金融資產報酬序列的波動性叢聚、厚尾與極端事件常產生資產報酬尾部行為相依結構的改變，因而借助適當計量模型建構避險組合，可增加投資風險管理效益。本研究以2001年1月2日至2017年12月29日的臺灣加權股價指數現貨每日收盤價與指數期貨的價格為研究對象。在移動視窗法(Rolling-Window)架構下，利用GARCH-EVT-COPULA模型，探討Normal分配與學生t分配下，臺灣加權股價指數與指數期貨的最小變異數避險組合之動態風險管理績效。本研究發現學生t分配的GARCH-EVT-COPULA模型比Normal分配的GARCH-EVT-COPULA模型更能捕捉尾部行為且提高風險管理效益。此外，本研究發現319槍擊案與福島核事件發生後的學生t分配的動態風險管理績效較Normal分配準確。另外，金融海嘯發生後的Normal分配與學生t分配的最小變異數避險組合的預期不足額有顯著差異。本研究的研究成果可提供投資人參考。

The volatility clustering, fat-tail due to rare events, and the dependence structure of tail between of financial asset returns time series may change. Hence, the construction of a minimum variance hedging portfolio(MVHP) from a proper econometric model can increase the effectiveness of risk management. The study examined Taiwan index spot daily close price and Taiwan index futures transaction price occurred close to 13:30 from January 2, 2001 to December 22, 2017. The rolling-window framework and GARCH-EVT-COPULA model are used to measure Value-at-Risk and expected shortfall of the MVHP for the effectiveness of dynamic risk management. The effectiveness of risk management is also compared between Normal distribution and Student t distribution on GARCH-EVT-COPULA model. 
　　The empirical results show that the model of Student t distribution is better than the one of Normal distribution. Moreover, results also show that 319-shooting incident and Japan's Fukushima nuclear explosion are compared between GARCH-EVT-COPULA model with Normal distribution and Student t distribution. In addition, the GARCH-EVT-COPULA model with Student t distribution is different from the one with Normal distribution after financial crisis. The evidences have direct implications for investors and risk managers during extreme index futures market comovements.

目錄	I
圖目錄	III
表目錄	IV
第一章　緒論	1
1.1 研究背景與動機	1
1.2 研究目的	6
1.3 研究流程	7
1.4 研究範圍與限制	8
第二章　資料與實證方法	10
2.1 樣本資料與來源	10
2.2 實證模型	10
第三章　最小變異數避險組合的動態風險管理績效	24
3.1 基本敘述統計量分析	24
3.2 單根檢定	26
3.3 極值檢定	28
3.4 實證模型的參數估計	29
3.5 最小變異數平均避險比率	32
3.6 最小變異數避險組合的風險值之估計	33
3.7 最小變異數避險組合的預期不足額之風險管理分析	42
第四章　結論與建議	47
4.1 結論	47
4.2 建議	48
參考文獻	50

1.	Ab Razak, R., and Ismail, N. (2016). Portfolio risks of bivariate financial returns using Copula-VaR approach: A case study on Malaysia and US stock markets. Global Journal of Pure and Applied Mathematics, 12(3), 1947-1964
2.	Acatrinei, M. (2015). A Copula-GARCH model for a proxy portfolio for bet-fi index. Studii Financiare (Financial Studies), 19(2), 8-16.
3.	Acerbi, C., and Tasche, D. (2002). On the coherence of expected shortfall. Journal of Banking & Finance, 26(7), 1487-1503.
4.	Aloui, C., and Jammazi, R. (2015). Dependence and risk assessment for oil prices and exchange rate portfolios: A wavelet based approach. Physica A: Statistical Mechanics and its Applications, 436, 62-86.
5.	Álvarez-Díez, S., Alfaro-Cid, E., and Fernández-Blanco, M. O. (2016). Hedging foreign exchange rate risk: multi-currency diversification. European Journal of Management and Business Economics, 25(1), 2-7.
6.	Antonakakis, N., Cunado, J., Filis, G., Gabauer, D., and De Gracia, F. P. (2018). Oil volatility, oil and gas firms and portfolio diversification. Energy Economics, 70, 499-515.
7.	Artzner, P., Delbaen, F., Eber, J. M., and Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203-228.
8.	Balcılar, M., Demirer, R., Hammoudeh, S., and Nguyen, D. K. (2016). Risk spillovers across the energy and carbon markets and hedging strategies for carbon risk. Energy Economics, 54, 159-172.
9.	Beckers, B., Herwartz, H., and Seidel, M. (2017). Risk forecasting in (T)GARCH models with uncorrelated dependent innovations. Quantitative Finance, 17(1), 121-137.
10.	Bentes, S. R. (2018). Is stock market volatility asymmetric? A multi-period analysis for five countries. Physica A: Statistical Mechanics and its Applications, 499, 258-265.
11.	Bhattacharyya, M., and Ritolia, G. (2008). Conditional VaR using EVT–towards a planned margin scheme. International Review of Financial Analysis, 17(2), 382-395.
12.	Blomvall, J., and Ekblom, J. (2017). Corporate hedging: An answer to the “how” question. Annals of Operations Research, 1-35.
13.	Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307-327.
14.	Bollerslev, T. (1987). A conditionally heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics, 542-547.
15.	Bollerslev, T., Engle, R. F., and Wooldridge, J. M. (1988). A capital asset pricing model with time-varying covariances. Journal of Political Economy, 96(1), 116-131.
16.	Brandtner, M. (2018). Expected shortfall, spectral risk measures, and the aggravating effect of background risk, or: risk vulnerability and the problem of subadditivity. Journal of Banking & Finance, 89, 138-149.
17.	Byström, H. N. (2004). Managing extreme risks in tranquil and volatile markets using conditional extreme value theory. International Review of Financial Analysis, 13(2), 133-152.
18.	Cai, J. J., Chavez-Demoulin, V., and Guillou, A. (2017). Modified marginal expected shortfall under asymptotic dependence. Biometrika, 104(1), 243-249.
19.	Chan, K. F., and Gray, P. (2006). Using extreme value theory to measure value-at-risk for daily electricity spot prices. International Journal of Forecasting, 22(2), 283-300.
20.	Chau, F., Deesomsak, R., and Wang, J. (2014). Political uncertainty and stock market volatility in the Middle East and North African (MENA) countries. Journal of International Financial Markets, Institutions and Money, 28, 1-19.
21.	Chen, X. B., Silvapulle, P., and Silvapulle, M. (2014). A semiparametric approach to value-at-risk, expected shortfall and optimum asset allocation in stock–bond portfolios. Economic Modelling, 42, 230-242.
22.	Chi, Y., and Tan, K. S. (2011). Optimal reinsurance under VaR and CVaR risk measures: A simplified approach. ASTIN Bulletin: The Journal of the IAA, 41(2), 487-509.
23.	Christoffersen, P. F. (1998). Evaluating interval forecasts. International Economic Review, 841-862.
24.	Chuang, C. C., Wang, Y. H., Yeh, T. J., and Chuang, S. L. (2015). Hedging effectiveness of the hedged portfolio: The expected utility maximization subject to the value-at-risk approach. Applied Economics, 47(20), 2040-2052.
25.	Clark, E., and Baccar, S. (2018). Modelling credit spreads with time volatility, skewness, and kurtosis. Annals of Operations Research, 262(2), 431-461.
26.	Degiannakis, S., and Potamia, A. (2017). Multiple-days-ahead value-at-risk and expected shortfall forecasting for stock indices, commodities and exchange rates: Inter-day versus intra-day data. International Review of Financial Analysis, 49, 176-190.
27.	Dickey, D. A., and Fuller, W. A. (1981). Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica: Journal of the Econometric Society, 49(4)1057-1072.
28.	Du, Z., and Escanciano, J. C. (2016). Backtesting expected shortfall: Accounting for tail risk. Management Science, 63(4), 940-958.
29.	Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the Econometric Society, 987-1007.
30.	Ewing, B. T., and Malik, F. (2017). Modelling asymmetric volatility in oil prices under structural breaks. Energy Economics, 63, 227-233.
31.	Fama, E. F. (1965). The behavior of stock-market prices. Journal of Business, 38(1), 34-105.
32.	Fei, F., Fuertes, A. M., and Kalotychou, E. (2017). Dependence in credit default swap and equity markets: Dynamic Copula with markov-switching. International Journal of Forecasting, 33(3), 662-678.
33.	Ferraty, F., and Quintela-Del-Río, A. (2016). Conditional VaR and expected shortfall: A new functional approach. Econometric Reviews, 35(2), 263-292.
34.	Girardi, G., and Ergün, A. T. (2013). Systemic risk measurement: Multivariate GARCH estimation of CoVaR. Journal of Banking & Finance, 37(8), 3169-3180.
35.	Glosten, L. R., Jagannathan, R., and Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48(5), 1779-1801.
36.	Gou, H., Chen, X., and Hua, Y. (2015). Measuring the value at risk of foreign exchange portfolio by a GARCH-EVT-COPULA based model. Journal of Industrial Engineering and Engineering Management, 29(1), 183-193.
37.	Herrera, R., Rodriguez, A., and Pino, G. (2017). Modeling and forecasting extreme commodity prices: A markov-switching based extreme value model. Energy Economics, 63, 129-143.
38.	Hill, J. B., and Prokhorov, A. (2016). GEL estimation for heavy-tailed GARCH models with robust empirical likelihood inference. Journal of Econometrics, 190(1), 18-45.
39.	Huang, C. K., North, D., and Zewotir, T. (2017). Exchangeability, extreme returns and value-at-risk forecasts. Physica A: Statistical Mechanics and its Applications, 477, 204-216.
40.	Huang, J. J., and So, L. C. (2018). Application of Copula-GARCH to Estimate VaR of a portfolio with credit default swaps. Journal of Mathematical Finance, 8(02), 382.
41.	Huang, J. J., Lee, K. J., Liang, H., and Lin, W. F. (2009). Estimating value at risk of portfolio by conditional Copula-GARCH method. Insurance: Mathematics and Economics, 45(3), 315-324.
42.	Huang, S. C., Chien, Y. H., and Wang, R. C. (2011). Applying garcg-EVT-Copula models for portfolio Value-at-Risk on G7 currency markets. International Research Journal of Finance and Economics, 74, 1450-2887.
43.	Hull, J., and White, A. (1998). Incorporating volatility updating into the historical simulation method for value-at-risk. Journal of Risk, 1(1), 5-19.
44.	Hull, J., and White, A. (1998). Value at risk when daily changes in market variables are not normally distributed. Journal of Derivatives, 5, 9-19.
45.	Jammazi, R., and Nguyen, D. K. (2017). Estimating and forecasting portfolio’s value-at-risk with wavelet-based extreme value theory: Evidence from crude oil prices and US exchange rates. Journal of the Operational Research Society, 68(11), 1352-1362.
46.	Jarque, C. M., and Bera, A. K. (1980). Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Economics Letters, 6(3), 255-259.
47.	Joe, H. (1997). Multivariate models and dependence concepts. London: Chapman and Hall.
48.	Jondeau, E., and Rockinger, M. (2006). The Copula-GARCH model of conditional dependencies: An international stock market application. Journal of international money and finance, 25(5), 827-853.
49.	Karmakar, M. (2013). Estimation of tail-related risk measures in the Indian stock market: An extreme value approach. Review of Financial Economics, 22(3), 79-85.
50.	Karmakar, M. (2017). Dependence structure and portfolio risk in Indian foreign exchange market: A GARCH-EVT-COPULA approach. Quarterly Review of Economics and Finance, 64, 275-291.
51.	Kellner, R., and Rösch, D. (2016). Quantifying market risk with Value-at-Risk or Expected Shortfall?–Consequences for capital requirements and model risk. Journal of Economic Dynamics and Control, 68, 45-63.
52.	Kim, J. S., and Ryu, D. (2015). Are the KOSPI 200 implied volatilities useful in value-at-risk models?. Emerging Markets Review, 22, 43-64.
53.	Klepáč, V., and Hampel, D. (2015). Assessing efficiency of D-vine Copula arma-GARCH method in value at risk forecasting: Evidence from PSE listed companies. Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis, 63(4), 1287-1295.
54.	Kole, E., Markwat, T., Opschoor, A., and Van Dijk, D. (2017). Forecasting value-at-risk under temporal and portfolio aggregation. Journal of Financial Econometrics, 15(4), 649-677.
55.	Kratz, M., Lok, Y. H., and McNeil, A. J. (2018). Multinomial VaR backtests: A simple implicit approach to backtesting expected shortfall. Journal of Banking & Finance, 88, 393-407.
56.	Kupiec, P. (1995). Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives, 3(2), 73-84.
57.	Kwiatkowski, D., Phillips, P. C., Schmidt, P., and Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root?. Journal of econometrics, 54(1-3), 159-178.
58.	Lai, Y. S., Sheu, H. J., and Lee, H. T. (2017). A multivariate markov regime-switching high‐frequency‐based volatility model for optimal futures hedging. Journal of Futures Markets, 37(11), 1124-1140.
59.	Lee, S. H., and Yeo, S. C. (2016). Performance analysis of EVT-GARCH-Copula models for estimating portfolio value at risk. Korean Journal of Applied Statistics, 29(4), 753-771.
60.	Li, L. (2017). A comparative study of GARCH and EVT model in modeling value-at-risk.  Journal of Applied Business and Economics, 19(7), 27-48.
61.	Liu, H. C., and Hung, J. C. (2010). Forecasting S&P-100 stock index volatility: The role of volatility asymmetry and distributional assumption in GARCH models. Expert Systems with Applications, 37(7), 4928-4934.
62.	Liu, H. C., and Hung, J. C. (2017). Minimum variance hedge performance of the realized-volatility-based GARCH model with alternative conditional correlation settings. Journal of Accounting, Finance & Management Strategy, 12(1), 35.
63.	Liu, N., Xia, E. J., and Zhang, Q. L. (2012). Model of credit risk assessment in commercial banks based on asymmetric GARCH and extreme value theory. Transactions of Beijing Institute of Technology, 32(5), 540-545.
64.	Lourme, A., and Maurer, F. (2017). Testing the gaussian and student's t Copulas in a risk management framework. Economic Modelling, 67, 203-214.
65.	Maheu, J. M., and McCurdy, T. H. (2004). News arrival, jump dynamics, and volatility components for individual stock returns. Journal of Finance, 59(2), 755-793.
66.	Mandelbrot, B. (1963). New methods in statistical economics. Journal of Political Economy, 71(5), 421-440.
67.	Martins-Filho, C., Yao, F., and Torero, M. (2018). Nonparametric estimation of conditional value-at-risk and expected shortfall based on extreme value theory. Econometric Theory, 34(1), 23-67.
68.	McNeil, A. J., and Frey, R. (2000). Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. Journal of Empirical Finance, 7(3-4), 271-300.
69.	Mensi, W., Hammoudeh, S., Shahzad, S. J. H., and Shahbaz, M. (2017). Modeling systemic risk and dependence structure between oil and stock markets using a variational mode decomposition-based Copula method. Journal of Banking & Finance, 75, 258-279.
70.	Neftci, S. N. (2000). Value at risk calculations, extreme events, and tail estimation. Journal of Derivatives, 7(3), 23-38.
71.	Nelson, R. B. (1999). An introduction to Copulas. New York: Springer.
72.	Park, J. S., and Shi, Y. (2017). Hedging and speculative pressures and the transition of the spot-futures relationship in energy and metal markets. International Review of Financial Analysis, 54, 176-191.
73.	Paul, S., and Sharma, P. (2017). Improved VaR forecasts using extreme value theory with the realized GARCH model. Studies in Economics and Finance, 34(2), 238-259.
74.	Phillips, P. C., and Perron, P. (1988). Testing for a unit root in time series regression. Biometrika, 75(2), 335-346.
75.	Pickands III, J. (1975). Statistical inference using extreme order statistics. Annals of Statistics, 119-131.
76.	Puccetti, G., Rüschendorf, L., Small, D., and Vanduffel, S. (2017). Reduction of value-at-risk bounds via independence and variance information. Scandinavian Actuarial Journal, 2017(3), 245-266.
77.	Righi, M. B., and Ceretta, P. S. (2015). Forecasting value at risk and expected shortfall based on serial pair-Copula constructions. Expert Systems with Applications, 42(17-18), 6380-6390.
78.	Sahamkhadam, M., Stephan, A., and Östermark, R. (2018). Portfolio optimization based on GARCH-EVT-COPULA forecasting models. International Journal of Forecasting, 34(3), 497-506.
79.	Sharma, R., and Mathur, S. K. (2016). Forecasting portfolio value-at-risk via dcc-mGARCH model: Impact of brexit on valuation of UK pound and Indian rupee. Journal of International Economics, 7(2), 56-63.
80.	Shi, W., Li, K. X., Yang, Z., and Wang, G. (2017). Time-varying Copula models in the shipping derivatives market. Empirical Economics, 53(3), 1039-1058.
81.	Sklar, A. (1973). Random variables, joint distribution functions, and Copulas. Kybernetika, 9(6), 449-460.
82.	Soltane, H. B., Karaa, A., and Bellalah, M. (2012). Conditional VaR using GARCH-EVT approach: Forecasting volatility in tunisian financial market. Journal of Computations & Modelling, 2(2), 95-115.
83.	Sowdagur, V., and Narsoo, J. (2017). Forecasting Value-at-Risk using GARCH and extreme-value-theory approaches for daily returns. International Journal of Statistics and Applications, 7(2), 137-151.
84.	Stavroyiannis, S., Makris, I., Nikolaidis, V., and Zarangas, L. (2012). Econometric modeling and value-at-risk using the Pearson type-IV distribution. International Review of Financial Analysis, 22, 10-17.
85.	Su, Y. C., Huang, H. C., and Lin, Y. J. (2011). GJR-GARCH model in value-at-risk of financial holdings. Applied Financial Economics, 21(24), 1819-1829.
86.	Taylor, J. W. (2017). Forecasting value at risk and expected shortfall using a semiparametric approach based on the asymmetric Laplace distribution. Journal of Business & Economic Statistics,35(4), 1-13.
87.	Walther, T. (2017). Expected shortfall in the presence of asymmetry and long memory: An application to Vietnamese stock markets. Pacific Accounting Review, 29(2), 132-151.
88.	Wang, Z. R., Chen, X. H., Jin, Y. B., and Zhou, Y. J. (2010). Estimating risk of foreign exchange portfolio: Using VaR and CVaR based on GARCH–EVT-Copula model. Physica A: Statistical Mechanics and its Applications, 389(21), 4918-4928.
89.	Weiß, G. N. (2013). Copula-GARCH versus dynamic conditional correlation: an empirical study on VaR and ES forecasting accuracy. Review of Quantitative Finance and Accounting, 41(2), 179-202.
90.	Wong, S. F., Tong, H., Siu, T. K., and Lu, Z. (2017). A new multivariate nonlinear time series model for portfolio risk measurement: The threshold Copula-based tar approach. Journal of Time Series Analysis, 38(2), 243-265.
91.	Xu, Q., Zhou, Y., Jiang, C., Yu, K., and Niu, X. (2016). A large CVaR-based portfolio selection model with weight constraints. Economic Modelling, 59, 436-447.
92.	Xu, R., and Li, X. (2017). Study about the minimum value at risk of stock index futures hedging applying exponentially weighted moving average-generalized autoregressive conditional heteroskedasticity model. International Journal of Economics and Financial Issues, 7(6), 104-110.
93.	Youssef, M., Belkacem, L., and Mokni, K. (2015). Value-at-risk estimation of energy commodities: A long-memory GARCH–EVT approach. Energy Economics, 51, 99-110.
94.	Yu, W., Yang, K., Wei, Y., and Lei, L. (2018). Measuring value-at-risk and expected shortfall of crude oil portfolio using extreme value theory and vine Copula. Physica A: Statistical Mechanics and its Applications, 490, 1423-1433.
95.	Zhang, Z., and Zhang, H. K. (2016). The dynamics of precious metal markets VaR: A GARCH-EVT approach. Journal of Commodity Markets, 4(1), 14-27.
96.	Zhou, J. (2012). Extreme risk measures for REITs: A comparison among alternative methods. Applied Financial Economics, 22(2), 113-126.
97.	Zhu, D., and Galbraith, J. W. (2011). Modeling and forecasting expected shortfall with the generalized asymmetric Student-t and asymmetric exponential power distributions. Journal of Empirical Finance, 18(4), 765-778.