DYNAMIC ECONOMETRIC MODELS Vol. 6 Nicolaus Copernicus University Toruń Piotr Fiszeder Nicolaus Copernicus University in Toruń

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1 DYNAMIC ECONOMETRIC MODELS Vol. 6 Nicolaus Copernicus Universiy Toruń Inroducion Pior Fiszeder Nicolaus Copernicus Universiy in Toruń Dynamic Hedging Porfolios Applicaion of Bivariae GARCH Models Fuures mares were originally developed o mee he needs of farmers and merchans, who were exposed o ris relaed o price flucuaions. Nowadays hedging is also one of he main areas applicaion of derivaives. Differen inds of derivaives can be used in hedging sraegies. This aricle deals only wih fuures conracs. When an individual or company chooses o use fuures conracs o hedge a ris, he objecive is usually o ae a posiion ha neuralizes he ris as far as possible. In his paper he hedging sraegy which minimizes he variance of he porfolio composed of he posiion aen in fuures conracs and he posiion being hedged is analysed. Tradiionally used mehods of esimaing hedge raios ignore one of he main properies of financial ime series, namely ime-varying condiional variances and covariances of reurns. Recenly, in many papers he GARCH models have been used o esimae imevarying hedge raios. This approach has been firs proposed by Cecchei, Cumby and Figlewsi (988), ha applied a univariae ARCH model in esimaing an opimal fuures hedge wih Treasury bonds, obaining a significan reducion of he ex-pos porfolio variance. Mulivariae GARCH models were used o esimae hedge raios among ohers by Baillie and Myers (99), Myers (99) for commodiy fuures, Kroner and Sulan (993) for currency fuures, Gagnon and Lypny (995) for ineres-rae fuures, Par and Swizer (995), Tong (996) for soc index fuures. Exising empirical wor on hedging performance has led o conflicing conclusions, however mos of hem indicae superior hedging performance of dynamic sraegy based on he GARCH framewor. In his aricle, firsly, performance of such sraegy is analysed using fuures conracs on WIG20 soc Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House

2 204 Pior Fiszeder index quoed on he Warsaw Soc Exchange (WSE). Secondly, hedging performance of differen specificaions of GARCH models is compared. In mos oher sudies only seleced formulaion is used. Thirdly, hedging effeciveness of an error correcion model wih a GARCH error srucure is also invesigaed. Fourhly, mehods used by praciioners of he financial mare o forecas variances and covariances of reurns are applied in esimaing hedge raios. The aricle is laid ou in four secions. Secion 2 oulines he compeing mehods of esimaing hedge raios. In Secion 3 hedging performance of presened sraegies is invesigaed using WIG20 soc index and fuures conracs on WIG 20 index quoed on he WSE. Secion 4 concludes. 2. Esimaion echniques for hedge raios The hedge raio is he raio of he size of posiion aen in fuures conracs o he size of he exposure (see Hull, 995). The reurn on a porfolio composed of cash and fuures posiions is given by: x, () + = s+ b f+ where s+ = ln S+ ln S, f + = ln F + ln F, S is spo price, F fuures price, b hedge raio. The opimal hedge raio which minimizes he variance of he hedged porfolio reurn can be expressed as: Cov ( s+, f + ψ ) b =, (2) Var ( f ψ ) + where ψ is he se of informaion available a ime. I can be easily shown ha formula (2) is also valid when invesor s preferences can be described by a quadraic uiliy funcion: E( U ( x+ ) ψ ) = E( x+ ψ ) ξ Var( x+ ψ ), (3) wih he assumpion ha log fuures price is a maringale, i.e. E ( lnf + ψ ) = ln F. If he condiional covariance marix is ime-invarian, hen he consan opimal hedge raio may be obained consrucing a linear regression: s = α + β + ε, (4) Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House f The opimal hedge raio, b, is equal o he esimaed OLS coefficien β. This approach is called OLS hedging.

3 Dynamic Hedging Porfolios 205 I has been shown by numerous sudies ha he daa do no suppor he assumpion ha he condiional covariance marix of reurns is consan over ime (see for insance Bollerslev, Chou and Kroner, 992, Bollerslev, Engle and Nelson, 994, Fiszeder, 2003). Therefore, we follow recen lieraure by employing a mulivariae GARCH model, which allows he condiional variances and covariances used as inpus o he hedge raio o be ime-varying. Given he bivariae GARCH model of spo and fuures prices, he ime-varying hedge raio can be expressed as: where hsf, + b =, (5) h f, + h sf, + is forecas from he GARCH model of condiional covariance beween reurns of spo and fuures prices and, + is a forecas from he GARCH model for condiional variance of reurn on fuure posiion a ime +. Specificaion of mulivariae GARCH model which is one of he mos frequenly used for several ime series is he BEKK represenaion: H = CC ' + q D ε i i ε ' i D ' + i j= Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House p E j H j h f E ', (6) where H is he N N symmeric condiional covariance marix, ε is he N vecor of errors, C is an upper riangular parameer marix, Di and E j are he N N parameer marices. The advanage of his formulaion is a posiive definieness of H and ime-varying condiional correlaions beween he reurns. Differen formulaions of mean equaions are analysed. The simples specificaion assumes ha spo and fuures reurns are consan: s α s0 + ε s, f α f 0 + ε f, ψ ~ N(0, H =, (7) =, (8) ε ), (9) H where ε = ( ε s,, ε f, )' and is given by (6). In order o capure shor erm relaions beween spo and fuure reurns he VAR BEKK model is considered: j s = α s0 + α si s i + β si f i + ε s,, (0)

4 206 Pior Fiszeder f = α f 0 + α fi s i + β fi f i + ε f, ~ N(0, H ), () ε ψ, (2) where H is given by (6). In he absence of ransacion coss, mare microsrucure effecs, or oher impedimens o heir free operaion, he efficien mares hypohesis and he absence of arbirage opporuniies imply ha he spo and corresponding fuures mares reac conemporaneously and idenically o new informaion. There has been some debae in he lieraure as o wheher his implies ha he wo mares mus be coinegraed. The condiional mean equaions of he model employed in his aricle represen a bivariae Vecor Error Correcion Mechanism, which may be wrien as: s = α s0 + α si s i + β si f i + γ s ( S γ F ) + ε s, f = α f 0 + α fi s i + β fi f i + γ f ( S γ F ) + ε f, ψ ~ N(0, H ), (3) Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House, (4) ε, (5) H where is given by (6). Tha is, we adop he coinegraion assumpion. Alhough he uncondiional disribuion for ε in he GARCH model wih condiional normal errors as given by (9), (2) and (5) has faer ails han he normal disribuion, for many financial ime series i does no adequaely accoun for lepourosis. Tha is why bivariae Suden- disribuion wih unnown degrees of freedom is addiionally applied insead of (9). Oher specificaions of he disribuion of ε are possible bu he GARCH model wih condiional Suden- disribuion of ε is adequae o accoun for he fa-ailed properies of he daa and is relaively easy o esimae. The hedging performance of differen specificaions of mulivariae GARCH models are also examined. One simple assumpion ha could be made o reduce he number of parameers is o specify ha he ( r, s) h elemen in H only depends on he corresponding ( r, s) h elemen in ε ε' and. This assumpion amouns o aing Di and E j o be diagonal marices. In he consan condiional correlaions model of Bollerslev (990), which is ouside he BEKK class, he ime-varying condiional covariances are parameerised o be proporional o he produc of he corresponding condiional sandard deviaions: i i H j H = D Γ D, (6)

5 Dynamic Hedging Porfolios 207 where D is he N N diagonal marix wih he condiional sandard deviaions and Γ is he N N marix of ime-invarian condiional correlaions. If he condiional variances are all posiive and he condiional correlaion marix Γ is posiive definie, hen he condiional covariance marix H is guaraneed o be posiive definie for all. Mehods used by praciioners of financial mare can also be applied o forecas variances and covariances of reurns in (2) (see Zangari, 996, Lierman and Winelmann, 998). Forecass of variances and covariances of reurns based on he moving average model are given by: 2 2 f, + = ( r f, i r f ) + σ, (7) sf, + = ( rs, i rs )( r f, i r f ) + σ, (8) 2 where and are forecass of variances and covariances of reurns a dae + respecively, and are reurns on spo and fuure posiions a dae i, σ f, + σ sf, + r = r i. r s, i r f, i Forecass of variances and covariances of reurns based on he exponenial smoohing model can be formed as follows: f, + = ( α) r f, α σ f, σ + σ + sf, + = ( α) r s, r f, α σ sf,, (9), (20) where 0 <α <. The choices of he moving average esimaion period () in (7) (8) and value of smoohing parameer (α ) in (9) (20) are arbirary and should be deermined empirically. 3. Hedging performance for he WIG20 soc index Hedging effeciveness of presened sraegies is invesigaed using he WIG 20 soc index and fuures conracs on he WIG 20 index quoed on he WSE. WIG20 index is a porfolio index of he 20 larges and mos acively raded socs. Fuures conracs on WIG 20 are he mos heavily raded fuures conracs quoed on he WSE. The period invesigaed is January 04, 999 o December 3, 2002 (996 daily reurns). Hedging performance of seleced sraegies is evaluaed for daa from he year 2002 (249 daily reurns). The mos acively raded conrac is always analysed. In order o avoid problems of scarce liquid- Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House

6 208 Pior Fiszeder iy and oher disorions, he rollover o he nex conrac is carried ou one wee before he las rading day. For each rading day in he year 2002 he porfolio composed of he WIG 20 soc index and fuures conracs on he WIG 20 index are consruced. Hedging sraegies, which are considered in his aricle, differ only in he mehod of esimaing hedge raios in (2). For each sraegy he mean and he sandard deviaion of he hedged porfolio reurns in he year 2002 were calculaed (ex pos). All resuls are presened in Table. For comparison he same resuls are presened for he unhedged porfolio (hedge raio equal o zero) and naive sraegy (hedge raio equal o one). As was o be expeced hedging a porfolio leads o significan ris reducion measured by he sandard deviaion of porfolio reurns. Convenional OLS hedging approach (equaion (4)) provides furher reducion of ris (see Table ). Oher hedging sraegies use he GARCH models o esimae ime-varying hedge raios (formula (5)). Firsly BEKK (p = q = ) represenaion which guaranees posiive definie of a condiional covariance marix and is a relaively simple specificaion of a mulivariae GARCH model is used. The influence of differen formulaions of mean equaions on hedging performance is analysed. The following specificaions are considered: consan spo and fuure reurns (equaions (7 9)), VAR model (equaions (0 2)), vecor error correcion model (equaions (3 5)). Capuring shor erm (VAR model) and long erm (VECM model) relaions beween he spo and fuure conracs resuls in ris reducion of hedged porfolio, however decrease is negligible. The GARCH model wih condiionally normal errors resuls in a lepouric uncondiional disribuion. However, he degree of lepourosis induced by he ime-varying condiional variance ofen does no capure all of he lepourosis presen in high frequency financial daa. Tha is why in formula (9) bivariae Suden- disribuion is applied. However he inroducion of -Suden disribuion does no improve hedging effeciveness. The BEKK model is a reasonable compromise beween generaliy and parsimony of he GARCH model. This is achieved by complicaed nonlinear resricions imposed on he general VECH GARCH(, ) model (see Osiewalsi and Pipień (2002) for a deailed presenaion and discussion). The influence of furher simplificaions of he mulivariae GARCH model on hedging performance is also analysed. The BEKK model in which D i and E j in equaion (6) are diagonal marices and consan condiional correlaions model (formula (6)) are applied. Simplificaion of he condiional covariance marix leads o he decrease of mean sandard deviaion esimaes of hedged porfolios. The highes reducion of ris is in he case forecass from he consan condiional correlaions are used o esimae a hedge raio. This model is compuaionally simple and is relaively easy o ensure he posiive definieness of he condiional covariance marix during he opimisaion. However none of he hedging srae- Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House

7 Dynamic Hedging Porfolios 209 gies based on he GARCH framewor are more effecive (in erms of ris reducion) han OLS hedging. Table. Effeciveness of differen hedging sraegies Esimaion echniques for Mean sandard Ris reducion Mean reurn hedge raios deviaion (in %) ( 0 4 ) Unhedged (b = 0) Naive (b = ) OLS BEKK N VAR BEKK N VECM BEKK N BEKK BEKK N diagonal marices Consan cond. correlaions N Moving variance and covariances Exponenial smoohing The mean reurn and mean sandard deviaion of he hedged porfolios are presened. Ris reducion is measured by mean sandard deviaion in comparison wih unhedged porfolio. Mehods used by praciioners of he financial mare can be applied o forecas variances and covariances of reurns namely: he moving average model (equaions (7) (8)) and exponenial smoohing model (equaions (9) (20)). The moving average esimaion period and he value of smoohing parameer are chosen o minimize he mean sandard deviaion of he hedged porfolios in a pre-sample. The applicaion of he moving average model and he exponenial smoohing model resuls in he increase of hedging performance. The lowes esimae of he mean sandard deviaion of he hedged porfolio is in case forecass of variances and covariances are from he moving average model. However i mus be emphasized ha such good performance of he moving average and he exponenial smoohing models resuls from applied mehod of selecion of he moving average esimaion period and he value of smoohing parameer. In Table he mean reurn in he year 2002 is also presened. The realized reurn can be an addiional crierion considered in he hedging sraegy selecion. Because he sraegy which minimizes he variance of he hedged porfolio is considered in his paper, ha is why he mean reurn is of secondary imporance. Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House

8 20 Pior Fiszeder Conclusions Differen mehods esimaing hedge raios are presened in he paper. Hedging performance of hese sraegies is invesigaed using he WIG20 soc index and fuures conracs on WIG 20 index quoed on he WSE. In our empirical example, none of he sraegies based on he GARCH framewor are more effecive han OLS hedging. Simplificaion of he model used o esimae hedge raios does no significanly decrease effeciveness of applied sraegies (i refers o boh mean and variance equaions). The lowes esimae of he mean sandard deviaion of he hedged porfolio is in case he forecass of variances and covariances are from he moving average model. However, i mus be remembered ha he moving esimaion period in he moving average model and he value of he smoohing parameer in he exponenial smoohing model are chosen for each forecasing period separaely o produce he bes fi in he pre-sample. If he esimaion period and value of parameer are chosen arbirarily, i does no provide such good resuls. All resuls presened in his paper should be reaed as inroducory and furher analysis for oher porfolios are necessary. References Baillie, R. T., Myers, R. J. (99), Bivariae GARCH Esimaion of he Opimal Commodiy Fuures Hedge, Journal of Applied Economerics, 6, Bollerslev, T. (990), Modelling he Coherence in Shor-Run Nominal Exchange Raes: A Mulivariae Generalized ARCH Approach, Review of Economics and Saisics, 72, Bollerslev, T., Chou, R. Y., Kroner, K. F. (992), ARCH Modelling in Finance: A Review of he Theory and Empirical Evidence, Journal of Economerics, 52, Bollerslev, T., Engle, R. F., Nelson, D. B. (994), ARCH Models, in: Engle R. F., McFadden D., (eds.), Handboo of Economerics, Vol. 4, Elsevier Science B. V., Amserdam. Cecchei, S. G., Cumby, R. E., Figlewsi, S. (988), Esimaion of Opimal Hedge, Review of Economics and Saisics, 50, Engle, R. F., Kroner, K. F. (995), Mulivariae Simulaneous Generalized ARCH, Economeric Theory,, Fiszeder, P. (2003), Tesy sałości współczynniów orelacji w wielorównaniowym modelu GARCH analiza orelacji między indesami giełdowymi: WIG, DJIA i Nasdaq Composie (Tess for Consan Correlaions in a Mulivariae GARCH Model Analysis of Correlaions beween Soc Indices: WIG, DJIA and Nasdaq Composie), Przegląd Saysyczny(Saisical Survey), 50, 2, Gagnon, L., Lypny, G. (995), Hedging Shor-Term Ineres Ris Under Time-Varying Disribuions, Journal of Fuures Mares, 995, 5, Hull, J. (997), Konray erminowe i opcje. Wprowadzenie, WIG Press Warszawa. Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House

9 Dynamic Hedging Porfolios 2 Kroner, K. F., Sulan, J. (993), Time-Varying Disribuions and Dynamic Hedging wih Foreign Currency Fuures, Journal of Financial and Quaniaive Analysis, 28, Lierman, R., Winelmann, K., Esimaing Covariance Marices, Ris Managemen Series, 998, Goldman Sachs. Myers, R. J. (99), Esimaing Time-Varying Opimal Hedge Raios on Fuures Mares, Journal of Fuures Mares,, Osiewalsi, J., Pipień M. (2002), Mulivariae -GARCH models Bayesian Analysis for Exchange Raes, MACROMODELS 2000 Conference Proceedings, Absolwen, Łódź 200; Correced prining in: Modelling Economies in Transiion Proceedings of he Sixh AMFET Conference, Absolwen, Łódź Par, T. H., Swizer, L. N. (995), Bivariae GARCH Esimaion of he Opimal Hedge Raios for Soc Index Fuures: A Noe, Journal of Fuures Mares, 5, Tong, W. H. S. (996), An Examinaion of Dynamic Hedging, Journal of Inernaional Money and Finance, 5, Zangari, P. (996), RisMerics Technical Documens, J. P. Morgan, New Yor. Copyrigh by The Nicolaus Copernicus Universiy Scienific Publishing House