本文使用 2010 至 2019 年之 S\&P500 及 費城半導體指數作為標的，以幾何布朗運動及四種 Copula 結構： Gaussian 、 Student-t 、 Clayton 、 Gumbel 進行模型配適後，以蒙地卡羅法針對配適之結果進行投資情境模擬。並且針對 10 年期及 20 年期下 GMDB 保本 、 GMMB 保證年化報酬率及 GMDB + GMMB 雙重保證三種附保證投資型商品，分析不同的資產配置策略下資產模型對風險值、準備金及期末帳戶價值的影響。

實證結果顯示 Student-t Copula 對標的資產之配適度最佳，而非一般常用的多元常態 Gaussian Copula。並且相較於其他 Copula ，以 Student-t Copula 做為模型之投資策略於後續計算之風險值及準備金較低。並且，於全期固定投資組合下，相較於考慮帳戶報酬率，選擇夏普比率較高的策略能使準備金最小。

This article use the price of S&P500 and Philadelphia Semiconductor Index from 2010-01-01 to 2019-12-31 as the target asset, and use Geometric Brownian Motion as the marginal distribution of two index with four types of copula as the joint distribution. After fitting above models, use Monte Carlo method to simulate the scenario of asset returns.

We use 10-year and 20-year GMDB, GMMB, and GMMB+GMDB product as the target and analyze the relation between investment strategy and the VaR, reserve and account value at maturity under different model.

The empirical result shows that Student-t Copula fit two stock index the most. Moreover, the investment strategy under student-t copula yield the lowest VaR and reserve compared to other copula include the common assumption of financial engineerring, Gaussian copula. On the other hand, we found that the investment strategy with higher sharpe ratio has the lowest VaR and reserve, instead of the highest annual return.

Ballotta, L., & Haberman, S. (2003). Valuation of guaranteed annuity conversion options. Insurance: Mathematics and Economics, 33(1), 87–108. doi: 10.1016/S0167­6687(03) 00146­X
Bauer, D., Kling, A., & Russ, J. (2008). A universal pricing framework for guaranteed minimum benefits in variable annuities. ASTIN Bulletin, 38(2), 621–651. doi: 10.1017/ s0515036100015312
Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of political economy, 81(3), 637–654. doi: 10.1086/260062
Brennan, M. J., & Schwartz, E. S. (1976). The pricing of equity­linked life insurance policies with an asset value guarantee. Journal of Financial Economics, 3(3), 195–213. doi: 10.1016/0304­405x(76)90003­9
Brennan, M. J., & Schwartz, E. S. (1979). Alternative investment strategies for the issuers of equity linked life insurance policies with an asset value guarantee. The Journal of Business, 52(1), 63. doi: 10.1086/296034
Brown, R. (1828). A brief account of microscopical observations made in the months of june, july and august 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. The Philosophical Magazine, 4(21), 161–173. doi: 10.1080/14786442808674769
Doan, B., Papageorgiou, N., Reeves, J. J., & Sherris, M. (2018). Portfolio management with targeted constant market volatility. Insurance: Mathematics and Economics, 83, 134– 147. doi: 10.1016/j.insmatheco.2018.09.010
Guo, N., Wang, F., & Yang, J. (2017). Remarks on composite bernstein copula and its application to credit risk analysis. Insurance: Mathematics and Economics, 77, 38–48. doi: 10.1016/ j.insmatheco.2017.08.007
Heston, S. L. (2015). A Closed­Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The Review of Financial Studies, 6(2), 327­ 343. doi: 10.1093/rfs/6.2.327
Hull, J. C. (2017). Options, futures, and other derivatives, global edition. Pearson. Retrieved from https://www.ebook.de/de/product/33013067/john_c_hull_options_futures_and_other_derivatives_global_edition.html
Itô, K. (1944). Stochastic integral. Proceedings of the Imperial Academy, 20(8), 519–524. doi: 10.3792/pia/1195572786
Li, D. X. (2000). On default correlation: A copula function approach. The Journal of Fixed Income, 9(4), 43–54. doi: 10.2139/ssrn.187289
Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77­91. doi: 10.1111/ j.1540­6261.1952.tb01525.x
Milevsky, M. A., & Posner, S. E. (2001). The titanic option: Valuation of the guaranteed minimum death benefit in variable annuities and mutual funds. The Journal of Risk and Insurance, 68(1), 93. doi: 10.2307/2678133
Milevsky, M. A., & Salisbury, T. S. (2006). Financial valuation of guaranteed minimum withdrawal benefits. Insurance: Mathematics and Economics, 38(1), 21–38. doi: 10.1016/j.insmatheco.2005.06.012
Ng, A. C.­Y., & Li, J. S.­H. (2011). Valuing variable annuity guarantees with the multivariate esscher transform. Insurance: Mathematics and Economics, 49(3), 393–400. doi: 10.1016/j.insmatheco.2011.06.003
Schönbucher, P. J., & Schubert, D. (2001). Copula­dependent default risk in intensity models.In Working paper, department of statistics, bonn university.
Sklar, A. (1959). Fonctions de reprtition an dimensions et leursmarges. Publ. inst. statist. univ.Paris, 8, 229–231.
Wang, C.­W., & Huang, H.­C. (2017). Risk management of financial crises: An optimal investment strategy with multivariate jump­diffusion models. ASTIN Bulletin: The Journal of the International Actuarial Association, 47(02), 501–525. doi: 10.1017/ asb.2017.2
Wang, C.­W., Yang, S. S., & Huang, J.­W. (2017). Analytic option pricing and risk measures under a regime­switching generalized hyperbolic model with an application to equity­ linked insurance. Quantitative Finance, 17(10), 1567­1581. doi: 10.1080/14697688.2017.1288297
Wei, J., & Wang, T. (2017). Time­consistent mean–variance asset–liability management with random coefficients. Insurance: Mathematics and Economics, 77, 84–96. doi: 10.1016/ j.insmatheco.2017.08.011
中華民國精算學會. (2019). 保險合約負債公允價值評價精算實務處理準則 (108 年版草案). Retrieved 2020­5­1, from http://www.airc.org.tw/rule/202
徐英豪. (2019). 附保證投資型保險商品資產配置之研究. 國立政治大學風險管理與保險學系碩士論文. Retrieved from https://hdl.handle.net/11296/83r94x
李振綱. (2007). 探討股票市場與債券市場的關聯結構­動態Copula 模型. 國立交通大學財務金融學系碩士論文. Retrieved from http://hdl.handle.net/11536/ 39361
林展源. (2019). 反向型ETF與波動型ETF之避險績效 ── 應用 Copula­-GJR-­GARCH模型. 國立政治大學國際經營與貿易學系碩士論文. Retrieved from https:// hdl.handle.net/11296/542phs
詹惟淳. (2013). 考慮保戶行為下對附保證投資型商品準備金之評估. 國立中央大學財務金融學系碩士論文. Retrieved from https://hdl.handle.net/11296/ jd3c3z
金管保一字第 09702503741 號. (2008). 人身保險業經營投資型保險業務應提存之各種準備金規範. Retrieved 2020­5­1, from https://law.fsc.gov.tw/law/ LawContent.aspx?id=FL046367