Assuming elliptical return distributions, we prove that minimum lower partial moments hedge ratios (according to Fishburn' s a-t model) are equal to or smaller than the minimum variance hedge ratio (strictly smaller for a=0, the target shortfall probability, and a=1 ). Therefore, if the latter is used instead of the first (1) downside risk can be reduced less and (2) the expected return of the hedged position is lower. Consequently, the minimum variance hedge strategy is not a very attractive one in hedging downside risk. As our empirical results show that minimum variance hedge ratios are less than 1, a 100% hedge strategy is even less attractive. Another downside risk measure we investigate is the mean semivariance. We show that the hedge ratio is equal to the minimum variance hedge ratio. Therefore, hedging the mean semivariance with the latter is appropriate. Using the Dutch FTI stock index futures contract, all findings are empirically confirmed.