對投資組合信用風險進行量化分析時，其因子模型不一定有封閉性型式，且因為投資組合間具相依關係，計算困難，大多數情況無法直接進行計算，因此使用模擬方法去估計其值，其中較常使用蒙地卡羅法，但在投資組合違約風險估計中，我們主要討論重大損失之風險違約事件，其屬於稀有事件，在蒙地卡羅法中對於估計稀有事件之模擬計算較缺乏效率，因此我們在此使用Glasserman and Li(2005) 所提之二階段重要性取樣法、 Chiang et al.(2007) 所提之改良式重要性取樣法，相較於傳統蒙地卡羅法，可以更有效率估計及達到變異數縮減之效果。
分配假設上，常使用常態分配當作系統風險因子之分配去進行模擬計算，但實際資料並非都符合此分配假設，像是匯率報酬率資料有些是雙峰或是偏態形式，因此我們使用偏斜常態分配(Skew Normal)、雙峰常態分配(Bimodal Normal)，去進行估計，同時也探討在此兩種方法是否可以推廣到非常態關聯結構之因子模型或其在不同模型下有何種限制。
在數值估計結果方面，改良式重要性取樣法在不同分配之因子模型都能使用，二階段重要性取樣法則是受到指數扭轉法(Exponential Twisting)之限制不能在雙峰常態分配之因子模型使用，兩方法應用在估計偏斜常態分配上，相較於蒙地卡羅法，皆能有效估計、達到變異數縮減，節省模擬時間，其中以改良式重要性取樣法為最佳。

The factor model uncertainly has a closed form when the portfolio credit risk is quantitatively analyzed. Because the investment portfolio has a dependent relationship, the calculation is difficult and most of the cases cannot be directly calculated. Therefore, the simulation method is applied to estimate the value, the most often applied method is Monte Carlo method. However, in the portfolio default risk estimation, we mainly discuss the risk default event of major losses, which belong to rare events. In the Monte Carlo method, the simulation calculation for estimating rare events is inefficient, so we apply the two-steps importance sampling method proposed by Glasserman and Li (2005) and the modified importance sampling method proposed by Chiang et al. (2007). These two methods can more effectively estimate and achieve better result of variance reduction than the traditional Monte Carlo method.
In the past, normal distribution was often applied as the distribution of system risk factors to perform simulation calculations. However, the actual data did not all fit this allocation hypothesis. Therefore, we also tried to use skew normal distribution and bimodal normal distribution to estimate, and also discussed Whether these two methods can be generalized to the factor model of the non-normal copula or what restrictions are there under different models.
In terms of numerical estimation results, the modified importance sampling method can be used in different distribution factor models. The two-steps importance sampling method is restricted by the exponential twisting method and cannot be applied in the bimodal normal distribution factor model. Comparing with the Monte Carlo method, the above two methods ,which are applied in the skew normal distribution, can effectively estimate and achieve better result of variance reduction, and save the simulation time. The modified importance sampling method is the best one among these methods.

誌謝 i
摘要 ii
Abstract iii
目錄 iv
圖目錄 vi
表目錄 viii
第一章 緒論 1
第二節 研究背景與動機 1
第二節 風險價值(Value at Risk,VAR) 1
第三節 研究目的 2
第二章 文獻回顧 4
第三章 分配介紹 7
第一節 偏斜常態(SN)分配介紹 7
第二節 雙峰常態(BN)分配介紹 10
第三節 Alpha偏斜常態(ASN)分配介紹 12
第四節 對稱分量Alpha偏斜常態(SCASN)分配介紹 16
第四章 研究方法 20
第一節 基本假設 20
第二節 因子模型介紹 21
第三節 分配應用於因子模型 22
I. 系統風險因子服從常態分配 22
II. 系統風險因子服從偏斜常態分配(SN) 22
III. 系統風險因子服從雙峰常態分配(BN) 24
第四節 兩階段重要性取樣法 25
I. 債務人之間彼此相互獨立 26
II. 債務人之間彼此相依 27
第五節 改良式重要性取樣法 31
第五章 不同因子模型之模擬方法 34
第一節 偏斜常態模型 34
I. 單因子模型 34
II. 多因子模型 37
第二節 雙峰常態模型 41
I. 單因子模型 41
第六章 估計結果比較與分析 44
第一節 偏斜常態因子模型估計結果 44
I. 單因子模型之估計結果 44
II. 多因子模型之估計結果 56
第二節 雙峰常態因子模型估計結果 59
I. 單因子模型之估計結果 59
第三節 不同分配因子模型之比較 68
I. 系統風險因子分別服從N(0,1)、SN(0,1,1)、BN之比較 68
第四節 總結分析與建議 72
參考文獻 74

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