臺灣博碩士論文加值系統

風險值(Value at Risk , VaR)是目前風險管理的熱門工具之一，顧名思義它是一個數值，衡量風險資產在特定期間內，特定信心水準下的最大預期損失。也可以說它是資產報酬分配尾部的一個分位數(quantile)。實證上發現財務數列大多存有厚尾(fat tails)和波動聚集(volatility clustering)的現象，用傳統風險值估計法可能會有低估的風險。
我們以極值理論(Extreme Value Theory , EVT)配合時間序列模型來估算股票市場風險值，比較條件風險值和非條件風險值回溯測試(Backtesting)的準確性，並以經驗法則、Peter Hall(1990)部份樣本自體抽樣法(bootstrap)，和Danielsson(1997a，2001)兩階段的部分樣本自體抽樣法，來探討最適門檻水準(Threshold Level)，最後用回溯測試比較預測出的風險值，以概似比檢定(LR test)衡量風險值的表現。
本文以道瓊工業指數、那斯達克指數、S&P500指數、香港恆生指數、日經225指數、韓國綜合指數、大陸上海A股指數、大陸深圳A股指數、倫敦金融指數、德國DAX指數和法國CAC40指數等11個指數的報酬分配作為實證分析資料，實證結果顯示配適ARMA(p,q)-GARCH( 1 , 1 )的條件風險值表現普遍較非條件風險值好，極值理論的風險值表現普遍也較傳統估算風險值的方法佳，此外，門檻水準以Hall1990和Danielsson1997方法決定的條件極值模型，表現也普遍比經驗法則的條件模型佳。

Value at Risk is a widespread tool of risk management recently. It is a value that measures the worst loss of asset under the particular confidence level and possessed of period. Moreover, it is a quantile describing the tail of distribution of financial return series in statistics. In empirical literature, most of financial data have some properties such as fat tails and volatility clustering. Thus, estimating Value at Risk by conventional method may underestimate the quantile as a result of fat tails.
We estimate Value at Risk in stock market by using extreme value theory combine with time series models and thereby compared the performance of the conditional Value at Risk with unconditional Value at Risk. In addition, we investigate optimal threshold level among past experience, method argued by hall and method proposed by Danielsson. Then we experiment backtesting on Value at Risk estimator and evaluate efficiency of estimator by LR statistic.
We backtest mentioned above on eleven stock indexes：Dow Jones industrial average, Nasdaq, S&P 500, Nikkei 225, Hang Seng index, A-Share, SSE A-Share,FTSE 100,CAC 40, DAX and KOSPI index. Our finding reveals that conditional Value at Risk fitted ARMA(p,q)-GARCH(1,1) performs better than unconditional Value at Risk, and extreme value theory performed better than traditional method. Furthermore, our empirical result displays the performance of conditional Value at Risk which threshold level is decided by Hall1990 and Danielsson1997, indeed improve on model which threshold level is decided by past experience.

摘要： ii
Abstract: iii
目錄 iv
表目錄 v
圖目錄 vi
第一章 緒論 1
第一節 研究動機與背景 1
第二節 研究目的 2
第三節 研究架構 3
第二章 文獻探討 5
第一節 風險值的介紹 5
第二節 國內外文獻 6
第三章 研究方法 17
第一節 非條件模型： 17
(一)歷史模擬法： 17
(二)非條件極值模型： 17
第二節 條件模型： 21
(一) GARCH模型： 21
(二) 條件極值模型： 22
第三節 最適門檻水準的抉擇 23
(一)Hall方法 24
(二)Danielsson et al.(1997b) 25
(三)Danielsson et al.(2001) 26
第四節 風險值績效的評估 27
(一)回溯測試(backtesting)： 27
第四章 實證結果分析 28
第一節 實證資料來源： 28
第二節 實證資料分析 28
第三節 全樣本極值模型的估計結果 29
第四節 樣本外回溯測試(Backtesting)結果 31
第五章 結論與建議 35
參考文獻： 37
附錄 41

1.李政峰、江明珠(2006)，「台灣短期利率的厚尾性質與風險值分析-極值理論的應用與比較」，working paper。
2.林楚雄、高子荃、邱瓊儀(2006)，｢結合GARCH模型與極值理論的風險值模型｣，管理學報，二十二卷，第四期，頁133-54。
3.林楚雄、王韻怡(2006)，「考慮GARCH效果下的尾部指數與風險值應用」，風險管理學報， 第八卷，第一期，頁49-70。
4.林楚雄、陳宜玫(2002)，「台灣股票市場風險值估測模型之實證研究｣，管理學報， 第十九卷，第四期，頁737-58。
5.黃玉娟、林帛靜(2002)，「期貨市場報酬分配之厚尾型態與風險值衡量模式之探討-臺灣臺指期貨與新加坡摩根臺指期貨」，台灣財務金融學會研討會。
6.楊奕農(2005) 「時間序列分析-經濟與財務上之應用」，雙葉書廊，台北。
7.Bali, T. G., 2003. An Extreme Value Approach to Estimating Volatility and Value at Risk, Journal of Business, 76(1)：83-107.
8.Bali, T. G., Neftci N. S., 2003. Disturbing extremeal behavior of spot rate dynamics, Journal of Empirical Finance, 10: 455-477
9.Bekiros, D. B, Georgoutsos, A. D.2005. Estimation of Value-at-Risk by extreme value and conventional methods : a comparative evaluation of their predictive performance. Int. Fin. Markets, Inst. and Money, 15:209-228.
10.Bond, S.A., (2006). Asymmetry and Downside Risk in Foreign Exchange Markets. The European Journal of Finance, Volume 12, Number 4, p. 333 - 345
11.Booth, G. G., J. P. Broussard, T. Martikainen, and V. Puttonen, 1997. Prudent Margin Levels in the Finnish Stock Index Futures Market, Management Science, 43(8)：1177-1188.
12.Brooks, c., A. D. Clare, J. W. Dalle Molle and G. Persand (2005), “A comparison of extreme value theory approaches for determining value at risk, ” Journal of Empirical Finance, 12, pp. 339-52.
13.Cotter, J., 2001. Margin Exceedences for European Stock Index Futures using Extreme Value Theory, Journal of Banking and Finance, 25(8)：1475-1502.
14.Danielsson J., D. W. Jansen, and C. G. de Vries, 1996. The Method of Moments Ratio Estimator for the Tail Shape Parameter, Communication in Statistics Theory and Methods, 25(4)：711-720.
15.Danielsson, J., and C. G. de Vries, 1997a. Beyond the Sample: Extreme Quantile and Probability Estimation, working papers, Erasmus University , Rotterdam.
16.Danielsson, J., and C. G. de Vries, 1997b. Tail Index and Quantile Estimation with Very High Frequency Data, Journal of Empirical Finance, 4：241-257.
17.Danielsson, J., de Vries, C., 1997c. Value-at-Risk and extreme returns, FMG-Discussion Paper NO 273, Financial Markets Group, London School of Economics.
18.Danielsson, J., de Haan, L., Peng, L., de Vries, C.G., 2001. Using a bootstrap method to choose the sample fraction in tail index estimation. Journal of Multivariate Analysis 76, 226– 248.
19.Diebold, F. X.,Schuermann, T.,Stroughair, J.D. 1999. Pitfalls and Opportunities in the Use of Extreme Value Theory in Risk Management. Working Paper 98-10, http://fic.wharton.upenn.edu/fic/.
20.Embrechts, P., C. Klüppelberg, and T. Mikosch, 2003. Modelling Extremal Events for Insurance and Finance, Springer-Verlag, London.
21.Feller, W. 1971, An introduction to probability theory and its applications. Wily, New York, vol ii. 2nd. ed. edn.
22.Goldie, C. M. and R. L. Smith, 1987. Slow Variation With Remainder: A Survey of the Theory and its Applications, Quarterly Journal of Mathematics, 38(2)： 45-71.
23.Hall, P., 1990. Using Bootstrap to Estimate Mean Square Error and Select Smoothing Parameters in Non-parametric Problems, Journal of Multivariate Analysis, 32：177-203.
24.Hans N.E. Bystrom, 2004. Managing extreme risks in tranquil and volatile markets using conditional extreme value theory, International Review of Financial Analysis,13:133-152.
25.Hill, B., 1975. A Simple General Approach to Inference About the Tail of a Distribution, Annuals of Mathematical Statistics, 3：1163-1174.
26.Hsing, T., 1991. On Tail Index Estimation using Dependent Data, Annals of Statistics, 19：1547-1569.
27.Jondeau, E., Rockinger, M., 2003. Testing for differences in the tails of stock-market returns. Journal of Empirical Finance, 10:559-581
28.Jorion, P.(2000). Value at Risk. The New Benchmark for Managing Financial Risk. New York McGraw Hill Trade.
29.Kearns P., and A. Pagan, 1997. Estimating the Density Tail Index for Financial Time Series, The Review of Economics and Statistics, 79：171-175.
30.Kupiec, P. H., 1995. Techniques for Verifying the Accuracy of Risk Measurement Models, Journal of Derivatives, Winter：73-84.
31.Longin, M. F., 2000. From Value at Risk to Stress Testing: the Extreme Value Approach,　Journal of Banking and Finance, 24：1097-1130.
32.Longin, M. F., 2005. The choice of the distribution of asset returns: How extreme value theory can help? Journal of Banking and Finance, 29：1017-1135.
33.McNeil, A. J. and R. Frey, 2000. Estimation of Tail-related Risk Measures for Heteroscedastic Financial Time Series: an Extreme Value Approach, Journal of Empirical Finance, 7：271-300.
34.Pickands, J., 1975. Statistical Inference using Extreme Order Statistics, Annals of Statistics, 3：119-131.
35.Ramazan, G. and Faruk, S, 2004. Extreme value theory and Value-at-Risk : Relative performance in emerging markets, International Journal of Forecasting, 20, 287-303.
36.Resnick, S. and C. Stărică, 1996. Testing the Covariance Stationarity of Heavy–tailed Time Series, Journal Empirical Finance, 1：211-248.
37.Segers, J., 2001. Abelian and Tauberian Theorems on the Bias of the Hill Estimator, Working Paper, University of Leuven.
38.Smith, R.L., 1985. Maximum likelihood estimation in a class of non-regular cases. Biometrika 72, 67–90.
39.Viviana Fernandez, 2003. Extreme Value Theory and Value at Risk, Working Paper, Georgetown University.
40.Eric Z., 2002. Modelling Financial Time Series with S-PLUS. Insightful Corporation, Seattle Washington.