The effect of asymmetries on optimal hedge ratios

Transcription

1 The effec of asymmeries on opimal hedge raios Aricle Acceped Version Brooks, C., Henry, O.T. and Persand, G. (2002) The effec of asymmeries on opimal hedge raios. Journal of Business, 75 (2). pp ISSN Available a hp://cenaur.reading.ac.uk/24151/ I is advisable o refer o he publisher s version if you inend o cie from he work. Published version a: hp://ideas.repec.org/a/ucp/jnlbus/v75y2002i2p hml Publisher: Universiy of Chicago Press All oupus in CenAUR are proeced by Inellecual Propery Righs law, including copyrigh law. Copyrigh and IPR is reained by he creaors or oher copyrigh holders. Terms and condiions for use of his maerial are defined in he End User Agreemen. CenAUR Cenral Archive a he Universiy of Reading

2 Reading s research oupus online

3 This is he auhors acceped manuscrip of an aricle published in he Journal of Business. The definiive version is available a: hp:// 1

4 The Effec of Asymmeries on Opimal Hedge Raios Chris Brooks The ISMA Cenre, Deparmen of Economics, Universiy of Reading Ólan T. Henry Deparmen of Economics, Universiy of Melbourne, Gia Persand The ISMA Cenre, Deparmen of Economics, Universiy of Reading 2

5 Absrac There is widespread evidence ha he volailiy of sock reurns displays an asymmeric response o good and bad news. This paper considers he impac of asymmery on ime varying hedges for financial fuures. An asymmeric model which allows forecass of cash and fuures reurn volailiy o respond differenly o posiive and negaive reurn innovaions gives superior in-sample hedging performance. However, he simpler symmeric model is no inferior in a hold-ou sample. A mehod for evaluaing he models in a modern risk managemen framework is presened, highlighing he imporance of allowing opimal hedge raios o be boh ime-varying and asymmeric. 3

6 1. Inroducion Over he pas wo decades, increases in he availabiliy and usage of derivaive securiies has allowed agens who face price risk he opporuniy o reduce heir exposure. Alhough here are many echniques available for reducing and managing risk, he simples and perhaps he mos widely used, is hedging wih fuures conracs. A hedge is achieved by aking opposie posiions in spo and fuures markes simulaneously, so ha any loss susained from an adverse price movemen in one marke should o some degree be offse by a favourable price movemen in he oher. The raio of he number of unis of he fuures asse ha are purchased relaive o he number of unis of he spo asse is known as he hedge raio. Since risk in his conex is usually measured as he volailiy of porfolio reurns, an inuiively plausible sraegy migh be o choose ha hedge raio which minimises he variance of he reurns of a porfolio conaining he spo and fuures posiion; his is known as he opimal hedge raio. There has been much empirical research ino he calculaion of opimal hedge raios (see, for example, Cecchei e al., 1988; Myers and Thompson, 1989; Baillie and Myers, 1991; Kroner and Sulan, 1991; Lien and Luo, 1993; Lin e al., 1994; Srong and Dickinson, 1994; Park and Swizer, 1995). The general consensus is ha he use of mulivariae generalised auoregressive condiionally heeroscedasic (MGARCH) models yields superior performances evidenced by lower porfolio volailiies, han eiher ime-invarian or rolling ordinary leas squares (OLS) hedges. Cecchei e al (1988), Myers and Thompson (1989) and Baillie and Myers (1991), for example, argue ha commodiy prices are characerised by ime-varying 4

7 covariance marices. As news abou spo and fuures prices arrives o he marke, he condiional covariance marix, and hence he opimal hedging raio, becomes ime-varying. Baillie and Myers (1991) and Kroner and Sulan (1993), iner alia, employ MGARCH models o capure ime-variaion in he covariance marix and he resuling hedge raio. On he oher hand, here is also evidence ha he benefis of a ime varying hedge are subsanially diminished as he duraion of he hedge is increased (e.g Lin e al., 1994). Moreover, here is evidence ha he use of volailiy forecass implied by opions prices can furher improve hedging effeciveness (Srong and Dickinson, 1994). This paper has hree main aims. Firsly, we link he concep of he opimal hedge wih he noion of he News Impac Surface of Kroner and Ng (1998). The hedging surface of he OLS model is independen of news arriving o he marke and herefore could be sub-opimal. Secondly, we exend he models of Cecchei e al (1988), Myers and Thompson (1989) and Baillie and Myers (1991) o allow for ime variaion and asymmery across he enire variance covariance marix of reurns. This means ha he hedge raio will be sensiive o he size and sign of he change in prices resuling from informaion arrival. Thirdly, we adap he mehods used by Hsieh (1993) o show how he effeciveness of hedges can be evaluaed by he calculaion of minimum capial risk requiremens (MCRRs). Such a procedure allows he hedging performance of he various models o be assessed using a relevan economic loss funcion as well as on pure saisical grounds. The paper is laid ou in six secions. Secion wo presens he heoreical framework for deriving he hedge raios, while secion hree 5

8 describes he daa. Secion four presens he empirical evidence on he performance of each hedging model, while he fifh secion oulines he boosrap mehodology used o calculae he MCRR for each of he porfolios. Secion six concludes. 2. The Derivaion of Opimal Hedge Raios Le C and F represen he logarihms of he sock index and sock index fuures prices respecively. The acual reurn on a spo posiion held from ime 1 o is C C C 1 similarly, he acual reurn on a fuures posiion is F F F 1. However, a ime 1, he expeced reurn, E 1 ( R ), of he porfolio comprising one uni of he sock index and unis of he fuures conrac may be wrien as E -1 (R ) = E -1 (C )- -1 E -1 (F ) (1) where -1 is he hedge raio deermined a ime -1, for employmen in period. 1 The variance of he porfolio may be wrien as 2 hp, hc, 1hF, 2 1 h (2) CF where h p,, h, and F h, represen he condiional variances of he porfolio C and he spo and fuures posiions respecively and h CF, represens he condiional covariance beween he spo and fuures posiion. If he agen has he wo momen uiliy funcion U ( E 1 R, hp, ) E 1( R ) hp, (3) hen he uiliy maximising agen wih degree of risk aversion seeks o solve max U( E E 1 1 R, h ) 2 C E F ( h h 2 h ) 1 p, C, 1 F, 1 CF, (4) 6

9 Solving (4) wih respec o under he assumpion ha F is a maringale process such ha F ) E F F F F 0 E yields 1 ( , he opimal number of fuures conracs in he invesor s porfolio hcf, 1 (5) h F, If he condiional variance-covariance marix is ime-invarian (and if C and F are no coinegraed) hen an esimae of, he consan opimal hedge raio, may be obained from he esimaed slope coefficien b in he regression C a bf u (6) The OLS esimae of b = h CF / h F is also valid for he muliperiod hedge in he case where he invesors uiliy funcion is ime separable. However, i has been shown by numerous sudies (see secion 1 above) ha he daa do no suppor he assumpion ha he variance-covariance marix of reurns is consan over ime. Therefore we follow recen lieraure by employing a bivariae GARCH model which allows he condiional variances and covariances used as inpus o he hedge raio o be ime-varying. In he absence of ransacions coss, marke microsrucure effecs or oher impedimens o heir free operaion, he efficien markes hypohesis and he absence of arbirage opporuniies implies ha he spo and corresponding fuures markes reac conemporaneously and idenically o new informaion. There has been some debae in he lieraure as o wheher his implies ha he wo markes mus be coinegraed. Ghosh (1993), for example, suggess ha marke efficiency should imply ha cash and fuures are coinegraed, while Baillie and Myers (1991) sugges ha, as a consequence of possible nonsaionariy of he risk free proxy employed in he cos of carry model, his 7

10 need no be he case. We do no wish o ener ino his debae from a heoreical viewpoin, bu suffice o say ha in all ensuing analysis, we allow for, bu do no impose, a [-1 1] coinegraing vecor for he wo series. The condiional mean equaions of he model employed in his paper are a bivariae Vecor Error Correcion Mechanism (VECM), which may be wrien as Y Y 4 i1 Y i i v 1 ( F F i, ; ; i ( C C i, F ) F C) F ( F ) i, C ( C) i, C F F ; ; C C,, (7) Under he assumpion ~ (0, H ), where represens he innovaion vecor in (6) and defining h as vech(h ), where vech(.) denoes he vecor-half operaor ha sacks he lower riangular elemens of an N N marix ino an ( N ( N 1) / 2) 1 vecor, he bivariae VECM(p) GARCH(1,1) vech model may be wrien h C, vec( H ) h hcf, C0 A1vec( 1 1') B1h 1 (8) h F, where c C c c ; a a a A1 21 a22 a23 ; 22 a a a a b B 1 b b b b b b b b Resricing he marices A 1 and B 1 o be diagonal gives he model proposed by Bollerslev, Engle and Wooldridge (1988) where each elemen of he condiional variance-covariance marix h ij, depends on pas values of iself and pas values of ' 1 1. There are 21 parameers in he condiional 8

11 variance-covariance srucure of he bivariae GARCH(1,1) vech model (8) o be esimaed, subjec o he requiremen ha H be posiive definie for all values of in he sample. The difficuly of checking, le alone imposing such a resricion led Engle and Kroner (1995) o propose he BEKK parameerisaion H C ' ' ' ' 0 C0 A11 1 1A11 B11H 1B11 (9) The BEKK parameerisaion requires esimaion of only 11 parameers in he condiional variance-covariance srucure and guaranees H posiive definie. I is imporan o noe ha he BEKK and vec models imply ha only he magniude of pas reurn innovaions is imporan in deermining curren condiional variances and covariances. This assumpion of symmeric imevarying variance-covariance marices mus be considered enuous given he exising body of evidence documening he asymmeric response of equiy volailiy o posiive and negaive innovaions of equal magniude (see Engle and Ng, 1993, Glosen, Jagannahan and Runkle, 1993, and Kroner and Ng, 1998, iner alia). Defining min,0, for j=fuures, cash, he BEKK model in (9) j, may be exended o allow for asymmeric responses as H C ' ' ' ' ' ' 0 C0 A A11 B11H 1B11 D11 1 1D11 (10) where 9

12 C B 0 11 c c c ; 12 ; 22 A 11 D ; 22 and F, 2 (11) C, The symmeric BEKK model (9) is given as a special case of (10) for m,n =0, for all values of m and n. 3. Daa Descripion The daa employed in his sudy comprises 3580 daily observaions on he FTSE 100 sock index and sock index fuures 2 conrac spanning he period 1 January April Days corresponding o UK public holidays are removed from he series o avoid he incorporaion of spurious zero reurns. The FTSE 100 comprises he 100 UK companies quoed on he London Sock Exchange wih he larges marke capialisaion, accouning for 73.2% of he marke value of he FTSE All Share Index as a 29 December 1995 (Sucliffe 1997). FTSE 100 fuures conracs are quoed in he same unis as he underlying index, excep ha he decimal is rounded o he neares The price of a fuures conrac (conrac size) is he quoed number (measured in index poins) muliplied by he conrac muliplier, which is 25 for he conrac. There are four delivery monhs: March, June, Sepember and December. Trading akes place in he hree neares delivery monhs alhough volume in he far conrac is very small. Each conrac is herefore raded for nine monhs. FTSE100 fuures conracs are cash-seled as opposed o physical delivery of he underlying. All conracs are marked o marke on he las rading day, which is he hird Friday in he delivery monh, a which poin 10

13 all posiions are deemed closed. For he FTSE100 fuures conrac, he selemen price on he las rading day is calculaed as an average of minueby-minue observaions beween 10:10AM and 10:30AM rounded o he neares 0.5. Summary saisics for he daa are displayed in panel A of able 1. Using Dickey Fuller (1979) uni roo ess, i is no possible o rejec he null hypohesis of non-saionariy for he cash and fuures price series. This nonsaionariy of he price series is consisen wih weak-form efficiency of he cash and fuures markes. The reurn series are calculaed as 100( C / C 1) and 100 ( F / F 1), respecively. The reurns are skewed o he lef, lepokuric and saionary. These feaures are enirely in accordance wih expecaions and resuls presened elsewhere. In he absence of a long run relaionship beween C and F, opimal inference based upon asympoic heory requires he use of reurns raher han price daa in calculaion of esimaion of dynamic hedge raios. Resuls for boh Engle-Granger (1987) and Johansen (1988) ess for coinegraion are displayed in able 1.The Engle-Granger resuls of panel B clearly demonsrae ha he null of non-saionariy in he residuals of he coinegraing regression is srongly rejeced, for he es boh wih and wihou a consan erm. Moreover he esimaed slope coefficien is very close o uniy wheher he spo or fuures price is he dependen variable. Similarly, he Johansen es saisics, for boh he race and he max forms, rejec he null of no coinegraing vecor, bu do no rejec he null of one coinegraing vecor. A resricion of he coinegraing relaionship beween he series o be [1-1] was marginally rejeced a he 5% level. However, afer normalising he 11

14 esimaed coinegraion vecor on C, he esimaed coefficien on F was suggesing ha his rejecion may no be economically imporan. On close examinaion of he shor run componens of he VECM i appears ha he fuures prices are weakly exogenous. A likelihood raio es suppors his resricion. Thus while he coinegraing equilibrium is defined by boh cash and fuures prices, equilibrium is resored hrough he cash markes. A es of he join hypohesis ha fuures prices are weakly exogenous and ha he parameers of he coinegraion vecor are [1,-1] was no rejeced a he 5% level of significance. Baillie and Myers (1991) argue ha a perfec 1:1 associaion does no exis in a commodiy fuures hedge due o he cos of carry, alhough his does no preclude some oher coinegraing relaionship from exising. On balance, he daa appear o be coinegraed wih a [1,-1] coinegraing vecor. 4. Hedging Model Esimaes, Tess and Performance Given he evidence of a long-run or coinegraing relaionship beween C and F he condiional mean equaions are parameerised as a VECM raher han a VAR o avoid loss of long run informaion. The parameer esimaes and associaed residual diagnosics for he mulivariae asymmeric GARCH model are presened in able 2. Again, he facor loading associaed wih he fuures prices is posiive indicaing ha he reurn o equilibrium is achieved via he cash markes. A high degree of persisence is variance in indicaed in boh markes. The persisence is 2 2 measured by for i=1,2. The saisical significance of he elemens of ii ii 12

15 he D 11 marix indicaes he presence of asymmeries in he variancecovariance marix. Kroner and Ng (1998) analyse he asymmeric properies of imevarying covariance marix models, idenifying hree possible forms of asymmeric behaviour. Firsly, he covariance marix displays own variance asymmery if h,, he condiional variance of C h C F, F, is affeced by he sign of he innovaion in C F. Secondly, he covariance marix displays cross variance asymmery if he condiional variance of C F is affeced by he sign of he innovaion in C F. Finally if he covariance of reurns h CF, is sensiive o he sign of he innovaion in reurn for eiher model is said o display covariance asymmery. C or F hen he The residual diagnosics indicae ha he model was able o capure all of he dependence on pas values in boh he condiional mean and condiional variances for boh he spo and fuures equaions. The coefficiens of skewness and excess kurosis are much reduced relaive o heir values on he raw daa, again indicaing a reasonable fi of he model o he wo series. The robus likelihood raio ess suggesed by Kroner and Ng (1998) o deec such asymmery in MGARCH models indicae ha he asymmeric model provides a superior daa characerisaion o he symmeric MGARCH(1,1). The final row of able 2 ess he resricion of he asymmeric model o be symmeric; ha is, a resricion ha good and bad news affec he volailiy of he spo and fuures markes equally. This resricion is clearly rejeced, suggesing ha he pursui of an asymmeric model is imporan and may yield superior hedging 13

16 performance relaive o a model which ignores his feaure which is manifes in he daa. The price innovaions, C C 1 C, and F -F- 1 ε F,, represen changes in informaion available o he marke (ceeris paribus). Kroner and Ng (1998) rea such innovaions as a collecive measure of news arriving o marke j beween he close of rade on period -1 and he close of rade on period. They define he relaionship beween innovaions in reurn and he condiional variance-covariance srucure as he news impac surface, a mulivariae form of he news impac curve of Engle and Ng (1993). Figures 1 o 3 display he variance and covariance news impac surfaces from he esimaes displayed in Table 2. Following Engle and Ng (1993) and Kroner and Ng (1998) each surface is evaluaed in he region j, 5,5 for j= fuures, cash. There are relaively few exreme ouliers in he daa, which suggess ha some cauion should be exercised in inerpreing he news impac surfaces for larger values of j,. Despie his cavea, he asymmery in variance and covariance is clear from each figure. The reurns and variances for he various hedging sraegies are presened in able 3. The simples approach, presened in he second column, is ha of no hedge a all. In his case, he porfolio simply comprises a long posiion in he cash marke. Such an approach is able o achieve significan posiive reurns in sample, bu wih a large variabiliy of porfolio reurns. Alhough none of he alernaive sraegies generae reurns ha are significanly differen from zero, eiher in sample or ou of sample, i is clear from columns 3-5 of able 3 ha any hedge generaes significanly less reurn variabiliy han none a all. 14

17 The naïve or coinegraing hedge, which akes one shor fuures conrac for every spo uni, bu does no allow he hedge o ime-vary, generaes a reducion in variance of he order of 80% in sample and nearly 90% ou of sample relaive o he unhedged posiion. Allowing he hedge raio o be ime-varying and deermined from a symmeric mulivariae GARCH model leads o a furher reducion as a proporion of he unhedged variance of 5% and 2% on he in- and hold-ou samples respecively. Allowing for an asymmeric response of he condiional variance o posiive and negaive shocks yields a very modes reducion in variance (a furher 0.5% of he iniial value) in sample, and virually no change ou of sample. Figure 4 graphs he ime varying hedge raio from he symmeric and asymmeric MGARCH models. The opimal hedge raio is never greaer han fuures conracs per index conrac, wih an average value of fuures conracs sold per long index conrac. The variance of he esimaed opimal hedge raio is Moreover he opimal hedge raio series obained hrough he esimaion of he asymmeric GARCH model appears saionary. An ADF es (see, for example, Fuller, 1976) of he null hypohesis 1 ~I(1) was srongly rejeced by he daa (ADF= , 5% Criical value = ). The ime varying hedge requires he sale (purchase) of fewer fuures conracs per long (shor) index conrac. 4 The opimal hedge raio 1 may be linked o he arrival of news o he marke using (5) and he relevan fuures price and covariance news impac surfaces. Evaluaing in he range 1 j, 5,5 for j=fuures, cash as before gives us he response of he opimal hedge o news. Noe ha he surface is drawn under he assumpion ha he porfolio is long he sock index 15

18 and he opimal hedge raio is wrien in erms of he number of fuures conracs o sell. A negaive opimal hedge raio hus implies he purchase of fuures conracs. Figure 5 graphs he response of 1 o news. I is worh noing ha 1 responds far more dramaically o bad news abou he cash marke index han o news abou he fuure price. Negaive innovaions in he cash price cause he opimal hedge raio o increase in magniude owards 1. Large posiive innovaions in he cash price sugges a negaive hedge raio. This migh appear couner inuiive, however he surface is drawn holding pas informaion consan. The implicaion of he asymmery is ha he hedge has very low value in bull marke siuaions. In conras, he coinegraing hedge implies ha he hedging surface is a plane a 1 = =1. One possible inerpreaion of he beer performance of he dynamic sraegies over he naive hedge is ha he dynamic hedge uses shor run informaion, while he coinegraing hedge is driven by long run consideraions. The performance evaluaion in able 3 is in erms of one-day-ahead hedges. In he nex secion we use a new crierion o judge hedging over various horizons, including he one-day horizon. 5. Evaluaing Hedging Effeciveness by Calculaing Minimum Capial Risk Requiremens Ensuring ha banks hold sufficien capial o mee possible fuure losses has been a opic of grea impor for regulaors and risk managers in recen years. A very popular approach involves he calculaion of he insiuion s value a risk (VaR) inheren in is rading book posiions. VaR is an esimaion of he probabiliy of likely losses which migh occur from 16

19 changes in marke prices from a paricular securiies posiion, and he minimum capial risk requiremen (MCRR) is defined as he minimum amoun of capial required o absorb all bu a pre-specified percenage of hese possible losses. We address an approach o he calculaion of MCRRs which is similar in spiri o he approach adoped in many Inernal Risk Managemen Models (IRMM), proposed by Hsieh (1993). 5 Capial risk requiremens are esimaed for 1 day, 10 day, 30 day, 3 monh and 6 monh invesmen horizons by simulaing he condiional densiies of price changes, using Efron s (1982) boosrapping mehodology based upon he mulivariae GARCH(1,1) model presened in equaions (7) and (9), boh wih and wihou asymmeries, for comparison. The simulaed errors are generaed by drawing randomly, wih replacemen, from he sandardised residuals and hence a pah of fuure Y s can be generaed, using he esimaes of,,, C 0, A 11, and B 11 from he sample and muli-sep ahead forecass of H. A securiies firm wishing o calculae he VaR of a porfolio conaining he cash and fuures asses 7 would have o simulae he price of he asses when i iniially opened he posiion. To calculae he appropriae capial risk requiremen, i would hen have o esimae he maximum loss ha he posiion migh experience over he proposed holding period. 6 For example, by racking he daily value of a long cash and shor fuures posiion and recording is lowes value over he sample period, he firm can repor is maximum loss for his paricular simulaed pah of cash and fuures prices. Repeaing his procedure for 20,000 simulaed pahs generaes an empirical disribuion of he maximum loss. This maximum loss (Q) is given by: 17

20 Q = (x 0 - x 1 ) (12) Where x 0 is he iniial value of he porfolio and x 1 is he lowes simulaed value of he porfolio (for a long fuures posiion) or he highes simulaed value (for a shor fuures posiion) over he holding period. We can express he maximum loss as a proporion of he iniial value of he porfolio as follows: Q x 0 x 1 x 1 0 (13) In his case, since x 0 is a consan, he disribuion of Q will depend on he disribuion of x 1. From expression (13), i can be seen ha he disribuion of Q x 0 will depend on he disribuion of x 1. Hence, he firs sep is o find he 5 h x 0 Quanile of x 1 Ln : x0 x 1 Ln m x0 Sd (14) Where is he 5 h Quanile from a sandard normal disribuion, m is he Mean of x 1 Ln and Sd is he Sandard deviaion of x0 x Ln x 1 0. Cross- muliplying and aking he exponenial, herefore x x Exponenial [( Sd) m] (15) 1 0 Q x { Exponenial [( Sd) m]} (16)

21 In his paper, we compare he MCRRs generaed by he porfolios consruced using he hedge raios derived from he models described above. The asymmeric mulivariae GARCH model appears well specified and able o capure he salien feaures of he daa. Given his, we now deermine wha would be an appropriae amoun of capial o cover he cash and fuures porfolio derived from he hedge raio as implied by he model. In paricular, we consider wheher his porfolio minimises he need for capial, given ha all such capial is ied up in an unproducive and unprofiable fashion. The esimaed minimum capial risk requiremens are presened in ables 4 and 5 for each of he models, ignoring and allowing for asymmeries, respecively, and are given in unis of index poins 8. Panel A of Tables 4 and 5 presen he MCRR for a shor hedge (long cash, shor fuures). While Panel B of he ables presens he resuls for a long hedge (long fuures, shor cash). The mos imporan feaure of he resuls is ha any ype of hedge, even a naïve hedge, is beer han a naked exposure. Moreover, a shor invesmen horizons, here are large gains o be made by allowing he hedge o vary over ime. For example, he shor hedge porfolio MCRR is 22.2 index poins for a naïve hedge, bu only 11.8 for a Mulivariae GARCH hedge. The long hedge posiions seem o be more risky overall over our ou of sample period, generaing higher values a risk han he corresponding shor hedges. The gain from he use of an asymmeric model, as opposed o a consrained symmeric model, which does no allow good and bad news o effec reurns differenly, is large a shor ime horizons. For example, for he symmeric ime-varying shor hedge, he porfolio MCRR is 11.8, while modelling he asymmeries reduces his o 2.0. However, he benefi of hese 19

22 more complex asymmeric and ime-varying hedges, and moreover, he benefis of hedging per se, are considerably reduced as he ime horizon is exended beyond one monh. For example, he MCRR for a long hedge calculaed using asymmeric MGARCH is less han 10% of ha using no hedge a he one day horizon, bu rises o more han 25% over a 6 monh hedging period. This resul is in agreemen wih he findings of Lin e al. (1994). 6. Conclusions This paper sough o advance he exan lieraure in his field by considering he impac of asymmeries on he hedging of sock index posiions using financial fuures conracs 9. We found ha asymmeric models, which allow posiive and negaive price innovaions o affec volailiy forecass differenly, yielded improvemens in forecas accuracy in sample, bu no ou of sample, when evaluaed using he radiional variance of realised reurns meric. The paper also demonsraed how such hedging mehodologies could be evaluaed in a modern risk managemen conex, using a echnique based on he esimaion of value a risk. Our primary finding was ha allowing for asymmeries led o considerably reduced porfolio risk a he shores forecasing horizons, and modes benefis when he duraion of he hedge was increased. Our resuls have a leas wo imporan implicaions for hose financial marke ransacors who wish o reduce heir exposure o risk using fuures conracs, and for furher research in his area. Firs, hedge raios which are 20

23 deermined aking ino accoun asymmeries in volailiy are expeced, in general, o be more effecive han hose which do no. Second, since recen changes in legislaion in Europe have allowed marke risk o be deermined using value a risk echnologies under he second EC Capial Adequacy Direcive (CAD II), i is surely desirable for hedgers o measure he risk inheren in heir hedged porfolios in a similar fashion. Such procedures are already now in widespread use in Europe as well as he US. The value a risk approach is (or soon will be) used o assess he risk of he books of securiies firms as a whole. The use of radiional mehods for assessing hedging effeciveness, such as porfolio reurn variances, could be incompaible wih, and give very differen resuls o, hose based on value a risk mehods. 21

24 Foonoes This paper was wrien while he second auhor was on sudy leave a he ISMA Cenre, Deparmen of Economics, The Universiy of Reading. The developmen of his paper benefied from commens by he anonymous referees and discussions wih Salih Nefci, Simon Burke and Peer Summers. The responsibiliy for any errors or omissions lies solely wih he auhors. 1 Noe ha we are no requiring a his sage ha he hedge raio, -1, be imevarying, bu raher ha i is deermined using informaion o ime Since hese conracs expire 4 imes per year - March, June, Sepember and December - o obain a coninuous ime series we use he closes o mauriy conrac unless he nex closes has greaer volume, in which case we swich o his conrac. 3 The reason for his is ha he minimum price movemen (known as ick) for he fuures conrac is i.e. a change of 0.5 in he index. 4 Alhough, of course, a ime-varying hedge may resul in considerably increased ransacions coss in he likely even ha such a hedge requires daily adjusmens of he fuures posiion. We herefore canno sae caegorically ha he ime-varying hedge would be cheaper. 5 See also Brooks e al. (2000) for a more deailed descripion of his mehodology and issues in is implemenaion. 22

25 6 See Dimson and Marsh (1997) for a discussion of a number of poenial issues which a financial insiuion may face when calculaing appropriae levels of capial for muliple posiions during periods of sress. 7 The curren BIS rules sae ha he MCRR should be he higher of he: (i) average MCRR over he previous 60 days or (ii) he previous rading days MCRR. 8 See secion 3 above. Alhough Hsieh (1993) and Brooks e al. (2000) measure MCRRs as a proporion of he iniial value of he posiion, his is no sensible in our case since by definiion an appropriaely hedged porfolio will have a zero value. 9 Alhough he mehodology could, of course, be equally applied o hedging a posiion in any financial asse using fuures conracs. 23

26 References Baillie, R.T. and Myers, R.J Bivariae GARCH esimaion of he opimal commodiy fuures hedge. Journal of Applied Economerics 6 (April): Bollerslev, T., Engle, R.F. and Wooldridge, J.M A capial asse pricing model wih ime-varying covariances. Journal of Poliical Economy 96 (February): Brooks, C., Clare, A.D. and Persand, G A word of cauion on calculaing marke-based capial risk requiremens. Journal of Banking and Finance forhcoming. Cechei, S.G., Cumby, R.E., Figlewski, S Esimaion of opimal fuures hedge. Review of Economics and Saisics 70 (November): Dimson, E. and P. Marsh Sress ess of capial requiremens. Journal of Banking and Finance. 21 (July): Efron, B The Jackknife, he Boosrap, and Oher Resampling Plans, Philadelphia, PA: Sociey for Indusrial and Applied Mahemaics. Engle, R.F. and Granger, C.W.J Coinegraion and error correcion: represenaion, esimaion and esing. Economerica 55 (March):

27 Engle, R.F. and Ng, V Measuring and esing he impac of news on volailiy. Journal of Finance, 48 (December): Engle, R.F. and Kroner, K Mulivariae simulaneous generalised ARCH. Economeric Theory, 11 (March), Fuller, W.A Inroducion o Saisical Time Series Wiley, N.Y. Garcia, P., Roh, J-S, and Leuhold, R.M Simulaneously deermined, ime-varying hedge raios in he soybean complex. Applied Economics 27 (November): Ghosh, A Coinegraion and error correcion models: Ineremporal causaliy beween index and fuures prices. Journal of Fuures Markes 13 (April): Glosen, L.R., Jagannahan, R. and Runkle, D.E On he relaion beween he expeced value and he volailiy of he nominal excess reurn on socks. The Journal of Finance 48 (December): Hodgson, A. and Okunev, J An alernaive approach for deermining hedge raios for fuures conracs. Journal of Business Finance and Accouning 19 (January):

28 Hsieh, D.A Implicaions of non-linear dynamics for financial risk managemen. Journal of Financial and Quaniaive Analysis 28 (March): Johansen, S Saisical analysis of coinegraion vecors Journal of Economic Dynamics and Conrol 12, Kroner, K.F., and Ng, V.K Modelling asymmeric co-movemens of asse reurns. Review of Financial Sudies, 11 (Winer): Kroner, K.F. and Sulan, J Exchange rae volailiy and ime-varying hedge raios. In Rhee, S.G. and Chang, R.P. (eds.) Pacific Basin Capial Markes Research. Elsevier, Norh Holland. Lien, D. and Luo, X Esimaing muliperiod hedge raios in coinegraed markes. Journal of Fuures Markes 13 (December): Lin, J.W., Najand, M., and Yung, K Hedging wih currency fuures: OLS versus GARCH. Journal of Mulinaional Financial Managemen 4 (January): Myers, R.T. and Thompson, S.R Generalised opimal hedge raio esimaion. American Journal of Agriculural Economics 71 (November):

29 Park, T.H. and Swizer, L.N Bivariae GARCH esimaion of he opimal hedge raios for sock index fuures: A noe. Journal of Fuures Markes 15 (February): Srong, R.A. and Dickinson, A Forecasing beer hedge raios. Financial Analyss Journal. 50 (January-February): Sucliffe, C Sock Index Fuures: Theories and Inernaional Evidence. Second Ediion, Thompson Business Press, London. 27

30 Table 1: Summary Saisics and Coinegraion Tess Panel A: Summary Saisics for he daa ADF() ADF F C Series Mean Variance Skewness Excess Kurosis ΔF ΔC

31 Panel B: Engle Granger Coinegraion Tess F as dependen variable 0 1 ADF() ADF (0.0039) (0.0005) C as dependen variable 0 1 ADF() ADF (0.0039) (0.0005) Panel C: Johansen Coinegraion Tess M T r = r = Likelihood Raio Tess H 0 : =[-1,1] H 0 : =[1,0] H 0 : =[-1,1] =[1,0] [0.0300] [0.4000] 5.51 [0.06] 29

32 30 Table 2: Esimaes of he Mulivariae Asymmeric GARCH Model Condiional Mean Equaions C F C F C C i C F i F C i F F i i C F i i i P F Y v Y Y,, ) (, ) (, ) (, ) (, ; ; ; ; (0.0061) (0.0053) (0.0072) (0.0060) (0.0110) (0.0089)

33 Table 2 Coninued: Esimaes of he Mulivariae Asymmeric GARCH Model Residual Diagnosics Mean Variance Skewness Excess Q(10) Q 2 (10) Kurosis F, [0.0000] [0.0000] [0.2055] [0.9946] C, [0.0000] [0.0000] [0.2820] [0.6607] Noes: Sandard errors displayed as (.). Marginal significance levels displayed as [.]. Q(10) and Q 2 (10) are are Ljung_Box ess for enh order serial correlaion in 2 z j, and z j, respecively for j = F,C. 31

34 Table 2 Coninued: Esimaes of he Mulivariae Asymmeric GARCH Model Condiional Variance-Covariance Srucure H C C 1 ' 0 F C 0, 1, 1 A ' 11 ; 1 1 ' 1 A 11 B min( F min( C ' 11, 1, 1 H 1,0),0) B 11 D ' 11 1 ' 1 D 11 C (0.0184) (0.0151) (0.0036) B (0.0077) (0.0067) (0.0072) A (0.0208) (00170) (0.0239) D (0.0885) (0.0717) (0.0892) H 0 : ij =0 for i,j=1, [0.0000] 32

35 Table 3: Porfolio Reurns In Sample Unhedged Naïve Hedge Symmeric Asymmeric = 0 = -1 Time Varying Time Varying (C.I. HEDGE) Hedge Hedge h FC, h F, h FC, h F, Reurn {2.3713} { } {0.9562} {0.9580} Variance Ou of Sample Unhedged Naïve Hedge Symmeric Asymmeric = 0 = -1 Time Varying Time Varying (C.I. HEDGE) Hedge Hedge h FC, h F, h FC, h F, Reurn {1.4958} {0.0216} {0.7761} {0.9083} Variance Noes: -Raios displayed as {.} 33

36 Table 4: MCRR Esimaes - Symmeric Hedging Models Panel A: Long Cash and Shor Fuures Days Unhedged Naïve Hedge Time-Varying Hedge

37 Table 4: MCRR Esimaes - Symmeric Hedging Models Panel B: Shor Cash and Long Fuures Days Unhedged Naïve Hedge Time-Varying Hedge

38 Table 5: MCRR Esimaes - Asymmeric Hedging Models Panel A: Long Cash and Shor Fuures Days Unhedged Naïve Hedge Time-Varying Hedge

39 Table 5: MCRR Esimaes - Asymmeric Hedging Models Panel B: Shor Cash and Long Fuures Days Unhedged Naïve Hedge Time-Varying Hedge

40 Figure 1: News Impac Surface for Fuures Marke Volailiy 38

41 Figure 2: News Impac Surface for Cash Marke Volailiy 39

42 Figure 3: Covariance News Impac Surface 40

43 1.00 Time Varying Hedge Raios Symmeric BEKK Asymmeric BEKK Figure 4: The opimal dynamic hedge raio, 1 41

44 Figure 5: Hedging Surface: The response of o News 42