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.

目 錄
第一章	緒論
第一節研究動機………………………………………………………..1
第二節研究目的………………………………………………………..1
第三節研究架構………………………………………………………..2
第四節研究流程………………………………………………………..4

第二章	理論基礎與相關文獻探討
第一節指數期貨探討．………………………………………………..5
第二節避險理論與基楚．……………………………………………..7
第三節避險實證之文獻探討…………………………………．….…18

第三章	研究方法
第一節資料說明．．…………………………………………………25
第二節單根檢定………………………………………………………27
第三節ARCH模型………………………………………………………30
第四節GARCH模型…………………………………………………….33
第五節雙變量GARCH模型避險……………………………………….44
第六節不同報酬型態下的避險比率…………………………………46

第四章	實證結果與分析
第一節資料處理………………………………………………………49
第二節單根檢定………………………………………………………52
第三節ARCH效果檢定……………………………………………....54
第四節固定相關係數檢定……………………………………………55
第五節不同報酬型態下的避險績效…………………………………56

第五章	結論與建議
第一章結論……………………………………………………………67

參考文獻……………………………………………………………..68

表 目 錄
【表2-1-1】世界主要股價指數之計算方法…………………………………………5 
【表2-1-2】世界主要指數期貨契約簡介……………………………………………6 
【表2-2-1】不同的避險比率…………………………………………………………11 
【表3-4-1】多變量GARCH 模型之比較………………………………………………44 
【表4-1-1】股價指數、原油期貨與現貨報酬基本統計量…………………………49 
【表4-2-1】指數與原油的期貨現貨時間序列資料之單根檢定(水準項)…………53 
【表4-2-2】指數與原油的期貨現貨報酬率時間序列資料之單根檢定(差分項)…54 
【表4-3-1】指數與原油的期貨現貨ARCH效果檢定…………………………………55 
【表4-4-1】指數與原油的期貨現貨標準化殘差之固定相關係數檢定……………56 
【表4-5-1】近月與遠月估計參數……………………………………………………60 
【表4-5-2】近月與遠月期貨契約報酬率變異數……………………………………61 
【表4-5-3】MVHR與傳統避險比較……………………………………………………62 
【表4-5-4】直接與交叉估計參數……………………………………………………64 
【表4-5-5】直接與交叉避險的報酬變異數…………………………………………65 
【表4-5-6】MVHR與傳統避險風險比較………………………………………………66 


圖 目 錄 

【圖4-1-1】現貨與期貨資料報酬圖………………………………50 
【圖4-1-2】現貨與期貨時間序列圖………………………………51 
【圖4-5-1】估計期間（500天）與預估1天之移動視窗方法……57 
【圖4-5-2】近月到期的避險比率…………………………………58 
【圖4-5-3】六個月到期的避險比率………………………………58 
【圖4-5-4】直接避險的避險比率…………………………………62 
【圖4-5-5】交叉避險的避險比率…………………………………63

參 考 文 獻

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3.   郭同境，最適避險操作策略-台灣股價指數之時證研究，私立淡江大學財務金融所碩士論文，民國八十七年六月。

4.   陳衍龍，國際指數期貨市場避險與避險比率之研究，國立交通大學財務金融所未士論文，民國九十三年六月。

5.   魏志良，國際股價指數期貨與現貨直接避險策略之研究，私立淡江大學財務金融所碩士論文，民國九十一年六月。

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