Optimal Hedge Ratio for Brent Oil Market; Baysian Approach

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1 Internationa Letters of Socia and Humanistic Sciences Onine: ISSN: , Vo. 37, pp doi: / SciPress Ltd., Switzerand Optima Hedge Ratio for Brent Oi Maret; Baysian Approach Mohsen Mehrara a, Monire Hamdar b Facuty of Economics, University of Tehran, Kargar-e-shomai, Po Box , Tehran, Iran a,b E-mai address: mmehrara@ut.ac.ir, hamdar@ut.ac.ir ABSTRACT This paper examines the optima hedging ratio (OHR) for the Brent Crude Oi Futures using daiy data over the period 1990/17/8-2014/11/3. To gain OHR, it is empoyed a Vector Autoregressive (VAR) and Vector Error Correction (VEC) and Baysian Vector Autoregressive (BVAR) modes. At ast, the efficiency of these cacuated OHR are compared through Edrington's index. Keywords: Optima Hedging Ratio; OHR; VAR; VECM; BVAR; Edrington's efficiency index 1. INTRODUCTION Price ris management, using hedging toos ie futures and options and their effectiveness, is an active area of research. Hedging decisions based on futures contracts have to dea with finding optima hedge ratio and hedging effectiveness. Roe of hedging using mutipe risy assets and using futures maret for minimizing the ris of spot maret fuctuation has been extensivey researched. Traditionay, hedging is envisaged using a hedge ratio of -1, i.e., taing a position in futures contract which is equa in magnitude and opposite in sign to the position in spot maret. If the movement of changes in spot prices and futures prices is exacty the same, then such a strategy eiminates the price ris. Such a perfect correation between spot and futures prices is rarey observed in marets and hence a need was fet for a better strategy. Johnson (1960) proposed minimum variance hedge ratio (MVHR), which factored in ess than perfect reationship between spot and futures prices. Ris was defined as the variance of returns on a two-asset hedged position. Foowing Doan, Litterman and Sims (1984), the Bayesian approach to the estimation of vector autoregressive (VARs) is empoyed. The forecasting modes have traditionay been formuated as simutaneous equation structura modes. However, for a variety of reasons structura modes have proved unreiabe for forecasting. The VAR modes have aso been criticized insofar as they ac strong theoretica justification over and above the use of theory as a guide in deciding which variabes to incude in the anaysis. Doan, Litterman and Sims (1984) in an attempt to improve the forecasting performance of unrestricted VARs suggested that they coud be estimated using Bayesian techniques which tae account of any prior information which may be avaiabe to the modeer. It is this Bayesian approach to parameter estimation in vector autoregressive which is empoyed in this study. SciPress appies the CC-BY 4.0 icense to wors we pubish:

2 Internationa Letters of Socia and Humanistic Sciences Vo METHODOLOGY AND EMPIRICAL RESULTS In this section we empoy the VAR, VEC and BVAR to cacuate the optima hedge ratio(ohr). The financia variabes used in the mode are spot and futures price of Brent Oi. The data series are obtained from Energy Information Agency (EIA). The data are daiy from 1990/17/8-2014/11/ Hedge Ratio and Hedging Effectiveness The optima hedge ratio is defined as the ratio of the size of position taen in the futures maret to the size of the cash position which minimizes the tota ris of portfoio. The return on an unhedged and a hedged portfoio can be written as: R U = S t+1 S t R H = (S t+1 S t ) H(F t+1 F t ) Variances of an unhedged and a hedged portfoio are: Var(U) = σ s 2 Var(H) = σ s 2 + H 2 σ F 2 2Hσ S,F where, S t and F t are natura ogarithm of spot and futures prices, H is the hedge ratio, R H and R U are return from unhedged and hedged portfoio, σ S and σ F are standard deviation of the spot and futures return and σ S,F is the covariance. Hedging effectiveness is defined as the ratio of the variance of the unhedged position minus variance of hedge position over the variance of unhedged position. Edrington s efficiency index = Var(U) Var(H) Var(U) (1) Modes for Cacuating Hedging Effectiveness and Hedge Ratio Severa modes have been used to cacuate hedge ratio and hedging effectiveness such as Vector Autoregressive regression (VAR) mode, Vector Error Correction mode (VEC), Baysian Vector Autoregressive regression (BVAR) mode. The VAR modes eiminates probems of autocorreation but it does not consider the possibiity of ong term integration. The advantage of the Bayesian approach to statistics is that provides a genera method for combining a modeer s beiefs with the evidence contained in the data The VAR Mode The bivariate VAR Mode is preferred over the simpe OLS estimation because it eiminates probems of autocorreation between errors and treat futures prices as endogenous variabe. The VAR mode is represented as

3 84 Voume 37 R st = α s + i=1 β si R st i + j=1 γ fj R ft j + ε st (2) R ft = α f + i=1 β fi R ft i + j=1 γ sj R st j + ε ft (3) The error terms in the equations, ε St, and ε Ft are independenty identicay distributed (IID) random vector. The minimum variance hedge ratio are cacuated as h = σ sf σ ff (4) σ sf = cov(ε st, ε ft ) σ ff = var(ε ft ) The VAR mode does not consider the possibiity of ong term integration between spot and futures returns The Error Correction Mode VAR mode does not consider the possibiity that the endogenous variabes coud be cointegrated in the ong term. If two prices are co-integrated in ong run then Vector Error Correction mode is more appropriate which accounts for ong-run co-integration between spot and futures prices (Lien & Luo, 1994; Lien, 1996). If the futures and spot series are cointegrated of the order one, then the Vector error correction mode of the series is given as: R st = α s + β s S t 1 + γ f F t 1 + i=1 β si R st i + j=1 γ fj R ft j + ε st (5) R ft = α f + β f F t 1 + γ s S t 1 + i=1 β fi R ft i + j=1 γ sj R st j + ε ft (6) where, S t and F t are natura ogarithm of spot and futures prices. The assumptions about the error terms are same as for VAR mode. The minimum variance hedge ratio and hedging effectiveness are estimated by foowing simiar approach as in case of VAR mode The BVAR Mode A Bayesian approach to vector autoregressive has in particuar been put forward by Doan, Litterman and Sims (1984). The origina Litterman or Minnesota prior was based on the idea that each series is best described as a random wa around an unnown deterministic component. Consider the n variabe vector autoregressive of order p, VAR(p), given by (7) yt = 1 yt py t-p + + (7) where yt is an (n x 1) vector of non-stationary time series, m is an (n x 1) vector of constants coefficients and et is an n x 1 vector of error terms. Г 1 through Г p represent (n x n) matrices of parameters to be estimated. Hence the prior distribution is centered around the random wa specification for variabe n given by (8) beow. Y(n,t) = n) + y(n,t-1) + n,t) (8)

4 Internationa Letters of Socia and Humanistic Sciences Vo As described in Litterman (1986), the standard error on the coefficient estimate for ag of variabe j in equation i is given by a standard deviation of the coefficient on ag of variabe j in equation i given by a standard deviation function of the form S(i, j, ) given by equation (9) beow. where S(i, j, )= [γ.g().f(i,j)]s i s j (9) f (i, j) = 1 if i = j and wij otherwise The hyperparameter and functions g() and f(i, j) determine the tightness or weight attaching to the prior in (8) above. Given the functiona specifications of g() and f(i, j), can simpy be interpreted as the standard deviation on the first own ag. It is aso often termed the overa tightness of the prior. The function g() determines the tightness on ag one reative to ag. The tightness around the prior mean is normay assumed to increase with increasing ag ength. This is achieved by aowing g() decay harmonicay with decay factor d, i.e. g() = d. The tightness of the prior on variabe j reative to variabe i in the equation for variabe i is determined by the function f(i, j); this can be the same across a equations in which case wij is equa to a constant (w) and the prior is said to be symmetric. Aternativey, the tightness of the prior for variabe j reative to variabe i (in the equation for variabe i) can vary depending upon the particuar equation and/or variabe in question (this is nown as a genera prior). However, the fexibiity inherent in the specification of a genera prior may not aways be desirabe. On the one hand, as argued by Doan (1990), it simpy transfers the probem of over-parameterization to one of having to estimate or search over too many hyperparameters. However, in a situation where the anayst has strong prior views that one of the variabes is exogenous, the genera prior may improve forecasting performance. In particuar, the equations for exogenous variabes may best be specified as univariate autoregressive with no feedbac from the other variabes in the system. This can be achieved by setting very ow vaues for the off-diagona eements in f(i, j) which correspond to that particuar variabe. Finay, the mutipicative ratio si/sj in equation (9) refects the fact that in genera the prior cannot be competey specified without reference to the data. In particuar it corrects for differences in the scae used in the measurement of each variabes incuded in the system. For exampe, how tight a standard deviation of 0.5 is on the ags of prices in an equation for the interest rate wi depend on whether the price index is based to equa unity or 100 in the base period. Litterman (1986) argues that the scae of the response of one variabe to another is a function of the reative size of unexpected movements in the two variabes rather than the reative sizes of their overa standard errors. Hence, he suggests scaing the standard error on the prior by the ratio of the standard deviations of the residuas (si) from a univariate autoregressive for variabe i to the standard deviation of the residuas (sj) from a univariate autoregressive for variabe j (both with p ags) The VAR Estimates To cacuate the hedge ratio and hedging effectiveness, system of equations is soved. We used covariance and variance errors from the equation [2, 3] to cacuate hedge ratio and

5 86 Voume 37 hedging effectiveness (equation [1]) of futures contracts. The covariance and variance errors and OHR equations are given in Tabe 1 and hedge effectiveness is presented in Tabe 2. Tabe 1 Cov (ε s, ε f ) Var (ε s ) Var (ε f ) OHR Tabe2 Var (U) Var (H) Hedging Effectivenss, E The VECM Estimates Using the same approach as in case of VAR mode, errors are estimated and hedging effectiveness and hedge ratio are cacuated for VECM mode. Resuts of the equation [5, 6] and OHR are presented in Tabe 3. Tabe 4 iustrates the estimates hedging effectiveness of futures contracts. Tabe 3 Cov (ε s, ε f ) Var (ε s ) Var (ε f ) OHR Tabe 4 Var (U) Var (H) Hedging Effectivenss, E The BVAR Estimates Errors are estimated through Baysian Vector Autoregressive (BVAR) mode and hedging effectiveness and hedge ratio are cacuated. OHR are presented in Tabe 5. Tabe 6 iustrates the estimate of hedging effectiveness of futures contracts.

6 Internationa Letters of Socia and Humanistic Sciences Vo Tabe 5 Cov (ε s, ε f ) Var (ε s ) Var (ε f ) OHR Tabe 6 Cov (U) Var (H) Hedging Effectiveness, E CONCLUSION To hedging ris, it is important to evauate the hedging effectiveness of derivatives. In the present paper, we report hedge ratios of Brent Oi futures through three aternative modeing framewors: VAR mode, VECM mode and BVAR mode. We compare the hedging effectiveness of the contacts using these modes, ex post (in-sampe) and ex ante (out-ofsampe) introduced by Edrington. The resuts show the VEC mode is more effective than the other modes used in this paper. References [1] Chen S. S., Lee C. F., Shrestha K., The Quartery Review of Economics and Finance 43 (2003) [2] Ederington L. H., Journa of Finance 34 (1979) [3] Litterman R. B., Journa of Business & Economic Statistics 4 (1986) [4] Sims C.A., Tao Zha, Internationa Economic Review 39(4) (1998) ( Received 03 August 2014; accepted 11 August 2014 )