臺灣博碩士論文加值系統

隨著時代的變遷與進步，越來越多的金融商品的創新，供公投資大眾或法人更多的選擇去做投資或避險，也因此使得財務槓桿上的風險越來越受到重視；不論新舊巴塞爾協定，均對金融業的投資風險、流動性風險、信用風險…等，多加規範，風險值的計算也日趨廣泛應用，蒙地卡羅模擬即為其中常用的衡量方法，而其中最重要的步驟就是分解相關係數矩陣；目前最常使用的是Cholesky分解法，然而，Cholesky分解法仍對被分解的相關係數矩陣多所限制，僅能分解正定矩陣，為了克服這個問題，我們可以採用Spectral分解法來替代Cholesky分解法，本篇論文即在探討Spectral分解法在風險值的應用，欲證明Spectral分解法能解決Cholesky分解法上的限制問題，應為計算風險值時的較佳選擇。

Since more and more financial products have been invented, we have more ways to get more variety payoff and hedging. As a result, how to control financial risk becomes more and more important. Monte Carlo Simulation is the most widely used method to conduct value-at-risk. The most important part of computing value-at-risk of an asset portfolio is to derive the default correlation matrix to apply into Monte Carlo Simulation. Generally, we use the Cholesky decomposition to decompose the correlation matrix. However, there are some limitations to the Cholesky decomposition. The Cholesky decomposition cannot be used to decompose a non-positive correlation matrix. Under this circumstance, we may adopt the Spectral decomposition. This paper will show the efficiency of Spectral decomposition when facing the non-positive correlation matrix. Due to having fewer limitations, the Spectral decomposition could be more widely used rather than the Cholesky decomposition.

I. Introduction: 1
1-1 Motivation: 1
1-2 Purpose: 2
1-3 Reference Review: 3
II. Methodology: 6
2-1 Data: 6
2-2 Decomposition methodology: 9
2-2-1 Spectral decomposition: 9
2-3 VaR evaluation methods: 10
2-3-1 Equally Weighted Moving Average approach: 11
2-3-2 Exponential Weighted Moving Average approach: 12
2-3-3 Monte Carlo Simulation: 13
2-3-4 History Simulation Approach: 14
2-4 VaR Tests: 15
2-4-1 Likelihood Ratio Test (LR Test): 15
2-4-2 Z test: 15
2-4-3 Uncovered Quadratic Value(UQV): 16
III. Applications to Equity Portfolio: 16
3-1 Statistics of Data: 16
3-2 Positive default correlation case—Cholesky decomposition & Spectral decomposition: 19
3-3 Non-positive default correlation case—Spectral decomposition: 21
IV. Conclusion: 25
Reference: 27

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