Conditional OLS Minimum Variance Hedge Ratio

Transcription

1 Condiional OLS Minimum Variance Hedge Raio Joëlle Miffre Ciy Universiy Business School Frobisher Crescen, Barbican, London, ECY 8HB Unied Kingdom Tel: +44 (0) Fax: +44 (0)

2 Condiional OLS Minimum Variance Hedge Raio Absrac The paper proposes a new mehodology o esimae ime dependen minimum variance hedge raios ha capure he predicabiliy of reurns. Compared o he exising lieraure on hedging, he advanages of he so called condiional OLS hedge raio are hreefold. Firs, i recognizes he less han perfec correlaion beween spo and fuures prices. Second, i capures he sochasic movemens in hedge raios arising from he ime variaion in expeced reurns. Third, i is easy o esimae and produces virually insan esimaes of he hedge raio. These characerisics make he condiional OLS hedge raio poenially superior o he radiional naïve, saic OLS, and GARCH hedges as a ool for risk managemen. These heoreical consideraions moivaed he empirical analysis of he aricle. The abiliy of he condiional OLS hedge raio o minimize he risk of a hedged porfolio is compared o convenional saic and dynamic approaches, such as he naïve hedge, he roll-over OLS hedge and he bivariae GARCH(1,1) error correcion model. The paper analyzes a cross secion of six currencies and concludes ha modeling he sochasic movemens in he hedge raio via condiional OLS subsanially reduces in-sample volailiy. Wih he noiceable excepion of Briish Pound, he condiional OLS hedge raio also enhances ou-of-sample hedging effeciveness. This suggess ha he proposed mehodology is a poenially superior way o manage foreign exchange risk. Keywords: Minimum Variance Hedge Raio, Condiional OLS, Hedging Effeciveness

3 1. Inroducion There is now mouning evidence o sugges ha reurns on risky asses ime vary. This phenomenon seems o prevail across a wide cross secion of asses and across boh developed and emerging markes. 1 Mos imporanly, he evidence indicaes ha condiional asse pricing models wih ime varying risk and risk premia capure his predicabiliy (Ferson and Harvey, 1993; Evans, 1994; Ferson and Korajczyk, 1995; Harvey, 1995; Miffre, 001). I follows herefore ha he predicable variaions in reurns are probably consisen wih raional pricing in efficien markes and mos likely resul from variaions in he reurns required by invesors over ime o compensae hem for risk and deferred consumpion (see Fama, 1991, for an early review). If he predicable variaions in reurns are raional, hedging sraegies ha ignore his predicabiliy migh lead o sub-opimal hedging decisions. The purpose of his paper is o propose a new mehodology ha akes he predicabiliy of reurns ino accoun while esimaing ime dependen minimum variance hedge raios. The approach modifies he radiional OLS hedge raio o incorporae condiioning informaion. I relies on previous work on condiional asse pricing and on porfolio performance evaluaion (Ferson and Warher, 1996; Ferson and Schad, 1996). This hedge raio, referred o as he condiional OLS (ordinary leas squares) hedge raio, presens hree advanages compared o he radiional naïve, saic OLS and GARCH (generalized auoregressive condiional heeroscedasiciy) hedge raios 1 Among ohers, he asses esed for ime varying risk premia include long-erm US corporae bonds (Chang and Huang, 1990), inernaional equiy indices (Ferson and Harvey, 1993; Harvey, 1995) and fuures conracs (Bessembinder and Chan, 199).

4 ypically encounered in he lieraure on hedging. Firs, unlike he naïve hedge raio, he condiional OLS hedge raio recognizes he less han perfec correlaion beween spo and fuures prices. Second, as opposed o he naïve and OLS hedge raios, he condiional OLS hedge raio is ime dependen. I akes ino accoun he sochasic movemens in hedge raios arising from he ime variaion in expeced reurns. Third, he condiional OLS hedge raio is simple o esimae and does no suffer from he problems of convergence ypically run ino wih he family of GARCH models. Like GARCH, i readjuss hedging posiions as new informaion becomes available. However i does no require he edious maximizaion of he GARCH log likelihood funcion and produces virually insan esimaes of he hedge raio. For hese reasons he invesmen communiy could welcome he condiional OLS hedge raio as an alernaive ool for risk managemen. These heoreical consideraions moivaed he empirical par of he paper. The hypohesis esed here is ha hedging effeciveness improves when one allows via condiional OLS for some sochasic movemens in he minimum variance hedge raio. The abiliy of he condiional OLS hedge raio o minimize porfolio risk is compared o convenional saic and dynamic approaches, such as he naïve hedge, he roll-over OLS hedge and he bivariae error correcion model wih a GARCH(1,1) error srucure (Kroner and Sulan, 1993). The paper looks a he exchange rae beween he US Dollar and he following six currencies: Ausralian Dollar (AD), Briish Pound (BP), Canadian Dollar (CD), Deusche Mark (DM), Japanese Yen (JY), and Swiss Franc (SF). For each of he six currencies, modeling he sochasic movemens in he hedge raio via condiional OLS subsanially reduces in-sample volailiy. Wih he noiceable

5 excepion of BP, he condiional OLS hedge raio also enhances ou-of-sample hedging effeciveness. This suggess ha he proposed mehodology is a poenially superior way o manage foreign exchange risk. For BP however, adjusing he hedge raio on a weekly basis, via eiher GARCH or condiional OLS, does no decrease ou-of-sample volailiy. For his currency, a naïve one-o-one hedge achieves he desired oucome beer. The remainder of he aricle is organized as follows. Secion reviews he lieraure on hedging. Secion 3 presens he heory ha underpins he condiional OLS minimum variance hedge raio and describes he mehodologies used o esimae compeing hedge raios. Secion 4 inroduces he daa se. Secion 5 presens he empirical resuls and secion 6 concludes.. A Brief Review of he Lieraure on Hedging Tradiionally, hree approaches have been proposed as a way o minimize he risk of a cash posiion. The firs one, called he naïve or one-o-one hedge, assumes ha he correlaion beween he spo and he fuures is perfec and ses he hedge raio equal o one over he period of he hedge. The problems wih his hedge raio are wofold. Firs, i fails o recognize ha he correlaion beween spo and fuures prices is less han perfec. Second, i fails o consider he sochasic naure of fuures and spo prices and he resuling ime variaion in hedge raios. The second approach, he saic OLS hedge, accuraely recognizes ha he correlaion beween fuures and spo prices is less han perfec and esimaes he hedge raio as he

6 OLS coefficien of a regression of spo reurn on fuures reurn (Ederingon, 1979; Anderson and Danhine, 1981; Figlewski, 1984). However, he main shorcoming of his measure is ha i imposes he resricion of a consan join disribuion of spo and fuures price changes. As such i could lead o sub-opimal hedging decisions in periods of high basis volailiy. The naïve and OLS approaches are saic risk managemen sraegies ha involve a one-ime decision abou he bes hedge and do no require any adjusmen of he hedge raio once his decision has been aken. More recenly, quie a large lieraure has been developed o overcome hese problems. The aricles ha challenge he limiaions of he naïve and OLS hedge raios model he dynamics in he join disribuion of reurns and he ime variaion in he opimal hedge raio. The hedge raio is hen esimaed using he family of GARCH models inroduced by Engle (198) and Bollerslev (1986). These papers sudy a variey of commodiies (Baillie and Myers, 1991; Myers, 1991), Treasury securiies (Cecchei, Cumby, and Figlewski, 1988), foreign exchange insrumens (Kroner and Sulan, 1991, 1993; Gagnon, Lypny, and McCurdy, 1998), and sock indices (Park and Swizer, 1995). The conclusions ha emerge from hese sudies are ha opimal hedge raios are ime dependen and ha dynamic hedging reduces porfolio variance marginally beer han saic hedging. 3. Theoreical Discussion Condiional OLS Hedge Raio

7 Consider a U.S. invesor who is long foreign asses. He wans o minimize his exposure o exchange rae risk by going shor β dollars of fuures conracs. The ime reurn on his hedged porfolio p equals p = s - β f where s and f are he changes in he fuures and spo prices respecively and β is he ime minimum variance hedge raio. The variance of he reurns on his hedged porfolio, condiional upon a se of insrumens Z available a ime 1, equals σ ( p Z ) σ ( s Z ) ( β Z ) σ ( f Z ) ( β Z ) σ ( s, f Z ) s ( ) and ( ). Z 1 1 = s.,. Z 1 denoe variance and covariance condiional upon Z 1. The minimum variance hedge raio is radiionally defined as he value of ( β Z 1 ) ha ses he firs derivaive of σ ( ) wih respec o ( ) p Z 1 β equal o zero (see, for Z 1 example, Figlewski, 1984, for an OLS esimaion or Kroner and Sulan, 1991, for a GARCH esimaion). I is he opimal proporion of he spo asse ha should be hedged. I is equal o ( Z ) ( s, f Z 1 ) ( f Z ) σ β 1 =. σ 1 A radiional approach o measuring minimum variance hedge raio is o regress he reurn on he spo asse ono he reurn on he fuures conrac, implicily assuming β is consan.

8 s = α + β + ε (1) f The slope coefficien is he OLS minimum variance hedge raio. The hedger in his case makes no use of publicly available informaion while forming expecaions of fuure hedge raios. Alernaively he hedger could model he ime variaion in he hedge raio by assuming a linear relaionship beween β and a se of L mean zero insrumens available a ime 1, z 1. β hen equals ( z 1) = β0 + β1z 1 β () where β 0 measures he mean hedge raio, β 1 is a L -vecor of parameer esimaes, z ( Z ) 1 = Z E is a L -vecor of normalized deviaion from Z 1. β 1z 1 capures he 1 ime variaion in he hedge raio. I measures he deparure from he mean hedge raio β 0. In his case he risk-minimizing hedge raio is ime-dependen and changes as new informaion arrives o he marke. The ime-varying hedge raio will reduce o he saic OLS hedge raio if no informaion is conveyed o he marke. Subsiuing β in (1) by ( β Z 1 ) in () yields s = α + β0 f + β1 f z 1 + ε (3) Equaion (3) is a regression of he spo reurn ono a consan, he fuures reurn, and he produc of he fuures reurn wih he lagged informaion variables. The condiional model (3) adds a vecor of L predicors f z 1 o he regression radiionally used o This assumpion is in line wih convenional research on condiional asse pricing models (see, among ohers, Harvey, 1989).

9 measure he saic OLS hedge raio. These regressors pick up he variaions hrough ime in he hedge raios ha are relaed o changing economic condiions. The resuling model is referred o as he condiional OLS model wih consan α. 3 We could equally well relax he resricion ha α is consan. A linear relaionship similar o () is used o model he ime variaion in α ( Z 1) = α 0 + α1z 1 α (4) Subsiuing α in (3) by ( α Z 1 ) in (4) leads o he following regression s = α 0 + α1z 1 + β 0 f + β1 f z 1 + ε (5) This model is referred o as he condiional OLS model wih ime-varying α. Noe also ha he saic OLS model (1) is nesed ino he condiional OLS models (3) and (5). In paricular he condiional models can be assessed agains he saic model by esing he resricions β 1 = 0 in (3) and α 1 = β1 = 0 in (5). The ime one-sep ahead hedge raios are esimaed using he ex-ane model (3) over he in-sample period. This produces esimaes of he coefficiens β 0 and β 1 in (3) and hence an esimae of he condiional hedge raio ( β Z 1 ) in (). The sample is hen 3 Z 1 is an approximaion of Ω 1, he acual informaion se used by hedgers while esimaing ime varying hedge raios. The omission of some insrumens of Ω 1 in Z 1 could resul in serial correlaion and heeroscedasiciy in he regression errors. To correc for his, he sandard errors in (3) are adjused for serial correlaion and heeroscedasiciy using Newey and Wes (1987) correcion wih Parzen weighs. The bandwidh is se equal o one-hird of he sample size (see Andrews, 1991).

10 rolled over o he nex weekly observaion and he model is re-esimaed over he new sample o produce a new esimae of he condiional hedge raio. Given he sample, his generaes a ime series of 180 rolling one-sep ahead hedge raios for each hedged porfolio over he ou-of-sample period June, November, The assumpion is herefore ha hedgers reconsider heir posiion on a weekly basis. They updae he parameer esimaes each Wednesday as new informaion becomes available. 3.. Alernaive Hedge Raios The abiliy of he condiional OLS hedge raio o minimize porfolio risk is compared o convenional saic and dynamic approaches, such as he naïve hedge, he saic OLS hedge and he bivariae error correcion model wih a GARCH(1,1) error srucure (Kroner and Sulan, 1993). Each is defined in urn below. The naïve approach simply ses he hedge raio equal o one over he period of he hedge. I herefore assumes a perfec correlaion beween he fuures and spo prices and no ime variaion in he hedge raio. The saic OLS hedge raio is defined as β in (1). The bivariae GARCH error correcion model capures he dynamics in he second momens of he disribuion of reurns via a GARCH(1,1) error srucure and imposes he long-run coinegraing relaionship beween fuures and spo reurns via an error correcion erm ( ln F ) S 1 ln 1 (see, Kroner and Sulan, 1993, for more deails). The specificaion of he model is as follows: ( ln S 1 ln F 1) us s α + (6a) = S + β S ( ln S 1 ln F 1) u F f α + (6b) = F + β F 4 The same procedure is applied o he condiional OLS model wih ime-varying α.

11 h (6c) = + + S cs as hs 1 bsus 1 h (6d) F = c + + F af hf 1 bf uf 1 h (6e) = + + SF csf asfhsf, 1 bsfus 1uF 1 h where h SF, F measures he GARCH(1,1) condiional hedge raio. 4. Daa The daa comprise weekly US spo and fuures exchange raes beween he US Dollar and he following currencies: Ausralian Dollar (AD), Briish Pound (BP), Canadian Dollar (CD), Deusche Mark (DM), Japanese Yen (JY), Swiss Franc (SF). This cross secion was chosen because i has been he subjec of an imporan scruiny in he lieraure (see, for example, Kroner and Sulan, 1991, 1993; Gagnon, Lypny, and McCurdy, 1998). To compile he ime series of fuures prices, he closing prices on he neares mauriy fuures conrac are colleced, excep in he mauriy monh when he prices on he second neares fuures are used (Bessembinder, 199). 5 Reurns are compued as 100 imes he difference in he naural logarihms of he prices a he end of Wednesday. The daase spans he period July, November 9, 000. The sample is spli ino wo sub-samples. The in-sample period covers he period July, June, , approximaely wo-hird of he daase, and is used for esimaion. The ou-of- 5 Rolling he hedge forward avoids hin rading and expiraion effecs. However i inroduces some uncerainy abou he difference beween he fuures price of he conrac being closed ou and he fuures price of he new conrac ha is enered ino. A ha specific ime, hedgers face roll-over basis risk as well as hedge risk.

12 sample period, used for forecasing and hedging decisions, covers he remaining onehird. A oal of 54 observaions is considered for each series. Table I presens he correlaion beween he changes in he spo and fuures prices for each currency. I is clear from his able ha he correlaion beween he spo and fuures prices is less han perfec. The correlaion is paricularly low for Canadian Dollar (0.9639). A leas for his asse, basis risk migh be subsanial and a naïve one-o-one hedge migh no be opimal. Inser Table I The insrumens used o capure he ime variaion in he spo and fuures reurns follow from he lieraures on reurn predicabiliy and condiional asse pricing (see, for example, Fama and French, 1989; Ferson and Harvey, 1993; Ferson and Schad, 1996). These insrumens include: he change in he Sandard & Poor s composie price index, he dividend yield on he Sandard & Poor s composie index, and he erm srucure of ineres raes, defined as he difference in he yields on 30 year Treasury bond and 3 monh Treasury bill. The insrumens have a mean of zero so ha he mean of he condiional hedge raio equals he saic OLS hedge raio. 5. Empirical Resuls The saisical significance of condiioning informaion Valid conclusions regarding he abiliy of he condiional OLS hedge o reduce porfolio volailiy can only be drawn wihin a well-specified condiional model. To ensure ha his condiion is me, model (3) is esimaed via OLS over he in-sample period and he hypohesis of a consan hedge raio is esed using a χ es wih L degrees of

13 freedom. Table II repors probabiliy values for he χ es of he marginal explanaory power of he condiioning informaion variables, along wih he parameer esimaes for β 1 in (3). In he lef-hand columns, we also repor esimaes of he saic OLS hedge raios (equaion (1)), hereby imposing he resricion β 1 = 0 in (3). The informaion variables are joinly significan a sandard levels of saisical significance. The χ es consisenly indicaes rejecion of he hypohesis of a consan hedge raio a 5 percen level wih an average probabiliy value across currencies of Clearly, according o able II, i is imporan o allow he risk minimizing hedge raio o be ime-dependen insead of resricing i o be consan. Inser Table II In able III, ime variaions in boh α and β are considered. The able repors esimaes of α 1 and β 1 in (5) and also ess he hypoheses of consan α and β in (1). Alhough marginally for Ausralian Dollar (wih a probabiliy-value of 0.086), he resuls indicae ha he hypohesis of a consan α is rejeced. β also appears o be ime dependen: he hedge raios are indeed correlaed wih he public informaion variables. The hypohesis ha he coefficiens α 1 and β 1 are joinly zero produces probabiliy values ha never exceed 0.0 and average 0.00 across he six currencies. I follows from ables II and III ha he condiional models (3) and (5) are correcly specified and can be used for risk managemen. Inser Table III

14 The paper compares hedging effeciveness across a wide range of hedge raios. In paricular special aenion is given o esimaing he minimum variance hedge raio via a bivariae error correcion model wih a GARCH(1,1) error srucure (sysem 6). The parameer esimaes are repored in able IV. The resuls indicae ha he parameers a i, b i, i c for i { S, F, SF} = are saisically significan for mos currencies, suggesing ha he GARCH(1,1) error srucure capures he dynamics in he second momens of he join disribuion of reurns. Ulimaely his implies ha he condiional variances ( h S and h F ) and covariances h SF are changing over ime and herefore ha he riskminimizing hedge raios are indeed ime-varying. Similar resuls were repored in Kroner and Sulan (1991,1993) or Gagnon, Lypny, and McCurdy (1998) among ohers. As opposed o he evidence presened in hese papers however, he parameer esimaes for β F and β S are insignifican, suggesing ha here is no major error correcion mechanism operaing beween he spo and fuures markes. Inser Table IV Figures 1 o 6 display plos of he esimaed hedge raios for each currency boh over he in- and ou-of-sample periods. I is clear from hese graphs ha he condiional hedge raios are changing over ime. I follows ha resricing i o be consan migh lead o hedging decision ha are less han opimal. Inser Figures 1 o 6

15 5.. Hedging Effeciveness The abiliy of he condiional OLS hedge (wih consan and ime-varying α ) o minimize porfolio risk is compared o convenional mehods, such as he naïve hedge, he saic OLS hedge and he GARCH(1,1) hedge. The abiliy of he compeing alernaives o minimize porfolio volailiy depends on how close he esimaed hedge raio is o he hedge raio ha would se he porfolio volailiy equal o zero. The lower he variance of he hedged porfolio, he beer he hedge raio is a miminizing porfolio risk. For each of he respecive hedge raios he variance of he hedged porfolio is compued each week as σ ( s β f ) The in-sample resuls are summarized in Table V. Panel A repors he variance of he unhedged asse and he hedged porfolios under differen esimaes of he hedge raio. Panel B presens measures of hedging effeciveness, esimaed as he percenage reducion in variance compared o an unhedged posiion. Inser Table V The in-sample resuls in able V clearly indicae ha a hedger who uses he condiional OLS hedge can subsanially reduce he volailiy of his porfolio. For example, he variance of he reurns on he condiional hedge wih consan α equals for CD. This compares favorably wih he variance of he unhedged posiion (0.355), of he naïve hedge (0.033), of he saic OLS hedge (0.074), and of he GARCH(1,1) hedge (0.07). Looking a panel B, he condiional OLS hedge indeed eliminaes 9.5 percen of he variabiliy in CD price changes (agains 90.63, 9.4, and 9.8 percens

16 for he naïve, he saic OLS, and he GARCH(1,1) hedges respecively). Noe also ha allowing for ime variaion in α does no improve hedging effeciveness. The resuls for he oher currencies follow he same rend: hedging effeciveness is improved when one allows for ime dependency in he hedge raio via he condiional OLS model wih consan α. Finally i is surprising o noe ha he resuls in Table V indicae ha he OLS hedge in mos insances performs beer han he naïve and GARCH(1,1) hedges. The magniude of variance reducion varies across currencies. Hedging effeciveness ranges from percen for CD (wih he naïve hedge) o 98.0 percen for SF (wih he condiional OLS hedges) over he in-sample period. As menioned in able I, he low correlaion beween fuures and spo price changes for CD also hins owards a low hedging performance of he naïve one-o-one hedge raio for his currency. How much variance reducion can be achieved if one uses hisorical hedge raios o deermine appropriae hedging sraegies for fuure ime periods? 6 Table VI repors he resuls. Panel A repors he variance of he unhedged asse and he hedged porfolios under differen esimaes of he hedge raio. Hedging effeciveness is presened in Panel B. Inser Table VI 6 Ou-of-sample condiional OLS and GARCH(1,1) hedge raios are esimaed by using only informaion available a he ime he hedge is placed. Roll-over esimaes of he saic OLS hedge raios are used as a basis for risk managemen in he ou-of-sample period.

17 Wih he excepion of BP, he condiional OLS approach o risk managemen reduces ou-of-sample volailiy more subsanially han he alernaive compeing approaches. For example he resuls for SF clearly indicae ha ou-of-sample hedging effeciveness is improved when one allows via condiional OLS for ime variaion in he esimaed hedge raio. For his currency he condiional OLS approach increases hedging effeciveness by 0.44, 0., and 0.8 percens compared o he naïve, roll-over OLS and GARCH(1,1) measures respecively. The conclusions are similar for AD, CD, DM and JY. For BP however, he coss involved in readjusing he fuures posiion weekly migh have o be weighed agains he relaively marginal increase in hedging effeciveness obained by weekly adjusing he hedge raio. The simpliciy of he naïve mehod and he very low ransacion coss i involves migh be more appealing o a hedger han an increase in hedging effeciveness of 0.01 percen obained by rolling over he OLS esimae. For BP, he condiional OLS hedges perform worse han any alernaive measures. Wih herefore he noiceable excepion of BP ou-of-sample, he condiional OLS hedge wih consan α improves hedging effeciveness more han any compeing alernaives. In paricular, i reduces he variance of he hedged porfolio a leas as well as he condiional OLS hedge wih ime-varying α boh in- and ou-of-samples. Alhough uncondiional α and condiional α are significanly differen (able III), relying on uncondiional α does no lead o sub-opimal hedging decisions (ables IV and V). Since condiioning α on pas informaion does no improve hedging effeciveness, he remainder of he paper exclusively focuses on he condiional OLS model wih consan α.

18 Finally he hypohesis ha he condiional OLS hedge raio subsanially differs from is naïve, roll-over OLS and GARCH(1,1) counerpars is esed. A casual look a figures 1 o 6 already suggess ha he correlaion beween he dynamic hedge raios is less han perfec. This hypohesis is formally esed in able VII. Panel A repors correlaion esimaes beween he condiional OLS hedge raios and he compeing alernaives for he six currencies. In panel B we formally es he hypohesis ha he condiional OLS and he alernaive hedge raios are equal by regressing he difference on a consan and esing he saisical significance of he esimaed coefficien. Looking firs a panel A, he average correlaion beween he condiional OLS and roll-over OLS (GARCH(1,1)) hedge raios equals (0.156). As he compeing hedge raios seem o differ, he inferences drawn from one model migh be a odds wih he one advocaed by a compeing approach. The resuls repored in panel B confirm his view. For all currencies, he condiional OLS hedge raio differs from he naïve hedge raio. I is saisically differen from he roll-over OLS esimae for five ou of six currencies. I is saisically differen from he GARCH(1,1) esimae for four ou of six currencies. I follows ha in mos cases esimaing ime dependen hedge raios via condiional OLS produces esimaes of he hedge raio ha are significanly differen from heir naïve, roll-over OLS, and GARCH(1,1) counerpars. Inser Table VII 6. Conclusions This aricle proposes a new mehodology o esimae minimum variance hedge raios ha relies on previous work on condiional asse pricing (Ferson and Warher, 1996;

19 Ferson and Schad, 1996). The advanages of his approach are ha (1) i recognizes he less han perfec correlaion beween spo and fuures prices, () i capures he sochasic movemens in hedge raios arising from he ime variaion in expeced reurns and (3) i is easy o esimae and produces virually insan esimaes of he hedge raio. These characerisics make he condiional OLS hedge raio poenially superior o he radiional naïve, saic OLS, and GARCH hedges as a ool for risk managemen. These heoreical consideraions moivaed he empirical analysis of he aricle. The hypohesis esed here is ha he variance of he hedged porfolio is minimized when one allows via condiional OLS for some sochasic movemens in hedge raios. The abiliy of he condiional OLS hedge raio o minimize porfolio risk is compared o convenional saic and dynamic approaches; such as he naïve hedge, he roll-over OLS hedge and he bivariae GARCH(1,1) model wih an error correcion erm. Special aenion is given o measuring hedging effeciveness in- and ou-of-samples. The aricle demonsraes ha, a leas in-sample, modeling he sochasic movemens in he hedge raio via condiional OLS subsanially reduces porfolio volailiy. Wih he noiceable excepion of BP, he condiional OLS hedge raio also enhances ou-ofsample hedging effeciveness. For BP however, adjusing he hedge raio on a weekly basis, via eiher GARCH or condiional OLS, does no decrease ou-of-sample volailiy. For his currency, a naïve one-o-one hedge achieves he desired oucome beer.

20 References Anderson R, Danhine J-P Cross hedging. Journal of Poliical Economy 89: Andrews, D. W. K. (1991). Heeroscedasiciy and auocorrelaion consisen covariance marix esimaion. Economerica, 59, Baillie R, Myers R Bivariae GARCH esimaion of he opimal commodiy fuures hedge. Journal of Applied Economerics 6: Bessembinder H Sysemaic risk, hedging pressure, and risk premiums in fuures markes. Review of Financial Sudies 5: Bessembinder H, Chan K Time varying risk premia and forecasable reurns in fuures markes. Journal of Financial Economics 3: Bollerslev T Generalized auoregressive condiional heeroscedasiciy. Journal of Economerics 31: Cecchei S, Cumby R, Figlewski S Esimaion of opimal fuures hedge. Review of Economics and Saisics 70: Chang E, Huang R Time varying reurn and risk in he corporae bond marke. Journal of Financial and Quaniaive Analysis 5: Ederingon L The hedging performance of he new fuures markes. Journal of Finance 34: Engle R Auoregressive condiional heeroscedasiciy wih esimaes of he variance of U.K. inflaion. Economerica 50: Evans M Expeced reurns, ime-varying risk and risk premia. Journal of Finance 49, : Fama E, French K Business condiions and expeced reurns on socks and bonds. Journal of Financial Economics 5: 3-49.

21 Fama E Efficien capial markes: II. Journal of Finance 46: Ferson W, Harvey C The risk and predicabiliy of inernaional equiy reurns. Review of Financial Sudies 6: Ferson W, Korajczyk R Do arbirage pricing models explain he predicabiliy of sock reurns? Journal of Business 68: Ferson W, Schad R Measuring fund sraegy and performance in changing economic condiions. Journal of Finance 51, : Ferson W, Warher V Evaluaing fund performance in a dynamic marke. Financial Analyss Journal, November-December, 0-8. Figlewski S Hedging performance and basis risk in sock index fuures. Journal of Finance 39: Gagnon L., Lypny G., McCurdy T Hedging foreign currency porfolios. Journal of Empirical Finance 5: Harvey C Time varying condiional covariances in ess of asse pricing models. Journal of Financial Economics 4: Harvey C Predicable risk and reurns in emerging markes. Review of Financial Sudies 8: Kroner K, Sulan J Exchange rae volailiy and ime varying hedge raios. Pacific Basin Capial Markes Research : Kroner K, Sulan J Time varying disribuion and dynamic hedging wih foreign currency fuures. Journal of Financial and Quaniaive Analysis 8: Miffre, J Efficiency in he pricing of he FTSE 100 fuures conrac. European Financial Managemen 7, 1: 9.

22 Myers R Esimaing ime varying opimal hedge raios on fuures markes. Journal of Fuures Markes 11: Newey, W. K., & Wes, K. D. (1987). Hypohesis esing wih efficien mehod of momens esimaion. Inernaional Economic Review 8: Park T, Swizer L Bivariae GARCH esimaion of he opimal hedge raios for sock index fuures: a noe. Journal of Fuures Markes 15:

23 Table I. Correlaion beween Spo and Fuures Price Changes Correlaion AD BP CD DM JY SF

24

25 Table II: Saisical Significance of Condiioning Informaion: Condiional OLS Model wih Consan a The coefficiens α and β are esimaed from he following uncondiional OLS regression: s = α + β + ε () f where s and f are he spo and fuures reurns respecively. For he condiional OLS model wih consan α, he regression is as follows: s = α + β0 f + β1 f z 1 + ε (3) or alernaively s = + β 0 f + β1, RM f RM 1 + β1, DY f DY 1 + β1, TS f TS 1 α + ε z is a vecor of mean zero pre-deermined insrumens available a ime 1, consising of he lagged reurn on he Sandard & Poor s 1 composie index RM 1, he lag in he dividend yield on he Sandard & Poor s composie index DY 1, and he lagged erm srucure of ineres raes TS 1. α, β 0, β 1, β 1, RM, 1, DY β, and β 1, TS are he esimaed parameers. p-value is he probabiliy value of a χ es for he significance of β 1. The figures in parenhesis are heeroscedasiciy and serial correlaion consisen -raios. R is he adjused R-squared of he regression. The daa are weekly and he model is esimaed over he in-sample period: July, June,

26 Table II - Coninued Uncondiional OLS () Condiional OLS model wih consan α (3) α β R α β0 β1, RM β1, DY β1, TS p-value: β 1 = 0 R AD (-1.16) (100.04) (-0.46) (48.16) (-.5) (-1.41) (0.01) BP < (-1.) (6.04) (-0.15) (105.57) (-0.53) (-3.18) (.86) CD < (-3.41) (37.73) (-1.11) (37.41) (-0.8) (-3.06) (.10) DM (-0.56) (101.6) (-0.06) (85.47) (-.38) (-1.46) (3.49) JY < (-0.09) (8.00) (-0.9) (61.38) (-1.46) (0.05) (4.7) SF (-1.3) (15.90) (-0.6) (96.81) (-.01) (-1.64) (.86)

27 Table III: Saisical Significance of Condiioning Informaion: Condiional OLS Model wih Time-Varying a The condiional OLS regression wih ime-varying α is as follows: s = α 0 + α1z 1 + β 0 f + β1 f z 1 + ε (5) or alernaively s = 0 + α1, RM RM 1 + α1, DY DY 1 + α1, TS TS 1 + β 0 f + β1, RM f RM 1 + β1, DY f DY 1 + β1, TS f TS 1 α + ε where s and f are he spo and fuures reurns respecively, z 1 is a vecor of mean zero pre-deermined insrumens available a ime 1, consising of he lagged reurn on he Sandard & Poor s composie index RM 1, he lag in he dividend yield on he Sandard & Poor s composie index DY 1, and he lagged erm srucure of ineres raes TS 1. α 0, β 1,TS are he esimaed parameers. p-value is he probabiliy value of a α 1, RM, 1, DY α, α 1, TS, β 0, β 1, β 1, RM, β 1, DY, and χ es for he significance of α 1 and/or β 1. The figures in parenhesis are heeroscedasiciy and serial correlaion consisen -raios. R is he adjused R-squared of he regression. The daa are weekly and he model is esimaed over he in-sample period: July, June,

28 Table III - Coninued α0 α1, RM Condiional α α1, DY α1, TS p-value: α 1 = 0 β0 β1, RM Condiional β β1, DY β1, TS p-value: β 1 = 0 p-value: α = β = R AD (-0.19) (.09) (-0.56) (-0.5) (99.64) (-1.06) (-1.10) (0.03) BP <0.001 < (-0.84) (.34) (1.4) (-.1) (1.95) (-0.55) (-8.93) (5.17) CD < (-4.01) (.60) (.45) (-1.85) (35.69) (-0.81) (-.81) (1.88) DM < < (-.70) (1.14) (5.33) (-.9) (13.41) (-.00) (-.9) (.69) JY < <0.001 < (-4.40) (-0.13) (5.37) (-0.45) (13.60) (-1.41) (-0.14) (4.37) SF (-4.07) (1.11) (3.1) (-1.96) (37.78) (-1.8) (-1.56) (.84)

29 Table IV: Bivariae error correcion model wih a GARCH(1,1) error srucure The able repors parameer esimaes from he following sysem of regressions: ( ln S 1 ln F 1) us ( ln S 1 ln F 1) u F s + = α S + β S = α F + β F f + h h = + + S cs as hs 1 bsus 1 = + + F cf af hf 1 bf uf 1 = + + SF csf asfhsf, 1 bsfus 1uF 1 h. where s and f are he spo and fuures reurns respecively, S 1 and F 1 are he spo and fuures exchange raes a ime 1, ln is he naural logarihm operaor, u S and u F are residuals, respecively, α S, h S, α F, S h F, and h SF are condiional variances and covariance β, β F, a i, b i, c i for i = { S, F, SF} are he esimaed parameers. -raios in parenhesis. The daa are weekly and he model is esimaed over he in-sample period: July, June,

30 Table IV - Coninued AD BP CD DM JY SF α S (0.3) (1.11) (-0.89) (-0.47) (1.55) (0.05) α F (0.16) (1.06) (-0.40) (-0.50) (1.48) (0.1) β S (-1.47) (0.17) (-0.6) (-0.10) (0.7) (-0.08) β F (-1.) (0.69) (-0.7) (-0.5) (0.71) (-0.15) c S (13.06) (3.36) (.74) (6.67) (7.75) (4.66) c F (1.53) (7.70) (16.0) (7.15) (8.9) (3.34) c SF (1.96) (5.80) (1.) (6.97) (8.05) (3.97) a S (0.05) (15.89) (.30) (8.04) (-3.30) (3.40) a F (-3.79) (0.84) (3.4) (14.4) (-.9) (3.16) a SF (-.54) (13.07) (4.60) (10.74) (-3.05) (3.31) b S (3.64) (5.37) (10.84) (8.73) (3.58) (31.08) b F (3.18) (1.6) (-3.58) (5.95) (4.7) (30.91) b SF (3.) (6.00) (-4.86) (7.41) (4.14) (31.79)

31 Table V. Hedging Effeciveness: In-Sample Resuls (July, June, ) AD BP CD DM JY SF Panel A: Variance No Hedge Naïve One-o-One Hedge Saic OLS Hedge GARCH (1,1) Hedge wih an Error Correcion Term Condiional OLS Hedge wih Consan Alpha Condiional OLS Hedge wih Time-Varying Alpha Panel B: Hedging Effeciveness Naïve One-o-One Hedge 93.99% 97.15% 90.63% 96.87% 96.35% 98.17% Saic OLS Hedge 94.37% 97.6% 9.4% 96.87% 96.56% 98.18% GARCH (1,1) Hedge wih an Error Correcion Term 94.8% 96.73% 9.8% 96.77% 96.53% 98.17% Condiional OLS Hedge wih Consan Alpha 94.47% 97.39% 9.5% 96.9% 96.64% 98.0% Condiional OLS Hedge wih Time-Varying Alpha 94.47% 97.39% 9.5% 96.9% 96.64% 98.0% The variance of he porfolio reurns is used as a measure of dispersion. Hedging effeciveness is compued as he percenage change in he variance of he hedged porfolio reurns compared o he unhedged posiion.

32 Table VI. Hedging Effeciveness: Ou-of-Sample Resuls (June, November, 9 000) AD BP CD DM JY SF Panel A: Variance No hedge Naïve One-o-One Hedge Roll-over OLS Hedge GARCH (1,1) Hedge wih an Error Correcion Term Condiional OLS Hedge wih Consan Alpha Condiional OLS Hedge wih Time-Varying Alpha Panel B: Hedging Effeciveness Naïve One-o-One Hedge 96.36% 96.79% 95.50% 95.46% 94.94% 93.50% Roll-over OLS Hedge 96.34% 96.80% 95.58% 95.54% 95.6% 93.74% GARCH (1,1) Hedge wih an Error Correcion Term 96.1% 96.69% 95.55% 94.99% 95.8% 93.66% Condiional OLS Hedge wih Consan Alpha 96.40% 96.69% 95.64% 95.58% 95.63% 93.94% Condiional OLS Hedge wih Time-Varying Alpha 96.39% 96.69% 95.64% 95.58% 95.59% 93.93% The variance of he porfolio reurns is used as a measure of dispersion. Hedging effeciveness is compued as he percenage change in he variance of he hedged porfolio reurns compared o he unhedged posiion.

33 Table VII. The Condiional OLS Hedge Raios wih Consan a versus he Compeing Hedge Raios Panel A: Correlaion beween he condiional OLS hedge raio and he hedge raio esimaed via Roll-Over OLS GARCH(1,1) AD BP CD DM JY SF Mean Panel B: Regression of he difference in hedge raios on a consan Naïve (1) Roll-over OLS () GARCH(1,1) (3) AD (-9.60) (3.64) (-0.60) BP (-14.53) (5.31) (5.57) CD (-43.13) (5.06) (0.37) DM (-6.13) (0.16) (4.15) JY (-30.44) (-.0) (-4.96) SF (-19.63) (-9.56) (-.94) The able repors coefficien esimaes for c in he following regressions: (1) HRCond. OLS, 1 = c + ε () HRCond. OLS, HRRoll over OLS, = c + ε (3) HRCond. OLS, HR GARCH(1,1), = c + ε HR Cond. OLS, is he condiional OLS hedge raio wih consan α, HR Roll overols, is he roll-over OLS hedge raio, HR GARCH( 1,1), is he GARCH(1,1) hedge raio. -raios in parenhesis. The models are esimaed over he whole sample (Augus, November 9, 000).

34 Figure 1: Ausralian Dollar Hedge Raios GARCH(1,1) Condiional OLS Hedge Raio Naïve Roll-over OLS 0.7 In-Sample Esimaion: 01 Aug Jun Ou-of-Sample Forecasing: 5 Jun Nov Time Figure : Briish Pound Hedge Raios Roll-over OLS Condiional OLS Hedge Raio Naïve 0.7 GARCH(1,1) In-Sample Esimaion: 01 Aug Jun Ou-of-Sample Forecasing: 5 Jun Nov Time Figure 3: Canadian Dollar Hedge Raio GARCH(1,1) Hedge Raio 1.1 Naïve Condiional OLS Roll-over OLS In-Sample Esimaion: 01 Aug Jun Ou-of-Sample Forecasing: 5 Jun Nov Time

35 Figure 4: Deusche Mark Hegde Raios GARCH(1,1) Condiional OLS Naïve Hedge Raio Roll-over OLS 0.7 In-Sample Esimaion: 01 Aug Jun Ou-of-Sample Forecasing: 5 Jun Nov Time Figure 5: Japanese Yen Hedge Raios Condiional OLS GARCH(1,1) Naïve Roll-over OLS Hedge Raio In-Sample Esimaion: 01 Aug Jun Ou-of-Sample Forecasing: 5 Jun Nov Time Figure 6: Swiss Franc Hedge Raios Roll-over OLS Hedge Raio Naïve 0.7 GARCH(1,1) Condiional OLS In-Sample Esimaion: 01 Aug Jun Ou-of-Sample Forecasing: 5 Jun Nov Time