Hedging: Scaling and the Investor Horizon* John Cotter and Jim Hanly

Transcription

1 Hedging: Scaling and he Invesor Horizon* John Coer and Jim Hanly Keywords: Hedging Effeciveness; Scaling; Volailiy Modelling, Forecasing. JEL classificaion: G10, G12, G15. Augus 2009 John Coer, Direcor of Cenre for Financial Markes, School of Business Universiy College Dublin, Blackrock, Co. Dublin, Ireland, Tel , john.coer@ucd.ie. Jim Hanly, School of Accouning and Finance, Dublin Insiue of Technology, Dublin 2, Ireland. el , james.hanly@di.ie. *The auhors would like o hank paricipans a a Universiy College Dublin seminar for heir commens on an earlier draf. Par of his sudy was carried ou while Coer was visiing he UCLA Anderson School of Managemen and is hankful for heir hospialiy. Coer s conribuion o he sudy has been suppored by a Universiy College Dublin School of Business research gran. The auhors hank an anonymous referee for helpful commens bu he usual cavea applies.

2 Hedging: Scaling and he Invesor Horizon Absrac This paper examines he volailiy and covariance dynamics of cash and fuures conracs ha underlie he Opimal Hedge Raio (OHR) across differen hedging ime horizons. We examine wheher hedge raios calculaed over a shor erm hedging horizon can be scaled and successfully applied o longer erm horizons. We also es he equivalence of scaled hedge raios wih hose calculaed direcly from lower frequency daa and compare hem in erms of hedging effeciveness. Our findings show ha he volailiy and covariance dynamics may differ considerably depending on he hedging horizon and his gives rise o significan differences beween shor erm and longer erm hedges. Despie his, scaling provides good hedging oucomes in erms of risk reducion which are comparable o hose based on direc esimaion. 1

3 Hedging: Scaling and he Invesor Horizon I Inroducion Much of he large body of work on hedging has focused on shor ime horizons such as a 1-day frequency 1. This ignores he fac ha he OHR is dependen on he ime horizon and ha a hedge calculaed from daa a one frequency may no provide good risk reducion oucomes over lower frequencies. Also, here is no definiive answer o he quesion of wha consiues he relevan horizon for risk managemen since invesors have differing invesmen horizons. Research suggess a variey of horizons ranging from 1-day for raders, o 1-monh or even 12 monhs for invesors and corporae risk managemen respecively. 2 However, he esimaion of hedge sraegies over longer erm ime horizons such as weekly or monhly can be problemaic. The key problem is ha for lower frequency daa, fewer observaions are available on which o base he esimaion (Hwang and Valls Pereira, 2006). For example, an esimaion period of 5- years of daily daa will yield around 1300 observaions; however, his number drops o around 260 for he weekly frequency and jus 60 if daa a he monhly frequency are used. Using a longer esimaion period such as 20 years may no yield beer esimaes since OHR s are ime varying (Lien and Tse, 2002). This means ha using a very long esimaion period may be subopimal, as some of he daa may no be relevan given heir emporal disance. 3 Also, ime varying hedge sraegies such as hose calculaed using GARCH models, require large numbers of observaions if he model is o mee he 1 See Lien and Tse (2002), for a review of he developmen of he lieraure on opimal hedging. 2 See Locke (1999), Smihson and Minon (1996), for evidence on he variey of ime horizons ha are relevan o invesors. 3 See for example Meron (1980) who finds ha beer volailiy esimaes are o be had from using a large number of high frequency daa raher han a small number of low frequency daa over a longer ime period. 2

4 non-negaiviy consrains which are a ypical feaure of hose models 4. The esimaion and saisical problems relaed o low sample size, have led o he adopion by risk managers, of models ha esimae volailiy a one frequency (say daily) and hen scale he volailiy esimae o obain low frequency volailiies (say monhly). A number of scaling laws have been applied in financial applicaions. Iniially hese were based on Gaussian disribuions and random walks which allowed scaled sums of Gaussians o follow he same disribuion. Furher work by Mandelbro (1963) proved ha daily prices were no from a Gaussian disribuion and a number of papers, including Dacarogna, Muller, Pice and De Vries (1998) provided evidence of scaling and power laws in many financial ime series 5. Scaling has been used in many applicaions in financial economics, mos noably in he Black-Scholes-Meron opion pricing framework and in risk managemen and banking regulaion, where scaling is based on he Square-Rooof-Time (SQRT) rule 6. For example, a ypical daily volailiy esimae for equiy index daa using he sandard deviaion would be abou 1.3%. Applying he SQRT rule o obain a monhly esimae would yield a volailiy of 5.8%, or an annual esimae of abou 21%. While scaling has been exensively applied in he lieraure on volailiy and high frequency finance, lile work has been done on scaling and hedging. However, a number of papers have examined hedging for differen ime horizons (see Malliaris and Urruia (1991), Gepper (1995), Chen, Lee and Shresha (2004) and In and Kim (2006)), and he findings show ha as he hedging horizon increases, boh he OHR and he insample hedging effeciveness increase. There are conrasing findings ou-of-sample 4 See Hwang and Valls Pereira (2006) for a discussion. 5 For an exensive discussion on scaling laws in finance see Brock (1999). 6 The Basel agreemens on banking supervision require ha daily volailiy esimaes be scaled using he SQRT rule. 3

5 wih some papers reporing decreased effeciveness a longer ime horizons (Malliaris and Urruia, 1991). However, mos papers repor a posiive relaionship beween hedge horizon and hedging effeciveness (Chen e al, 2004, Lien and Shresha, 2007). A variey of mehods have been used o calculae hedges over longer ime horizons. These include he use of models based on he underlying daa generaing process and more recenly he use of wavele analysis 7. In his paper we calculae a hedge raio a a relaively high frequency using a 1-day (daily) ime horizon. We hen apply his hedge raio by scaling i up o boh 5-day (weekly) and 20-day (monhly) frequencies. We compare he scaled hedge raios wih OHR s ha are esimaed by maching he frequency of he daa o he hedging horizon 8. We also re-examine he issue of hedging effeciveness across differen ime horizons using he performance evaluaion crieria of Value a Risk (VaR) and Condiional Value a Risk (CVaR). To our knowledge, hese crieria have no been applied in he lieraure on hedging o evaluae hedges over differen ime horizons. Indeed he findings on he relaionship beween hedging effeciveness and ime horizon are based on a single evaluaion crierion which is variance. By applying addiional performance evaluaion crieria, we will provide addiional evidence on he issue of hedging effeciveness across differen ime horizons. A furher conribuion of his paper is an analysis of he levels of persisence in volailiy for differen hedging horizons. Dros and Nijman (1993), Diebold e al (1998), and Chrisofferson, Diebold and Schuermann (1998), all address he issue of how he 7 For example, Geppar (1995) use a Daa Generaing Process o generae reurns a differen ime horizons while In and Kim (2006) apply waveles o decompose he variance and covariance over differen ime scales. 8 OHR s esimaed in his way are hereafer referred o as acual hedges or equivalenly direc esimaion. 4

6 accuracy of volailiy forecass change wih he ime horizon. The findings of hese papers show ha forecasabiliy will decrease as we move from shor o long ime horizons. We invesigae he implicaions of his finding for opimal hedging by examining he emporal aggregaion properies of GARCH models ha have been successfully used o esimae ime varying OHR s. This allows us o draw a link beween he lieraure on volailiy persisence and hedging performance. Our findings show ha acual hedge sraegies saisically ouperform scaled hedge sraegies; however he differences are only marginal when viewed from an economic perspecive. Furhermore, we find ha scaled hedges are effecive in ha hey provide accepable reducions in risk as measured by Variance, VaR and CVaR. We also provide evidence ha ex-pos hedging effeciveness increases as we move from high o low frequency hedging. Finally, we show ha lower levels of volailiy persisence, does no maerially affec he ex-pos hedging effeciveness a lower frequencies implying ha GARCH models can sill provide good forecas oucomes over longer hedging horizons. The remainder of he paper proceeds as follows. In secion II we ouline he hedging models. In Secion III we deail he scaling approach. In Secion IV we describe he merics for measuring hedging effeciveness. Secion V describes he daa followed by our empirical findings in Secion VI. Finally, Secion VII summarises and concludes. 5

7 II Hedging Models Hedging using fuures involves combining a fuures conrac wih a cash posiion in order o reduce he risk of a posiion. The OHR is he raio ha minimises he risk of he payoff of he hedged porfolio which is given by: + r β r (1) s f where rs and r f are he reurns on he cash and fuures respecively, β is he esimaed OHR and is he subscrip denoing ime. We use wo differen hedging models in his sudy o esimae β. The firs mehod is an OLS regression of he cash on he fuures reurns which yields he minimum variance hedge raio (MVHR). This mehod has been applied exensively in he lieraure since (Ederingon, 1979) and has been found o yield reasonably good performance. Is key advanage is is simpliciy and ease of esimaion. This is given as: r s = α + β r + ε (2) f where β is he MVHR. This can also be calculaed as he covariance beween he cash and fuures reurn divided by he variance of he fuures reurn. A criicism of his approach is ha he OLS HR is effecively consan whereas is has been empirically esablished ha volailiy and correlaions upon which he OHR are based are boh imevarying (Bollerslev, Chou and Kroner, 1992). We herefore also esimae a ime varying OHR using a mulivariae GARCH model. We use he Diagonal Vech GARCH (1,1) model proposed by Bollerslev, Engle and Wooldridge (1988). This provides a useful benchmark from which o examine hedging 6

8 over differen ime horizons given is abiliy o represen he dynamics of variances and covariance s (see Bauwens e al, 2006, for example) The model is specified as follows: rs = µ s + ε s rf µ f + ε f ε ε s =, ~ N ( 0, H ) Ω 1 f (3) m n α s, jε s + j β s, k j= 1 k = 1 2 s k H s = ω H (4) m n α f, jε f + j β f, k j= 1 k = 1 2 f k H f = ω H (5) m n 3 + α sf, jε s ε + j f j β sf, k j= 1 k = 1 H sf = ω H (6) where Ω 1 is he informaion se a ime 1, ε s, ε f are he residuals, H s, H f sf k denoes he variance of cash and fuures and H sf is he covariance beween hem. ω = ( ω, ω ω ) is a 3x1 vecor, and ( α, α α ) 1 2, 3 α = and ( β, β β ) j s, j f, j, sf, j β = are 3x1 k s, k f, k, sf, k vecors. The model conains 3 +3m+3n parameers. The marices α j and β k are resriced o be diagonal. This means ha he condiional variance of he cash reurns depends only on pas values of iself and pas values of he squared innovaions in he cash reurns. The condiional variance of he fuures reurns and he condiional covariance beween cash and fuures reurns have similar srucures. Because of he diagonal resricion we use only he upper riangular porion of he variance covariance marix, he model is herefore parsimonious, wih only nine parameers in he condiional variance-covariance srucure of he Diagonal VECH (1,1) model o be esimaed. We esimae he GARCH models using boh Maximum Likelihood and Quasi Maximum Likelihood Esimaors o obain GARCH coefficiens a he differen frequencies 9. 9 QMLE provides more consisen esimaes of GARCH coefficiens for he lower frequency daa given he differen disribuional characerisics of lower frequency daa which end o approximae normaliy. 7

9 III Scaling The deerminaion of he OHR requires an esimae of he variance of he fuures reurn a whaever frequency is being examined, ogeher wih an esimae of he covariance or correlaion beween he cash and fuures reurn. The problem of esimaing volailiy over longer erm ime horizons has been examined in some deail in he risk managemen lieraure. Risk managers have good high frequency daa bu require reliable esimaes of volailiy a low frequencies corresponding o he respecive holding periods of invesors. An alernaive approach is o esimae volailiy using high frequency daa and hen scale i o obain low frequency esimaes. The mos popular mehod of scaling volailiies is based upon he SQRT rule. The heoreical jusificaion and background for he SQRT rule is as follows: Consider S, he log price of an asse a ime, where he changes in he log price are independen and idenically disribued (i.i.d.). Then he price a ime can be expressed as S 2 = 1 + ε ε, i ~ ( 0, σ ) (7a) S and he 1-day reurn, is r = S S 1 = ε (7b) wih varianceσ 2. Aggregaing h-day reurns resuls in r h = S S h = 1 n= 0 ε = ε + n h ε (7c) 2 wih variance hσ and sandard deviaion h σ, which implies he square-roo-of-ime rule. This rule can be considered a special case of he more general empirical scaling law discussed by Dacarogna e al (2001), which gives a direc relaion beween ime 8

10 inervals Λ and he average volailiy as measured by a cerain power P of he absolue reurns observed over hose inervals. This is given as 1 p p D( p) { E[ r ]} c( p) = (8) where E is he expecaion operaor and c ( p) and ( p) D are deerminisic funcions of p. D is he drif exponen which deermines he scaling behaviour across differen daa frequencies. For p = 2, he sandard deviaion will scale according o he following rule D { E r ]} = c [ (9) where c is a consan depending on he underlying ime series. For he Gaussian random walk model he drif exponen is D = 0. 5 which yields he SQRT rule for scaling volailiy. This scaling law has been widely applied boh by praciioners and academics o obain scaled volailiy esimaes for use in opion pricing via he Black-Scholes- Meron model (eg. see Hull, 2008, chaper 13) where he h-period volailiy is given byσ h. I has also been widely used for esimaing quaniles and in paricular for risk measures such as VaR. (see Danielsson and Zigrand, 2006). For example, he 1-day VaR can be muliplied by 10 o obain he 10-day VaR. This mehod of scaling by he SQRT rule has been widely used wihin he financial services indusry as recommended by he Basel Commiee on Banking Supervision (2004). I is broadly used because i is easy o undersand and apply and because here are no simple alernaives. However, here are a number of objecions o he use of he SQRT rule. In he firs insance i requires he assumpion ha he log reurns are i.i.d. However, high frequency financial reurns are no i.i.d as evidenced by he numerous papers documening srong volailiy persisence in financial reurns 10. Also, because he SQRT rule magnifies volailiy 10 See for example he survey on ARCH and GARCH effecs in Bollerslev, Chou and Kroner (1992). 9

11 flucuaions when hey should be damped, i ends o produce overesimaes of longhorizon volailiy (see Diebold e al, 1998 for a discussion). Despie hese issues, he SQRT mehod has been widely adoped in risk managemen and banking as a means of obaining low frequency volailiies. Dowd and Oliver (2006) provide limied suppor for he use of he SQRT rule. They sugges ha he key o using he SQRT rule is o apply i o he uncondiional volailiy as opposed o he mos recen esimae 11. Furhermore, Anderson e al (2001) repor he means of he popular measure realized variance esimaors grow linearly wih ime, which would be consisen wih he SQRT rule. Dacarogna e al (2001) find ha his scaling law is appropriae for a wide range of financial daa and for ime inervals ranging from 10 minue o more han a year. They also esimae scaling exponens for foreign exchange daa and find values of D very close o 0.5 for USD/GBP and oher exchange raes. Addiional suppor for he SQRT rule is pu forward by Brummelhuis and Kaufman (2007) who apply i for scaling quaniles and conclude ha i provides reasonable esimaes for risk managemen purposes. Therefore, he use of he SQRT rule should provide a good approximaion in erms of convering volailiy from one frequency o anoher frequency even where he disribuion is no sricly normal. An alernaive o scaling volailiy by ime is he use of formal model based aggregaion as proposed in Dros and Nijman (1993) (henceforh DN), who sudy he emporal aggregaion of GARCH processes. They propose he following approach: Suppose we 11 This finding is based on a simulaions based approach which indicae ha if he daily volailiy used as he basis of exrapolaion is greaer han he uncondiional volailiy i resuls in he SQRT rule overesimaing he GARCH or rue volailiy and vice versa. Therefore hey conclude ha he SQRT rule is appropriae only where he daily volailiy o be used as he basis for exrapolaion is equal o uncondiional volailiy. 10

12 begin wih have a sample pah of one-day reurns { R (1, 1) process as follows: } T ( 1) = 1 which follow a weak GARCH R = σ ε 2 ε ~ NID(0,1) σ = ω + αy βσ 2 1 (10) 2 for = 1,..., T where σ is he variance and ω, α, β are he esimaed parameers on he GARCH model. The following saionariy and regulariy condiions are imposed, 0 < ω <, and α 0, β 0, α + β < 1.Then DN show ha, under regulariy condiions, he corresponding sample pah of h-day reurns, { GARCH (1, 1) process wih } T / R h ( h ) = 1 also follows a σ ( h) = ω( h) + α ( h) y( h) 1 + β ( h) σ ( h) 1 (11) where h ( β + α ) ( β + α ) 1 ω( h) = h ω (12) 1 α h ( β + α ) β, = (13) ( h) ( h) and β 1 is he soluion of he quadraic equaion, ( h) < where h ( β + α ) ( β + ) β ( h) a = 2 1+ β a α a = + 4 ( h) 2h ( 1+ ) b, 2b ( ) ( ) ( ) 2 ( 2 ) 2 1 β α 1 β 2βα h 1 β + 2h h 1 2 ( κ 1) ( 1 ( β + α ) ) h ( h 1 h( β + α ) + ( β + α ) )( α βα ( β + α )) 1 ( β + α ) 2 (14) (15) 2h ( β + α ) 2 ( β + α ) 1 b = ( α βα( β + α )), (16) 1 11

13 and κ is he kurosis of y. This approach allows us o fi a GARCH model a one frequency (eg. 1-day) and he model coefficiens obained ω, α and β can hen be subsiued ino he DN equaions o obain scaled coefficiens a anoher frequency (E.g. 5-day). Also his formula can be used o conver 1-day covariance o h-day covariance if we subsiue he covariance parameers from a mulivariae GARCH model 12. This approach is useful a generaing parameer values a long ime horizons from shor ime horizons. From he formulas for α h and β h, α 0, β 0as h.this means ha asympoically he volailiy becomes consan, h h and herefore he weak GARCH(1,1) process behaves in he limi like a random walk. This implies ha condiional forecass will have poor performance over long periods as predicabiliy decreases. A number of papers have examined he usefulness of he DN approach. Diebold e al (1998) show ha if he shor horizon reurn model is correcly specified as a GARCH (1, 1) process, hen he DN approach can be used for he correc conversion of 1-day o h-day volailiy. Kaufman and Paie (2003) use he DN formula in a simulaion based approach and conclude ha i provides good parameer esimaes based on daily daa scaled o horizons of up o 1-monh. A shorcoming of he approach is ha i assumes an exac fi for he model whereas in general i is an approximaion 13. In his paper we provide furher evidence on he applicabiliy of he DN approach. We measure volailiy persisence by fiing a GARCH (1, 1) model direcly from he daa a he relevan ime horizon. We hen use he DN formula o obain scaled parameers from he 1-day daa for boh 5-day and 20-day horizons. This allows us o compare he 12 Therefore he DN approach can be used o scale hedge raios which are composed of boh variance and covariance componens. 13 See Chrisoffersen, Diebold, & Schuermann (1998) for a deailed discussion of he condiions under which emporal aggregaion formulae may by used. 12

14 volailiy parameers esimaed direcly from he daa wih hose obained by scaling. This also allows us o examine he persisence of volailiy across differen ime horizons and in so doing, o draw inferences abou he relaive forecasing abiliy of GARCH models across differen ime horizons. Our ex-ane expecaion is ha volailiy persisence will decrease wih he hedging horizon, and ha his may affec he ex-pos forecasing performance of hedges a lower frequencies. The deerminans of he opimal hedge are he covariance beween he cash and fuures reurn and he variance of he fuures reurn. Similar o he variance, he covariance can also be scaled by ime under he i.i.d and normaliy assumpions. However, whaever he scaling mehod applied, he composiion of he OHR effecively means ha boh he numeraor and he denominaor are scaled by he same facor. This means ha when we scale he OHR, we are effecively applying a hedge raio calculaed a one frequency o hedges applied a a differen frequency. We now urn o mehods employed o measure he effeciveness of our hedges. IV Hedging Effeciveness We compare hedging effeciveness using he percenage reducion in he variance of he cash (unhedged) posiion as compared o he variance of he hedged porfolio. This is given as: VARIANCE folio % Variance Reducion = 1 HedgedPor (17) VARIANCEUnhedgedPorfolio This measure of effeciveness has been used in he lieraure on hedging over differen ime horizons, however hedgers may seek o minimise some measure of risk oher han 13

15 he variance. For his reason, we use wo addiional hedging effeciveness merics ha will allow us o compare he hedging performance of scaled hedges wih hedges calculaed wih daa mached o he ime horizon. While hese merics have been applied before o he hedging problem (see Coer and Hanly, 2006), hey have no been used o examine he relaionship beween ime horizon and hedging effeciveness. The second hedging effeciveness meric is VaR. For a porfolio his is he loss level over a cerain period ha will no be exceeded wih a specified probabiliy. The VaR a he confidence levelα is VaR α = q α (18) where qα is he relevan quanile of he loss disribuion. The performance meric employed is he percenage reducion in he VaR of he hedged as compared wih he unhedged posiion. VaR1% HedgedPorfolio HE2 = 1 (19) VaR1% UnhedgedPorfolio The hird hedging effeciveness meric is he CVaR 14 which is he average of he wors ( 1 α ) 100% of losses. In effec his means aking he average of quaniles in which ail quaniles are equally weighed and non-ail quaniles have zero weigh. 15 The performance meric we use o evaluae hedging effeciveness is he percenage reducion in CVaR as compared wih a no hedge posiion. 14 This is also called he Expeced Shorfall. 15 For more deail on he properies of he CVaR see Coer and Dowd (2006) 14

16 CVaR1% HedgedPorfolio HE4 = 1 (20) CVaR1% UnhedgedPorfolio Boh he VaR and he CVaR are measures of economic or moneary risk given ha hey provide explici measures of he poenial money loss on a porfolio as well as a probabiliy. None of he sudies ha have examined hedging over differen ime horizons have used hedging effeciveness merics oher han he variance; herefore in applying boh VaR and CVaR, we augmen earlier sudies and add some new findings o he lieraure on hedging and ime-horizon. V Daa Descripion We examine hedging over differen ime horizons using equiy index (FTSE100), commodiy (Crude Oil) 16 and foreign exchange (USD vs GBP) daa. Cash and fuures closing prices were obained from Daasream. Three differen frequencies were examined; 1-day (daily), 5-day (weekly) and 20-day (monhly). In each case, he reurns were calculaed as he differenced logarihmic prices over he respecive frequencies. We obain hedge raios and hedged porfolios for each frequency based on direc esimaion. We also obain hedge raios and hedged porfolios for boh 5-day and 20- day by using he SQRT rule o scale he variance and covariances as described in secion III. The hedged porfolios obained in his way are labelled as scaled o disinguish hem from he hedging porfolios calculaed from daa over he relevan inerval. 16 The Crude Oil conrac used is he Wes Texas Inermediae Ligh Swee Crude which rades on he Nymex in New York. This is he dominan conrac for Oil rading. 15

17 Because we are examining low frequency daa, we required a daase ha would be large enough o allow us o be able o fi he ime varying GARCH model 17 and o carry ou esimaion wih a reasonable degree of saisical accuracy. The full sample runs from March 29, 1993 hrough March 6, The esimaion sample runs from March 29, 1993 hrough March 17, 2003 or abou 10 years of daa. This provides us wih day, day and day observaions. The remaining observaions were used as a hold ou sample. [INSERT TABLE I HERE] Descripive saisics for he daa are displayed in Table I. The following properies of he daa are worh noing. As we move from he 1-day frequency o lower frequencies, he mean and volailiy, as measured by he sandard deviaion increase. One of he more imporan resuls from a hedging perspecive is ha he correlaion beween cash and fuures increases as we move from high o low frequency 18. Also, correlaions a he 1- day frequency are very differen for he differen asses. For example, he correlaion is for he FTSE100 bu jus for Oil and for USD/GBP. When we look a he 20-day frequency, however, here is lile difference beween he correlaions for he differen asses which are 0.992, and for FTSE100, Oil and USD/GBP respecively. This means hedging effeciveness should increase as we hedge longer ime horizons, bu also ha for shorer ime horizons here may be significan 17 Hwang and Valls Pereira (2006) in an examinaion of he small sample properies of GARCH models repor ha very small numbers of observaions can cause unreliable parameer esimaes. Our daase is large enough o generae reasonable esimaes from he GARCH model. 18 This finding was discussed a lengh in Gepper (1995) who poined ou ha given a coinegraed relaionship beween spo and fuures prices which is made up of boh permanen and ransiory componens, over longer horizons he permanen componen ies he fuures and spo ogeher while he effec of he ransiory componen becomes negligible. 16

18 differences beween he differen asses in erms of hedging effeciveness. For his reason, he hedging model choice may be more imporan a he 1-day frequency. The disribuion of he daa is significanly non-normal a he 1-day inerval whereas a lower frequencies he daa are more symmeric. For boh Oil and USD/GBP, he daa can be characerised as Gaussian a he 20-day frequency, as we fail o rejec he hypohesis of normaliy a convenional significance levels. This implies ha scaling using he SQRT rule may be more appropriae using lower frequency daa 19. We find significan ARCH effecs a he 1% level a he 1-day frequency for each of he asses wih he excepion of USD/GBP which is significan a he 5% level, however, hese diminish a lower frequencies. We can also observe a difference beween he characerisics of he FTSE100 as compared wih boh he Oil and USD/GBP daa. For example he FTSE100 exhibis significan ARCH effecs across each hedging horizon (p-values of 0.01 or lower) whereas he Oil and USD/GBP daa which have significan ARCH effecs a he 1-day frequency only, have insignifican ARCH effecs a he 5-day and 20-day frequencies. This finding agrees wih he well known resul from he lieraure ha volailiy persisence decreases as we move from high o low frequency daa (see, for example, Poon and Granger, 2003). The implicaion is ha GARCH models will generae beer forecass a shorer ime horizons. However, in his paper we are esimaing he ex-pos hedge raio using 1-sep-ahead forecass so i remains o be seen wheher he lower volailiy persisence a lower frequency will affec he efficiency of he hedges 20. Saionariy is esed using boh he Phillips Peron and 19 This finding means ha scaling may be more applicable from say 20-Day o 1-Year han from 1-Day o 20-Day however he problem of having enough observaions a even he 20-Day inerval remains. 20 Chrisofferson and Diebold (2000) find ha volailiy forecass are unreliable beyond 10 days. However hey used daily daa which means in effec ha volailiy is forecasable up o 10-seps. In his sudy alhough we are generaing a forecass of up o 20-days ahead for he 20-Day horizon, i is based on a 1-sep forecas. 17

19 Kwiakowski, Phillips, Schmid and Shin (KPSS) ess 21. The resuls of boh ess indicae ha he log reurns series is saionary irrespecive of he sampling frequency. This indicaes ha he OLS esimaes should be reliable across each of he ime horizons. VI Empirical Findings [INSERT TABLE II HERE] Table II repors he GARCH parameers using direc esimaion a he 1-day, 5-day and 20-day frequencies ogeher wih he scaled parameers esimaed from he 1-day daa using he Dros Nijman approach. Examining firs he resuls of direc esimaion, as we move from high frequency o low frequency he level of persisence in volailiy is significanly reduced. This is mos pronounced for he Oil and USD/GBP daa. For example, as we move from 1-day o 5-day, volailiy persisence 22 for he FTSE drops jus over 1% from o for cash 23. In he case of Oil, persisence decreases by abou 16% from o while for USD/GBP he drop is even more pronounced going from o When we examine he resuls for he 20-day frequency, persisence is sill relaively high for he FTSE a 0.875, whereas for boh Oil and USD/GBP i is in he region of jus These findings are supporive of previous work such as Chrisofferson Diebold and Schuermann (1998) ha have found ha volailiy persisence decreases as he ime horizon increases, or alernaively as we move from high frequency o low frequency daa. The implicaions of his relae o he forecasing abiliy of he GARCH models a lower frequency, where we may see a performance 21 For breviy we have only included he resuls from he KPSS es in Table I. 22 As measured by he sum of he A or ARCH coefficien and he B or GARCH coefficien. 23 For breviy we base he parameer comparisons on he cash asse unless oherwise saed. 18

20 differenial in he ex-pos hedging effeciveness of he hedges for he differen asses a differen frequencies. We also repor he scaled parameers obained using he DN approach based on inpus from he GARCH model a he 1-day frequency. Comparing firs he scaled parameers for he FTSE, hey are broadly in line wih he parameers obained from direc esimaion a he 5-day frequency bu when we move o he 20-day frequency larger differences emerge. For example, he scaled parameers a he 5-day frequency are lower by abou 2% 3% as compared wih he 5-day acual. When we move o he 20- day frequency, however, here are significan differences. For example, persisence as measured using he scaled coefficiens a is significanly lower han he acual esimae of would sugges. For boh Oil and USD/GBP, he scaled parameers are significanly differen from he acual parameers a boh he 5-day and 20-day frequencies. For Oil, acual volailiy persisence a he 5-day frequency is and a he 20-day frequency. This compares wih scaled persisence of and jus for he 5-day and 20-day frequencies respecively. Similarly, he acual coefficiens for he USD/GBP daa are very differen from hose obained using he DN approach. These findings indicae ha he DN approach may provide a reasonable approximaion for he volailiy daa generaing process for equiy reurns using a scaling facor in he region of 5, from 1-day o 5-day. This finding agrees wih Kaufmann and Paie (2003) who found ha he DN approach provided reasonable esimaes of coefficiens when scaling from 1-day frequency up o 1-week. However for oher asses such as Oil or foreign exchange raes, he coefficien esimaes obained by scaling do no approximae hose obained by direc esimaion. [INSERT TABLE III HERE] 19

21 [INSERT FIGURES 1A, 1B AND 1C HERE] The esimaed opimal hedges are presened in Figures 1a, 1b and 1c ogeher wih heir associaed saisics in Table III. All hedges are less han 1 wih he excepion of he USD/GBP a he 20-day frequency 24. For he FTSE and USD/GBP, he OHR s increase in line wih he ime horizon and end o approach he Naïve hedge raio which is 1. For example, he mean OHR for he FTSE goes from o o as we move from he 1-day frequency o he 5-day and 20-day frequencies respecively. This means ha he invesor would sell a larger number of fuures conracs o achieve he opimal hedge as he frequency of he hedge increases. This would have implicaions on he cos of he hedging sraegy. For example, i would mean ha hedging a single 20-day period would be more expensive han a single 1-day period. For Oil, he OHR s increase as we move from he 1-day o he 5-day horizon from o However, while here is hen a sligh decrease wih he 20-day OHR a he resuls are broadly similar o hose for he FTSE and USD/GBP. Also he mean OHR s for he differen ime horizons are significanly differen based on sandard -ess a he 1% level. These findings reflec he fac ha he correlaion beween cash and fuures increases as we move from high frequency o low frequency daa. The OLS hedges also end o increase wih he hedging horizon wih he excepion of Oil for he 20-day frequency. These resuls are consisen wih he lieraure (see, for example, Chen Lee and Shresha, 2004) in ha for longer invesmen horizons, he opimal hedge raio converges owards he Naïve hedge raio. In erms of dispersion of he OHR s, here are also large differences in he sandard deviaions of he OHR s a 24 A hedge raio in excess of 1 implies ha an invesor hold a naked shor posiion in fuures which in some cases would be necessary in order o achieve he minimum variance porfolio. 20

22 he differen ime horizons. This is mos pronounced for he Oil hedges. For example, he sandard deviaion of he 1-day ime varying OHR is bu his drops o for he 5-day OHR s and o jus for he 20-day hedge. This can be confirmed wih a quick glance a figure 1b. This finding highlighs he fac ha a we move from high o low frequency hedges, he gap beween he ime-varying and he consan hedges narrows. We now urn o he hedging effeciveness of he hedge sraegies and compare he effeciveness of hedges calculaed direcly using daa a he relevan hedging horizon wih he hedges based on scaling. Table IV presens he hedging effeciveness merics for boh OLS and GARCH models across hedging horizons. Hedging models are compared in erms of he percenage reducion in a given risk measure as compared wih a no-hedge posiion. The effec of hedging horizon on hedging effeciveness is apparen in ha in-sample hedging effeciveness increases wih he hedging horizon. For example, using he variance as he performance crierion, boh he OLS and GARCH hedges for USD/GBP yield around a 67% reducion in risk a he 1-day horizon. This increases o around 90% a he 5-day horizon and around 98% a he 20-day horizon. Boh FTSE and Oil exhibi similar resuls 25. The differences beween he hedging performances a he differen ime horizons are also saisically significan a he 1% level i.e. he 5-day performance is saisically significanly beer han he 1-day performance, similarly he 20-day is significanly 25 Resuls are rounded o wo decimal places, however he similariy beween he OLS and GARCH hedging performance in no surprising, given ha he resuls are based on averages of he hedging effeciveness of individual hedges. This finding furher suppors Coer and Hanly (2006) who find lile difference in hedging performance beween OLS and GARCH hedging sraegies. 21

23 beer han he 5-day. Comparisons are based on -ess of he difference in he average performance of he differen hedges. These were esimaed using sandard errors based on Efrons (1979) boosrap mehodology. The findings are robus in ha hey apply o all asses. Furhermore, boh he VaR and CVaR merics confirm he findings based on he Variance performance crierion. This suppors he broad findings in he lieraure in relaion o in-sample hedging performance a differen ime horizons. In addiion, our findings make a furher conribuion in ha he lieraure has hihero based is resuls on saisical performance crierion (he variance) alone, whereas by using economic crieria such as VaR and CVaR, we find furher evidence of he posiive relaionship beween hedging effeciveness and ime horizon. We now compare he hedging effeciveness of he scaled hedges wih hose based on direc esimaion. Table V presens -Saisics for differences of mean hedging effeciveness for in-sample hedges obained from 5-day acual and 5-day scaled esimaion periods and similarly for he 20-day esimaion period. The null hypohesis ha he hedging effeciveness is he same for acual and scaled hedges can be rejeced across all asses and boh hedging horizons in 89% of cases indicaing ha here is a saisical difference in he hedging effeciveness beween acual and scaled hedges. If we are o look a jus he Variance and VaR, here are differences beween he acual and scaled hedges in all cases. In erms of which hedging sraegy performs beer, differences emerge beween asses and depending on hedging horizon. [INSERT TABLE V HERE] For he 5-day frequency, if we use jus he variance as he measure of hedging effeciveness, he acual hedges perform beer for all asses. If we also ake ino accoun he VaR and CVaR, he scaled hedges end o perform relaively beer for he 22

24 USD/GBP conrac. Looking a he 20-day frequency, he variance measure indicaes ha he acual hedges perform bes, however he VaR and CVaR of he scaled hedges are lower for he Oil conrac. These resuls indicae ha acual hedges end o perform significanly beer han scaled hedges. However, despie he saisical difference, i would appear ha he hedging performance may no be significanly differen from an economic perspecive. To demonsrae his, consider ha he CVaR for he 5-Day acual hedge on he FTSE esimaed using he GARCH model for a $1,000,000 exposure would be $15,100. This compares wih an exposure of jus $16,300 if he scaled hedge were used. The difference is jus $1,200 whereas he relevan -sa is Similar differences apply o he Oil conrac however hey are more pronounced for he USD/GBP conrac. A similar comparison for he USD/GBP conrac yields a difference in he CVaR of jus $500 in favour of he scaled hedge (-sa 18.53). A he 20-day frequency however here are larger economic differences in he effeciveness of he acual and scaled hedges. For example, again using he CVaR yields a figure of $10,100 for he acual hedge as compared wih $16,500 for he scaled hedge, which is 63% larger. This reflecs boh he difference in hedging sraegy bu also he fac ha he exposure is over a longer period. The differences in hedging effeciveness for he FTSE are no so large while for Oil, he scaled hedges end o perform beer when VaR and CVaR are used alhough no by an economically significan amoun. The implicaion of hese findings is ha using a 1-day hedge scaled up o a 5-day hedging horizon may be a relaively good soluion, especially for he asses examined here. When we increase he hedging horizon o 20-days, however, differences emerge depending on he asse being examined. Scaled hedges would 23

25 appear o be reasonable in performance erms for boh FTSE and Oil bu do no provide effecive hedging soluions for he USD/GBP asse. These comparisons have been based on in-sample resuls however a more sringen es would be o examine scaled hedges in an ou-of-sample seing. To obain ou-ofsample hedges, we use one-sep ahead forecass of he variance and covariances obained from he GARCH model. Tables VI and VII presen he ou-of-sample hedging effeciveness. [INSERT TABLES VI AND VII] The ou-of-sample resuls broadly suppor he in-sample findings. There is a significan increase in hedging effeciveness as we move from he 1-day hedges o he 5-day and 20-day hedges. There are also large differences in erms of he differen asses a he differen frequencies. The FTSE has he bes hedging performance a he 1-day hedge horizon however he performance of boh he Oil and USD/GBP conracs increases dramaically as we move he hedge horizon up o 5-day and 20-day. This finding addresses a gap he lieraure by providing more evidence on ou-of-sample effeciveness by using effeciveness crieria such as VaR and CVaR in addiion o he variance reducion measure. The implicaion of hese resuls is ha hedging over longer ime horizons would be a preferable sraegy compared wih hedging shorer ime spans and rolling he hedges over. The differences in hedging effeciveness beween asses a higher frequencies end o converge a lower frequencies indicaing ha hedging effeciveness is broadly similar for differen asses a longer hedging horizons. Looking now a comparisons beween he acual and scaled hedges for he ou-ofsample hedges, we find ha here are significan saisical differences in 83% of cases across asses and frequencies. A he 5-Day hedging horizon, he acual hedges 24

26 ouperform he scaled hedges for FTSE and USD/GBP, whereas for Oil he scaled hedges generally yield lower risk. However he differences again are only economically significan for he USD/GBP conrac. When we examine he 20-Day hedges, we find ha he acual hedges are he bes performers for all asses bu again he differences are only economically significan for he USD/GBP conrac. For example, using a $1,000,000 exposure o illusrae, he difference beween he CVaR of he USD/GBP hedge calculaed direcly is $6,800 lower han he CVaR of he scaled hedge. This represens a difference of 81% in favour of he acual hedge. In erms of hedging models, here are generally no significan differences beween he OLS and he GARCH models, irrespecive of wheher saisical or economical evaluaion is used. The bes model may change for a given asse or hedge horizon. We also expeced ex-ane ha he ou-of-sample performance of he GARCH model would be relaively beer as compared wih he OLS model for shorer horizons because volailiy persisence is greaer for high frequency daa. However here is no conclusive evidence ha his is he case as he performance differenial is no significanly differen for differen hedge horizons. Looking a he acual risk measures, boh he OLS and GARCH models provide broadly similar performance. For he FTSE he GARCH model appears o have he edge. I ouperforms he OLS model for seven ou of nine cases across he difference frequencies and he differen risk measures. The excepions are he VaR a he 5-day frequency and he Variance a he 20-day frequency. For he Oil and he USUK asses, he performance is more even, and depends on he paricular risk measure and he frequency. Taking Oil, for example, he OLS yields a marginally lower CVaR of 3.19 a he 20-day frequency as compared wih 3.20 for he GARCH model. The similariy in performance of he models is no surprising given ha i relaes 25

27 o he average performance of he hedges across ime, and herefore performance differences end o be quie small. Noe also ha he variance of he hedged porfolios ends o increase as we increase he hedging horizon. The only excepion o his is for Oil a he 5-day and 20-day frequencies and his relaes o very similar performance for hese paricular hedges. VII Summary and Conclusion This paper compares hedge sraegies across hree differen invesor ime horizons. We calculae hedges using volailiy and covariance esimaes based on direc esimaion and compare hese wih hedges obained using scaled daa. Significan differences emerge beween he hedge sraegies, indicaing ha scaled hedges end o be lower in absolue value and less volaile han hose obained from direc esimaion. These differences can be raced back o he correlaion properies of cash and fuures which increase as he hedging horizon lenghens. Despie hese differences, scaled hedges provide good oucomes in erms of absolue hedging effeciveness across all asses. In erms of he relaive performance of scaled versus acual hedges, for Equiy Index and Oil hedges, paricularly when scaling 1-day hedges up o 1-week, he relaive performance is broadly similar. For foreign exchange hedges especially a he 20-day frequency, he resuls of scaling are poor when compared wih he acual hedges. We conclude herefore ha only for he foreign exchange (USD/GBP) hedges can economic and saisical differences be called significan, herefore he broad finding is ha scaling provides reasonably good oucomes in reducing risk. 26

28 We also examine he emporal aggregaion properies of a GARCH model using he Dros Nyman approach which allows us o compare volailiy persisence for high frequency and low frequency daa. The resuls show ha lower levels of volailiy persisence do no maerially affec he ex-pos hedging effeciveness a lower frequencies. We also provide furher evidence ha ex-pos hedging effeciveness increases as we move from high o low frequency hedging. The implicaions of our findings are wofold. Firsly, scaling provides good absolue, and reasonable relaive hedging oucomes vis a vis direc esimaion while avoiding some of he saisical problems associaed wih direc esimaion a low frequencies. Furhermore, i is paricularly useful for asses such as equiy index hedges and for relaively shor ime scales such as 1-day o 5-day. For ime scales such as 20-day or longer, he findings seem o sugges ha consan or average OHR s approach he Naïve hedge raio of 1, and herefore i may be ha his approach may prove suiable for hedges based on ime horizons longer han 1-monh. 27

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33 Malliaris, A., & Urruia, J. (1991). The impac of he lenghs of esimaion periods and hedging horizons on he effeciveness of a hedge: evidence from foreign currency fuures. The Journal of Fuures Markes, 11, Mandelbro, B. (1963). The variaion of cerain speculaive prices. Journal of Business, 36, Meron, R., (1980). On esimaing he expeced reurn on he marke: an exploraory invesigaion. Journal of Financial Economics, 8, Poon, S., & Granger, W. (2003). Forecasing volailiy in financial markes: A review. Journal of Economic Lieraure, 61, Smihson, C., & Minon, L. (1996). Value-a-Risk. Risk, 9,

34 Table I: Descripive Saisics This able presens summary saisics for he log reurns of each cash and fuures series. The Mean and Sandard Deviaion (SD) are expressed as percenages. A comparison of he scaled and he acual daa shows ha as we scale he 1-day sandard deviaions o he 5-day frequency using he SQRT rule, he average deviaion of he scaled sandard deviaion as compared wih he acual sandard deviaion across all asses would be jus 1.14% however a he 20-day frequency he average difference rises o 7.7%. This indicaes ha scaling provides a good approximaion up o a facor of five bu ha i declines rapidly hereafer. The excess skewness saisic measures asymmery where zero would indicae a symmeric disribuion. The excess kurosis saisic measures he shape of a disribuion where a value of zero would indicae a normal disribuion. The Jarque and Bera (J-B) saisic measures normaliy. LM wih 4 lags is he Lagrange Muliplier es for ARCH effecs proposed by Engle 1982 and is disribued as χ 2. Saionariy is esed using he KPSS es which ess he null of saionariy agains he alernaive of a uni roo. Criical Values for he KPSS es a he 1% level are and for he consan and rend saisics respecively. The correlaion coefficien beween each se of cash and fuures is also given. Associaed p-values are given in brackes. 1-Day 5-Day 20-Day Cash Fuures Cash Fuures Cash Fuures FTSE 100 Mean SD SD Scaled Skewness (0.06) (0.06) Kurosis J-B LM (0.01) (0.01) KPSS - Consan - Trend Correlaion OIL Mean SD SD Scaled Skewness (0.83) (0.48) Kurosis (0.42) (0.32) J-B (0.69) 1.51 (0.46) LM (0.35) 3.79 (0.43) 5.74 (0.21) 6.83 (0.14) KPSS - Consan - Trend Correlaion USD/GBP Mean SD SD Scaled Skewness (0.71) (0.74) (0.33) (0.09) (0.90) (0.92) Kurosis (0.15) 0.25 (0.23) 0.65 (0.13) 0.88 (0.04) J-B LM KPSS - Consan 9.18 (0.05) (0.22) (0.10) (0.12) 9.67 (0.10) (0.31) 3.14 (0.53) Trend Correlaion (0.12) 3.44 (0.48)

35 Table II: GARCH (1, 1) Esimaes This able repors he maximum likelihood esimaes for FTSE, Oil and USD/GBP reurns for 1-Day and 5-Day frequencies for he period 29/03/1993 o 17/03/2003. Also presened are he GARCH coefficiens ha are based on he 1-Day parameers scaled up using he Dros Nijman formula. Volailiy persisence is measured by he sum of he GARCH parameers α and β. The numbers in parenheses are robus sandard errors. m n = + 2 ω α + jε s j j= 1 k = 1 H s β H m n = + 2 ω α + jε f j j = 1 k = 1 H f β H m j= 1 n j s ε + j f j k = 1 k k 2 s k 2 f k H sf = ω + α ε β H 1-DAY 5-DAY 20-DAY 5-DAY 20-DAY ACTUAL ACTUAL SCALED SCALED FTSE ω s ω sf ω f α s (0.004) (0.013) (0.081) α sf (0.004) (0.013) (0.080) α f (0.004) (0.013) (0.078) β s (0.004) (0.015) (0.090) β sf (0.004) (0.015) (0.091) β f (0.005) (0.015) (0.093) OIL ω s ω sf ω f α s (0.011) (0.042) α sf (0.010) (0.038) α f (0.011) (0.036) β s (0.010) (0.028) β sf (0.006) (0.024) β f (0.006) (0.026) k, sf k

36 Table II coninued 1-DAY 5-DAY 20-DAY 5-DAY 20-DAY ACTUAL ACTUAL SCALED SCALED USD/GBP ω s ω sf ω f α s (0.007) (0.030) α sf (0.007) (0.032) α f (0.009) (0.036) β s (0.019) (0.096) β sf (0.021) β f (0.022) (0.090) (0.086) VOLATILITY PERSISTENCE FTSE α s + β s α f + β f OIL α s + β s α f + β f USD/GBP α s + β s α f + β f

37 Table III: Opimal Hedge Raios Descripive Saisics This able presens descripive saisics for he ime-varying GARCH Opimal Hedge Raios a 1-Day, 5-Day and 20-Day frequencies ogeher wih saisics for he scaled hedge raios a 5- Day and 20-Day frequencies. Saionariy is esed using he Phillips Peron uni roo es wih associaed p-values in brackes. For all asses he OHR s are saionary. OLS Hedge Raios are also presened. ACTUAL SCALED 1-Day 5-Day 20-Day 5-Day 20-Day GARCH Hedges FTSE Mean SD Minimum Maximum Saionariy (0.01) OIL Mean SD Minimum Maximum Saionariy USD/GBP Mean SD Minimum Maximum Saionariy OLS Hedges FTSE OIL USD/GBP

38 Table IV: Hedging Performance of Acual and Scaled Hedge Sraegies This able presens in-sample hedging performance measures for hedge sraegies a he 1-Day, 5-Day and 20-Day ime horizons calculaed from Acual daa ogeher wih performance measures for hedges based on Scaled daa for boh he 5-Day and 20-Day horizons. The performance measures are he Variance, VaR a he 1% level and CVaR a he 1% level. Hedging effeciveness is measured as he percenage reducion in he relevan performance measure as compared wih a no-hedge sraegy and is repored as he figure in brackes. For example, examining he resuls for he FTSE using he Acual daa a he 5-day horizon, he GARCH model yields a 97% (0.97) reducion in he variance of a hedging porfolio as compared wih a nohedge sraegy IN-SAMPLE ACTUAL SCALED 1-DAY 5-DAY 20-DAY 5-DAY 20-DAY OLS GARCH OLS GARCH OLS GARCH OLS GARCH OLS GARCH FTSE VARIANCE (x10-4 ) (0.94) (0.95) (0.97) (0.97) (0.98) (0.98) (0.97) (0.97) (0.98) (0.98) VaR (x10-2 ) (0.73) (0.76) (0.85) (0.85) (0.88) (0.88) (0.84) (0.83) (0.87) (0.85) CVaR (x10-2 ) (0.72) (0.74) (0.83) (0.84) (0.87) (0.87) (0.84) (0.82) (0.87) (0.85) OIL VARIANCE (x10-4 ) (0.77) (0.76) (0.94) (0.94) (0.99) (0.99) (0.94) (0.93) (0.98) (0.97) VaR (x10-2 ) (0.41) (0.40) (0.67) (0.68) (0.88) (0.87) (0.65) (0.64) (0.90) (0.90) CVaR (x10-2 ) (0.31) (0.30) (0.71) (0.72) (0.84) (0.83) (0.70) (0.71) (0.85) (0.90) USD/GBP VARIANCE (x10-4 ) (0.67) (0.67) (0.90) (0.90) (0.98) (0.98) (0.87) (0.88) (0.91) (0.92) VaR (x10-2 ) (0.38) (0.36) (0.53) (0.57) (0.84) (0.84) (0.59) (0.58) (0.67) (0.72) CVaR(x10-2 ) (0.31) (0.31) (0.56) (0.57) (0.82) (0.82) (0.59) (0.59) (0.71) (0.71) 37

39 Table V: Comparison of Acual vs Scaled Hedging Performance This able presens -Saisics for difference of mean hedging effeciveness for in-sample hedges obained from 5-Day acual and 5-Day scaled esimaion periods and similarly for he 20-Day esimaion period. * denoes no significan a 5% level. H O : HE scaled = HE, H : HE acual A scaled HE acual IN-SAMPLE 5-DAY 20-DAY OLS GARCH OLS GARCH FTSE Variance VAR CVAR 1.57* * 9.07 OIL Variance VAR CVAR 0.30* 0.35* USD/GBP Variance VAR CVAR

40 Table VI: Hedging Performance of Acual and Scaled Hedge Sraegies This able presens ou-of-sample hedging performance measures for hedge sraegies a he 1-Day, 5-Day and 20-Day ime horizons calculaed from Acual daa ogeher wih performance measures for hedges based on Scaled daa for boh he 5-Day and 20-Day horizons. The performance measures are he Variance, VaR a he 1% level and CVaR a he 1% level. Hedging effeciveness is measured as he percenage reducion in he relevan performance measure as compared wih a no-hedge sraegy and is repored as he figure in brackes. For example, examining he resuls for he FTSE, using he Scaled daa a he 20-day horizon, he GARCH model yields a 98% (0.98) reducion in he variance of a hedging porfolio as compared wih a nohedge sraegy OUT-OF-SAMPLE ACTUAL SCALED 1-DAY 5-DAY 20-DAY 5-DAY 20-DAY OLS GARCH OLS GARCH OLS GARCH OLS GARCH OLS GARCH FTSE VARIANCE (x10-4 ) (0.96) (0.97) (0.99) (0.99) (0.98) (0.98) (0.98) (0.99) (0.98) (0.98) VaR (x10-2 ) (0.81) (0.81) (0.89) (0.89) (0.86) (0.87) (0.87) (0.89) (0.88) (0.86) CVaR (x10-2 ) (0.79) (0.80) (0.89) (0.90) (0.88) (0.88) (0.86) (0.90) (0.90) (0.88) OIL VARIANCE (x10-4 ) (0.83) (0.83) (0.94) (0.94) (0.98) (0.98) (0.94) (0.94) (0.97) (0.98) VaR (x10-2 ) (0.44) (0.42) (0.62) (0.63) (0.79) (0.80) (0.65) (0.63) (0.77) (0.78) CVaR (x10-2 ) (0.27) (0.26) (0.69) (0.69) (0.80) (0.80) (0.70) (0.70) (0.77) (0.79) USD/GBP VARIANCE (x10-4 ) (0.78) (0.79) (0.91) (0.92) (0.98) (0.98) (0.87) (0.90) (0.90) (0.93) VaR (x10-2 ) (0.56) (0.57) (0.65) (0.65) (0.85) (0.84) (0.55) (0.58) (0.71) (0.75) CVaR(x10-2 ) (0.42) (0.44) (0.67) (0.67) (0.83) (0.83) (0.59) (0.62) (0.69) (0.73) 39

41 Table VII: Comparison of Acual vs Scaled Hedging Performance This able presens T-Saisics for difference of mean hedging effeciveness for ou-of-sample hedges obained from 5-Day acual and 5-Day scaled esimaion periods and similarly for he 20-Day esimaion period. * denoes no significan a 5% level. H : HE = HE, H : HE HE O scaled acual OUT-OF-SAMPLE 5-DAY 20-DAY OLS GARCH OLS GARCH FTSE Variance * VAR * * CVAR * OIL Variance * VAR * CVAR USD/GBP Variance VAR CVAR A scaled acual 40

42 Figure 1a: Opimal Hedge Raios

43 Figure 1b: Opimal Hedge Raios 42

44 Figure 1c: Opimal Hedge Raios 43