Estimating optimal hedge ratio: a multivariate skew-normal distribution approach

Abstract

In this article, we adopt Multivariate Skew-Normal (MSKN) distributions to test for the joint normality of spot and futures returns and to estimate optimal hedge ratios. Using daily data for 22 different commodities, we reject the joint normality hypothesis in favour of Skew-Normal (SKN) distributions for all commodities at less than 1% significance level. In the out-of-sample performance comparison, the MSKN hedge ratio is found to outperform the conventional Minimum Variance (MV) hedge ratio for about half of the 22 commodities considered. On the other hand, the Lower Partial Moment (LPM) hedge ratio based on the MSKN dominates the LPM hedge ratio based on the multivariate normal distribution for almost all commodities in the out-of-sample comparison.

Notes

¹ This is an important issue in the hedging literature. As pointed out earlier, there are many optimal hedge ratios depending on the objective functions being optimized. However, if the two returns are jointly normally distributed and if both returns are pure martingales, then the optimal hedge ratios derived based on the mean-variance approach, the expected utility maximization approach, the mean extended-Gini approach and the generalized semi-variance approach will all converge to the well-known MV hedge ratio (Chen et al., 2008).

² See Azzalini and Capitanio (1999) for the discussion on the maximum likelihood estimation of the parameters of the model.

³ See Lien and Tse (2000) and Chen et al. (2001) for discussions on hedge ratios based on LPMs and semi-variance.

⁴ The difference between MV hedge ratio and SKN hedge ratio is that MV hedge ratio is obtained by dividing the sample covariance of futures and spot returns by the sample variance of the futures returns (or the slope coefficient in the regression of spot return on futures return) where as SKN hedge ratio is computed from the parameters of the SKN distribution as given in Equation 14. The description of FIC hedge ratio is given in Lien and Shrestha (2005).