Wavelet Estimation of Asymmetric Hedge Ratios: Does Econometric Sophistication Boost Hedging Effectiveness?

Transcription

1 Inernaional Journal of Business and Economics, 2008, Vol. 7, No. 3, Wavele Esimaion of Asymmeric Hedge Raios: Does Economeric Sophisicaion Boos Hedging Effeciveness? Elizabeh A. Maharaj Deparmen of Economerics and Business Saisics, Monash Universiy, Ausralia Imad Moosa * Deparmen of Accouning and Finance, Monash Universiy, Ausralia Jonahan Dark Deparmen of Finance, Universiy of Melbourne, Ausralia Param Silvapulle Deparmen of Economerics and Business Saisics, Monash Universiy, Ausralia Absrac This paper uilises wavele analysis, which is becoming popular in economics and finance, o esimae he hedge raios for spo posiions on he Wes Texas Inermediae crude oil, soybeans and he S&P500 index. This echnique is combined wih a wo-sage regime swiching hreshold model o esimae asymmeric hedge raios corresponding o posiive and negaive reurns on fuures conracs. Oher simple and sophisicaed echniques are also used as a benchmark for he purpose of comparison, including he naïve model and he asymmeric error correcion GJR-GARCH model. On he basis of he variance raio es and variance reducion, i is revealed ha economeric sophisicaion does no boos hedging effeciveness. Key words: asymmeric hedge raios; variance raio; variance reducion; waveles JEL classificaion: G30; C22; C53 1. Inroducion Financial hedging is he covering risk resuling from price changes by aking an opposie posiion on a hedging insrumen, which is ypically a derivaive (such as a fuure or opion). The hedging process consiss of seps ha are implied by he Received Ocober 1, 2008, revised May 5, 2009, acceped May 18, * Correspondence o: Deparmen of Accouning and Finance, Monash Universiy, PO Box 197, Caulfield Eas, Vicoria 3145, Ausralia. imad.moosa@buseco.monash.edu.au. We hank wo anonymous referees for heir insighful and consrucive commens, which improved he presenaion of he paper considerably, and Ron Ripple for providing he crude oil daa used in his sudy.

2 214 Inernaional Journal of Business and Economics following quesions. (i) Should we hedge a paricular posiion or leave i uncovered? (ii) If he decision o hedge is aken, wha hedging insrumen should be used? (iii) Having seleced he hedging insrumens, how much of he posiion should be hedged? The hird quesion, which is addressed in his paper, perains o he deerminaion of he hedge raio. A significan srand of lieraure is concerned wih he mehods used o esimae he hedge raio and wheher or no he esimaion echnique and/or he specificaion of he underlying model have implicaions for hedging effeciveness. Examples of differences in model specificaion are linear versus nonlinear models, saic versus dynamic models, symmeric versus asymmeric models, firs difference versus level models, and firs difference versus error correcion models. Esimaion mehods refer o procedures such as ordinary leas squares (OLS), maximum likelihood, Kalman filers, and mehods for models wih ARCH/GARCH errors. The same model specificaion may produce differen esimaes for he hedge raio when differen esimaion mehods are used. This paper exends he lieraure on he esimaion of he hedge raio by using wavele analysis and a wo-sage regime swiching process o esimae he opimal hedge raio using fuures conracs. Wavele analysis is becoming popular in economics and finance, as i has been used o sudy a number of issues. The echnique has been used, for example, o sudy sock reurns and economic aciviy (Gallegai, 2008), he relaion beween financial variables and real economic aciviy (Kim and In, 2003), and he decomposiion of he relaion beween money and income (Ramsey and Lampar, 1998). Oher applicaions can be found in Gencay e al. (2001) who describe he use of waveles and oher filering mehods in finance and economics. Wavele analysis allows he use of long ime inervals when more precise lowfrequency informaion is needed and shor ime inervals when more precise highfrequency informaion is needed. Using a paricular filer, a ime series is ransformed ino muliple series consising of wavele coefficiens and o a single series consising of scaling coefficiens. The scaling coefficiens reflec long-erm variaion, which would exhibi a similar rend o he original series. To our knowledge, his is he firs paper in which wavele analysis is employed o esimae asymmeric hedge raios using daily daa. Boh symmeric and asymmeric models are used o esimae he hedge raio by wavele analysis. The resuls obained by applying wavele analysis are hen compared wih hose obained from oher esimaion mehods and models, such as he naïve model and he EC-GJR-GARCH mehod. Anoher advanage of wavele decomposiion is ha high-frequency daa can be used o creae he wavele deail ha broadly maches he hedge horizon. This procedure solves he problem of small sample size resuling from aemps o mach daa frequency wih he hedge horizon. More specifically, his paper invesigaes hree issues. The firs issue is wheher or no he specificaion of he model and/or he esimaion mehod used o calculae he hedge raio has a significan effec on risk reducion and hence hedging effeciveness. The second issue is wheher or no he calculaion of separae hedge

3 Elizabeh A. Maharaj, Imad Moosa, Jonahan Dark, and Param Silvapulle 215 raios for rising and falling markes leads o improvemen in he performance of he hedge. As par of his exercise we conduc formal esing of he equaliy of he separae hedge raios. The hird issue is an invesigaion ino he relaion beween hedge horizon and daa frequency. Chen e al. (2003, 2004) sugges ha he daa frequency should be mached wih he hedge horizon. When hedging over long horizons, his procedure resuls in a subsanial decrease in sample size. Wavele analysis does no suffer from his limiaion, because high-frequency (e.g., daily) daa can be used o uncover long-run dynamics. Shedding some ligh on hese issues should be rewarding, no only in erms of he academic value added o he lieraure bu also in erms of he pracical implicaions of he resuls, given ha risk managemen commands increasing imporance as a corporae aciviy. The paper is organised as follows. Following he presenaion of a brief lieraure survey, a descripion of wavele analysis is presened in general erms. This is followed by an exposiion of he wo-sage regime swiching hreshold model and how wavele analysis is applied o he measuremen of he hedge raio. Then we will describe a daase and presen empirical resuls. The paper ends wih concluding remarks. 2. A Look a he Lieraure The lieraure on esimaing he hedge raio is exensive (e.g., Moosa, 2003a, Ch. 5; Chen e al., 2003). The early lieraure focused on he esimaion of he hedge raio by applying OLS o a firs difference model in which he response variable is he rae of reurn on he unhedged posiion and he predicor variable is he rae of reurn on he hedging insrumen (e.g., Ederingon, 1979). This model has been criicised as being misspecified, consequenly leading o erroneous hedging decisions. For example, Lien (1996) has shown analyically ha if he prices of he asse underlying he hedged posiion and ha of he hedging insrumen are coinegraed, he opimal hedge raio mus be esimaed from an error correcion model raher han a sraigh firs difference model. Lien concludes ha a hedger acing on he hedge rae esimaed from an error correcion model will be underhedged. However, Moosa (2003b) has shown empirically ha he esimaed hedge raio does no appear o be sensiive o model specificaion. Specifically, he poins ou ha while Lien s analyical resuls are sound, he quaniaive difference beween he hedge raios derived from a firs difference model and an error correcion model is negligible o he exen ha here are hardly any pracical ramificaions. Anoher issue ha ofen arises in he hedging lieraure perains o he assumpion of a consan hedge raio, as some economiss argue for ime-varying hedge raios derived from ARCH/GARCH models or sae space models (e.g., Kroner and Sulan, 1993). Ye anoher modificaion o he basic model used o calculae he hedge raio is he inroducion of nonlineariy, as suggesed by Broll e al. (2001). Chen e al. (2003) provide a review of differen heoreical approaches o he esimaion of he opimal hedge raio, based on minimum variance, mean-

4 216 Inernaional Journal of Business and Economics variance, expeced uiliy, mean exended-gini coefficien, and semivariance. They also discuss various ways of esimaing he hedge raio, ranging from OLS o complicaed heeroscedasic coinegraion mehods. An imporan issue ha is relaed o nonlineariy (and also o ime-varying hedge raios), which is no deal wih exensively in he lieraure, is ha of asymmery. If he resricive assumpion of risk aversion is relaxed, hen hedgers would aim a maximising uiliy (raher han minimising risk), in which case boh risk and reurn will be aken ino accoun. In a bull marke, hedgers may reduce he hedge raio in a rising marke o benefi from any rise in prices. Conversely, hey end o use a higher hedge raio in a declining marke o proec hemselves from adverse marke moves. This proposiion is suppored by he noion of one-sided risk. For example, Adams and Monesi (1995) found ha corporae managers are mosly concerned wih downside risk. Pey and Sco (1981) suggesed ha many Forune 500 firms idenify risk as he possibiliy of falling below a arge reurn. Bu even if a risk-minimising hedge raio is chosen, here is no reason o assume a priori ha his raio has he same value in a rising marke as in a falling marke. Symmeric hedge raios would be obained only if he rae of reurn on he unhedged posiion reacs in a similar manner o a rise as o a fall in he rae of reurn on he hedging insrumen. Given ha he opimal hedge raio depends on correlaion beween he raes of reurn on he unhedged posiion and ha on he hedging insrumen, correlaion asymmery would necessarily imply hedge raio asymmery. Correlaion asymmery means ha correlaion beween wo raes of reurn in a bull marke is differen from wha is found in a bear marke. This is imporan for he hedging decision because unsable correlaion makes i difficul o hedge exposure by aking an offseing posiion on oher asses. From he risk manager s poin of view, knowledge of he paricular paern of correlaion under differen marke condiions is a major concern. This is because he effeciveness of hedging based on esimaed correlaion depends on how accurae he esimae is, and how represenaive he esimaion period is, of he period when hedging is mos needed. Failure o ake correlaion asymmery ino consideraion may lead o subopimal posiions for he hedger. Demirer and Charnes (2003) argue ha analysis of correlaion asymmery is even more imporan for risk managers who are concerned wih downside risk. For a hedger who aims o avoid falling beyond a arge rae of reurn, failing o ake higher downside correlaion ino accoun leads o lower han opimal hedge posiions, hus reducing he effeciveness of he hedge. Recen research shows ha correlaion asymmery does exis in financial markes, specifically ha correlaion is sronger in bear markes han in bull markes (e.g., Booksaber, 1997; Lorean and English, 2000; Bekaer and Wu, 2000; Ang and Bekaer, 2000; Ang and Chen; 2002). These sudies invariably deal wih asymmeric correlaion beween individual socks and he marke as a whole. Only a few sudies have exended his lieraure o he derivaives marke and hence o he hedging problem. Brooks e al. (2002) analysed he impac of asymmery on ime-varying hedge raios involving daily FTSE 100 sock index and sock index fuures conracs over he period 1985 o They concluded ha a model allowing for ime

5 Elizabeh A. Maharaj, Imad Moosa, Jonahan Dark, and Param Silvapulle 217 variaion and asymmery gives superior hedging performance. Demirer and Charnes (2003) sudied he correlaion srucure of spo and fuures reurns across 10 markes and analysed he implicaions of asymmeric correlaion for opimal hedge posiions. For sock index and oil fuures conracs, hey found correlaion beween spo and fuures reurns o be much greaer on he downside han on he upside, paricularly for exreme moves. Their analysis also indicaes ha hedge raios ha minimise downside risk yield generally beer resuls han he radiional minimum variance hedge raios. Chen e al. (2003, 2004) argue ha daa frequency should be mached wih he hedge horizon. For example a hedge horizon of one week should employ weekly daa o esimae hedge raios. The maching of daa frequency wih he hedge horizon clearly becomes problemaical when hedging over (say) a hree-monh horizon, because of he subsanial reducion in sample size. One approach o dealing wih his problem is o creae overlapping daases while using sophisicaed modelling approaches o correc for he effec of overlapping daa on he esimaors. Chen e al. (2004) employ an alernaive approach whereby shor-run and long-run hedge raios are esimaed. However, he esimaion of shor-run hedge raios (hedging over a period ranging beween one week and eigh weeks) sill suffers from he need o mach daa frequency wih he hedge horizon, resuling in smaller sample size. This problem moivaes he use of wavele decomposiion, which is employed in his paper. High-frequency (e.g., daily) daa can be used o creae he wavele deail ha broadly maches he hedge horizon. Having gone hrough a brief lieraure review, i seems ha here is some room for exending he lieraure by using he echnique of wavele analysis o esimae he hedge raio. This is paricularly so in view of he imporance of he issue of he sensiiviy of he resuls, and herefore hedging effeciveness, o he underlying economerics. The nex sep is o presen a descripion of wavele analysis. 3. Wavele Analysis Wavele analysis is a echnique ha can be applied o any (e.g., saionary or nonsaionary, linear or nonlinear) ime series. I is a windowing echnique wih variable size regions ha allows he use of long ime inervals when more precise low-frequency informaion is needed, and shor ime inervals when more precise high-frequency informaion is needed. If, for example, we wan o examine he behaviour of daily ime series over differen ime periods (such as weeks or monhs), emporal aggregaion would resul in he loss of useful informaion. Wih wavele analysis, his can be done wihou aggregaion and hence wihou any loss of informaion (Gencay e al., 2003). In wha follows, a brief descripion of wavele analysis is presened (see Percival and Walden (2000) for more deails). Given a signal represened by { x (): < < }, he collecion of coefficiens { W( λ, ): λ > 0, < < } is known as he coninuous wavele ransform of x() such ha:

6 218 Inernaional Journal of Business and Economics W( λ, ) = ψ λ, ( u) x( u) du (1) and 1 u ψλ, ( u) ψ λ λ, (2) where λ is he scale associaed wih he ransformaion and is is locaion. The funcion () ψ is a wavele filer ha saisfies he properies: and ψ ( u) du = 0 (3) ψ 2 ( u) du = 1. (4) I is admissible if is Fourier ransform: iωu Ψ( ω) ψ( u) e dω (5) is such ha 2 Ψ( ω) 0 < dω <. (6) ω Many ypes of wavele filers are available, he mos commonly used of which are he Haar and Daubechies filers (Percival and Walden, 2000, Ch. 4). By applying wavele analysis o ime series observed over discree poins in ime ( = 1, K, T ), we would be ineresed in he discree wavele ransform (DWT), which can be hough of as a sensible sub-sampling of W( λ, ), in which we deal wih dyadic scales (Percival and Walden, 2000, Ch. 1). This means ha we need o pick λ i o be of he form 2 j 1, j = 1, K, J and hen wihin a given dyadic scale 2 j 1, we pick poins in ime ha are separaed by muliples of 2 j. For he scale λ = 2 j1, here are T 2 j i j = T wavele coefficiens ha can be defined. For example, 8 if T = 256 = 2, eigh dyadic scales 2, 0 K,2 7 will be available a which here are 128, 64, 32, 16, 8, 4, 2 and 1 wavele coefficiens respecively. Hence, he wavele coefficiens for he eigh scales accoun for he DWT coefficiens, he number of which is equal o one less he lengh of he ime series. The single remaining coefficien (which, ogeher wih he oal number of DWT coefficiens, is equal o

7 Elizabeh A. Maharaj, Imad Moosa, Jonahan Dark, and Param Silvapulle 219 he lengh of he ime series) is known as he scaling coefficien. In pracice, we may choose o decompose a ime series using a fewer number of scales, depending on he 0 4 lengh of he series. For example, if we pick 5 scales 2, K,2, hen he wavele coefficiens corresponding o hese scales would be 128, 64, 32, 16 and 8 respecively, whereas he eigh remaining coefficiens would be he scaling coefficiens. The wavele coefficiens are proporional o he differences in averages of he ime series observaions a each scale, whereas he scaling coefficiens are proporional o he averages of he original series over he larges scale. The scaling coefficiens reflec long-erm variaions, which would exhibi a similar rend o he original series. Long ime scales give more low-frequency informaion abou he series, whereas shor ime scales give more high frequency informaion abou he ime series. The DWT re-expresses a ime series in erms of coefficiens ha are associaed wih a paricular ime and paricular dyadic scale. These coefficiens are fully equivalen o he informaion conained in he original series in ha a ime series can be perfecly reconsruced from is DWT coefficiens. The reconsrucion of a ime series X from is wavele and scaling coefficiens defines wha is known as a muli-resoluion analysis (MRA) of X (Percival and Walden, 2000, Ch. 4). The non-decimaed wavele ransformaion, or he saionary wavele ransform (MODWT), is a modificaion of he DWT in ha i deals wih all poins in ime and no jus muliples of 2 j. For example, if a series of lengh T = 256 is considered, here would be 256 wavele coefficiens a each scale. Reaining all possible imes a each scale of he MODWT decomposiion has he advanage of reaining he ime invarian propery of he original series. In general, he wavele coefficiens a scale λ i are associaed wih frequencies ( j 1) j in he inerval [2,2 ]. Hence, he wavele coefficiens a he firs scale, λ 1, are associaed wih frequencies in he inerval [2 2, 2 1 ], whereas he coefficiens a he second scale λ 2 are associaed wih frequencies in he inerval [2 3, 2 2 ], and so on. If he ime series under consideraion are daily, hen he firs scale capures he dynamics of he ime series wihin a 2- o 4-day period, he second scale capures he dynamics of he ime series wihin a 4- o 8-day period, and so on. Here λ 1, which is he shores ime scale, conains he highes frequency informaion abou he ime series. The level of informaion of he ime series conveyed from one scale o he nex decreases as he ime scale increases. 4. The Threshold Model and Hedge Raios In his secion we presen he specificaion of he models used o esimae symmeric and asymmeric hedge raios, applying hese models o oal series and he wavele ransformaion of he reurn series. The simples model used o calculae he hedge raio is he firs difference model: s α hδf u, (7) Δ = 0

8 220 Inernaional Journal of Business and Economics where s and f are he logs of he spo and fuures prices and h is he riskminimising hedge raio. A limiaion of (7) is ha i does no capure he asymmeric naure of he response of s o f, which makes i invalid if he response of s o f depends on wheher f is rising or falling. In order o capure hese asymmeric effecs, Δ f is decomposed as follows: Δf if Δf 0 f = 0 oherwise (8) Δf if Δf 0 f = 0 oherwise. (9) Δ Δ Equaion (7) can herefore be rewrien as: Δs = α h Δf h Δf u. (10) 0 The asymmeric hedge raio from o 1 requires he hedger o form expecaion a ime of he direcion of he fuures price change from ime o 1. An expecaion of a posiive (negaive) fuures price change implies he implemenaion of h ( h ). Here, we make he assumpion ha he hedger is able o predic accuraely he direcion, bu no he magniude, of he fuures price change each period. Given ha reurns are largely unpredicable, i is acknowledged ha his assumpion may be unrealisic and ha is adopion produces oversaemen of he risk reducion resuling from hedging. If he proposed approach (assuming perfec foresigh) produces superior risk reducion o some of he alernaive approaches used in he lieraure, his assumpion is relaxed in he second sage of he analysis where a model is used o predic he direcion of price changes. If, however, he proposed approach does no produce superior risk reducion, hen i can be concluded ha even he bes case scenario fails o deliver improvemen in hedging effeciveness relaive o exising approaches. Hedging effeciveness is measured by he variance raio ( VR ) and variance reducion (VD ). In he absence of asymmery, he variance raio is calculaed as 2 σ ( Δs ) VR =, 2 (11) σ ( Δs hδf ) 2 2 in which case he null hypohesis σ ( Δs ) = σ ( Δs hδf ) is rejeced if VR > F( n 1, n 1), where n is he number of observaions. If here are asymmeric raios, he VR is calculaed as: 2 σ ( Δs ) VR = σ 2 ( Δs I h Δf I h Δf, (12) ) where I = 1 if Δf 0, I = 0 if Δf 0, I = 1 if Δf 0, and I = 0 if Δf 0. Variance reducion can be calculaed from he variance raio as:

9 Elizabeh A. Maharaj, Imad Moosa, Jonahan Dark, and Param Silvapulle VD = 1. (13) VR In his sudy we decompose spo and fuures reurns ino wavele deail series a differen resoluions using he MODWT. The symmeric model represened by (7) is esimaed for reurns and wavele deail series. As such he esimaed slope * * * * * coefficien is h = cov( Δs, Δf ) var( Δf ), where Δ s and Δ f are he appropriae wavele deail series. To obain he corresponding asymmeric model (10), posiive and negaive fuures reurns and posiive and negaive wavele deail values are separaed a each level. Likewise, he asymmeric h and h coefficiens are obained for each wavele scale. A es for symmery will hen be conduced. 5. Daa and Empirical Resuls The empirical resuls are derived from he analysis of hree ses of daily daa. 1. The NYMEX fuures price for he near-monh conrac on ligh, swee crude oil and he spo price of he Wes Texas Inermediae (WTI) crude, which is he primary deliverable crude oil agains he NYMEX conrac. Fuures prices were obained from NYMEX, whereas spo prices were obained from he US EIA daabase. Daily daa are used in his analysis covering he period 7 June 1989 o 12 Ocober 2005 (a oal of 4096 observaions). 2. A daase on soybeans spo and fuures prices ha consiss of 2048 daily observaions from 26 April 1999 o 28 February Boh series were obained from Daasream. 3. A daase ha consiss of 4096 daily observaions on he S&P500 index and fuures prices, covering he period from 5 January 1988 o 19 March The index daa were obained from IRESS, whereas he floor selemen price for fuures conracs was obained from he Chicago Mercanile Exchange. The splicing of fuures conracs o form a coninuous ime series is an issue ha arises in all sudies of his kind. When modelling spo and fuures dynamics, any jump on rollover is commonly ignored (e.g., Koumos and Tucker, 1996; Baillie and Myers, 1991; Bhar, 2001). Furhermore, visual examinaion of he daa reveals no significan jumps in he fuures reurn series on rollover. The splicing of crude oil daa is based on he procedure used by Ripple and Moosa (2007), which is based on he use of price and associaed rading aciviy. As a conrac approaches mauriy, he marke shifs aenion away from he near-monh conrac o he nex-o-nearmonh conrac before he near-monh conrac reaches is las rading day. When boh he daily rading volume and open ineres for he nex-o-near monh conrac exceed hose for he near-monh conrac, his is aken as evidence ha he marke s aenion has shifed away from he near-monh conrac. A his poin, he series is shifed o he nex-o-near monh conrac. For he S&P500 series, a rollover of 10 rading days prior o expiry is used, whereas a 20-rading day rollover is employed for he soybeans daa.

10 222 Inernaional Journal of Business and Economics Table 1 shows descripive saisics of reurns measured as he firs log differences of prices. For each daase, he sample means of he series are close o 0. The reurn disribuions are non-normal, exhibiing excess kurosis and negaive skewness. Non-normaliy is implied by he high values of he Jarque-Bera saisic. Table 1. Descripive Saisics of Daily Fuures and Spo Reurn Series Panel A: Crude Oil Reurn Series Mean Sandard Deviaion Excess Kurosis Skewness Jarque-Bera Saisic Δ f Δ s Panel B: Soybeans Reurn Series Mean Sandard Deviaion Excess Kurosis Skewness Jarque-Bera Saisic Δ f Δ s Panel C: S&P500 Reurn Series Mean Sandard Deviaion Excess Kurosis Skewness Jarque-Bera Saisic Δ f Δ s Noes: Δ f and Δ s are fuures and spo reurns respecively. For crude oil and S&P500 daases, wavele decomposiion resuls in an scale MRA because he lengh of he daily series is equal o 4096 = 2. For he soybean daase, wavele decomposiion produces a 10-scale MRA because he 11 lengh of he daily series is equal o 2048 = 2. In pracice, however, we do no need o consider all possible scales in he decomposiion because larger scales resul in very smooh deail series, hus failing o provide much useful informaion abou he original ime series. Consequenly, we use an 8-scale decomposiion for each daase and he leas asymmeric wavele filer of lengh 8 o decompose he series. The leas asymmeric filer is a variaion of he Daubechies filer, which has been chosen because i has good alignmen properies (see Percival and Walden, 2000). The resuls presened in Tables 2 and 3 cover he reurn series (wihou wavele decomposiion) and 8 wavele deail series where wavele decomposiion is performed on boh he spo and fuure reurn series of each daase. Table 2 displays he resuls of esimaing asymmeric and symmeric models over he whole sample period for daily reurns and for he wavele deail series. The hedge coefficiens for reurns are obained by esimaing (7) for he symmeric model and (10) for he asymmeric model. The resuls include he esimaed hedge raios (wo in he case of he asymmeric model) and he associaed p-values for judging heir saisical significance, as well as he p-value for esing he null of symmery (i.e., h = h ). Also repored are he variance raio (VR ) and variance reducion ( VD ) o assess he effeciveness of he hedge, using he hedge raios derived from asymmeric and symmeric models. Saring wih he asymmeric model, he esimaed hedge raios are significan and very close o 1. The null

11 Elizabeh A. Maharaj, Imad Moosa, Jonahan Dark, and Param Silvapulle 223 hypohesis of symmery is rejeced only in he cases of deail 7 for he crude oil daa, deail 5 for he soybeans daa, and for he daily reurn series of he S&P500 daa. The variance raio is significan in all cases, producing variance reducion of over 75% for he crude oil daa, over 70% for he soybeans daa, and over 90% for he S&P500 daa. For all hree daases, very close resuls are produced by he symmeric model for he esimaed hedge raios, variance raios and variance reducions. One would end o hink ha if here is no evidence for asymmery, hen he asymmeric and symmeric models would produce similar resuls in erms of hedging effeciveness. Wha is more imporan perhaps is ha he hedging effeciveness produced by a simple OLS esimaion of he hedge raio (daily reurn series) is no significanly differen from wha is produced by using wavele decomposiion. Table 2. Hedge Raio Esimaes and Oher Measures for Daily Reurn Series and Wavele Decomposiion: Toal Sample Scale of Wavele Decomposiion ( j ) Panel A: Crude Oil Dynamics in days Asymmeric Model Daily Reurns Series Deail 1 Deail 2 Deail 3 Deail 4 Deail 5 Deail 6 Deail 7 Deail 8 Hedge Raio h Hedge Raio h Tes for Symmery p-value VR VD Symmeric Model Hedge Raio VR VD Noes: VR and VD are he variance raio and variance reducion respecively. h and h are hedge raios corresponding o posiive and negaive reurns as in (10).

12 224 Inernaional Journal of Business and Economics Scale of Wavele Decomposiion ( j ) Table 2. (Coninued) Panel B: Soybeans Dynamics in days Asymmeric Model Daily Reurn Series Deail 1 Deail 2 Deail 3 Deail 4 Deail 5 Deail 6 Deail 7 Deail 8 Hedge Raio h Hedge Raio h Tes for Symmery p-value VR VD Symmeric Model Hedge Raio VR VD Scale of Wavele Decomposiion ( j ) Table 2. (Coninued) Panel C: S&P Dynamics in days Asymmeric Model Daily Reurn Series Deail 1 Deail 2 Deail 3 Deail 4 Deail 5 Deail 6 Deail 7 Deail 8 Hedge Raio h Hedge Raio h Tes for Symmery p-value VR VD Symmeric Model Hedge Raio VR VD

13 Elizabeh A. Maharaj, Imad Moosa, Jonahan Dark, and Param Silvapulle 225 Table 3. In-Sample and Ou-of-Sample Hedge Raio Esimaes and Oher Measures for Daily Reurn Series and Wavele Decomposiion Scale of Wavele Decomposiion ( j ) Panel A: Crude Oil Dynamics in days Asymmeric Model Daily Reurns Series Deail 1 Deail 2 Deail 3 Deail 4 Deail 5 Deail 6 Deail 7 Deail 8 Hedge Raio h Hedge Raio h Tes for Symmery p-value VR (in-sample) VR (ou-of-sample) VD (in-sample) VD (ou-of-sample) Symmeric Model Hedge Raio VR (in-sample) VR (ou-of-sample) VD (in-sample) VD (ou-of-sample) Noes: VR and VD are he variance raio and variance reducion respecively. h and h are hedge raios corresponding o posiive and negaive reurns as in (10). I is arguable ha hedging effeciveness should be assessed on an ou-of-sample basis. For his reason, we spli he samples ino wo hirds for he esimaion period and one hird for he esing period as follows. 1. Crude oil: he esimaion period is 7 June 1989 o 25 April 2000 whereas he esing period is 26 April 2000 o 12 Ocober Soybeans: he esimaion period is 26 April 1999 o 16 July 2004 whereas he esing period is 17 July 2004 o 28 February S&P500: he esimaion period is 5 January 1988 o 19 Ocober 1998 whereas he esing period is 20 Ocober 1998 o 19 March 2004.

14 226 Inernaional Journal of Business and Economics Scale of Wavele Decomposiion ( j ) Table 3. (Coninued) Panel B: Soybeans Dynamics in days Asymmeric Model Hedge Raio Daily Reurn Series Deail 1 Deail 2 Deail 3 Deail 4 Deail 5 Deail 6 Deail 7 Deail 8 h Hedge Raio h Tes for Symmery p- value VR (in-sample) VR (ou-of-sample) VD (in-sample) VD (ou-of-sample) Symmeric Model Hedge Raio VR (in-sample) VR (ou-of-sample) VD (in-sample) VD (ou-of-sample) This sample spli enables us o measure he hedging effeciveness in-sample and ou-of-sample. We implemen ou-of-sample hedging as follows. 1. Hedge raios are esimaed using he in-sample daa. These hedge raios hen remain consan hroughou he ou-of-sample period (i.e., hey are no updaed each period by adding an addiional observaion o he esimaion period). 2. A ime (he firs ou-of-sample period), he hedger deermines he likely direcion of he fuures price change from ime o ime 1. As saed above, we assume he bes case scenario in he firs sage, where he hedger correcly predics he direcion each period. 3. The hedger akes he appropriae posiion in he fuures conracs, hen he porfolio reurn from o 1 is calculaed. 4. A 1, he hedger deermines he direcion of he fuures price change and if necessary updaes he fuures posiion, hen he porfolio reurn is calculaed.

15 Elizabeh A. Maharaj, Imad Moosa, Jonahan Dark, and Param Silvapulle 227 This one-sep-ahead procedure is performed hrough he end of he ou-of-sample period, where oal porfolio reurns can be calculaed. Scale of Wavele Decomposiion ( j ) Table 3. (Coninued) Panel C: S&P Dynamics in days Asymmeric Model Daily Reurn Series Deail 1 Deail 2 Deail 3 Deail 4 Deail 5 Deail 6 Deail 7 Deail 8 Hedge Raio h Hedge Raio h Tes for Symmery p-value VR (in-sample) VR (ou-of-sample) VD (in-sample) VD (ou-of-sample) Symmeric Model Hedge Raio VR (in-sample) VR (ou-of-sample) VD (in-sample) VD (ou-of-sample) The spli-sample resuls are repored in Table 3 for he asymmeric and symmeric models and for he daily reurn series and he 8 wavele deails. Evidence for asymmery was found in he daily reurn and deail 7 series for he crude oil daa, in deails 5 and 7 series for he soybean daa, and in he daily reurn and deails 5 and 7 series for he S&P500 daa. Bu while here is some evidence of asymmery, we find ha hedging effeciveness is no affeced by using he hedge raios esimaed from he symmeric or asymmeric models. Furhermore, hedging effeciveness is generally no improved by using wavele decomposiion irrespecive of wheher we use in-sample or ou-of-sample analysis. The excepion is he (in-sample) soybeans daase, where he wavele approach resuls in he bes variance reducion. The resuls of Table 2 are confirmed by he resuls repored in Table 3. The resuls also largely suppor he empirical evidence produced by Chen e al. (2004), which demonsraes ha as daa frequency is reduced (from, say, daily o quarerly), he hedge raio approaches uniy, which is he value implied by he naïve model. This resul is consisen wih he hedge raios for crude oil and soybeans, which approach uniy as he scale increases. Chen e al. (2004) also demonsrae ha

16 228 Inernaional Journal of Business and Economics hedging effeciveness increases wih hedge horizon. The resuls presened in his paper, however, are no supporive of his finding. I seems, herefore, ha he economeric sophisicaion inroduced by using wavele analysis does no lead o any improvemen in hedging effeciveness. One possible reason for he poor performance of he wavele hedge raios is heir inabiliy o mach he hedge horizon o a wavele deail series. If implemening a hedge over he course of a week, should one employ deail 1 (2 o 4 days) or deail 2 (4 o 8 days)? The proposed approach, herefore, involves a rade-off beween no having o reduce he sample (paricularly for long-erm hedges) and he inabiliy of he procedure o mach exacly he hedge horizon wih an appropriae scale. Table 4. In-Sample and Ou-of-Sample Resuls for Oher Mehods Naive Panel A: Crude oil OLS EC-GARCH (Symmeric) EC-GJR-GARCH (Asymmeric) VR (in-sample) VR (ou-of-sample) VD (in-sample) VD (ou-of sample) Panel B: Soybeans Naive OLS EC-GARCH (Symmeric) EC-GJR-GARCH (Asymmeric) VR (in-sample) VR (ou-of-sample) VD (in-sample) VD (ou-of sample) Panel C: S&P500 Naive OLS EC-GARCH (Symmeric) EC-GJR-GARCH (Asymmeric) VR (in-sample) VR (ou-of-sample) VD (in-sample) VD (ou-of sample) Noes: VR and VD are he variance raio and variance reducion respecively. The quesion ha arises here is wheher or no hedging effeciveness can be improved by inroducing anoher kind of economeric sophisicaion. For his reason, we measure VR and VD based on hedge raios esimaed using symmeric EC- GARCH and asymmeric EC-GJR-GARCH models. The in-sample and ou-ofsample resuls are repored in Table 4, which also includes he resuls of using he naïve model and OLS (corresponding o he oal series in Tables 2 and 3). Again, here is no evidence showing ha more sophisicaed approaches boos hedging effeciveness significanly. Consider, for example, he ou-of-sample variance reducion for crude oil: i is 0.76 for he naïve model, 0.77 for he OLS model, 0.77 for he wavele asymmeric and symmeric models (deail 1), and 0.78 for he EC- GARCH symmeric model and for he EC-GJR-GARCH asymmeric model. I is

17 Elizabeh A. Maharaj, Imad Moosa, Jonahan Dark, and Param Silvapulle 229 clear ha model specificaion and economeric sophisicaion do no conribue o he improvemen of hedging effeciveness. 6. Conclusion This paper uilises wavele analysis o esimae hedge raios. This echnique allows he use of long ime inervals when more precise low-frequency informaion is needed and shor ime inervals when more precise high-frequency informaion is needed. This makes i possible o examine he feaures of a daily ime series in differen ime inervals, such as weekly or monhly, wihou he need for aggregaion, which would resul in loss of useful daa. By using wavele analysis in conjuncion wih symmeric and asymmeric models, we calculae various hedge raios ha are subsequenly used o assess he hedging effeciveness of crude oil fuures conracs, soybeans fuures conracs and he S&P500 fuures conrac. For comparaive purposes, oher models and mehods are used o esimae hedge raios and corresponding hedging effeciveness. Based on he variance raio, we find ha wavele analysis does no lead o an improvemen in hedging effeciveness, irrespecive of wheher symmeric or asymmeric hedge raios are used for consrucing he hedge and irrespecive of he conrac used (crude oil, soybeans or S&P500). The general conclusion reached by his sudy is ha economeric sophisicaion does no boos hedging effeciveness. References Adams, J. B. and C. J. Monesi, (1995), Major Issues Relaed o Hedge Accouning, Newark (Connecicu): Financial Accouning Sandard Board. Ang, A. and G. Bekaer, (2000), Inernaional Asse Allocaion wih Time-Varying Correlaions, Working Paper, Columbia Universiy. Ang, A. and J. Chen, (2002), Asymmeric Correlaions of Equiy Porfolios, Journal of Financial Economics, 63, Baillie, R. and R. Myers, (1991), Bivariae GARCH Esimaion of he Opimal Commodiy Fuures Hedge, Journal of Applied Economerics, 6, Bekaer, G. and G. Wu, (2000), Asymmeric Volailiy and Risk in Equiy Markes, Review of Financial Sudies, 13, Bhar, R., (2001), Reurn and Volailiy Dynamics in he Spo and Fuures Markes in Ausralia: An Inervenion Analysis in a Bivariae EGARCH-X Framework, Journal of Fuures Markes, 21, Booksaber, R., (1997), Global Risk Managemen: Are We Missing he Poin? Journal of Porfolio Managemen, 23(3), Broll, U., K. W. Chow, and K. P. Wong, (2001), Hedging and Nonlinear Risk Exposure, Oxford Economic Papers, 53, Brooks, C., O. T. Henry, and G. Persand, (2002), The Effec of Asymmeries on Opimal Hedge Raios, Journal of Business, 75, Chen, S. S., C. Lee, and K. Shresha, (2003), Fuures Hedge Raios: A Review, Quarerly Review of Economics and Finance, 43,

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