期貨避險的變異數極小公式是期貨現貨間報酬率的條件共變異數除以期貨報酬的條件變異數。這標準的結果可在許多衍生性商品風險管理的教科書中找到但這個結論必須要在某些條件下才可成立。但實際上傳統的避險比率只有以報酬金額來計算才是變異數最小的。 本文在不同報酬型式下的極小變異數避險比率公式，將會發現傳統的避險比率只在以金額表示下的報酬型態才是變異數極小，其餘的報酬型態則否。然而在實證方面對數報酬、報酬百分比上使用傳統的避險比率會得到較佳結果，但在交叉避險下使用報酬百分比、對數報酬計算MVHR與傳統的避險比率下會有顯著的不同，經由模擬交叉避險中若使用傳統的避險比率會使避險績效顯著的降低。 It is widely known that the variance-minimizing futures hedge is given by the ratio of the conditional covariance of the futures and spot returns to the conditional variance of the futures return. This standard result can be found in virtually every leading derivatives or risk management textbook. There is, however, much confusion over the conditions under which this result holds. This result has been asserted either explicitly or implicitly when returns are measured in dollar terms. In this article, we examine the minimum-variance hedge ratio (MVHR) under alternative return specifications. Formulas for the MVHR are derived for cases in which returns are measured in dollar terms, percentage terms, and log terms.. It is found that the conventional hedge ratio given by the ratio of the conditional covariance of the futures and spot returns to the conditional variance of the futures return is variance-minimizing when computed from returns measured in dollar terms but not from returns measured in percentage or log terms. the MVHR can vary significantly from the conventional hedge ratio computed from percentage or log returns when used in cross-hedging situations. Simulation analysis shows that the incorrect application of the conventional hedge ratio can substantially reduce hedging performance in cross-hedging situations.
