A Simplified Approach to Modelling the Comovement of Asset Returns

Transcription

1 A Appoach o Modelling he Comovemen of Asse Reuns Pape Numbe 04/08 Richad D.F. Hais* Evais Soja Jon Tucke Xfi Cene fo Finance and Invesmen Univesiy of Exee Ocobe 004 Absac This pape poposes a simplified mulivaiae GARCH model ha involves he esimaion of only univaiae GARCH models, boh fo he individual eun seies and fo he sum and diffeence of each pai of seies. The covaiance beween each pai of eun seies is hen impued fom hese vaiance esimaes. The model ha we popose is consideably easie o esimae han exising mulivaiae GARCH models and does no suffe fom he convegence poblems ha chaaceize many of hese models. Moeove, he model can be easily exended o include moe complex dynamics o alenaive foms of he GARCH specificaion. We use he simplified mulivaiae GARCH model o esimae he minimum-vaiance hedge aio fo he FTSE 100 index pofolio, hedged using index fuues, and compae i o fou of he mos widely used mulivaiae GARCH models. The simplified mulivaiae GARCH model pefoms a leas as well as he ohe models ha we conside, and in some cases bee han hem. Keywods: Mulivaiae GARCH; Hedging; Minimum-vaiance hedge aio; FTSE 100 index. Acknowledgemens: We would like o hank Mak Feeman, Cheif Guema and Jian Shen fo useful commens and suggesions. Coesponding auho: Pofesso Richad D. F. Hais, Xfi Cene fo Finance and Invesmen, Univesiy of Exee, Exee EX4 4ST, UK. Tel: +44 (0) Fax: +44 (0) R.D.F.Hais@exee.ac.uk.

2 1. Inoducion Thee ae many applicaions in finance ha ely on an esimae of he mulivaiae condiional covaiance maix of euns. Such applicaions include condiional asse picing models, pofolio opimizaion, minimum-vaiance hedging, value a isk and he picing of opions ha depend on moe han one undelying asse. Pehaps he mos widely used appoach o modeling he condiional covaiance maix of euns is he mulivaiae GARCH class of models. 1 A numbe of diffeen mulivaiae GARCH models have been poposed, each imposing a diffeen se of esicions on he dynamic pocess ha govens he covaiance maix of euns. These models include he Vech and Diagonal Vech models of Bolleslev, Engle and Woolidge (1988), he BEKK model of Engle and Kone (1995), he Consan Coelaion model of Bolleslev (1990), he Faco ARCH model of Engle, Ng and Rohschild (1990) and he Dynamic Condiional Coelaion model of Engle and Sheppad (001). While commonly employed in he academic lieaue, mulivaiae GARCH models suffe fom a numbe of poblems in pacice. Fis, hey end o be compuaionally budensome, ypically involving he simulaneous esimaion of a lage numbe of paamees. This is paiculaly ue of he Vech and BEKK models, boh of which impose elaively few esicions on he dynamic pocess ha govens he evoluion of he covaiance maix. Despie ecen advances in echnology, hee ae many insances when compuaional cos is impoan such as when esimaing ou-of-sample foecass of he condiional covaiance maix using a olling window ove a lage sample, o whee foecass of he condiional covaiance maix of euns mus be compued fo a lage numbe of asses in a sho peiod of ime (such as when esimaing ina-day VaR fo a deivaives ading desk). In hese insances, mulivaiae GARCH models ae ofen eschewed by paciiones in favou of simple alenaives such as exponenially weighed esimaos of he covaiance maix. Second, owing o he lage numbe of paamees ha mus be esimaed simulaneously, and he non-concaviy of he 1 Ohe appoaches o esimaing he condiional covaiance maix include olling esimaos of he sample covaiance maix, exponenially weighed esimaos (JP Mogan, 1994) and mulivaiae sochasic volailiy models (Havey, Ruiz and Shephad, 1994). Fo a summay of mulivaiae GARCH models see Bolleslev, Chou and Kone (199) and Kone and Ng (1998).

3 likelihood funcion, maximum likelihood esimaion of mulivaiae GARCH models can be poblemaic. Esablishing ha esimaion has popely conveged (i.e. o paamee values ha epesen a global maximum of he likelihood funcion ahe han a local maximum) involves a poenially compuaionally inensive gid-seach ove all of he paamees in he model. Thid, compaed wih hei univaiae counepas, i is elaively difficul o consuc muli-peiod foecass of he covaiance maix using mulivaiae GARCH models (see, fo example, Kone and Ng, 1998). Fouh, owing o hei compuaional complexiy, i is ofen difficul o exend mulivaiae GARCH models o include moe complicaed dynamics such as longe lag specificaions, he asymmeic esponse of volailiy o eun shocks, and dummy vaiables o capue seasonaliy, oulies and sucual beaks. In an aemp o ovecome hese compuaional issues, a numbe of simple specificaions of he mulivaiae GARCH model have been poposed. Howeve, he simplificaions ha hese models enail geneally come a he cos of imposing sevee, and ofen implausible, coss-equaion esicions on he elemens of he covaiance maix. Fo example, in he Diagonal Vech model, each elemen of he covaiance maix is assumed o evolve independenly, meaning ha shocks o he vaiances of individual asses have no impac on he fuue covaiance beween hem. In he Consan Coelaion model, he covaiance beween individual asses is deemined solely by hei individual vaiances. The Faco ARCH model assumes ha he covaiance beween any wo asses deives solely fom a common covaiance wih one o moe undelying facos. None of hese models is able o capue, fo example, he well documened feaue ha coelaion among financial asses ends o incease as volailiy inceases (see, fo example, Longin and Solnik, 1995). In his pape, we popose an alenaive, simplified mulivaiae GARCH model. The model ha we popose involves he esimaion of only univaiae GARCH models, boh fo he individual eun seies and fo he sum and diffeence of each pai of seies. The covaiance beween each pai of eun seies is hen impued fom hese vaiance esimaes. The model ha we popose is consideably moe saighfowad o esimae han he Vech and BEKK models, and because he esimaion involves only univaiae GARCH models and hence only a small numbe of paamees in any single esimaion i does no suffe fom he convegence poblems ha ypically chaaceize 3

4 hese models. Moeove, i is easily exended o include he moe complex dynamics ha ae commonly found in he univaiae GARCH lieaue, o o use alenaive foms of he GARCH specificaion. The model ha we popose is less esicive han he Diagonal Vech, Consan Coelaion and Faco ARCH models, allowing he covaiance beween wo asses o depend on he hisoy of boh hei covaiance and hei individual vaiances, wihou imposing he esicion ha he coelaion coefficien beween hem is consan ove ime o ha hei covaiance deives solely fom a common covaiance wih an undelying faco. We illusae he simplified mulivaiae GARCH model by esimaing he minimumvaiance hedge aio fo he FTSE 100 index pofolio, hedged using index fuues. We compae he simplified model o fou of he mos widely used mulivaiae GARCH models, namely he Diagonal Vech, Consan Coelaion, BEKK and Dynamic Condiional Coelaion models. We evaluae he pefomance of each model boh saisically, using a egession of each elemen of he ealized covaiance maix on he coesponding elemen of he esimaed covaiance maix, and economically, by consideing he pefomance of he hedged pofolio. We find ha he simplified mulivaiae GARCH model pefoms a leas as well as he ohe models ha we conside, and in some cases bee han hem. Moeove, he compuaion ime fo he simplified model is consideably lowe han fo he ohe models. The es of he pape is oganised as follows. The following secion inoduces he simplified mulivaiae GARCH model. Secion 3 pesens he empiical applicaion. Secion 4 concludes.. The Mulivaiae GARCH Model Conside wo asses, i and j, whose pe-peiod abnomal euns ae given by i, i, µ i, = and j, = j, µ j,, whee i and j ae acual euns and µ i, and µ j, ae condiional mean euns. Given he ime 1 infomaion se, Ω 1, an esimae of he condiional covaiance maix equies esimaes of he condiional vaiances σ i, = va( i, Ω 1) and σ j, = va( j, Ω 1), and an esimae of he condiional covaiance σ ij = cov( i,, j, Ω 1)., 4

5 The mulivaiae GARCH model ha we popose involves fisly esimaing he condiional vaiances, and, using a univaiae GARCH model. We hen σ i, σ j, consuc he new seies +, = i, + j, and, = i, j, and use a univaiae GARCH model o esimae σ = va(, 1) and σ = va(, Ω 1). An esimae +, + Ω, of he condiional covaiance, σ ij,, can hen be obained using he following ideniies. σ +, σ i, + σ j, + σ ij, (1) σ, σ i, + σ j, σ ij, () In paicula, combining (1) and (), we have σ (1/ 4)( σ σ ) (3) ij, +,, The ideniy given by (3) is commonly used in he saisics lieaue in ode o deive (uncondiional) covaiance esimaos fom (uncondiional) vaiance esimaos when no obvious mulivaiae exension of he vaiance esimao exiss such as in he case of obus esimaion of he covaiance maix (see, fo example, Hube, 1981). In he conex of condiional volailiy, Hais and Shen (003) employ his ideniy o genealise a univaiae obus EWMA esimao o he mulivaiae case. In his pape, we exend he applicaion of his ideniy o he mulivaiae GARCH model. The simplified mulivaiae GARCH model involves he esimaion of only univaiae GARCH models and is heefoe consideably easie o implemen han he Vech and BEKK models. In paicula, because only a few paamees ae esimaed in each model, i is moe likely ha maximum likelihood esimaion will convege popely and hence much less expeimenaion is equied wih diffeen saing values fo he model paamees. In he empiical example below, we use he simples GARCH(1,1) model in ode o esimae,, and. Howeve, i would be saighfowad o σ i, σ j, σ +, σ, exend he model o allow fo moe complicaed dynamics in boh he mean and 5

6 volailiy of euns using, fo example, a moe geneal ARMA(p,q)-GARCH(,s) specificaion, o one of he many alenaive specificaions of he univaiae GARCH model, such as he EGARCH model of Nelson (1990). In paicula, i would be saighfowad o include ems ha capue he asymmeic esponse of volailiy o eun shocks due o changes in financial leveage, o dummy vaiables ha capue seasonaliy, oulies and sucual beaks, in eihe he mean o he volailiy of euns. In he simplified model, is deemined by and, which ae funcions of σ ij, σ +, σ, +, 1 and. Fom he definiions of and i can be seen ha is a, 1 +, 1, 1 σ ij, funcion of boh and, and i, 1 j, 1 i, 1 j, 1. Theefoe he specificaion of he simplified model allows shocks o he vaiances of and o affec hei fuue covaiance (as well as hei especive vaiances), wihou imposing he esicion ha hei coelaion is consan ove ime. In his espec, he simplified model is consideably moe flexible han he Diagonal Vech model (which assumes ha he covaiance beween i, and j, i, j, is deemined solely by hei lagged covaiance) and he Consan Coelaion model (which assumes ha he covaiance beween i, and j, is deemined solely by hei lagged vaiances). 3. Empiical Illusaion: Esimaion of he Minimum Vaiance Hedge Raio In his secion, we illusae he simplified mulivaiae GARCH model by esimaing he minimum-vaiance hedge aio fo he FTSE 100 index pofolio, hedged using he FTSE100 index fuues conac. When he condiional covaiance maix of spo and fuues euns is ime-vaying, he minimum-vaiance hedge aio a ime is equal o h σ = (4) σ sf, f, whee σ is he condiional covaiance of spo and fuues euns and is he sf, σ f, condiional vaiance of fuues euns (see, fo example, Kone and Sulan, 1993). 6

7 Daa We obained daily closing pices fo he FTSE 100 index fom Daaseam, and fo he FTSE 100 index fuues conacs fom LIFFE, fo he peiod 04 May 1984 o 03 May 00, which is he longes common sample available. A any one ime, hee ae fou fuues conacs ousanding. On each day, we use he neaes conac o delivey bu ollove o he nex neaes conac on he fis day of he delivey monh in ode o avoid hin ading and expiaion effecs. Using he daily closing spo and fuues pices, we compued coninuously compounded euns. We emoved he euns on he ollove daes fom he sample in ode o avoid spuious jumps in he fuues pice ha aise fom suddenly inceasing he mauiy of he fuues conac. Table 1 gives summay saisics fo he spo and fuues euns. The mean euns ae almos idenical fo he wo seies, and vey close o zeo. The volailiy of fuues euns is somewha highe han he volailiy of spo euns, which is a common empiical finding (see, fo example, Kone and Sulan, 1993). The ARCH(4) pomaneau es fo up o fouh ode seial coelaion in squaed euns shows ha boh spo and fuues euns display significan volailiy cluseing. Boh spo and fuues euns ae highly lepokuic (which is consisen wih he exisence of ime-vaying volailiy) and negaively skewed. The Jaque-Bea saisic vey songly ejecs he null hypohesis of nomaliy. [Table 1] Mehodology In ode o implemen he mulivaiae GARCH model, we fis esimae he condiional vaiances of and. In pincipal, any condiional volailiy model could be used, s, f, bu o illusae ou appoach, we use he simples GARCH(1,1) specificaion. Since ou inees is in modeling he condiional covaiance maix of spo and fuues euns, and no expeced euns, we include only a consan in he mean equaion fo boh spo and fuues euns. The model fo s, is heefoe given by s, µ s + s, = (5) 7

8 σ (6) = + + s, β s,0 β s,1σ s, 1 β s, s, 1 The model fo f, is given by f, µ f + f, = (7) σ (8) = + + f, β f,0 β f,1σ f, 1 β f, f, 1 Afe esimaing hese wo models, we compue he esiduals ˆ ˆ s, and f,, and using hese, compue he new seies ˆ ˆ ˆ +, = s, + f, and ˆ ˆ ˆ, = s, f,. We hen esimae he condiional vaiances of equaion specified (since ˆ+, and ˆ, using a GARCH(1,1) model wih no mean ˆ s, and f, ˆ, and hence ˆ+, and ˆ,, have zeo mean by consucion). 3 The models fo ˆ+, and ˆ, ae heefoe given by σ (9) ˆ +, = β +,0 + β +,1σ +, 1 + β +, +, 1 σ (10) ˆ, = β,0 + β,1σ, 1 + β,, 1 The esimaed condiional vaiances of ˆ+, and ˆ, ae hen used o compue he condiional covaiance of and using equaion (3): s, f, σ = (1/ 4)( σ σ ) (11) sf, +,, The esimaed condiional vaiance of fuues euns, of spo and fuues euns, σ sf, σ f,, and condiional covaiance, ae hen used o compue he minimum-vaiance hedge 3 The seies ˆ+, and ˆ, ae subjec o measuemen eo bu since he infomaion maix is block diagonal, maximum likelihood esimaion is consisen, and when he condiional disibuion is nomal, fully efficien (see, fo example, Kone and Ng, 1998). 8

9 aio using equaion (4). We esimae each of he univaiae GARCH models above by maximum likelihood wih a condiional nomal disibuion, using he BHHH algoihm wih a convegence cieion of applied o he coefficien values. Noe ha if f, and s, ae boh condiionally nomally disibued, hen +, and, will also be condiionally nomally disibued by consucion. If and ae condiionally nonnomal hen we can ely on he consisency esuls of Quasi-Maximum Likelihood (see Bolleslev and Woolidge, 199). 4 f, s, Evaluaion We compae ou esuls wih fou of he mos commonly used mulivaiae GARCH models, namely he Diagonal Vech (DVech) model of Bolleslev, Engle and Woolidge (1988), he BEKK model of Engle and Kone (1995), he Consan Coelaion (CC) model of Bolleslev (1990) and he Dynamic Condiional Coelaion (DCC) model of Engle and Sheppad (001). As wih he simplified model, we esimae each of hese mulivaiae GARCH models by maximum likelihood wih a condiional nomal disibuion, using he BHHH algoihm wih a convegence cieion of applied o he coefficien values. In each case, he mean equaion fo boh spo and fuues eun is specified o include only a consan. In ode o evaluae he pefomance of he five mulivaiae GARCH models, we employ wo appoaches. The fis is a saisical evaluaion. If a mulivaiae GARCH model is coecly specified hen i should geneae esimaes of he ealized covaiance maix ha ae condiionally unbiased. Fo each elemen of he covaiance maix, we es his using a egession of he ealized vaiance (o covaiance) on he esimaed vaiance (o covaiance). As a measue of he ealized covaiance maix, we use he squaes and coss-poduc of equaions. ˆ ˆ s, and f,. We heefoe esimae he following hee 4 Alhough Quasi-Maximum Likelihood esimaion is consisen if and ae condiionally non-nomal, moe efficien esimaos of he condiional covaiance maix can be obained by using, fo example, he APARCH model of Ding, Gange and Engle (1993) (see Nelson and Fose, 1996). This is easily accommodaed in he simplified mulivaiae GARCH model. f, s, 9

10 ˆ = δ + δ σ + (1) s, s,0 s,1 s, v s, ˆ = δ + δ σ + (13) f, f,0 f,1 f, v f, ˆ + (14) s, ˆ f, = δ sf,0 + δ sf,1 σ sf, v sf, If he esimaed covaiance maix is condiionally unbiased, he inecep in each egession should be zeo and he slope coefficien should be uniy (see, fo example, Andesen and Bolleslev, 1998). We esimae hese egessions using OLS. The null hypohesis of condiional unbiasedness is esed fo each egession using an F-saisic. The second appoach is an economic evaluaion. In paicula, we epo he sandad deviaion of he hedged pofolio fo each model. The moe accuae he esimaed condiional covaiance maix, he lowe should be he sandad deviaion of he hedged pofolio. We also epo he sandad deviaion of he hedge aio iself, which gives some indicaion of he likely ansacion coss associaed wih a dynamic hedging saegy based on each model. Resuls Table epos he coelaion maices fo, σ and acoss he five models. As expeced, owing o he diffeen esicions ha hey impose, he five mulivaiae GARCH models yield quie diffeen esimaes of he covaiance maix of spo and fuues euns. Fo all hee elemens of he covaiance maix, he lowes coelaion is beween he DCC and CC models, while he highes coelaion is beween he simplified model and he CC model, excep fo, whee he coelaion beween he DCC and DVech models is maginally highe han beween he simplified model and he CC model. Oveall, he lowes coelaion is 0.81 (beween he DCC model and he CC model fo σ f, and he CC model fo ). σ s, σ f, f, σ sf, ), while he highes coelaion is 0.99 (beween he simplified model σ s, [Table ] 10

11 Table 3 epos he esuls of he egessions o es he condiional unbiasedness of, σ s, σ f, and fo each of he five mulivaiae GARCH models. Fo, he null σ sf, σ s, hypohesis of condiional unbiasedness is ejeced a he five pecen significance level fo all of he models excep he simplified model, alhough fo he CC and DCC models, he ejecion is maginal. The ejecion is paiculaly song fo he BEKK model. In conas, fo σ f,, he null hypohesis of condiional unbiasedness is ejeced fo all five models, alhough he ejecion is weakes fo he CC model, followed by he DVech model, he simplified model and he DCC model. Again, he null hypohesis of condiional unbiasedness is vey songly ejeced fo he BEKK model. Fo, he null hypohesis of condiional unbiasedness is ejeced a he five pecen level fo all of he models excep he CC and BEKK models, alhough fo he emaining models, he ejecion is vey maginal. Oveall, i would appea ha in ems of condiional unbiasedness, hee is lile o choose beween he simplified model and he DVech, CC and DCC models, bu ha he BEKK model pefoms significanly wose han hese models. σ sf, [Table 3] The fis ow of Table 4 epos he sandad deviaion of he esimaed hedge aio fo he five mulivaiae GARCH models. The CC model yields he lowes hedge aio sandad deviaion, eflecing he fac ha i imposes he esicion ha he coelaion coefficien beween spo and fuues euns is consan, while he ohe models allow he coelaion coefficien o vay ove ime. The BEKK model yields he highes hedge aio sandad deviaion. The simplified model yields a hedge aio sandad deviaion ha is maginally highe han he DCC model. The second ow of Table 4 epos he sandad deviaion of he hedge pofolio daily eun fo he five mulivaiae GARCH models. The DVech and DCC models yield he lowes hedge pofolio sandad deviaion, alhough he simplified model yields a sandad deviaion ha is only maginally highe han hese. The sandad deviaion fo he CC and BEKK models is significanly highe han fo he ohe models. Thus, in ems of hedge pofolio 11

12 sandad deviaion, he simplified model pefoms almos as well as he DVech and DCC models, and subsanially bee han he CC and BEKK models. [Table 4] Finally, Table 5 epos he esimaion ime fo each of he five mulivaiae GARCH models. Even in his simple bivaiae case, i is clea ha esimaion of mulivaiae GARCH models can be elaively ime-consuming, wih he BEKK model aking almos six minues o esimae. 5 Howeve, he esimaion ime fo he simplified model is less han half ha of he ohe models. Moeove, hese figues significanly undesae he ue compuaional advanage of he simplified model since hey ae based on he final esimaion fo each model afe expeimening wih diffeen saing values fo he model paamees. The simplified model conveged diecly, iespecive of he saing values fo he paamees, while, in conas, esimaion of he ohe models equied consideable expeimenaion wih diffeen saing values in ode o achieve convegence. The DVech and BEKK models poved o be paiculaly poblemaic in his espec. [Table 5] 4. Conclusion While commonly employed in he academic lieaue, mulivaiae GARCH models suffe fom a numbe of poblems in pacice owing o he complexiy of hei specificaion. In paicula, because of he lage numbe of paamees ha mus be esimaed simulaneously, mulivaiae GARCH models end o be compuaionally budensome. Moeove, because he likelihood funcion of hese models is no globally concave, hee is no guaanee ha maximum likelihood esimaion will convege o he coec paamee values, paiculaly when he models ae supplemened by moe complicaed dynamics, asymmeic ems o dummy vaiables. 5 The models wee esimaed using he RATS 5.01 package on a Penium IV.8 GHz PC. 1

13 In his pape, we popose a simple bu effecive mulivaiae GARCH model ha ovecomes hese poblems. The model ha we popose involves he esimaion of only univaiae GARCH models, boh fo he individual eun seies and fo he sum and diffeence of each pai of seies. The covaiance ems in he covaiance maix ae hen impued fom hese vaiance esimaes. Since he esimaion involves only univaiae GARCH models, i is consideably moe saighfowad o esimae han exising mulivaiae GARCH models and does no suffe fom he convegence poblems ha ypically chaaceise many of hese models. We illusae he simplified mulivaiae GARCH model, and compae i o fou of he mos widely used mulivaiae GARCH models he Diagonal Vech model, he Consan Coelaion model, he BEKK model and he Dynamic Condiional Coelaion model by esimaing he minimum-vaiance hedge aio fo he FTSE 100 index pofolio, hedged using index fuues. We evaluae he pefomance of each model boh saisically, using a egession of each elemen of he ealized covaiance maix on he coesponding elemen of he esimaed covaiance maix, and economically, by consideing he pefomance of he hedged pofolio. We find ha by boh measues, he simplified mulivaiae GARCH model pefoms a leas as well as he ohe models ha we conside, and in some cases bee han hem. 13

14 Refeences Andesen, T., and T. Bolleslev, 1998, Answeing he Skepics: Yes, Sandad Volailiy Models Do Povide Accuae Foecass, Inenaional Economic Review 39, Bolleslev, T., 1990, Modelling he Coheence in Sho Run Nominal Exchange Raes: A Mulivaiae Genealized ARCH Model, Review of Economics and Saisics 7, Bolleslev, T., R. Chou and K. Kone, 199, ARCH Modeling in Finance: A Review of he Theoy and Empiical Evidence, Jounal of Economeics. Bolleslev, T, R. Engle and J. Wooldidge, 1988, A Capial Asse Picing Model wih Time-Vaying Covaiances, Jounal of Poliical Economy 96, Bolleslev, T. and J. Wooldidge, 199, Quasi-Maximum Likelihood Esimaion and Infeence in Dynamic Models wih Time-Vaying Covaiances, Economeic Reviews 11, Ding, Z., C. Gange and R. Engle, 1993, A Long Memoy Popey of Sock Make Reuns and a New Model, Jounal of Empiical Finance 1, Engle, R., and K. Kone, 1995, Mulivaiae Simulaneous Genealised ARCH, Economeic Theoy 11, Engle, F., and K. Sheppad, K., 001, Theoeical and Empiical Popeies of Dynamic Condiional Coelaion Mulivaiae GARCH, NBER Woking Pape Engle, F., V. Ng and M. Rohschild, 1990, Asse Picing wih a Faco ARCH Covaiance Sucue: Empiical Esimaes fo Teasuy Bills, Jounal of Economeics 45, Glosen, L., R. Jagannahan and D. Runkle, 1993, On he Relaion Beween he Expeced Value and he Volailiy of he Nominal Excess Reun on Socks, Jounal of Finance 48, Hais, R. D. F., and J. Shen, 003, Robus Esimaion of he Opimal Hedge Raio, Jounal of Fuues Makes 3, Havey, A., E. Ruiz and N. Shephad, 1994, Mulivaiae Sochasic Vaiance Models, Review of Economic Sudies 61, Hube, P., 1981, Robus Saisics, Wiley. JP Mogan, 1996, RiskMeics Technical Documen, fouh ediion, New Yok. Kone, K., and V. Ng, 1998, Modelling Asymmeic Co-Movemens of Asse Reuns, Review of Financial Sudies 11,

15 Kone, K., and J. Sulan, 1993, Time Vaying Disibuion and Dynamic Hedging wih Foeign Cuency Fuues, Jounal of Finance and Quaniaive Analysis 8, Longin, F., and B. Solnik, 1995, Is he Coelaion in Inenaional Equiy Reuns Consan: ?, Jounal of Inenaional Money and Finance 14, 3-6. Nelson, D., 1990, Condiional Heeoskedasiciy in Asse Reuns: A New Appoach, Economeica 59, Nelson, D., and D. Fose, 1996, Asympoic Fileing Theoy fo Univaiae ARCH Models, Economeica 6,

16 Table 1 Summay Saisics of FTSE 100 Spo and Fuues Reuns s, f, Mean 0.034% 0.034% Sandad Deviaion 1.00% 1.144% Skewness Excess kuosis Jaque-Bea ARCH(4) Noes: The able epos summay saisics fo coninuously compounded spo and fuues euns fo he FTSE 100 fo he peiod 04 May 1984 o 03 May 00. The Jaque-Bea saisic ess he null hypohesis of zeo skewness and excess kuosis, and has a χ () disibuion wih a ciical value of 5.99 a he 5% significance level. The ARCH(4) saisic ess he null hypohesis ha he fis fou paial auocoelaions of squaed euns ae zeo, and has a χ (4) disibuion wih a ciical value of 9.49 a he 5% significance level. 16

17 Table Coelaion Maices of, and σ s, σ f, σ sf, Panel A: Coelaion Maix of σ s, DVech CCR BEKK DCC Model DVech CCR BEKK DCC Model Panel B: Coelaion Maix of σ f, DVech CCR BEKK DCC Model DVech CCR BEKK DCC Model Panel C: Coelaion Maix of σ sf, DVech CCR BEKK DCC Model DVech CCR BEKK DCC Model Noes: The able epos he coelaion maices of, and acoss he five mulivaiae GARCH models. σ s, σ f, σ sf, 17

18 Table 3 Resuls fo Condiional Bias Regessions Panel A ˆ = δ + δ σ + s, s,0 s,1 s, v s, DVech CCR BEKK DCC Model ˆs,0 δ 6.4E E E E E-06 ˆs,1 δ F-saisic Panel B ˆ = δ + δ σ + f, f,0 f,1 f, v f, DVech CCR BEKK DCC Model ˆf δ,0 1.9E E E-05.0E-05.0E-05 ˆf δ, F-saisic Panel C ˆ + s, ˆ f, = δ sf,0 + δ sf,1 σ sf, v sf, DVech CCR BEKK DCC Model δ 3.5E E E-05.9E E-05 ˆsf,0 ˆsf δ, F-saisic Noes: The able epos he esuls of esimaing he condiional bias egessions fo σ, and σ. The F-saisic ess he null hypohesis ha δ 0 and δ i, = 1 s, σ f, sf, ˆ i, 0 = ( i = s, f, sf ), and has an F(, 4518) disibuion wih a ciical value of 3.00 a he 5% significance level. ˆ 1 18

19 Table 4 Resuls fo Minimum Vaiance Hedge Pofolio DVech CC BEKK DCC Model σ ĥ σ 0.400% 0.481% 0.51% 0.400% 0.403% p Noes: The able epos he sandad deviaion of he esimaed hedge aio, σ, and he ĥ sandad deviaion of he hedge pofolio eun, σ, fo each of he five mulivaiae GARCH models. p 19

20 Table 5 Esimaion Time DVech CC BEKK DCC Model 5mins 3secs 4mins 31secs 5mins 49secs 4mins 36secs mins 15secs Noes: The able epos he esimaion ime fo each of he five mulivaiae GARCH models. The models wee esimaed using RATS 5.01 on a Penium IV.8 GHz PC. 0