Hedging with stock index futures: downside risk versus the variance

In this paper we investigate hedging a stock portfolio with stock index futures. Instead of defining the hedge ratio as the minimum variance hedge ratio, we consider several measures of downside risk: the semivariance according to Markowitz [ 19591 and the various lower partial moments according to Fishburn's [ 1977] alpha-t model (alpha>O). Analytically we show that for normal returns and biased futures markets there is an extra cost associated with hedging lower partial moments if the minimum variance hedge ratio instead of the optimal hedge ratio is used. We prove that the extra cost is different from zero if a is bigger than or equals 1. Furthermore, in case futures markets are positively biased minimum lower partial moment hedge ratios are smaller than the minimum variance hedge ratio (strictly smaller in case a is bigger than or equals 1). We used the Dutch FTI contract to hedge three Dutch stock market indexes. The in-sample analysis shows that (i) minimum semivariance and minimum variance hedge ratios are almost the same in size, (ii) minimum lower partial moment hedge ratios are smaller than minimum variance hedge ratios (only slightly smaller for a is bigger than or equals 1) and (iii) except for the lower partial moment with a=0.5, hedging downside risk using the minimum variance hedge ratio instead of the optimal hedge ratio is appropriate. For both strategies risk can be reduced in the same proportion whereas the extra cost of using the minimum variance hedge strategy is negligible. In contrast to (iii), out-of-sample results show that the extra cost of hedging lower partial moments with the minimum variance hedge strategy can be significant (statistically as well as in