本文提出一個針對罕見事件模擬的動態重點抽樣方法 (簡稱DIS)，該方法可以快速且精確估計兩個重要且基礎的投資組合信用風險議題：大違約機率以及高於特定門檻值的期望損失。DIS透過巧妙的解析，將複雜的多維模擬問題降階為簡單的一維零變異之模擬問題。DIS主要過程係由三大步驟組成：首先，選擇具有最大負載係數的因子作為關鍵共同因子 (簡稱KCF)；其後，透過排序統計量的搜尋指標指引後續重點抽樣的區間；最後，再利用截斷抽樣技術進行重點抽樣。DIS除了要求KCF中的負載係數不可有零元素外，並沒有其它的假設前提。因此，DIS可以應用在任何多因子關聯結構模型上。
在高斯 (常態) 關聯結構模型上，本文以 Glasserman, Kang and Shahabuddin (2008) 的數值範例做為基礎，在高市場影響條件下 (關鍵共同因子的負載係數設為0.8)，DIS無論在變異數縮減比率或計算效率比率上，均遠遠優於Glasserman et al. (2008) 所提方法。兩種方法的估計效率與效果，隨著關鍵共同因子的負載係數降至0.5及0.25時逐漸趨同。雖然如此，DIS相較仍具有概念簡單與易於實作的優勢。此外，數值結果亦顯示 DIS 估計式具有有界相對誤差特性，此意謂DIS估計式具有穩健的特性。
綜言之，在多因子關聯結構模型上，DIS 在數值上具有估計的效率性與穩健性。且由於DIS 的概念簡單及假設條件少，因此在實務上易於實作且應用性廣。DIS特別適用在高市場條件，尤其是在市場發生危機的情況。

This study develops a dynamic importance sampling method (DIS) for numerical simulations of rare events. The DIS method is flexible, fast, and accurate. The most importance is that it is very easy to implement. It could be applied to any multifactor copula models, which conduct by arbitrary independent random variables. First, the key common factor (KCF) is determined by the maximum value among the coefficients of factor loadings. Second, searching the indicator by the order statistics and applying the truncated sampling techniques, the probability of large losses (PLL) and the expected excess loss above threshold (EELAT) can be estimated precisely. Except for the assumption that the factor loadings of KCF do not exit zero elements, we do not impose any restrictions on the composition of the portfolio. The DIS method developed in this study can therefore be applied to a very wide range of credit risk models. Comparison of the numerical experiment between the method of Glasserman, Kang and Shahabuddin (2008) and the DIS method developed in this study, under the multifactor Gaussian copula model and the high market impact condition (the factor loadings of marketwide factor of 0.8), both variance reduction ratio and efficient ratio of the DIS model are much better than that of Glasserman et al. (2008)’s. And both results approximate when the factor loadings of marketwide factor decreases to the range of 0.5 to 0.25. However, the DIS method is superior to the method of Glasserman et al. (2008) in terms of the practicability. Numerical simulation results demonstrate that the DIS method is not only feasible to the general market conditions, but also particularly to the high market impact condition, especially in credit contagion or market collapse environments. It is also noted that the numerical results indicate that the DIS estimators exit bounded relative error.

1. Introduction ... 1
2. Portfolio Credit Risk in the Multifactor Copula Models ... 5
2.1. Problem Formulation ... 5
2.1.1. The Portfolio Structure and Loss Distribution ... 5
2.2. The Multifactor Copula Models ... 7
3. Importance Sampling Methods for Multifactor Portfolio Credit Risk ... 9
3.1. Notations Setting ... 9
3.2. Multifactor Gaussian Copula Model ... 11
3.2.1. The CMC Algorithm for the probability of large losses ... 12
3.2.2. The DIS Algorithm for the probability of large losses ... 12
3.2.3. The CMC Algorithm for the expected excess loss above threshold ... 13
3.2.4. The DIS Algorithm for the expected excess loss above threshold ... 13
4. Numerical Experiments ... 15
4.1. Performance Evaluation Criterions ... 15
4.1.1. Efficient Measures ... 15
4.1.2. Robust Measures ... 16
4.2. Numerical Examples and Settings ... 19
4.3. The Results of Case 1 ... 20
4.4. The Results of Case 2 ... 22
4.4.1. High Market Condition ... 22
4.4.2. Medium Market Condition ... 23
4.4.3. Low Market Condition ... 24
4.4.4. KMV Market Condition ... 25
5. Concluding Remarks ... 27
References ... 29
Appendix ... 32

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