April 7 exchange raes) increased by 58% and 69% beween wo consecuive riennial surveys. Comparing he average daily urnovers of US $5 billion in 988 and US $3. rillion in 7 indicaes ha rading volume in FX markes increased by more han five imes over he pas wo decades. The FX marke has become he world s larges financial marke, and i is no uncommon o see over US $3 rillion raded each day. By conras, he New York Sock Exchange (NYSE) he world s larges equiy marke wih daily rading volumes in he US $6 o $8 billion dollar range is posiively dwarfed when compared o he FX marke. Daily urnover in FX markes is now more han en imes he size of he combined daily urnover on all he world s equiy markes. Even when combining he US bond and equiy markes, oal daily volumes sill do no come close o he values raded on he currency marke. The FX marke is unique because of is rading volumes, he exreme liquidiy of he marke, he large number of, and variey of, raders in he marke, is geographical dispersion, is long rading hours (4 hours a day excep on weekends), he variey of facors ha affec exchange raes, he low margins of profi compared wih oher markes of fixed income (bu profis can be high due o very large rading volumes), and he use of leverage. Earlier sudies have so far focused on he U.S. sock marke when invesigaing he ICAPM. However, wih an average daily rading volume of US $3 rillion per day, Forex is far and away he mos enormous financial marke in he world, dwarfing he rading volumes of oher markes. We conribue o he exising lieraure by examining for he firs ime he significance of an ineremporal relaion beween 3 As FX rading has evolved, several locaions have emerged as marke leaders. Currenly, London conribues he greaes share of ransacions wih over 3% of he oal rades. Oher rading ceners lised in order of volume are New York, Tokyo, Zurich, Frankfur, Hong Kong, Paris, and Sydney. Because hese rading ceners cover mos of he major ime zones, FX rading is a rue 4-hour marke ha operaes five days a week. 4 In addiion o radiional urnover of US $3. rillion in global foreign exchange marke, US $. rillion was raded in currency derivaives.
6 3 expeced reurn and risk on currency. We also es wheher aggregae risk aversion in he FX marke changes hrough ime. We uilize 5-minue reurns on he spo exchange raes of he U.S. dollar vis-à-vis six major currencies (he Euro, Japanese Yen, Briish Pound Serling, Swiss Franc, Ausralian Dollar, and Canadian Dollar) o consruc he daily reurns, realized volailiy and range volailiy esimaors. Then, using he inraday daa-based daily reurns as well as he GACH, realized, and range-based volailiy measures we es for he presence and significance of a risk-reurn radeoff in he FX marke. By sampling he reurn process more frequenly, we improve he accuracy of he condiional volailiy esimae and measure he risk-reurn relaionship a he daily level. When we assume a consan risk-reurn radeoff in currency markes, we find a posiive bu saisically weak relaion beween expeced reurn and risk on currency. We esimae he dependence of expeced reurns on he lagged realized variance over ime using rolling regressions. This also allows us o check wheher our resuls are driven by a paricular sample period. Two differen rolling regression approaches provide srong evidence on he ime-variaion of risk aversion parameers for all currencies considered in he paper. However, he direcion of a relaionship beween expeced reurn and risk is no clear for he enire FX marke. The paper is organized as follows. Secion II provides he descripive saisics for he daily and five-minue reurns on exchange raes as well as he daily realized and range-based volailiy measures. Secion III explains he esimaion mehodology. Secion IV presens he empirical resuls on a consan risk-reurn radeoff in he FX marke. Secion V examines he significance of ime-varying risk aversion. Secion VI invesigaes wheher he covariances of individual exchange raes wih he FX marke are priced in currency marke. Secion VII concludes he paper. II. Daa To es he significance of a risk-reurn radeoff in currency markes, we use daily reurns on he spo exchange raes of he U.S. dollar vis-à-vis six major currencies: he Euro (EU), Japanese Yen (JPY), Briish Pound Serling (GBP), Swiss Franc (CHF), Ausralian Dollar (AUD), and Canadian Dollar (CAD). According o he BIS (7) sudy, on he spo marke he mos heavily raded currency pairs were EU/USD (7%), JPY/USD (3%), GBP/USD (%), AUD/USD (6%), CHF/USD (5%) and CAD/USD (4%). As repored in Table, he U.S. dollar has been he dominan currency in boh he spo and he forward and he swap ransacions. Specifically, he U.S. currency was involved in 88.7% of ransacions, followed by he Euro (37.%), he Japanese Yen (.3%), he Pound Serling (6.9%), he Swiss Franc (6.%), Ausralian Dollar (5.5%), and Canadian Dollar (4.%). The sum of he six major
7 4 currencies (EU, JPY, GBP, CHF, AUD, CAD) accouns for a marke share approximaely equal o ha of he US Dollar (9.%). 5 The raw 5-minue daa on six exchange raes (EU/USD, JPY/USD, GBP/USD, CHF/USD, AUD/USD and CAD/USD) are obained from Olsen and Associaes. The full sample covers,8 days, from January, o March 3, 8. Following Bollerslev and Domowiz (993), and Andersen, Bollerslev, Diebold, and Labys (), we define he day as saring a :5 pm on one nigh and ending a : pm he nex nigh. The oal number of 5-minue observaions for each exchange rae is herefore equal o,8 88 = 657,6. However, we are no able o use all of hese observaions because he rading aciviy in FX markes slows down subsanially during he weekends and he major US official holidays. Following Andersen, Bollerslev, Diebold, and Labys (), along wih he weekends, we removed he following holidays from our sample: Chrismas (December 4-6), New Year s (December 3-January ), July 4 h, Good Friday, Easer Monday, Memorial Day, Labor Day, Thanksgiving Day and he day afer. In addiion o official holidays and weekends, we removed hree days (March 4,, April 4, 3, and January 3, 4) from our sample as hese days conained he longes zero or consan 5- minue reurn sequences ha migh conaminae he daily reurn and variance esimaes. As a resul, we end up wih a oal of,556 daily observaions. Panel A of Table 3 presens he mean, median, maximum, minimum, sandard deviaion, skewness, kurosis, and auoregressive of order one, A(), saisics for daily reurns on he six exchange raes. The sandard errors of he skewness and kurosis esimaes provide evidence ha he empirical disribuions of reurns on exchange raes are generally symmeric and fa-ailed. More specifically, he skewness measures are saisically insignifican for all currencies, excep for he Japanese Yen. The kurosis measures are saisically significan wihou any excepion. The Jarque-Bera, JB = n[(s /6) + (K 3) /4], is a formal saisic wih he Chi-square disribuion for esing wheher he reurns are normally disribued, where n denoes he number of observaions, S is skewness and K is kurosis. The JB saisics indicae significan deparures from normaliy for he empirical reurn disribuions of six exchange raes. As expeced, daily reurns on exchange raes are no highly persisen, as shown by he negaive A() coefficiens which are less han. in absolue value. Alhough he economic significance of he A() coefficiens is low, hey are saisically significan a he 5% or % level for all currencies, excep for he Briish Pound and Ausralian Dollar. The daily ineremporal relaion beween expeced reurn and risk on currency is esed using he daily realized variance of reurns on exchange raes. In very early work, he daily realized variance of 5 Noe ha volume percenages should add up o %; % for all he sellers and % for all he buyers. As shown in Table, he marke shares of seven major currencies add up o 8%. The remaining % of he oal (%) marke urnover has been accouned by oher currencies from Europe and from oher pars of he world.
8 5 asse reurns is measured by he squared daily reurns, where he asse reurn is defined as he naural logarihm of he raio of consecuive daily closing prices. A series of papers by Andersen, Bollerslev, Diebold, and Ebens (), Andersen, Bollerslev, Diebold, and Labys (, 3), and Andersen, Bollerslev, and Diebold (4) indicae ha hese radiional measures are poor esimaors of day-by-day movemens in volailiy, as he idiosyncraic componen of daily reurns is large. They demonsrae ha he realized volailiy measures based on inraday daa provide a dramaic reducion in noise and a radical improvemen in emporal sabiliy relaive o realized volailiy measures based on daily reurns. Andersen, Bollerslev, Diebold, and Labys (3) show formally ha he concep of realized variance is, according o he heory of quadraic variaion and under suiable condiions, an asympoically unbiased esimaor of he inegraed variance and hus i is a canonical and naural measure of daily reurn volailiy. Following he recen lieraure on inegraed volailiy, we use he high-frequency inraday daa o consruc he daily realized variance of exchange raes. To se forh noaion, le P denoe he ime ( ) exchange rae wih he uni inerval corresponding o one day. The discreely observed ime series process of logarihmic exchange rae reurns wih q observaions per day, or a reurn horizon of /q, is hen defined by ( q), ln P ln P / q =, () where = /q, /q,... We calculae he daily realized variance of exchange raes using he inraday highfrequency (five-minue) reurn daa as where realized q i i= ( q), i / q q i, is he number of five-minue inervals on day and VA =, (3) i, is he logarihmic exchange rae reurn in five-minue inerval i on dae. On a regular rading day, here are 88 five-minue inervals. The exchange rae of he mos recen record in a given five-minue inerval is aken o be he exchange rae of ha inerval. A fiveminue reurn is hen consruced using he logarihmic exchange rae difference for a five-minue inerval. Wih,556 days in our full sample, we end up wih using a oal of, = 448,8 fiveminue reurn observaions o calculae daily reurn and variance esimaes. Panel B of Table 3 presens he summary saisics of he daily realized variances of exchange rae reurns. The average daily realized variance is 6-5 for AUD/USD, for CHF/USD, for JPY/USD, for CAD/USD, for EU/USD, and.77-5 for GBP/USD. These measures correspond o an annualized volailiy of.3% for AUD/USD,.4% for CHF/USD,.3% for JPY/USD, 9.5% for CAD/USD, 9.35% for EU/USD, and 8.35% for GBP/USD.
9 6 A noable poin in Panel B is ha he daily realized variances are highly persisen, as shown by he A() coefficiens which are in he range of.49 o.64. Consisen wih Andersen, Bollerslev, Diebold, Ebens () and Andersen, Bollerslev, Diebold, Labys (), he disribuions of realized variances are skewed o he righ and have much hicker ails han he corresponding normal disribuion. Marke microsrucure noises in ransacion daa such as he bid-ask bounce may influence our risk measures based on he realized volailiy and GACH volailiy forecass, even hough he daa we use conain very liquid financial ime series and hus are leas subjec o biases creaed by marke microsrucure effecs. An alernaive volailiy measure ha uilizes informaion conained in he high frequency inraday daa is Parkinson s (98) range-based esimaor of he daily inegraed variance: where max P and VA range max min [ P ) ln( P )] =.36 ln(, (4) min P are he maximum and minimum values of he exchange rae on day. Alizadeh e al. () and Brand and Diebold (6) show ha he range-based volailiy esimaor is highly efficien, approximaely Gaussian and robus o cerain ypes of microsrucure noise such as bid-ask bounce. In addiion, range daa are available for many asses over a long sample period. Panel C of Table 3 presens he summary saisics of he daily range variances of exchange rae reurns. The average daily range variance is for AUD/USD, for CHF/USD, for JPY/USD,.63-5 for CAD/USD,.76-5 for EU/USD, and.3-5 for GBP/USD. These measures correspond o an annualized volailiy of.9% for AUD/USD, 9.47% for CHF/USD, 8.9% for JPY/USD, 8.4% for CAD/USD, 8.34% for EU/USD, and 7.6% for GBP/USD. These resuls indicae ha he daily realized volailiy esimaes are somewha higher han he daily range volailiies. Anoher noable poin in Panel C is ha he daily range variances are less persisen han he daily realized variances. Specifically, he A() coefficiens are in he range of.9 o.34 for he daily range variances. Similar o our findings for he daily realized variances, he disribuions of range variances are skewed o he righ and have much hicker ails han he corresponding normal disribuion. III. Esimaion Mehodology The following GACH-in-mean process is used wih he condiional normal densiy o model he ineremporal relaion beween expeced reurn and risk on currency: ε + z β σ + + ε + α, (5) = σ, z + ~ (,), E ( ) =, (6) E N ε + ( ε Ω ) = σ + = γ + γε + γ σ +, (7)
10 7 f ( + + µ + ; µ = +, σ + ) exp, (8) πσ + σ + where + is he daily reurn on exchange raes for period +, mean for period + based on he informaion se up o ime, + α + β σ + µ is he condiional ε + z + + = σ is he error erm wih E ( ε + ) =, σ + is he condiional sandard deviaion of daily reurns on currency and z + ~ N(,) is a random variable drawn from he sandard normal densiy and can be viewed as informaion shocks in he FX marke. + σ is he condiional variance of daily reurns based on he informaion se up o ime denoed by Ω. The condiional variance, σ +, follows a GACH(,) process as defined by Bollerslev (986) o be a funcion of he las period s unexpeced news (or informaion shocks), z, and he las period s variance, σ. ; µ, ) is he condiional normal densiy funcion of + wih he f ( + + σ + condiional mean of µ and condiional variance of σ + +. Our focus is o examine he magniude and saisical significance of he risk aversion parameer β in equaion (5). Campbell (987) and Scruggs (998) poin ou ha he approximae relaionship in equaion () may be misspecified if he hedging erm in ICAPM is imporan. To make sure ha our resuls from esimaing equaion (5) are no due o model misspecificaion, we added o he specificaions a se of conrol variables ha have been used in he lieraure o capure he sae variables ha deermine changes in he invesmen opporuniy se. Several sudies find ha macroeconomic variables associaed wih business cycle flucuaions can predic he sock marke. 6 The commonly chosen variables include Treasury bill raes, federal funds rae, defaul spread, erm spread, and dividend-price raios. We sudy how variaions in he fed funds rae, defaul spread, and erm spread affec he ineremporal risk-reurn relaion. 7 Earlier sudies also conrol for he lagged reurn in he condiional mean specificaion. We obain daily daa on he federal funds rae, 3-monh Treasury bill, -year Treasury bond yields, BAA-raed and AAA-raed corporae bond yields from he H.5 daabase of he Federal eserve Board. The federal funds (FED) rae is he ineres rae a which a deposiory insiuion lends immediaely available funds (balances a he Federal eserve) o anoher deposiory insiuion overnigh. I is a closely wached baromeer of he ighness of credi marke condiions in he banking sysem and he sance of moneary policy. In addiion o he fed funds rae, we use he erm and defaul spreads as conrol variables. The erm spread (TEM) is calculaed as he difference beween he yields on he - 6 See Keim and Sambaugh (986), Chen, oll, and oss (986), Campbell and Shiller (988), Fama and French (988, 989), Campbell (987, 99), Ghysels, Sana-Clara, and Valkanov (5), and Guo and Whielaw (6). 7 We could no include he aggregae dividend yield (or he dividend-price raio) because he daa on dividends are available only a he monhly frequency while our empirical analyses are based on he daily daa.
11 8 year Treasury bond and he 3-monh Treasury bill. The defaul spread (DEF) is compued as he difference beween he yields on he BAA-raed and AAA-raed corporae bonds. We es he significance of he risk aversion parameer, β, afer conrolling for macroeconomic variables and lagged reurn: + + β σ + + λ FED + λ DEF + λ3 TEM + λ4 + ε + α, (9) = σ, z + ~ (,), E ( ) =, () ε + z + + E N ε + ( ε Ω ) = σ + = γ + γε + γ σ +. () Earlier sudies ha invesigae he daily risk-reurn radeoff generally rely on he GACH-inmean mehodology. In risk-reurn regressions, i is no common o use he realized variance measures obained from he inraday daa. In his paper, we firs generae he daily realized variance based on he 5- minue reurns on exchange raes and hen esimae he following he risk-reurn regression: realized + = + β E[ VA + ] + ε + α, () where + is one-day ahead reurn on exchange rae and E [ VA + ] is proxied by he lagged realized realized realized variance measure, i.e., E [ VA + ] = VA defined in eq. (3). As repored in Panel B of Table 3, realized realized VA has significan persisence measured by he firs-order serial correlaion ha makes realized VA reasonable proxy for he one-day ahead expeced realized variance. The slope coefficien β in eq. (), according o Meron s (973) ICAPM, is he relaive risk aversion coefficien which is expeced o be posiive and saisically significan. To conrol for macroeconomic variables and lagged reurns ha may poenially affec he flucuaions in he FX marke, we esimae he risk aversion coefficien, β, afer conrolling for he federal funds rae, erm spread, defaul spread, and lagged reurn: realized + + β VA + λ FED + λ DEF + λ3 TEM + λ4 + ε + α, (3) and es he saisical significance of β. In addiion o he GACH-in-mean and realized volailiy models, we use he range-based volailiy esimaor wih and wihou conrol variables o es he significance of risk aversion β: range + = + β + ε + α VA, (4) range + + β VA + λ FED + λ DEF + λ3 TEM + λ4 + ε + α, (5) range where VA is he Parkinson s (98) range-based esimaor of he daily inegraed variance defined in eq. (4).
12 9 The uncovered ineres rae pariy indicaes ha he appreciaion (or depreciaion) rae of a currency is relaed o he ineres rae differenial of wo counries. 8 Therefore, he hedging demand componen of he ICAPM is proxied by he shor-erm ineres raes of he wo counries. Specifically, he ineremporal relaion is esed based on he GACH-in-mean, realized, and range volailiy esimaors along wih he London Inerbank Offer ae (LIBO) for he US and he corresponding foreign counry: where US foreign + + β σ + + λ LIBO + λ LIBO + λ3 + ε + α, (6) realized US foreign + + β VA + λ LIBO + λ LIBO + λ3 + ε + US LIBO and α, (7) range US foreign + + β VA + λ LIBO + λ LIBO + λ3 + ε + α, (8) foreign LIBO are he LIBO raes for he US and he corresponding foreign counry. To conrol for a poenial firs-order serial correlaion in daily reurns on exchange raes, we include he lagged reurn ( ) o he condiional mean specificaions. IV. Empirical esuls Table 4 presens he maximum likelihood parameer esimaes and he -saisics in parenheses for he GACH-in-mean model. The risk aversion parameer (β) is esimaed o be posiive for all currencies considered in he paper, bu he parameer esimaes are no saisically significan, excep for he Briish Pound and he Canadian Dollar. Specifically, β is esimaed o be 5.8 for he Euro, 4.4 for he Japanese Yen, 9.7 for he Briish Pound,.87 for he Swiss Franc,.4 for he Ausralian Dollar, and.4 for he Canadian Dollar. Based on he Bollerslev-Wooldridge (99) heeroscedasiciy consisen covariance -saisics repored in Table 4, he risk aversion coefficien has a -saisic of.83 for he Canadian Dollar and -saisic of.77 for he Briish Pound. Alhough we do no have a srong saisical significance, we can inerpre his finding as a posiive risk-reurn radeoff in he US/Canadian Dollar and US Dollar/Pound exchange rae markes. Overall, hese resuls indicae a posiive, bu saisically weak relaion beween expeced reurn and risk on currency. Anoher noable poin in Table 4 is he significance of volailiy clusering. For all currencies, he condiional volailiy parameers (γ, γ ) are posiive, beween zero and one, and highly significan. The resuls indicae he presence of raher exreme condiionally heeroskedasic volailiy effecs in he exchange rae process because he GACH parameers, γ and γ, are found o be no only highly significan, bu also he sum (γ + γ ) is close o one for all exchange raes considered in he paper. This implies he exisence of subsanial volailiy persisence in he FX marke. 8 Assuming ha he ineres rae is 5% per annum in he U.S. and % per annum in Japan, he uncovered ineres rae pariy predics ha he U.S. dollar would depreciae agains he Japanese Yen by 3%.
13 Table 5 repors he daily risk aversion parameer esimaes and heir saisical significance for each currency afer conrolling for macroeconomic variables and lagged reurn. The risk-reurn coefficien esimaes are similar o our earlier findings in Table 4. The relaionship beween expeced reurn and condiional risk is posiive bu saisically weak for all exchange raes, excep for he Briish Pound and he Canadian Dollar where we have a risk aversion parameer of 36.5 wih -sa. =.89 for he Briish Pound and 7.86 wih -sa. =.5 for he Canadian Dollar. These resuls indicae ha conrolling for he hedging demand componen of he ICAPM does no aler our findings. Table 5 shows ha he slope coefficien (λ 4 ) on he lagged reurn is negaive for all currencies, bu i is saisically significan only for he Euro (wih -sa. =.4) and he Swiss Franc (wih -sa. =.48). 9 We find a negaive bu insignifican firs-order serial correlaion for he Japanese Yen, Briish Pound, Ausralian Dollar, and Canadian Dollar. Table 6 presens he parameer esimaes and heir Newey-Wes (987) adjused -saisics from he risk-reurn regressions wih daily realized variance. Panel A repors resuls wihou he conrol variables and ess wheher he realized variance obained from he sum of squared 5-minue reurns can predic one-day ahead reurns on exchange raes. The risk aversion parameer (β) is esimaed o be posiive for five ou of six currencies considered in he paper, bu only wo of hese parameer esimaes are saisically significan a he % level. Specifically, β is esimaed o be.39 for he Euro, 7.9 for he Japanese Yen, 7.9 for he Briish Pound, 3.78 for he Swiss Franc,.9 for he Ausralian Dollar, and.89 for he Canadian Dollar. Based on he Newey-Wes (987) -saisics repored in Table 6, he Swiss Franc wih a risk aversion parameer of 3.78 (-sa. =.) and he Euro has a risk aversion coefficien of.39 (-sa. =.77). These resuls indicae ha he daily realized variance measure obained from inraday daa posiively predic fuure reurns on exchange raes, bu he link beween risk and reurn is generally saisically insignifican. Panel B of Table 6 presens he risk aversion coefficien esimaes afer conrolling for he federal funds rae, erm spread, defaul spread, and lagged reurn. Similar o our findings in Panel A, he risk aversion parameer is esimaed o be 8.76 wih -sa. =.4 for he Euro, 8.77 wih -sa. =.8 for he Japanese Yen, and 8.89 wih -sa. =.7 for he Swiss Franc, indicaing a posiive and significan link beween he realized variance and he one-day ahead reurns on he US Dollar/Euro, US Dollar/Yen, and US Dollar/Swiss Franc exchange raes. There is also a posiive, bu saisically weak relaion for he Briish Pound and he Canadian Dollar. Table 7 repors he parameer esimaes and heir Newey-Wes -saisics from he risk-reurn regressions wih he daily range variance of Parkinson (98). As shown in boh panels, wih and wihou 9 Jegadeesh (99), Lehmann (99), and Lo and MacKinlay (99) provide evidence for he significance of shorerm reversal (or negaive auocorrelaion) in shor-erm sock reurns.
14 conrol variables, he risk aversion parameer (β) is esimaed o be posiive bu saisically insignifican, excep for he marginal significance of β for he Canadian dollar in Panel B. These resuls provide evidence ha he daily range volailiy obained from he inraday daa posiively predic fuure reurns on exchange raes, bu here is no significan relaion beween risk and reurn on currency. The esimaes in Tables 6 and 7 presen a negaive and significan auocorrelaion for he Euro, Japanese Yen, Swiss Franc, and Canadian Dollar. The firs-order auocorrelaion coefficien is negaive, bu saisically insignifican for he Briish Pound and he Ausralian Dollar. An ineresing observaion in Tables 5, 6, and 7 is ha he slope coefficiens (λ, λ, λ 3 ) on he lagged macroeconomic variables are found o be saisically insignifican, excep for some marginal significance for he erm spread in he regressions wih he Swiss Franc. Alhough one would hink ha unexpeced news in macroeconomic variables could be viewed as risks ha would be rewarded in he FX marke, we find ha he changes in federal funds rae, erm and defaul spreads do no affec ime-series variaion in daily exchange rae reurns. Our inerpreaion is ha i would be very difficul for macroeconomic variables o explain daily variaions in exchange raes. If we examined he risk-reurn radeoff a lower frequency (such as monhly or quarerly frequency), we migh observe significan impac of macroeconomics variables on monhly or quarerly variaions in exchange raes. Panel A of Table 8 presens he maximum likelihood parameer esimaes and he -saisics in parenheses for he GACH-in-mean model wih LIBO raes for he US and he corresponding foreign counry. The risk aversion parameer (β) is esimaed o be posiive for all currencies, bu he parameer esimaes are saisically significan only for he Briish Pound, Ausralian Dollar, and Canadian Dollar. Specifically, β is esimaed o be 3.87 for he Briish Pound, 7.75 for he Ausralian dollar, and 3.9 for he Canadian Dollar. Based on he Bollerslev-Wooldridge heeroscedasiciy consisen covariance - saisics repored in Table 8, he risk aversion coefficien has a -saisic of.8 for he Briish Pound,.84 for he Ausralian Dollar, and.36 for he Canadian Dollar. Alhough we do no have a srong saisical significance, we can inerpre his finding as a posiive risk-reurn radeoff in he US Dollar/Briish Pound, US/Ausralian Dollar, and US/Canadian dollar markes. Overall, he resuls indicae a posiive, bu saisically weak relaion beween expeced reurn and risk on currency. Anoher poin worh menioning in Panel A is ha he slope coefficiens on he US LIBO rae are esimaed o be posiive and saisically significan a he 5% level for he Euro, Japanese Yen, and Swiss Franc and significan a he % level for he Canadian Dollar. As expeced, he slope coefficiens on he LIBO raes of he corresponding foreign counry urn ou o be negaive, bu saisically insignifican. Panel B of Table 8 repors he parameer esimaes and heir Newey-Wes adjused -saisics from he risk-reurn regressions wih daily realized variance afer conrolling for he LIBO raes and he
15 lagged reurn. The resuls indicae a posiive and significan link beween he realized variance and he one-day ahead reurns on he Euro, Japanese Yen, Swiss Franc, and Canadian Dollar. There is also a posiive, bu saisically weak relaion for he Briish Pound. Panel C of Table 8 shows he parameer esimaes and heir Newey-Wes -saisics from he riskreurn regressions wih he daily range variance of Parkinson (98). Wih LIBO raes and he lagged reurn, he risk aversion parameer (β) is esimaed o be posiive for all currencies, bu saisically significan only for he Canadian Dollar. Overall, he resuls provide evidence ha afer conrolling for he ineres rae differenial of wo counries, here is a posiive bu saisically weak relaion beween risk and reurn on currency. Similar o our earlier findings from he GACH-in-mean model, Panels B and C of Table 8 show ha he slope coefficiens on he US LIBO rae are generally posiive, whereas he slopes on he corresponding foreign LIBO raes are negaive wih a few excepions. Many sudies fail o idenify a saisically significan ineremporal relaion beween risk and reurn of he sock marke porfolios. French, Schwer, and Sambaugh (987) find ha he coefficien esimae is no significanly differen from zero when hey use pas daily reurns o esimae he monhly condiional variance. Chan, Karolyi, and Sulz (99) employ a bivariae GACH-in-mean model o esimae he condiional variance, and hey also fail o obain a significan coefficien esimae for he Unied Saes. Campbell and Henchel (99) use he quadraic GACH (QGACH) model of Senana (995) o deermine he exisence of a risk-reurn radeoff wihin an asymmeric GACH-in-mean framework. Their esimae is posiive for one sample period and negaive for anoher sample period, bu neiher is saisically significan. Glosen, Jagannahan, and unkle (993) use monhly daa and find a negaive bu saisically insignifican relaion from wo asymmeric GACH-in-mean models. Based on semi-nonparameric densiy esimaion and Mone Carlo inegraion, Harrison and Zhang (999) find a significanly posiive risk and reurn relaion a one-year horizon, bu hey do no find a significan relaion a shorer holding periods such as one monh. Using a sample of monhly reurns and implied and realized volailiies for he S&P 5 index, Bollerslev and Zhou (6) find an insignifican ineremporal relaion beween expeced reurn and realized volailiy, whereas he relaion beween reurn and implied volailiy urns ou o be significanly posiive. Several sudies find ha he ineremporal relaion beween risk and reurn is negaive (e.g., Campbell (987), Breen, Glosen, and Jagannahan (989), Turner, Sarz, and Nelson (989), Nelson (99), Glosen, Jagannahan, and unkle (993), Harvey (), and Brand and Kang (4)). Some sudies do provide evidence supporing a posiive and significan relaion beween expeced reurn and When esing monhly risk-reurn radeoff, French, Schwer, and Sambaugh (987) use he monhly realized variance obained from he sum of squared daily reurns wihin a monh.
16 3 risk on sock marke porfolios (e.g., Bollerslev, Engle, and Wooldridge (988), Scruggs (998), Ghysels, Sana-Clara, and Valkanov (5), Bali and Peng (6), Guo and Whielaw (6), Lundblad (7), and Bali (8)). Meron s (973) ICAPM provides a heoreical model ha gives a posiive equilibrium relaion beween he condiional firs and second momens of excess reurns on he aggregae marke porfolio. However, Abel (988), Backus and Gregory (993), and Gennoe and Marsh (993) develop models in which a negaive relaion beween expeced reurn and volailiy is consisen wih equilibrium. As summarized above, here has been a lively debae on he exisence and direcion of a risk-reurn radeoff and empirical sudies are sill no in agreemen for he sock marke porfolios. The empirical resuls presened in Tables 4-8 indicae ha he ineremporal relaion beween expeced reurn and risk on currency is posiive, bu in mos cases saisically insignifican. Hence, our findings from he FX marke are in line wih some of he earlier sudies ha invesigaed he significance of a risk-reurn radeoff for he sock marke. V. Time-Varying isk Aversion in he Foreign Exchange Marke Chou, Engle and Kane (99), Harvey (), and Leau and Ludvigson (4) sugges ha he risk-reurn relaion for he sock marke may be ime-varying. In he exising lieraure, here is no sudy invesigaing he presence and significance of ime-varying risk aversion in he FX marke. We have so far assumed a consan risk-reurn radeoff in currency markes and found a posiive bu saisically insignifican relaion beween expeced reurn and risk on exchange raes. We now esimae he dependence of expeced reurns on he lagged realized variance over ime using rolling regressions. This also allows us o check wheher our resuls are driven by a paricular sample period. We esimae he risk-reurn relaion specified in equaions () and (3) for he six exchange raes wih rolling samples. We used wo differen rolling regression approaches. The firs one uses a fixed rolling window of 5 business days (i.e., approximaely year), whereas he second one sars wih he in-sample period of 5 business days and hen exends he sample by adding each daily observaion o he esimaion while keeping he sar dae consan. Figure plos he esimaed relaive risk aversion parameers (β) and heir saisical significance over ime from he fixed rolling window of 5 days. Specifically, he firs 5 daily reurn observaions of exchange raes and heir realized variances (from /3/ o /8/3) are used for esimaion of he relaive risk aversion parameer for /8/3. The sample is hen rolled forward by removing he firs Since he ime-varying risk aversion coefficiens from esimaing equaions () and (3) wih and wihou conrol variables urn ou o be very similar, we only repor resuls from he full specificaion of eq. (3). Time-varying risk aversion esimaes obained from he parsimonious specificaion of eq. () are available from he auhors upon reques.
17 4 observaion of he sample and adding one o he end, and anoher one-day ahead risk-reurn relaionship is measured. This recursive esimaion procedure is repeaed unil he sample is exhaused on March 3, 8. The esimaed relaive risk aversion parameer over he fixed rolling sample period represens he average degree of risk aversion over ha sample period. Compuaion of he relaive risk aversion parameers using a rolling window of daa allows us o observe he ime variaion in invesors average risk aversion. A common observaion in Figure is ha here is a srong ime-series variaion in he risk aversion esimaes for all currencies considered in he paper. The firs panel in Figure indicaes ha in he US dollar/euro FX marke, he aggregae risk aversion is generally posiive wih some excepions in he second half of 6 and from May o Augus 7. For he ou-of-sample period of January 3 o March 8, only 8 ou of,36 daily risk aversion esimaes are negaive. Based on he Newey-Wes adjused -saisics, all of hese negaive risk aversion esimaes are saisically insignifican. 43 (9) ou of,98 posiive risk aversion esimaes urn ou o be saisically significan a leas a he 5% level (% level). These resuls indicae a posiive bu saisically insignifican ime-varying risk aversion in he US dollar/euro marke. The second panel in Figure displays ha in he Japanese Yen marke, he aggregae risk aversion is generally posiive bu here are quie a lo of days in which we observe a negaive relaion beween expeced reurn and risk in he US dollar/yen marke. 43 ou of,36 daily risk aversion esimaes are negaive, bu abou one-hird is saisically significan a he % level. 85 (34) ou of 875 posiive risk aversion esimaes urn ou o be saisically significan a leas a he 5% level (% level). These resuls indicae ha here is a posiive bu no srong ime-varying risk aversion in he US Dollar/Yen exchange rae marke. Third panel in Figure shows ha in he US Dollar/Pound Serling marke, he risk aversion is generally posiive bu here is a long period of ime in which we observe a negaive relaion beween expeced reurn and risk in he US dollar/pound marke. Specifically, 87 ou of,36 daily risk aversion esimaes are posiive, bu only 5 ou of 87 are marginally significan. Similarly, only 46 ou of 434 negaive risk aversion esimaes urn ou o be saisically significan a he % level. These resuls provide evidence ha alhough here is a significan ime-variaion in he aggregae risk aversion, i is no clear wheher he currency rade generaes a larger or smaller risk premium a imes when he US dollar/pound FX marke is riskier. The fourh panel in Figure indicaes ha in he Swiss Franc marke, he risk aversion is esimaed o be posiive hroughou he sample period (-8), excep for a few monhs in 6. Only 7 ou of,36 daily risk aversion esimaes are negaive, bu none of hem is saisically significan. 353 (467) ou of,35 posiive risk aversion esimaes urn ou o be saisically significan a
18 5 leas a he 5% level (% level). These resuls indicae a posiive and relaively srong ime-varying risk aversion, implying ha he currency rade generaes a larger risk premium a imes when he US Dollar/Swiss Franc rade becomes riskier. The fifh panel in Figure indicaes ha in he Ausralian dollar marke 736 ou of,36 daily risk aversion esimaes are posiive, bu none of hem is saisically significan. Only 65 ou of 57 negaive risk aversion esimaes urn ou o be marginally significan a he % level. The figure indicaes a srong ime-varying risk aversion, bu here is no significanly posiive or negaive relaion beween risk and reurn in he US/Ausralian Dollar exchange rae marke. The las panel in Figure demonsraes ha in he US/Canadian Dollar marke, for slighly more han half of he sample, he risk aversion is esimaed o be posiive and slighly less han half of he sample i urns ou o be negaive. However, based on he -saisics of hese risk aversion esimaes, here is no evidence for a significanly posiive or negaive link beween expeced reurn and risk on currency. Only 35 ou of 757 posiive risk aversion coefficiens and only 46 ou of 549 negaive risk aversion parameers are found o be significan a he % level. Alhough here is a significan ime-series variaion in he aggregae risk aversion, rading in he US/Canadian Dollar FX marke does no provide clear evidence for a larger or smaller risk premium a imes when he marke is riskier. Figure plos he esimaed relaive risk aversion parameers (β) and heir saisical significance over ime from he rolling regressions wih a fixed saring dae. Specifically, he firs 5 daily reurn observaions of exchange raes and heir realized variances (from /3/ o /7/3) are used for esimaion of he relaive risk aversion parameer for /8/3. The sample is hen exended by adding one observaion o he end (from /3/ o /8/3), and he one-day ahead risk-reurn relaion is measured for /9/3. This recursive esimaion procedure is repeaed unil March 3, 8. Similar o our findings from he fixed rolling window regressions, Figure provides evidence for a significan ime-variaion in he risk aversion esimaes for all currencies considered in he paper. The firs panel in Figure shows ha in he US Dollar/Euro marke, he aggregae risk aversion is posiive wih a few excepions in January 3. Only 6 ou of,36 risk aversion esimaes are negaive bu none of hese esimaes is saisically significan based on he Newey-Wes -saisics. 795 (87) ou of,9 posiive risk aversion esimaes urn ou o be saisically significan a leas a he 5% level (% level). These resuls indicae a posiive and srong ime-varying risk aversion in he US Dollar/Euro marke. The second panel in Figure shows ha in he US Dollar/Yen FX marke, he aggregae risk aversion is posiive wih a few excepions from March o June 4. Only 68 ou of,36 risk aversion esimaes are negaive and all of hem are saisically insignifican. 9 ou of,38 posiive risk aversion esimaes urn ou o be marginally significan a he % level. These resuls imply a posiive bu saisically weak ime-varying risk aversion in he US Dollar/Yen marke.
19 6 The hird panel in Figure depics ha in he Pound Serling marke, he risk aversion is posiive hroughou he sample, excep for a shor period of ime in 3. Only 9 ou of,36 risk aversion esimaes are negaive, bu hey are no saisically significan. Alhough here is a significan imevariaion in he risk aversion and mos of he risk-reurn coefficiens is posiive, only 4 ou of,6 posiive risk aversion esimaes urn ou o be significan a he % level. Therefore, i is no clear wheher he currency rade generaes a larger or smaller risk premium a imes when he US Dollar/Pound marke is riskier. The fourh panel in Figure provides evidence ha in he Swiss Franc marke, he risk aversion is esimaed o be posiive hroughou he sample period (-8), excep for a few days in January 3. Only ou of,36 risk aversion esimaes are negaive bu saisically insignifican. 86 (934) ou of,86 posiive risk aversion esimaes urn ou o be saisically significan a leas a he 5% level (% level). These resuls sugges a posiive and srong ime-varying risk aversion, implying ha he currency rade generaes a larger risk premium a imes when he US Dollar/Swiss Franc exchange rae marke is riskier. The fifh panel in Figure shows ha in he Ausralian Dollar marke only 38 ou of,36 risk aversion esimaes are negaive wih no saisical significance even a he % level. Only 6 ou of,68 posiive risk aversion coefficiens are found o be marginally significan a he % level. Alhough here is significan ime-variaion in he aggregae risk aversion, he resuls do no sugges a srong posiive or negaive link beween expeced reurn and risk in he US/Ausralian Dollar marke. The las panel in Figure demonsraes ha in he US/Canadian Dollar marke, he risk aversion is esimaed o be posiive, excep for a few days in May, Ocober, and November 3. Similar o our earlier findings, only 4 ou of,38 risk aversion esimaes are negaive wih very low -saisics. However, based on he saisical significance of posiive risk aversion esimaes, here is no evidence for a srong posiive link beween expeced reurn and risk on currency eiher. Only 76 ou of,65 posiive risk aversion coefficiens are found o be significan a he % level. Alhough here is a significan imeseries variaion in he aggregae risk aversion, rading in he US/ Canadian Dollar FX marke does no provide clear evidence for a larger or smaller risk premium a imes when he marke is riskier. VI. Tesing Meron s (973) ICAPM in Currency Marke Meron s (973) ICAPM implies he following equilibrium relaion beween risk and reurn for any risky asse i: µ r = A σ + B σ, (9) i where r is he risk-free ineres rae, µ i r is he expeced excess reurn on he risky asse i, σ im denoes he covariance beween he reurns on he risky asse i and he marke porfolio m, and σ ix denoes a im ix
20 7 ( k ) row of covariances beween he reurns on risky asse i and he k sae variables x. A denoes he average relaive risk aversion of marke invesors, and B measures he marke s aggregae reacion o shifs in a k-dimensional sae vecor ha governs he sochasic invesmen opporuniy. Equaion (9) saes ha in equilibrium, invesors are compensaed in erms of expeced reurn for bearing marke risk and for bearing he risk of unfavorable shifs in he invesmen opporuniy se. Meron (98) shows ha he ineremporal hedging demand componen ( B σ ) is economically and saisically smaller han he marke risk componen ( A σ im ) of ICAPM. While esing he significance of A and B a daily frequency, Bali and Engle (7) provide supporing evidence for Meron (98) ha he condiional covariances of individual socks wih he marke porfolio have posiive and saisically significan loading, whereas he innovaions in sae variables are no priced in he sock marke. Tha is, he condiional covariances of sock reurns wih he unexpeced news in sae variables have insignifican loadings. We examine Meron s (973) ICAPM based on he following sysem of equaions: i, + = Ci + A im, + ε i, + m, + = Cm + A m, + ε m, + σ, () σ, where he expeced condiional covariance of individual exchange raes wih he currency marke, E ( σ im, + ), is represened by he one-day lagged realized covariance, i.e., E ( σ im, + ) = σ im,. Similarly, he expeced condiional variance of he currency marke, E ( σ ), is represened by he one-day lagged realized variance, i.e., E ( σ m, + ) = σ m,. The currency marke porfolio is measured by he value-weighed average reurns on EU, JPY, GBP, CHF, AUD, and CAD. The weighs are obained from he US Dollar Index. Jus as he Dow Jones Indusrial Average reflecs he general sae of he US sock marke, he US Dollar Index (USDX) reflecs he general assessmen of he US Dollar. USDX does i hrough exchange raes averaging of US Dollar and six mos radable global currencies. The weighs are 57.6% for EU, 3.6% for JPY,.9% for GBP, 9.% CAD, 4.% for AUD, and 3.6% for CHF. In our empirical analysis, daily reurns on he currency marke, m, +, are calculaed by muliplying daily reurns on he six exchange raes by he aforemenioned weighs. We esimae he sysem of equaions () using an ordinary leas square (OLS) as well as a weighed leas square mehod ha allows us o place consrains on coefficiens across equaions. We m, + ix Daily realized covariances beween he exchange raes and he currency marke and daily realized variance of he currency marke are compued using five-minue reurns in a day.
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