VaR- and CVaR-Minimal Futures Hedging Strategies: An Analytical Approach

Abstract

Although Value at Risk (VaR) and Conditional Value at Risk (CVaR) have been established as standard techniques in many fields of risk management and portfolio selection, the literature rarely applies these risk measures to futures hedging. The purpose of this paper is to characterize analytically VaR-and CVaR-minimal hedging strategies. We apply results about quantile derivatives to obtain first order conditions that hold under weak assumptions on the underlying return distribution. We then focus on conditionally elliptical return processes, which enables us to derive closed form expressions for these conditions. In the case of hedging with a single futures contract, these expressions can explicitly be solved for (C)VaR-minimal hedge ratios. Hedging strategies based on these results account for the risk caused by the fat tails of return distributions. In a further step, we extend our characterizations of optimal hedging strategies to mixtures of elliptical distributions. This generalization allows capturing distributional asymmetries, which was found to be highly important for tail based risk measurement. Overall, our findings can be used to implement (C)VaR-minimal hedging rules for most econometric models employed in the futures hedging literature, including multivariate GARCH and regime switching models.