本篇論文以 copulas 配合不同的邊際分配估計投資組合風險值，投資組合以三個股價指數為標的分別組成二商品和三商品的投資組合，並和時間序列模型的估計結果相互比較，結果發現在二商品組合中 copulas 對部分投資組合和部分信賴水準的風險值有比較好的估計能力，但大部分的情形下差異並不顯著，
被拒絕的模型都太過保守高估未來的市場風險，在三商品組合模型幾乎都無法掌握未來風險，而且反而是常態分配相對有比較好的估計能力。假設 t 分配為邊際分配稍微減少高估風險的情形，但並不影響模型被接受與否。

1 前言 1
2 風險值 3
2.1 風險值定義. . . . . . . . . . . . . . . . . . . . . . 3
2.2 以時間序列模型估計風險值. . . . . ....................4
3 Copula 6
3.1 Copula 理論 . . . . . . . . . . . . . . . . . . . . 6
3.2雙變量Copula . . . . . . . . . . . . . . . .......... 9
3.3 相關性. . . . . . . . . . . . .. . . . . . . . . . . 11
3.3.1 Concordance . . . . . . . . . . . . . . . . . . . 11
3.3.2 相關係數 (linear dependence measurement )..........12
3.3.3 序列相關係數 (rank correlation) . . . . . . . . . 13
3.4 以 Copulas 估計風險值 . . . . . . . . . . . . . . . .14
4 實證分析
4.1 資料分析與統計軟體的應用 . . . . . . . . . . . . . . .16
4.2 回溯測試 (backtesting) . . . . . . . . . . . . . . . .17
4.3 實證結果 . . . . . . . . . . . . . . . . . . . . . . .18
5 結論與未來研究方向
5.1 結論 . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 未來研究方向 . . . . . . . . . . . . . . . . . ... . . 24
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