為了能夠更精確地捕捉資產報酬率機率分配的厚尾特性，以避免常態假設下計算出來的VaR所可能發生的低估損失的情況，本文利用相關的極端值理論方法，以報酬率的極端樣本來推論其尾端分配，希望不必對整個報酬率分配作太多假設，而直接根據極端報酬率的資訊來估計百分位數，也就是VaR。
以台灣、日本、香港、美國及新加坡五個國家的股價指數組成一模擬投資組合，透過對此投資組合作常態假設及極端值理論方法兩種模型的比較，我們得到後者在VaR的估計準確性上有較佳的表現，當機率值愈極端時，兩者的差異愈大。

To capture the heavy-tail property of the empirical distributions of asset returns, and improve the possible underestimating of expected losses due to normal distribution assumption, this research applies the extreme value theory in estimating VaR. By making no assumption on the entire distribtuion, the tail of asset distribution is directly inferred from its extreme samples, and VaR, the extreme quantile, is estimated.
Using a portfolio composed of five countries stock indices, I compare the accuracy of the normal model and the extreme value theory approach. The comparison follows that the later has better performance in estimating the portfolio''s VaR, and there is obvious difference between two models especially when the probablility is extreme.

一、前言 ……………………………………………………… 1
二、研究動機與目的 ………………………………………… 3
三、文獻回顧 ………………………………………………… 4
A. VaR模型的簡介 ………………………………………………… 4
1. 常態分配模型 ……………………………………………… 4
2. 歷史模擬 …………………………………………………… 5
3. 極端值理論方法 …………………………………………… 5
B. 相關的極端值理論及應用 ……………………………………… 6
1. Extreme Value Distribution …………………………… 6
2. Tail Index ………………………………………………… 8
3. Estimating Tail Index …………………………………. 9
4. Selecting Threshold ……………………………………. 12
四、研究方法 ………………………………………………… 17
A. 研究架構 ………………………………………………………… 17
B. 以極端值理論計算VaR ………………………………………… 17
1. Bootstrapping MSE ………………………………………. 18
2. Reiss & Thomas Method …………………………………. 19
C. 以其他方式計算VaR …………………………………………… 20
1. Fixed Weight Var-Covariance Matrix ………………… 20
2. Exponential Weight Var-Covariance Matrix ………… 21
3. Constant-Correlation GARCH …………………………… 20
五、研究結果 ………………………………………………… 23
A. Case Study ……………………………………………………… 23
1. 人造資料 …………………………………………………… 23
2. 以Bootstrapping MSE估計尾部指標 …………………… 20
3. 以Reiss & Thomas Method估計尾端指標 ……………… 27
B. 估計投資組合的VaR ……………………………………………. 29
1. 所使用的資料 ……………………………………………… 30
2. 各國股價指數報酬的機率分配尾端 ……………………… 30
3. 投資組合VaR與模型的比較 ……………………………… 32
六、結論與建議 ……………………………………………… 36
七、參考資料 ………………………………………………… 38
附錄一 ………………………………………………………… 40
附錄二 ………………………………………………………… 42
附錄三 …………………………………………………………. 44

Dacorogna, M.M., Muller, U. A., Pictet, O. V. and de Vries, C. G. (1995), The Distribution of Extremal Foreign Exchange Rate Returns in Extremely Large Data Sets, Q & A Research Group Working Paper.
Danielsson, J. and de Vries, C. G. (1997), Beyond the Sample: Extreme Quantile and Probability Estimation, Tinbergen Institute Rotterdam Working Paper.
de Haan, L., Resnick, S. I., Rootzen, H. and de Vries, C. G. (1989), Extreme behaviour of solutions to a stochastic difference equation with applications to ARCH process, Stochastic Processes and their Applications, p.213-224.
de Haan, L. and Resnick, S. I. (1980), A Simple Asymptotics Estimate for the Index of A Stable Distribution, Journal of the Royal Statistical Society, Series B, p.83-87.
Falk, M., Husler, J. and Reiss, R. D. (1994), Law of Small Numbers: Extremes and Rare Events, Birkhauser-Verlag, Basel.
Hall, P. (1990), Using the Bootstrap to Estimate Mean Square Error and Select Smoothing Parameter in Nonparametric Problems, Journal of Multivariate Analysis, p.177-203.
Hill, B. M. (1975), A Simple General Approach to Inference About the Tail of A Distribution, Annals of Statistics, p.1163-1173.
Hols, M. C. A. B. and de Vries, C. G. (1991), The Limiting Distribution of Extremal Exchange Rate Returns, Journal of Applied Econometics, p.287-302.
Kesten, H. (1973), Random difference equations and renewal theory for products of random matrices, Acta Mathematica, p.207-248.
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983), Extremes and Related Properties of Random Sequences and Processes, Springer-Verlag, New York.
Pickands, J. (1975), Statistical Inference Using Extreme Order Statistics, Annals of Statistics, p.119-131.
Reiss, R. D. and Thomas, M. (1997), Statistical Analysis of Extreme Values, Birkhauser-Verlag, Basel.
Resnick, S. I. and Starica, C. (1997), Smoothing the Hill Estimator, Advances in Applied Probability, p.271-293.
Starica, C.and Pictet, O. (1997), The tales the tails of GARCH processes tell, Chalmers University of Technology Working Paper.