全球市場的報酬走勢根據過往的文獻並不符合常態分佈，極端行情出現的可能性高於預期而且頻繁，其分配具有厚尾且高峰的現象，並且因為隨著全球化，世界發生的大事在短時間內，市場間互相影響，因此資產間的關聯結構越來越被重視。過往像是Markowiz (1952) 提出Mean–Variance投資組合理論，其背後之假設便是資產需符合常態，因此如何解決非常態相關性問題是許多學者在意的問題。Copula在Sklar (1959) 提出後，聯合分配函數可以透過邊際分配和Copula函數所組成，也有效解決常態性假設的必須性，不過在過往的Copula文獻中，多數都是使用兩資產的建構，直到Oh and Patton (2018)提出之動態Factor Copula模型，成功將高維度變數轉成單因子估計，並使用GAS架構對時間序列進行估計，本文將使用不同的動態Factor Copula對不同的投資組合進行配適，接下來與不同的模型進行比較，並用模擬出來的情境帶入附保證投資型商品中，觀察對於保險公司提列準備金的影響。

According to the existing literature, the trend of the global market return didn’t follow the normal distribution. The distribution of stock returns may appear a thick tail and high peakness. Besides, because of the globalization, the correlation struc-ture between assets is getting more and more attention. In the past paper, for exam-ple, Markowiz (1952) proposed the Mean-Variance portfolio theory. The assumption behind it was that assets must follow normal distribution. Therefore, how to solve the problem of abnormal correlation is a problem that many scholars attempt to work with. After Copula proposed by Sklar (1959), the joint distribution function can be composed of marginal distribution and the Copula function, which solves the ne-cessity of the normality assumption. However, in the existing literature regarding Copula , most of them only constructed by two assets. Until the Dynamic Factor Copula proposed by Oh and Patton (2018) successfully converted high-dimensional variables into single factor estimation, and used the GAS framework to estimate the time series. This article will use different Dynamic Factor Copula to adapt to differ-ent portfolios. Compare with different models, we use the simulated scenarios with the application on GMXB proucts to observe the impact on insurance companies' reserve requirements.

[1]Ang, A., & Chen, J. (2002). Asymmetric correlations of equity portfoli-os. Journal of financial Economics, 63(3), 443-494.
[2]Andersson, M., Krylova, E., & Vähämaa, S. (2008). Why does the correlation between stock and bond returns vary over time?. Applied Financial Econom-ics, 18(2), 139-151.
[3]Alonso-García, J., Wood, O., & Ziveyi, J. (2018). Pricing and hedging guaran-teed minimum withdrawal benefits under a general Lévy framework using the COS method. Quantitative Finance, 18(6), 1049-1075.
[4]Brennan, M. J., & Schwartz, E. S. (1976). The pricing of equity-linked life in-surance policies with an asset value guarantee. Journal of Financial Economics, 3(3), 195-213.
[5]Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedastici-ty. Journal of econometrics, 31(3), 307-327.
[6]Connolly, R., Stivers, C., & Sun, L. (2005). Stock market uncertainty and the stock-bond return relation. Journal of Financial and Quantitative Analysis, 161-194.
[7]Creal, D., Koopman, S. J., & Lucas, A. (2013). Generalized autoregressive score models with applications. Journal of Applied Econometrics, 28(5), 777-795.
[8]Dai, M., Kuen Kwok, Y., & Zong, J. (2008). Guaranteed minimum withdrawal benefit in variable annuities. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 18(4), 595-611.
[9]Erb, C. B., Harvey, C. R., & Viskanta, T. E. (1994). Forecasting international equity correlations. Financial analysts journal, 50(6), 32-45.
[10]Engle, R. (2002). Dynamic conditional correlation: A simple class of multivari-ate generalized autoregressive conditional heteroskedasticity models. Journal of Business & Economic Statistics, 20(3), 339-350.
[11]Jondeau, E., & Rockinger, M. (2006). The copula-garch model of conditional dependencies: An international stock market application. Journal of interna-tional money and finance, 25(5), 827-853..
[12]Longin, F., & Solnik, B. (2001). Extreme correlation of international equity markets. The journal of finance, 56(2), 649-676.
[13]Meneguzzo, D., & Vecchiato, W. (2004). Copula sensitivity in collateralized debt obligations and basket default swaps. Journal of Futures Markets: Futures, Options, and Other Derivative Products, 24(1), 37-70.
[14]Oh, D. H., & Patton, A. J. (2017). Modeling dependence in high dimensions with factor copulas. Journal of Business & Economic Statistics, 35(1), 139-154.
[15]Oh, D. H., & Patton, A. J. (2018). Time-varying systemic risk: Evidence from a dynamic copula model of cds spreads. Journal of Business & Economic Statis-tics, 36(2), 181-195.
[16]Patton, A. J. (2004). On the out-of-sample importance of skewness and asym-metric dependence for asset allocation. Journal of Financial Econometrics, 2(1), 130-168.
[17]Patton, A. J. (2006). Modelling asymmetric exchange rate dependence. Inter-national economic review, 47(2), 527-556.
[18]Riccetti, L. (2010). The use of copulas in asset allocation: when and how a cop-ula model can be useful. LAP LAMBERT Academic Publishing.
[19]Schönbucher, P. J., & Schubert, D. (2001). Copula-dependent default risk in in-tensity models. In Working paper, Department of Statistics, Bonn University.