Crude Oil Hedging Sraegies Using Dynamic Mulivariae GARCH Roengchai Tansucha * Faculy of Economics Maejo Universiy Chiang Mai, Thailand Chia-Lin Chang Deparmen of Applied Economics Naional Chung Hsing Universiy Taichung, Taiwan Michael McAleer Economerics Insiue Erasmus School of Economics Erasmus Universiy Roerdam and Tinbergen Insiue The Neherlands January 2010 * For financial suppor, he firs auhor is mos graeful o he Faculy of Economics, Maejo Universiy, Thailand, he second auhor would like o acknowledge he Naional Science Council, Taiwan, and he hird auhor wishes o hank he Ausralian Research Council, Naional Science Council, Taiwan, and he Cener for Inernaional Research on he Japanese Economy (CIRJE), Faculy of Economics, Universiy of Tokyo. 1
Absrac The paper examines he performance of four mulivariae volailiy models, namely CCC, VARMA-GARCH, DCC and BEKK, for he crude oil spo and fuures reurns of wo major benchmark inernaional crude oil markes, Bren and WTI, o calculae opimal porfolio weighs and opimal hedge raios, and o sugges a crude oil hedge sraegy. The empirical resuls show ha he opimal porfolio weighs of all mulivariae volailiy models for Bren sugges holding fuures in larger proporions han spo. For WTI, however, DCC and BEKK sugges holding crude oil fuures o spo, bu CCC and VARMA-GARCH sugges holding crude oil spo o fuures. In addiion, he calculaed opimal hedge raios (OHRs) from each mulivariae condiional volailiy model give he ime-varying hedge raios, and recommend o shor in crude oil fuures wih a high proporion of one dollar long in crude oil spo. Finally, he hedging effeciveness indicaes ha DCC (BEKK) is he bes (wors) model for OHR calculaion in erms of reducing he variance of he porfolio. Keywords: Mulivariae GARCH, condiional correlaions, crude oil prices, opimal hedge raio, opimal porfolio weighs, hedging sraegies. JEL Classificaions: C22, C32, G11, G17, G32. 2
1. Inroducion As he srucure of world indusries changed in he 1970s, he expansion of he oil marke has coninually grown o have now become he world s bigges commodiy marke. This marke has developed from a primarily physical produc aciviy ino a sophisicaed financial marke. Over he las decade, crude oil markes have maured grealy, and heir range and deph could allow a wide range of paricipans, such as crude oil producers, crude oil physical raders, and refining and oil companies, o hedge oil price risk. Risk in he crude oil commodiy marke is likely o occur due o unexpeced jumps in global oil demand, a decrease in he capaciy of crude oil producion and refinery capaciy, peroleum reserve policy, OPEC spare capaciy and policy, major regional and global economic crises, and geopoliical risks. A fuures conrac is an agreemen beween wo paries o buy and sell a given amoun of a commodiy a an agreed upon cerain dae in he fuure, a an agreed upon price, and a a given locaion. Furhermore, a fuures conrac is he insrumen primarily designed o minimize one s exposure o unwaned risk. Fuures raders are radiionally placed in one of wo groups, namely hedgers and speculaors. Hedgers ypically include producers and consumers of a commodiy, or he owners of an asse, who have an ineres in he underlying asse, and are aemping o offse exposure o price flucuaions in some opposie posiion in anoher marke. Unlike hedgers, speculaors do no inend o minimize risk bu raher o make a profi from he inherenly risky naure of he commodiy marke by predicing marke movemens. Hedger wan o minimize risk, regardless of wha hey are invesing in, while speculaors wan o increase heir risk and hereby maximize profis. Concepually, hedging hrough rading fuures conracs is a procedure used o resrain or reduce he risk of unfavourable price changes because cash and fuures prices for he same commodiy end o move ogeher. Therefore, changes in he value of a cash posiion are offse by changes in he value of an opposie fuures posiion. In addiion, fuures conracs are favoured as a hedging ool because of heir liquidiy, speed and lower ransacion coss. 3
Among he indusries and firms ha are more likely o use a hedging sraegy is he oil and gas indusry. Firms will hedge only if hey expec ha an unfavourable even will arise. Knill e al. (2006) suggesed ha if an oil and gas company uses fuures conracs o hedge risk, hey hedge only he downside risk. When an indusry perspecive is good (bad), hey will scale down (up) on heir fuures usage, hereby pushing fuures prices higher (lower). Hedging by he crude oil producers normally involves selling he commodiy fuures because producers or refiners use fuures conracs o lock he fuures selling prices or a price floor. Thus, hey end o ake shor posiions in fuures. A he same ime, energy raders, invesors or fuel oil users focusing o lock in a fuures purchase price or price ceiling end o long posiions in fuures. Daniel (2001) shows ha hedging sraegies can subsanially reduce oil price volailiy wihou significanly reducing reurns, and wih he added benefi of greaer predicabiliy and cerainy. Theoreically, issues in hedging involve he deerminaion of he opimal hedge raio (OHR). One of he mos widely-used hedging sraegies is based on he minimizaion of he variance of he porfolio, he so-called minimum variance hedge raio (see Chen e al. (2003) for a review of he fuures hedge raio, and Lien and Tse (2002) for some recen developmens in fuures hedging). Wih he minimum-variance crierion, risk managemen requires deerminaion of he OHR (he opimal amoun of fuures bough or sold expressed as a proporion of he cash posiion). In order o esimae such a raio, early research simply used he slope of he classical linear regression model of cash on he fuures price, which assumed a ime-invarian hedge raio (see, for example, Ederingon (1979), Figlewski (1985), and Myers and Thomson (1989)). However, i is now widely agreed ha financial asse reurns volailiy, covariancec and correlaions are ime-varying wih persisen dynamics, and rely on echniques such as condiional volailiy (CV) and sochasic volailiy (SV) models. Baillie and Myers (1991) claim ha, if he join disribuion of cash prices and fuures prices changes over ime, esimaing a consan hedge raio may no be appropriae. In his paper, ime-varying hedge raios are esimaed and analysed. The widely used ARCH and GARCH models appear o be ideal for esimaing ime-varying OHRs, and a number of applicaions have concluded ha such raios seem o display considerable variabiliy over ime (see, for example, Cecchei e al. (1988), Baillie and Myers (1991), Myers (1991), and Kroner and Sulan (1993)). Typically, he hedging model is 4
consruced for a decision maker who allocaes wealh beween a risk-free asse and wo risky asses, namely he physical commodiy and he corresponding fuures. OHR is defined as p f F f F OHR cov, var, where p and f are fuures price and spo price, 1 1 respecively, and F 1 is he informaion se. Therefore, OHR can be calculaed given he knowledge of he ime-dependen covariance marix for cash and fuures prices, which can be esimaed using mulivariae GARCH models. In he lieraure, research has been conduced on he volailiy of crude spo, forward and fuures reurns. Lanza e al. (2006) applied he consan condiional correlaion (CCC) model of Bollerslev (1990) and he dynamic condiional correlaion (DCC) model of Engle (2002) for Wes Texas Inermediae (WTI) oil forward and fuures reurns. Manera e al. (2006) used CCC, he vecor auoregressive moving average (VARMA-GARCH) model of Ling and McAleer (2003), he VARMA- Asymmeric GARCH model of McAleer e al. (2009), and DCC o spo and forward reurn in he Tapis marke. Recenly, Chang e al. (2009a, 2009b, 2009c) esimaed mulivariae condiional volailiy and examined volailiy spillovers for he reurns on spo, forward and fuures reurns for Bren, WTI, Dubai and Tapis o aid risk diversificaion in crude oil markes. However, hese auhors did no focus on OHR or he design of an opimal hedging sraegy based on a wide range of models. For esimaed ime-varying hedge raios using mulivariae condiional volailiy models, Haigh and Hol (2002) modelled he ime-varying hedge raio among crude oil (WTI), heaing oil and unleaded gasoline fuures conracs in reducing price volailiy for an energy rader wih he BEKK model of Engle and Kroner (1995), and accouned for volailiy spillovers. Alizadeh e al. (2004) examined appropriae fuures conracs, and examine he effeciveness of hedging marine bunker price flucuaions in Roerdam, Singapore and Houson using differen crude oil and peroleum fuures conracs raded on he New York Mercanile Exchange (NYMEX) and he Inernaional Peroleum Exchange (IPE) in London, using he VECM and BEKK models. Jalali-Naini and Kazemi-Manesh (2006) examined hedge raios using weekly spo prices of WTI and fuures prices of crude oil conracs one monh o four monhs on NYMEX. The resuls from he BEKK model showed ha he OHRs are ime varying for all conracs, and higher duraion conracs had higher perceived risk, a higher OHR mean, and sandard deviaions. 5
The purpose of his paper is o esimae mulivariae condiional volailiy models, namely CCC, VARMA-GARCH, DCC and BEKK, for he reurns on spo and fuures prices for Bren and WTI markes, o calculae he opimal porfolio weighs and OHRs raio from he condiional covariance marices for effecive opimal porfolio designs and hedging sraegies, and o invesigae and compare he performance of OHRs from esimaed mulivariae condiional volailiy models by applying he hedging effeciveness index. The srucure of he remainder of he paper is as follows. Secion 2 discusses he mulivariae GARCH models o be esimaed, and he derivaion of he OHR and hedging effecive index. Secion 3 describes he daa, descripive saisics and uni roo es saisics. Secion 4 analyses he empirical esimaes from empirical modelling. Some concluding remarks are given in Secion 5. 2. Economeric Models 2.1 Mulivariae Condiional Volailiy Models This secion presens he CCC model of Bollerslev (1990), VARMA-GARCH model of Ling and McAleer (2003), VARMA-AGARCH model of McAleer e al. (2009), DCC model of Engle (2002), and BEKK model of Engle and Kroner (1995). The firs hree models assume consan condiional correlaions, while he las wo models accommodae dynamic condiional correlaions. Consider he CCC mulivariae GARCH model of Bollerslev (1990): y E y F 1, D (1) var F D D 1 Where, y,..., y1 y m,..., 1 m is a sequence of independenly and idenically disribued (i.i.d.) random vecors, F is he pas informaion available a ime, 6
12 12 1 m D diag h,..., h, m is he number of reurns, and 1,..., n, (see, for example, McAleer 92005) and Bauwens, e al. (2006)). As E F 1 E, where ij for i, j 1,..., m, he consan condiional correlaion marix of he uncondiional shocks,, is equivalen o he consan condiional covariance marix of he condiional shocks,, from, D diag Q 12 (1), D D condiional covariance marix., and 1 E F Q D D, where Q is he The CCC model of Bollerslev (1990) assumes ha he condiional variance for each reurn, h, i 1,.., m, follows a univariae GARCH process, ha is i h r s 2 i i iji, j ijhi, j j1 j1, (2) where represens he ARCH effec, or shor run persisence of shocks o reurn i, ij ij represens he GARCH effec, and r s ij ij denoes he long run persisence. j1 j1 In order o accommodae inerdependencies of volailiy across differen asses and/or markes, Ling and McAleer (2003) proposed a vecor auoregressive moving average (VARMA) specificaion of he condiional mean, and he following specificaion for he condiional variance: 1 Y E Y F (3) LY L (4) D (5) H W A BH r s l l l i, l l1 l1 (6) 7
where W, A l and B l are m m marices, wih ypical elemens ij and ij, respecively.,..., 2 2 1,,... 1 p m, L Im 1 L... pl and L Im 1 L... H h h m L q are polynomials in L, he lag operaor. I is clear ha when q A l and B are diagonal l marices, (6) reduces o (2) The VARMA-GARCH model assumes ha negaive and posiive shocks of equal magniude have idenical impacs on he condiional variance. McAleer e al. (2009) exended he VARMA-GARCH o accommodae he asymmeric impacs of he uncondiional shocks on he condiional variance, and proposed he VARMA-AGARCH specificaion of he condiional variance as follows: r r s H W A C I B H i i i i i j j i1 i1 j1, (7) where C i are m m marices for i 1,.., r wih ypical elemen ij is an indicaor funcion, given as, and I diag I,..., I, 1 m I i 0, i 0 1, i 0 (8). If m 1, (7) collapses o he asymmeric GARCH (or GJR) model of Glosen e al. (1992). Moreover, VARMA-AGARCH reduces o VARMA-GARCH when Ci 0 for all i. If Ci 0 and A i and B are diagonal marices for all i and j, hen VARMA-AGARCH reduces j o he CCC model. The srucural and saisical properies of he model, including necessary and sufficien condiions for saionariy and ergodiciy of VARMA-GARCH and VARMA- AGARCH, are explained in deail in Ling and McAleer (2003) and McAleer e al. (2009), respecively. The parameers of model (1)-(7) are obained by maximum likelihood esimaion (MLE) using a join normal densiy. When does no follow a join mulivariae normal disribuion, he appropriae esimaor is QMLE. 8
The assumpion ha he condiional correlaions are consan may seem unrealisic in many empirical resuls, paricularly in previous sudies abou crude oil reurns (see, for example, Lanza e al. (2006), Manera e al. (2006), and Chang e al. (2009a, 2009b, 2009c)). In order o make he condiional correlaion marix ime dependen, Engle (2002) proposed a dynamic condiional correlaion (DCC) model, which is defined as y (0, Q ), 1,2,..., n (9) 1 Q D D, (10) where D 12 12 diag h1,..., hm is a diagonal marix of condiional variances, and is he informaion se available a ime. The condiional variance, h i, can be defined as a univariae GARCH model, as follows: h p h i i ik i, k il i, l k1 l1 q. (11) If is a vecor of i.i.d. random variables, wih zero mean and uni variance, Q in (12) is he condiional covariance marix (afer sandardizaion, i y i h i ). The i are used o esimae he dynamic condiional correlaions, as follows: 1/2 1/2 ( diag( Q ) Q ( diag( Q ) (12) where he k k symmeric posiive definie marix Q is given by Q (1 ) Q Q, (13) 1 2 1 1 1 2 1 in which 1 and 2 are scalar parameers o capure he effecs of previous shocks and previous dynamic condiional correlaions on he curren dynamic condiional correlaion, and 1 and 2 are non-negaive scalar parameers. When 1 2 0, Q in (13) is equivalen 9
o CCC. As Q is a condiional on he vecor of sandardized residuals, (13) is a condiional covariance marix, and Q is he k k uncondiional variance marix of. DCC is no linear, bu may be esimaed simply using a wo-sep mehod based on he likelihood funcion, he firs sep being a series of univariae GARCH esimaes and he second sep being he correlaion esimaes (see Caproin and McAleer 92009) for furher deails and caveas). An alernaive dynamic condiional model is BEKK, which has he aracive propery ha he condiional covariance marices are posiive definie. However, BEKK suffers from he socalled curse of dimenionaliy (see McAleer e al. (2009) for a comparison of he number of parameers in various mulivariae condiional volailiy models). The BEKK model for mulivariae GARCH(1,1) is given as: H CC A A+BH B, (14) 1 1 1 where he individual elemen for he marices C, A and B marices are given as a A a a 11 12 a 21 22, b B b b 11 12 b 21 22, c C c 11 0 c 21 22 wih 1, i 1, 2 for saionariy. In his diagonal represenaion, he condiional 2 2 ii ii variances are funcions of heir own lagged values and own lagged reurns shocks, while he condiional covariances are funcions of he lagged covariances and lagged cross-producs of he corresponding reurns shocks. Moreover, his formulaion guaranees H o be posiive definie almos surely for all. For furher deails and a comparison beween BEKK and DCC, see Caporin and McAleer (2008, 2009). 2.2 Opimal Hedge Raios and Opimal Porfolio Weighs Marke paricipans in fuures markes choose a hedging sraegy ha reflecs heir aiudes oward risk and heir individual goals. Consider he case of an oil company, which usually 10
wans o proec exposure o crude oil price flucuaions. The reurn on he oil company s porfolio of spo and fuures posiion can be denoed as: R R R, (15) H, S, F, where R H, is he reurn on holding he porfolio beween 1 and, R S, and R F, are he reurns on holding spo and fuures posiions beween and 1, and is he hedge raio, ha is, he number of fuures conracs ha he hedger mus sell for each uni of spo commodiy on which price risk is borne. According o Johnson (1960), he variance of he reurns of he hedged porfolio, condiional on he informaion se available a ime 1, is given by 2 RH, 1 RS, 1 RS, RF, 1 RF, 1 var var 2 cov, var, (16) where var RS, 1, var RF, 1 and cov RS,, RF, 1 are he condiional variance and covariance of he spo and fuures reurns, respecively. The OHRs are defined as he value of which minimizes he condiional variance (risk) of he hedged porfolio reurns, ha is, min var R, 1 H. Taking he parial derivaive of (16) wih respec o, seing i equal o zero and solving for, yields he OHR condiional on he informaion available a 1 (see, for example, Baillie and Myers (1991)): 1 cov R, R var R S, F, 1 F, 1 (17) where reurns are defined as he logarihmic differences of spo and fuures prices. From he mulivariae condiional volailiy model, he condiional covariance marix is obained, such ha he OHR is given as: 11
h SF, 1, (18) hf, where h SF, is he condiional covariance beween spo and fuures reurns, and h F, is he condiional variance of fuures reurns. In order o compare he performance of OHRs obained from differen mulivariae condiional volailiy models, Ku e al. (2007) sugges ha a more accurae model of condiional volailiy should also be superior in erms of hedging effeciveness, as measured by he variance reducion for any hedged porfolio compared wih he unhedged porfolio. Thus, a hedging effecive index (HE) is given as: var HE unhedged var var unhedged hedged, (19) where he variances of he hedge porfolio are obained from he variance of he rae of reurn, R H,, and he variance of he unhedged porfolio is he variance of spo reurns (see, for example, Ripple and Moosa (2007)). A higher HE indicaes a higher hedging effeciveness and larger risk reducion, such ha a hedging mehod wih a higher HE is regarded as a superior hedging sraegy. Alernaively, in order o consruc an opimal porfolio design ha minimizes risk wihou lowering expeced reurns, and applying he mehods of Kroner and Ng (1998) and Hammoudeh e al. (2009), he opimal porfolio weigh of crude oil spo/fuures holding is given by: w SF, hf, hsf, h 2h h S, SF, F, (20) and 12
0, if wsf, < 0 wsf, wsf,, if 0 < wsf, 0 1, if wsf, > 0 (21) where w SF, ( 1 wsf, spo/fuures a ime. ) is he weigh of he spo (fuures) in a one dollar porfolio of crude oil 3. Daa Daily synchronous closing prices of spo and fuures crude oil prices from wo major crude oil markes, namely Bren and WTI, are used in he empirical analysis. The 3,132 price observaions from 4 November 1997 o 4 November 2009 are obained from he DaaSream daabase. The reurns of crude oil prices i of marke j a ime in a coninuous compound basis are calculaed as rij, log Pij, Pij, 1, where P ij, and Pij, 1 are he closing prices of crude oil price i in marke j for days and 1, respecively. Table 1 presens he descripive saisics for he reurns series of crude oil prices. The average reurns of spo and fuures in Bren and WTI are similar and very low, bu he corresponding variance of reurns is much higher. These crude oil reurns series have high kurosis, which indicaes he presence of fa ails. The negaive skewness saisics signify he series has a longer lef ail (exreme losses) han righ ail (exreme gains). The Jarque-Bera Lagrange muliplier saisics of crude oil reurns in each marke are saisically significan, hereby implying ha he disribuion of hese prices is no normal. Based on he coefficien of variaion, he hisorical volailiy among all crude oil reurns are no especially differen. [Inser Table 1 here] Figure 1 presens he plo of synchronous crude oil price prices. All prices move in he same paern, suggesing hey are conemporaneously highly correlaed. The calculaed conemporaneous correlaions beween crude oil spo and fuures reurns for Bren and WTI markes are boh 0.99. Figure 2 shows he plo of crude oil reurns. These indicae volailiy clusering, or periods of high volailiy followed by periods of relaive ranquiliy. Figure 3 13
displays he volailiies of crude oil reurns, where volailiies are calculaed as he square of he esimaed residuals from an ARMA(1,1) process. These plos are similar in all four reurns, wih volailiy clusering and an apparen oulier. [Inser Figures 1-3 here] Sandard economeric pracice in he analysis of financial ime series daa begins wih an examinaion of uni roos. The Augmened Dickey-Fuller (ADF) and Phillips-Perron (PP) ess are used o es for all crude oil reurns in each marke under he null hypohesis of a uni roo agains he alernaive hypohesis of saionariy. The resuls from uni roo ess are presened in Table 2. The ess yield large negaive values in all cases for levels, such ha he individual reurns series rejec he null hypohesis a he 1% significance level, so ha all reurns series are saionary. [Inser Table 2 here] 4. Empirical Resuls An imporan ask is o model he condiional mean and condiional variances of he reurns series. Therefore, univariae ARMA-GARCH models are esimaed, wih he appropriae univariae condiional volailiy model given as ARMA(1,1)-GARCH(1,1). These resuls are available upon reques. All mulivariae condiional volailiy models in his paper are esimaed using he RATS 6.2 economeric sofware package. Table 3 presens he esimaes for he CCC model, wih p qr s 1. The wo enries corresponding o each of he parameers are he esimae and he Bollerslev-Wooldridge (1992) robus -raios. The ARCH and GARCH esimaes of he condiional variance beween crude oil spo and fuures reurns in Bren and WTI are saisically significan. The ARCH ( ) esimaes are generally small (less han 0.1), and he GARCH ( ) esimaes are generally high and close o one. Therefore, he long run persisence, is generally close o one, indicaing a near long memory process. In addiion, since 1, all markes saisfy he second momen and log-momen condiion, which is a sufficien condiion for he QMLE o be consisen and asympoically normal (see McAleer, Chan and Marinova (2007)). The 14
CCC esimaes beween he volailiy of spo and fuures reurns of Bren and WTI are high, wih he highes being 0.923 beween he sandardized shocks o volailiy in he crude oil spo and fuures reurns of he WTI marke. [Inser Table 3 here] Table 4 repors he esimaes of he condiional mean and variance for VARMA(1,1)- GARCH(1,1) models. The ARCH ( ) and GARCH ( ) esimaes, which refer o he own pas shocks and volailiy effecs, respecively, are saisically significan in all markes. The degree of shor run persisence,, varies across hose reurns. In he case of he Bren marke, he shock dependency in he shor run of fuures reurns (0.100) is higher han ha of spo reurns (0.069). In he WTI marke, spo reurns (0.211) are higher han fuures reurns (0.066). However, he degree of long run persisence,, of fuures reurns in boh markes is higher han for spo reurns. This indicaes ha convergence o he long run equilibrium afer shocks o fuures reurns is faser han for spo reurns. Moreover, volailiy spillover effecs beween volailiy of spo and fuures reurns are found in boh markes, especially he inerdependency of spo and fuures reurns in he Bren marke, 0.712 and 0.212. This means ha he condiional variances of spo and fuures reurns of he Bren marke are affeced by he previous long run shocks from each oher, while he condiional variance of spo reurns is only affeced by he previous long run shocks from fuures reurns, 0.654 in he case of he WTI marke. [Inser Table 4 here] The DCC esimaes of he condiional correlaions beween he volailiies of spo and fuures reurns based on esimaing he univariae GARCH(1,1) model for each marke are given in Table 5. Based on he Bollerslev and Wooldridge (1992) robus -raios, he esimaes of he DCC parameers, ˆ 1 and ˆ 2, are saisically significan in all cases. This indicaes ha he assumpion of consan condiional correlaion for all shocks o reurns is no suppored empirically. The shor run persisence of shocks on he dynamic condiional correlaions is greaes for WTI a 0.139, while he larges long run persisence of shocks o he condiional correlaions is 0.986 (= 0.070 + 0.916) for Bren. 15
The ime-varying condiional correlaions beween spo and fuures reurns are given in Figure 4. I is clear ha here is significan variaion in he condiional correlaions over ime, especially he spo and fuures reurns of Bren. The esimaes for BEKK are given in Table 6. The elemens of he 2 2 parameer marices, A and B, are saisically significan. [Inser Tables 5 and 6 here] [Inser Figure 4 here] Table 7 gives he opimal porfolio weighs, OHRs and hedge effeciveness. The average value of w SF,, calculaed from (20) and (21), based on he Bren and WTI markes, are repored in he firs and second columns. In he case of he Bren marke, he opimal porfolio weighs from each model are no paricularly differen, suggesing ha he porfolio consrucions give similar resuls. For example, he larges average value of w SF, of he porfolio comprising crude oil spo and fuures from he CCC model is 0.383, meaning ha invesors should have more crude oil fuures han spo in heir porfolio in order o minimize risk wihou lowering expeced reurns. In addiion, he opimal holding of spo in one dollar of crude oil spo/fuures porfolio is 38.3 cens, and 61.7 cens for fuures. [Inser Table 7 here] In he case of he WTI marke, opimal porfolio weighs from consan condiional correlaion models, namely CCC and VARMA-GARCH, are differen and smaller han hose from he dynamic condiional correlaion models, namely DCC and BEKK. For example, he larges w SF, is 0.571 from he BEKK model, while he smalles w SF, is 0.350 from he CCC model, hereby signifying ha he dynamic condiional correlaion models sugges holding crude oil spo (57.1 cens for spo) more han fuures (42.9 cens for fuures), whereas he consan condiional correlaion models sugges holding crude oil fuures (65 cens for fuures) han spo (35 cens for fuures) of a one dollar spo/fuures porfolio. Figure 5 presens he calculaed ime-varying OHRs from each mulivariae condiional volailiy model. There are clearly ime-varying hedge raios. The hird and fourh columns in 16
Table 7 repor he average OHR values. The average OHR values of he Bren marke obained from several differen mulivariae condiional volailiy models are high and have similar paerns o hose of he WTI marke. Following from he hedge sraegy, for example, he larges average OHR values are 0.846 and 0.956 from VARMA-GARCH of Bren and WTI suggess ha one dollar long (buy) in he crude oil spo should be shored (sold) by abou 84.6 and 95.6 cens of fuures, respecively. In addiion, he consan condiional correlaions of boh markes recommend o shor fuures as compared wih he dynamic condiional correlaions. [Inser Figure 5 here] The hedging effeciveness in columns five and six in Table 7 shows ha all four mulivariae condiional volailiy models effecively reduce he variances of he porfolio, and perform beer in he WTI marke han he Bren marke (he HE indices are around 80% for WTI and 56% for Bren). Of he mulivariae GARCH models, he larges HE value of he Bren marke (WTI marke) is obained from DCC, such ha DCC is he bes model for OHR calculaion in erms of he variance of porfolio reducion. In conras, he lowes HE value in boh markes is obained from BEKK model. Therefore, he BEKK model is he wors model in erms of he variance of porfolio reducion. 5. Conclusion This paper esimaed four mulivariae volailiy models, namely CCC, VARMA-GARCH, DCC and BEKK, for he crude oil spo and fuures reurns of wo major benchmark inernaional crude oil markes, namely Bren and WTI. The esimaed condiional covariance marices from hese models were used o calculae he opimal porfolio weighs and opimal hedge raios, and o indicae crude oil hedge sraegies. Moreover, in order o compare he abiliy of variance porfolio reducion due o differen mulivariae volailiy models, he hedging effecive index was also esimaed. The empirical resuls for daily daa from 4 November 1997 o 4 November 2009 showed ha, for he Bren marke, he opimal porfolio weighs of all mulivariae volailiy models suggesed holding fuures in larger proporion han spo. On he conrary, for he WTI marke, 17
he dynamic condiional correlaions models, DCC and BEKK, recommended holding fuures o spo, bu he consan condiional correlaion models, CCC and VARMA-GARCH, suggesed holding spo o fuures. The calculaed OHRs from each mulivariae condiional volailiy model presened he ime-varying hedge raios, and recommended o shor in crude oil fuures, wih a high proporion of one dollar long in crude oil spo. The hedging effeciveness indicaed ha DCC (BEKK) was he bes (wors) model for OHR calculaion in erms of he variance of porfolio reducion. 18
References Alizadeh, A.H., M.G. Kavussanos and D.A. Menachof (2004), Hedging agains bunker price flucuaions using peroleum fuures conrac: Consan versus ime-varying hedge raios, Applied Economics, 36, 1337-1353. Baillie, R. and R. Myers (1991), Bivariae GARCH esimaion of he opimal commodiy fuures hedge, Journal of Applied Economerics, 6, 109-124. Bauwens, L., S. Lauren and J. Rombous (2006), Mulivariae GARCH models: A survey, Journal of Applied Economerics, 21, 79-109. Bollerslev, T. (1986), Generalised auoregressive condiional heeroscedasiciy, Journal of Economerics, 31, 307-327. Bollerslev, T. (1990), Modelling he coherence in shor-run nominal exchange rae: A mulivariae generalized ARCH approach, Review of Economics and Saisics, 72, 498-505. Bollerslev, T. and J. Wooldridge (1992), Quasi-maximum likelihood esimaion and inference in dynamic models wih ime-varying covariances, Economeric Reviews, 11, 143-172. Caporin, M. and M. McAleer (2008), Scalar BEKK and indirec DCC, Journal of Forecasing, 27, 537-549 Caporin, M. and M. McAleer (2009), Do we really need boh BEKK and DCC? A ale of wo covariance models, Available a SSRN: hp://ssrn.com/absrac=1338190. Cecchei, S., R. Cumby, and S. Figlewski (1988), Esimaion of he opimal fuures hedge, Review of Economics and Saisics, 70, 623-630. Chang, C.-L., M. McAleer and R. Tansucha (2009a), Modeling condiional correlaions for risk diversificaion in crude oil markes, o appear in Journal of Energy Markes, Available a SSRN: hp://ssrn.com/absrac=1401331. Chang, C.-L., M. McAleer and R. Tansucha (2009b), Forecasing volailiy and spillovers in crude oil spo, forward and fuures marke, Available a SSRN: hp://ssrn.com/absrac=1402164. Chang, C.-L., M. McAleer and R. Tansucha (2009c), Volailiy spillovers beween reurns on crude oil fuures and oil company socks, Available a SSRN: hp://ssrn.com/absrac=1406983. 19
Chen, S.-S., C.-F. Lee and K. Shresha (2003), Fuures hedge raios: A review, Quarerly Review of Economics and Finance, 43, 433-465. Daniel, J. (2001), Hedging governmen oil price risk, IMF Working Paper 01/185. Ederingon, L.H. (1979), The hedging performance of he new fuures markes, Journal of Finance, 34, 157-170. Engle, R. (2002), Dynamic condiional correlaion: A simple class of mulivariae generalized auoregressive condiional heeroskedasiciy models, Journal of Business and Economic Saisics. 20, 339-350. Engle, R.F. and K.F. Kroner (1995), Mulivariae simulaneous generalized ARCH, Economeric Theory, 11, 122-150. Figlewski, S. (1985), Hedging wih sock index fuures: esimaion and forecasing wih error correcion model, Journal of Fuures Markes, 13, 743-752. Glosen, L., R. Jagannahan and D. Runkle (1992), On he relaion beween he expeced value and volailiy of nominal excess reurn on socks, Journal of Finance, 46, 1779-1801. Haigh, M.S. and M. Hol (2002), Crack spread hedging: Accouning for ime-varying spillovers in he energy fuures markes, Journal of Applied Economerics, 17, 269-289. Hammoudeh, S., Y. Yuan, M. McAleer and M.A. Thompson (2009), Precious mealsexchange rae volailiy ransmission and hedging sraegies, Available a SSRN: hp://ssrn.com/absrac=1495748. Jalali-Naini, A. and M. Kazemi-Manesh (2006), Price volailiy, hedging, and variable risk premium in he crude oil marke, OPEC Review, 30(2), 55-70. Johnson, L.L. (1960), The heory of hedging and speculaion in commodiy fuures, Review of Economic Sudies, 27, 139-151. Knill, A., M. Krisina and A. Nejadmalayeri (2006), Selecive hedging, informaion, asymmery, and fuures prices, Journal of Business, 79(3), 1475-1501. Kroner, K. and V. Ng (1998), Modeling asymmeric movemens of asse prices, Review of Financial Sudies, 11, 871-844 Kroner, K. and J. Sulan (1993), Time-varying disribuions and dynamic hedging wih foreign currency fuures, Journal of Financial and Quaniaive Analysis, 28, 535-551. 20
Ku, Y.-H., H.-C. Chen and K.-H. Chen (2007), On he applicaion of he dynamic condiional correlaion model in he esimaing opimal ime-varying hedge raios, Applied Economics Leer, 14, 503-509. Lanza, A., M. Manera and M. McAleer (2006), Modeling dynamic condiional correlaions in WTI oil forward and fuure reurns, Finance Research Leers, 3, 114-132. Lien, D. and Y.K. Tse (2002), Some recen developmens in fuures hedging, Journal of Economic Surveys, 16(3), 357-396. Ling, S. and M. McAleer (2003), Asympoic heory for a vecor ARMA-GARCH model, Economeric Theory, 19, 278-308. Manera, M., M. McAleer and M. Grasso (2006), Modelling ime-varying condiional correlaions in he volailiy of Tapis oil spo and forward reurns, Applied Financial Economics, 16, 525-533. McAleer, M. (2005), Auomaed inference and learning in modeling financial volailiy, Economeric Theory, 21, 232-261. McAleer, M., F. Chan and D. Marinova (2007), An economeric analysis of asymmeric volailiy: Theory and applicaion o paens, Journal of Economerics, 139, 259-284 McAleer, M., S. Hoi and F. Chan (2009), Srucure and asympoic heory for mulivariae asymmeric condiional volailiy, Economeric Reviews, 28, 422-440. Myers, R. (1991), Esimaing ime varying hedge raio on fuures markes, Journal of Fuures Markes, 11, 39-53. Myers, R. and S. Thompson (1989), Generalized opimal hedge raio esimaion, American Journal of Agriculural Economics, 71, 858-868. Ripple, R.D. and I.A. Moosa (2007), Hedging effeciveness and fuures conrac mauriy: The case of NYMEX crude oil fuures, Applied Financial Economics, 17, 683-689. 21
Table 1. Descripive Saisics Reurns Mean Max Min SD CV Skewness Kurosis Jarque-Bera BRSP 0.0004 0.152-0.170 0.025 0.016-0.047 6.113 1265.547 BRFU 0.0004 0.129-0.144 0.024 0.017-0.142 5.576 876.642 WTISP 0.0004 0.213-0.172 0.027 0.015-0.002 7.932 3174.982 WTIFU 0.0004 0.164-0.165 0.025 0.016-0.120 7.164 2270.166 22
Table 2. Uni Roo Tess ADF es (-saisic) Phillips-Perron es Reurns Consan Consan None Consan None Consan and Trend and Trend BRSP -55.266-55.275-55.267-55.276-55.280-55.271 BRFU -59.269-59.281-59.273-59.239-59.252-59.244 WTISP -56.678-56.684-56.676-56.881-56.906-56.897 WTIFU -42.218-42.231-42.224-57.169-57.191-57.183 Noe: Enries in bold are significan a he 1% level. 23
Table 3. CCC Esimaes Panel a: BRSP_BRFU Reurns C AR MA BRSP BRFU 1.878e-03 (2.648) 1.343e-03 (2,930) Panel b: WTISP_WTIFU -0.841 (-8.338) -0.383 (-27.87) 0.859 (9.045) 0.309 (21.08) 6.871e-06 (5.636) 6.299 (5.691) 0.039 (13.72) 0.035 (9.693) 0.951 (256.6) 0.953 (204.2) Consan condiional correlaion 0.990 0.794 (159.65) 0.988 Reurns C AR MA WTISP WTIFU 1.086e-03 (3.560) 1.209e-03 (4.005) -0.093 (-0.768) -0.177 (-18.21) 0.029 (0.240) 0.100 (8.240) 2.069e-05 (15.58) 1.978e-05 (11.55) 0.083 (23.90) 0.083 (20.818) 0.888 (244.9) 0.888 (163.0) Consan condiional correlaion 0.971 0.923 (550.9) 0.971 Loglikelihood AIC 16291.932-10.399 Loglikelihood AIC 17421.123-11.1198 24
Table 4. VARMA-GARCH Esimaes Panel a: BRSP_BRFU Reurns C AR MA BRSP BRFU BRSP BRFU BRSP BRFU Panel b: WTISP_WTIFU -0.855 (-9.824) -0.384 (-25.856) 0.872 (10.673) 0.308 (21.408) 3.644e-06 (0.433) 7.749e-06 (2.481) 0.069 (4.590) -0.064 (-4.946) -0.037 (-2.424) 0.100 (6.867) 0.412 (3.327) 0.212 (2.364) 0.712 (4.585) 0.762 (9.794) Reurns C AR MA BRSP BRFU BRSP BRFU WTISP WTIFU 1.492e- 03 (2.066) 1.183e- 03 (2.634) 1.011e- 03 (3.414) 1.143e- 03 (3.914) -0.060 (-0.392) -0.179 (-18.196) 1.303e-03 (0.009) 0.105 (9.717) 1.641e-05 (3.382) 1.048e-05 (8.172) 0.211 (21.250) 1.679 (0.274) -0.138 (-20.464) 0.066 (11.045) 0.305 (10.436) 0.033 (1.387) 0.654 (19.323) 0.887 (40.623) Consan condiional correlaion 0.481 0.803 (158.566) 0.862 Consan condiional correlaion 0.516 0.928 (583.246) Noes: (1) The wo enries for each parameer are heir respecive parameer esimaes and Bollerslev and Wooldridge (1992) robus - raios. (2) Enries in bold are significan a he 5% level. 0.953 Loglikelihood AIC 16348.450-10.432 Loglikelihood AIC 17525.095-11.184 25
Panel a: BRSP_BRFU BRSP BRFU Table 5. DCC Esimaes C AR MA 1 2 1.821e-03 (2.671) 1.244e-03 (2.789) Panel b: WTISP_WTIFU WTISP 0.001 (1.580) WTIFU 0.0003 (1.796) -0.762 (-3.584) -0.346 (-21.576) 0.776 (3.765) 0.299 (18.131) 7.742e-06 (5.033) 6.012e-06 (5.156) 0.053 (13.851) 0.043 (11.195) 0.935 (189.947) 0.946 (195.999) 0.988 0.070 (18.766) 0.989 0.916 (183.616) C AR MA 1 2-0.259 (-5.655) 0.626 (6.871) 0.252 (5.160) -0.658 (-8.085) 3.34E-05 (2.118) 3.98E-05 (2.129) 0.151 (3.142) 0.151 (3.460) 0.774 (10.295) 0.789 (12.108) 0.925 0.139 (1.981) 0.458 (0.174) Noes: (1) The wo enries for each parameer are heir respecive parameer esimaes and Bollerslev and Wooldridge (1992) robus - raios. (2) Enries in bold are significan a he 5% level. 0.940 AIC 16424.565-10.483 Loglikelihood Loglikelihood AIC 17618.890-11.246 26
Table 6. BEKK Esimaes Panel a: BRSP_BRFU Reurns C AR MA C A B BRSP 0.002-0.715 0.724-0.001-0.320 0.153-0.151-0.878 (2.527) (-2.585) (2.481) (-1.286) (-7.800) (5.613) (-6.583) (-27.652) BRFU 0.001-0.331 0.285 0.005-0.0001 0.182-0.357-0.897-0.043 (2.386) (-19.748) (12.605) (6.616) (-0.063) (5.438) (-13.812) (-81.273) (-0.967) Panel b: WTISP_WTIFU Reurns C AR MA C A B WTISP 0.0004 0.310-0.387 0.002-0.911-0.018 0.494-0.079 (1.289) (2.370) (-3.186) (5.808) (-5.941) (-2.394) (9.593) (-62.908) WTIFU 0.0007-0.212 0.157-0.003 1.00E-06 0.938 0.192 0.500 1.053 (1.575) (-6.440) (4.043) (-7.358) (0.001) (6.758) (6.130) (10.064) (213.713) a11 a12 b11 b12 c11 0 Noe; (1) A a21 a, B 22 b21 b, C 22 c21 c are he coefficien marices from equaion (14). 22 (2) The wo enries for each parameer are heir respecive parameer esimaes and Bollerslev and Wooldridge (1992) robus - raios. (3) Enries in bold are significan a he 5% level. Loglikelihood AIC 16427.720-10.483 Loglikelihood AIC 18021.590-11.501 27
Table 7. Alernaive Hedging Saegies Opimal Porfolio Weighs Average OHR Variance of Porfolios Hedge Effeciveness (%) Model Bren WTI Bren WTI Bren WTI Bren WTI CCC 0.383 0.382 0.840 0.955 2.682e-04 1.349e-04 56.724 80.857 VARMA-GARCH 0.377 0.377 0.846 0.956 2.706e-04 1.373e-04 56.346 80.513 DCC 0.366 0.478 0.824 0.923 2.663e-04 1.342e-04 57.045 80.942 BEKK 0.355 0.571 0.827 0.922 2.710e-04 1.417e-04 56.294 79.886 Unhedged Porfolio 6.199e-04 7.046e-04 28
Figure 1. Crude Oil Spo and Fuures Prices for Bren and WTI 160 160 140 140 120 120 100 100 $/barrel 80 $/barrel 80 60 60 40 40 20 20 0 0 BRSP BRFU 160 160 140 140 120 120 100 100 $/barrel 80 60 $/barrel 80 60 40 40 20 20 0 0 WTIFU WTISP 29
Figure 2. Logarihm of Daily Crude Oil Spo and Fuures Prices for Bren and WTI.16.15.12.08.10.04.05 Reurns.00 -.04 Reurns.00 -.08 -.05 -.12 -.16 -.10 -.20 -.15.3 BRENT SPOT.20 BRENT FUTURES.15.2.10.1.05 Reurns.0 Reurns.00 -.05 -.1 -.10 -.15 -.2 -.20 WTISP WTIFU 30
Figure 3. Esimaed Condiional Volailiies of Reurns for Bren and WTI.030.024.025.020.020.016.015.012.010.008.005.004.000.000 BRSP BRFU.05.030.04.025.03.020.015.02.010.01.005.00.000 WTISP WTIFU 31
Figure 4. DCC Esimaes 1.0 1.0 0.9 0.8 0.7 0.8 0.6 0.6 0.4 0.5 0.2 0.4 0.3 0.0 0.2-0.2 0.1-0.4 BRSP_BRFU WTISP_WTIFU 32
Figure 5. Opimal Hedge Raios 1.2 1.3 1.1 1.2 1.1 1.0 1.0 OHR 0.9 OHR 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 CCC:BRSP_BRFU VARMA-GARCH:BRSP_BRFU 1.4 1.2 1.2 1.0 1.0 0.8 OHR 0.8 0.6 OHR 0.6 0.4 0.2 0.4 0.0 0.2-0.2 0.0-0.4 DCC:BRSP_BRFU BEKK:BRSP_BRFU 2.8 3.0 2.4 2.5 2.0 2.0 OHR 1.6 OHR 1.2 1.5 0.8 1.0 0.4 0.5 CCC:WTISP_WTIFU VARMA-GARCH:WTISP_WTIFU 2.5 2.5 2.0 2.0 1.5 1.5 OHR 1.0 OHR 1.0 0.5 0.5 0.0 0.0-0.5-0.5 DCC:WTISP_WTIFU 33 DCC:WTISP_WTIFU