Optimal hedge ratio estimation and hedge effectiveness with multivariate skew distributions

Abstract

This article proposes to use the three multivariate skew distributions (generalized hyperbolic distribution, multivariate skew normal distribution, and multivariate skew Student-t distribution) for estimating the minimum variance hedge ratio in a dynamic setting. Three criteria for measuring hedge effectiveness are employed: hedging instrument effectiveness, overall hedge effectiveness, and relative-to-optimal hedge ratio effectiveness (RHRE). Three portfolios of spot and futures series are formed for empirical analysis. The outcomes confirm that the three multivariate skew distributions are more helpful in deciding the minimum variance hedge ratio, especially the generalized hyperbolic distribution, than the symmetrical normal and Student-t distributions. This outperformance is significant especially at critical market moments and it is indicated by three hedge effectiveness measures. This advantage is held without the cost of lowering portfolio return. In addition, there is speculation possibility existing in the portfolio hedged by the traditional optimal hedge ratio and this potential can be detected especially by RHRE.

Keywords:

• dynamic optimal hedge ratio
• generalized hyperbolic distribution
• hedge effectiveness
• multivariate skew normal distribution
• multivariate skew-t distribution

JEL Classification:

• G11
• C13

Notes

¹ The other hedge ratios include minimum mean extended-Gini coefficient hedge ratio, optimum mean-extended Gini coefficient hedge ratio, minimum generalized semivariance hedge ratio, optimum mean-generalized semivariance hedge ratio, maximum mean-generalized semivariance hedge ratio.

² Under the martingale and joint-normality assumption, various optimal hedge ratios are identical to the minimum variance hedge ratio. Under mild assumptions, those optimal hedge ratios are identical: minimum variance, maximum Sharpe measure, and minimum generalized semivariance.

³ Although there are even higher orders, it is generally acceptable to focus on the first four orders of moment. Including more orders does not necessarily improve modelling performance.

⁴ So far, there is not a multivariate model equipped with a specific kurtosis parameter and its estimate can be built by a certain linear combination of the first three orders of moment.

⁵ Financial derivatives are subject to measurement at fair value through loss or gain. An accounting mismatch can occur when an entity uses a financial derivative to hedge against exposures to a market risk arising from an underlying asset or liability that is not measured at fair value through gain or loss. An accounting mismatch can occur due to the different basis of accounting between the hedged item and hedging instrument. Hedge accounting is intended to deal with this accounting mismatch.

⁶ There are several alternative parameterizations and it is feasible to switch between different parameterizations. Prause (1999) proposes three major types of parameterization. For simplicity, I choose for this article the one that eliminates the degree of freedom within and constrains the determinant of the dispersion matrix to be one. It simply requires the expected value of the generalized inverse Gaussian distributed mixing variable to be one. This parameterization makes the interpretation of the skewness parameter easier and the fitting procedure becomes faster.

⁷ The aforementioned five parametric distributions are all fitted via the maximum likelihood method, while the GH is estimated with the expectation maximization type algorithm. See Meng and Rubin (1993) for details.

⁸ http://online.thomsonreuters.com/datastream/

⁹ The Pearson product-moment correlation coefficient estimates for the three portfolios are .958032, .975289, .843419, and .7681650, respectively.

¹⁰ The significance level is specified as 5% throughout this article.