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題名 最低保證給付人壽保險附約之風險分析
Risk analysis for guaranteed minimum benefit life insurance riders
作者 李一成
貢獻者 張士傑
李一成
關鍵詞 最低保證給付
投資帳戶餘額不足機率
有限差分法
Guaranteed minimum benefit
Ruin probability
Numerical PDE solution
日期 2008
上傳時間 9-May-2016 11:51:42 (UTC+8)
摘要 保險人因提供最低保證給付之投資型商品，使公司亦涉入投資風險。本研究旨在探討最低保證給付人壽保險附約之風險分析。首先利用隨機模型建構投資者帳戶價值的動態過程，進而推導出在未來時點帳戶發生餘額不足之機率及其所符合的偏微分方程式。並藉由數值方法－有限差分法，求出投資帳戶餘額不足之機率。最終，以不同的參數選取之下，進行敏感度分析，探討參數值的設定對於帳戶發生餘額不足之機率的影響。本研究結果可以提供保險公司與監理機關，作為日後發行保證給付商品時，一項風險管理上的考慮因素。
研究結果可以歸納為兩點結論：
1. 在市場因素中，投資帳戶連結之標的報酬率與帳戶餘額不足機率呈現反向變動，而波動度則是與帳戶餘額不足機率呈現正向變動。在兩因素同時考慮下，當報酬率愈高且波動度愈低，投資帳戶發生餘額不足的機率會愈低。當波動度愈高且報酬率愈低時，帳戶餘額不足機率則會愈高。其兩者的力量會相互抵銷，對投資帳戶餘額不足之機率的影響需視何者的力量較強而定。

2. 在條款設計的因素中，保證附約相關費用率、保證提領比率與保證提領期間對於投資帳戶發生餘額不足機率的影響皆呈現正向的關係。而投資帳戶期初的價值則與帳戶餘額不足機率呈現反向變動。其中保證提領比率對於投資帳戶的價值影響最大，其帳戶餘額不足機率之變動百分比相較於其他因素而言，變動幅度較大，範圍皆大於4%以上，甚至高達37.11%。
Insurers have investment risks because they issue the guaranteed minimum benefit life insurance riders. Therefore, the purpose of this thesis is analyzing the risk for the riders. In the context, we implement numerical PDE solution to compute the ruin probability of separate account which is the probability that guaranteed minimum benefit life insurance riders will lead to financial insolvency under stochastic investment returns. Moreover, we will do sensitivity analyses to discuss the two aspects, market factors and contract designs, how to influence the ruin probability.

Finally, we conclude two main results:

1. For market factors, the rate of investment return is negatively related to ruin probability; however, the volatility is positive correlation.

2. For contract designs, the results show negative correlation between ruin probability and insurance fee, withdrawals, and withdrawal period. But the initial account value shows positive correlation.
參考文獻 1. Aase, K.K., and Persson, S.A., 1994, Pricing of Unit-linked Life Insurance Policies. Scandinavian Actuarial Journal 1, 26-52.
2. Aase, K.K., and Persson, S.A., 1997, Valuation of the Minimum Guaranteed Return Embedded in Life Insurance Contracts. Journal of Risk and Insurance 64 (4), 599-617.
3. Albrecht, P., and Maurer, R., 2002. Self-annuitization, Consumption Shortfall in Retirement and Asset Allocation: The Annuity Benchmark. Journal of Pension Economics and Finance 1 (2), 57–72.
4. Bacinello, A.R. and Ortu, F., 1993, Pricing Guaranteed Securities-linked Life Insurance under Interest Rate Risk. Actuarial Approach for Financial Risks, 35-55.
5. Bacinello, A.R. and Ortu, F., 1994, Single and Periodic Premiums for Guaranteed Equity-linked Life Insurance under Interest Rate Risk: the ＂Lognormal＋Vasicek＂ Case, Financial Modeling, L. Peccati and M. Viren (Eds.), Physica-Verlag, Heidelberg, Germany, 1-25.
6. Black, F. and Scholes, M., 1973, The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81, 637-654.
7. Boyle, P.P., Schwartz, E., 1977. Equilibrium Prices of Guarantees under Equity-linked Contracts. Journal of Risk and Insurance 44(2), 639–680.
8. Boyle, P.P. and Schwartz, E.S., 1997, Equilibrium Prices of Guarantees under Equity-linked Contracts. Journal of Risk and Insurance 44, 639-680.
9. Boyle, P.P. and Hardy, M.R., 1997, Reserving for Maturity Guarantees: Two Approaches. Insurance: Mathematics and Economics 21, 113-127.
10. Boyle, P.P. and Hardy, M.R. 2003, Guaranteed Annuity Options. Astin Bulletin 33 (2), 125-152
11. Brennan, M.J., and Schwartz, E.S., 1976. The Pricing of Equity-linked Life Insurance Policies with an Asset Value Guarantee. Journal of Financial Economics 3 (1), 195–213.
12. Brennan, M.J. and Schwartz, E.S., 1979, Alternative Investment Strategies for the Issuers of Equity-linked Life Insurance Policies with an Asset Value Guarantee, Journal of Business 52, 63-93.
13. Duffy, D.J., 2006, Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, 1st ed., Hoboken, NJ: John Wiley & Sons.
14. Hardy, M.R. 2000, Hedging and Reserving for Single-premium Segregated Fund Contracts. North American Actuarial Journal 4 (2), 63-74.
15. Hardy, M.R., 2003, Investment Guarantees: Modeling and Risk Management for Equity-linked Life Insurance. 1st ed., Hoboken, N.J.: Wiley.
16. Huang, H., Milevsky, M.A., Wang, J., 2004. Ruined Moments in Your Life: How Good are the Approximations? Insurance: Mathematics and Economics 34 (3), 421-448
17. London, J., 2005, Modeling Derivatives in C++, 1st ed., Hoboken, N.J.: J. Wiley.
18. Milevsky, M.A., 1997. The Present Value of a Stochastic Perpetuity and the Gamma Distribution. Insurance: Mathematics and Economics 20, 243–250.
19. Milevsky, M.A., 1998. Optimal Asset Allocation towards the end of the Life Cycle: To Annuitize or not to Annuitize? Journal of Risk and Insurance 65 (3), 401–426.
20. Milevsky, M.A., 1999. Martingales, Scale Functions and Stochastic Life Annuities: A Note. Insurance: Mathematics and Economics 24 (1–2), 149–154.
21. Milevsky, M.A., and Robinson, C., 2000. Self-annuitization and Ruin in Retirement. North American Actuarial Journal 4, 113–129.
22. Milevsky, M.A. and Posner, S., 2001, The Titanic Option: Valuation of Guaranteed Minimum Death Benefits in Variable Annuities and Mutual Funds. Journal of Risk and Insurance 68 (1), 55-79
23. Milevsky, M.A., Salisbury, T.S., 2006. Financial Valuation of Guaranteed Minimum Withdrawal Benefits. Insurance: Mathematics and Economics 38, 21-38
24. Morton, K.W., and Mayers, D.F., 2005, Numerical Solution of Partial Differential Equations: An Introduction, 2nd ed., England: Cambridge University Press.
25. Nielsen, J.A. and Sandmann, K., 1995, Equity-linked Life insurance: A Model with Stochastic Interest Rates. Insurance: Mathematics and Economics 16, 225-253.
26. Nielsen, J.A. and Sandmann, K., 1996, Uniqueness of the Fair Premium for Equity-linked Life Insurance Contracts. The Geneva Papers on Risk and Insurance Theory 21, 65-102.
27. Shreve, S., 2004. Stochastic Calculus for Finance II: Continuous-time Models. 1st ed., New York: Springer.
28. Smith, G.D., 1985, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed., New York: Oxford University Press.
29. Wilmott, P., Dewynne, J.N., Howison, S., 1993, Option Pricin : Mathematical Models and Computation, 1st ed., England: Oxford Financial Press.
30. Wilmott, P., Dewynne, J.N., Howison, S., 1995, The Mathematics of Financial Derivatives: A Student Introduction, 1st ed., New York: Cambridge University Press.
31. Young, V.R., 2004. Optimal Investment Strategy to Minimize the Probability of Lifetime Ruin. North American Actuarial Journal 4, 106–126.
描述 碩士
國立政治大學
風險管理與保險研究所
95358001
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0095358001
資料類型 thesis
dc.contributor.advisor 張士傑zh_TW
dc.contributor.author (Authors) 李一成zh_TW
dc.creator (作者) 李一成zh_TW
dc.date (日期) 2008en_US
dc.date.accessioned 9-May-2016 11:51:42 (UTC+8)-
dc.date.available 9-May-2016 11:51:42 (UTC+8)-
dc.date.issued (上傳時間) 9-May-2016 11:51:42 (UTC+8)-
dc.identifier (Other Identifiers) G0095358001en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/94773-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 風險管理與保險研究所zh_TW
dc.description (描述) 95358001zh_TW
dc.description.abstract (摘要) 保險人因提供最低保證給付之投資型商品，使公司亦涉入投資風險。本研究旨在探討最低保證給付人壽保險附約之風險分析。首先利用隨機模型建構投資者帳戶價值的動態過程，進而推導出在未來時點帳戶發生餘額不足之機率及其所符合的偏微分方程式。並藉由數值方法－有限差分法，求出投資帳戶餘額不足之機率。最終，以不同的參數選取之下，進行敏感度分析，探討參數值的設定對於帳戶發生餘額不足之機率的影響。本研究結果可以提供保險公司與監理機關，作為日後發行保證給付商品時，一項風險管理上的考慮因素。
研究結果可以歸納為兩點結論：
1. 在市場因素中，投資帳戶連結之標的報酬率與帳戶餘額不足機率呈現反向變動，而波動度則是與帳戶餘額不足機率呈現正向變動。在兩因素同時考慮下，當報酬率愈高且波動度愈低，投資帳戶發生餘額不足的機率會愈低。當波動度愈高且報酬率愈低時，帳戶餘額不足機率則會愈高。其兩者的力量會相互抵銷，對投資帳戶餘額不足之機率的影響需視何者的力量較強而定。

2. 在條款設計的因素中，保證附約相關費用率、保證提領比率與保證提領期間對於投資帳戶發生餘額不足機率的影響皆呈現正向的關係。而投資帳戶期初的價值則與帳戶餘額不足機率呈現反向變動。其中保證提領比率對於投資帳戶的價值影響最大，其帳戶餘額不足機率之變動百分比相較於其他因素而言，變動幅度較大，範圍皆大於4%以上，甚至高達37.11%。		zh_TW
dc.description.abstract (摘要) Insurers have investment risks because they issue the guaranteed minimum benefit life insurance riders. Therefore, the purpose of this thesis is analyzing the risk for the riders. In the context, we implement numerical PDE solution to compute the ruin probability of separate account which is the probability that guaranteed minimum benefit life insurance riders will lead to financial insolvency under stochastic investment returns. Moreover, we will do sensitivity analyses to discuss the two aspects, market factors and contract designs, how to influence the ruin probability.

Finally, we conclude two main results:

1. For market factors, the rate of investment return is negatively related to ruin probability; however, the volatility is positive correlation.

2. For contract designs, the results show negative correlation between ruin probability and insurance fee, withdrawals, and withdrawal period. But the initial account value shows positive correlation.		en_US
dc.description.tableofcontents 目錄
第一章 前言 1
第二章 文獻回顧 4
第三章 最低保證附約之介紹 7
第一節 最低身故給付保證附約(GMDB) 8
第二節 最低滿期金額保證附約(GMAB) 9
第三節 最低年金金額保證附約(GMIB) 12
第四節 最低提領金額保證附約(GMWB) 14
第五節 範例說明(GMWB) 17
第四章 模型架構 20
第一節 一般化模型－GMWB 20
第二節 GMAB與GMIB之模型 25
第五章 數值方法－有限差分法 30
第一節 偏微分方程式介紹與分類 30
第二節 差分公式 33
第三節 定義域之分割與符號說明 37
第四節 有限差分法－顯式、隱式與Crank-Nicolson 38
第五節 收斂性、穩定性與一致性 41
第六節 Crank-Nicolson法求解投資帳戶餘額不足之機率 43
第六章 帳戶餘額不足機率計算與敏感度分析 46
第一節 帳戶餘額不足機率之計算 46
第二節 敏感度分析 47
第七章 結論與建議 58
參考文獻 60





表目錄

表1.1 臺灣之變額壽險保費收入情形(2003~2007) 1
表3.1 最低身故給付保證附約(GMDB)商品介紹 8
表3.2 最低滿期金額保證附約(GMAB)商品介紹 10
表3.3 最低年金金額保證附約(GMIB)商品介紹 12
表3.4 最低提領金額保證附約(GMWB)商品介紹 15
表3.5 最低提領金額保證附約商品之現金流量 17
表5.1 偏微分方程式分類之判斷準則 31
表6.1 不同報酬率與波動度下，投資帳戶餘額不足之機率 46
表6.2 報酬率與波動度對投資帳戶餘額不足機率之影響 48
表6.3 投資帳戶期初價值對帳戶餘額不足之機率影響 51
表6.4 保證附約相關費用率對投資帳戶餘額不足之機率影響 52
表6.5 保證提領比率對投資帳戶餘額不足之機率影響 54
表6.6 保證提領期間對投資帳戶餘額不足之機率影響 55
表6.7 市場因素與條款設計對投資帳戶餘額不足之機率關係 57




圖目錄

圖1.1 臺灣之變額壽險保費收入情形(2003~2007) 2
圖3.1 最低保證附約之種類與演進 7
圖3.2 GMDB附約下之帳戶價值 9
圖3.3 GMAB附約下之帳戶價值：市場狀況表現不好的情境 10
圖3.4 GMAB附約下之帳戶價值：市場狀況表現良好的情境 11
圖3.5 GMIB附約下之帳戶價值：市場狀況表現不好的情境 13
圖3.6 GMIB附約下之帳戶價值：市場狀況表現良好的情境 13
圖3.7 GMWB附約下之帳戶價值：市場狀況表現不好的情境 15
圖3.8 GMWB附約下之帳戶價值：市場狀況表現良好的情境 16
圖3.9 GMWB附約下之帳戶價值：情境一 19
圖3.10 GMWB附約下之帳戶價值：情境二 19
圖5.1 偏微分方程式之分類與其相對應特別形式的範例 32
圖5.2 一階導函數近似之幾何意義 34
圖5.3 定義域之分割 37
圖5.4 在固定時間分割點下，不同資產分割點之函數圖形 38
圖5.5 顯式差分法 39
圖5.6 隱式差分法 40
圖6.1 帳戶價值報酬率與波動度對投資帳戶餘額不足之機率影響 49
圖6.2 投資帳戶期初價值對與帳戶餘額不足機率之影響 51
圖6.3 保證附約相關費用率對投資帳戶餘額不足之機率影響 53
圖6.4 保證提領比率對投資帳戶餘額不足之機率影響 54
圖6.5 保證提領期間對投資帳戶餘額不足之機率影響 56		zh_TW
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0095358001en_US
dc.subject (關鍵詞) 最低保證給付zh_TW
dc.subject (關鍵詞) 投資帳戶餘額不足機率zh_TW
dc.subject (關鍵詞) 有限差分法zh_TW
dc.subject (關鍵詞) Guaranteed minimum benefiten_US
dc.subject (關鍵詞) Ruin probabilityen_US
dc.subject (關鍵詞) Numerical PDE solutionen_US
dc.title (題名) 最低保證給付人壽保險附約之風險分析zh_TW
dc.title (題名) Risk analysis for guaranteed minimum benefit life insurance ridersen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) 1. Aase, K.K., and Persson, S.A., 1994, Pricing of Unit-linked Life Insurance Policies. Scandinavian Actuarial Journal 1, 26-52.
2. Aase, K.K., and Persson, S.A., 1997, Valuation of the Minimum Guaranteed Return Embedded in Life Insurance Contracts. Journal of Risk and Insurance 64 (4), 599-617.
3. Albrecht, P., and Maurer, R., 2002. Self-annuitization, Consumption Shortfall in Retirement and Asset Allocation: The Annuity Benchmark. Journal of Pension Economics and Finance 1 (2), 57–72.
4. Bacinello, A.R. and Ortu, F., 1993, Pricing Guaranteed Securities-linked Life Insurance under Interest Rate Risk. Actuarial Approach for Financial Risks, 35-55.
5. Bacinello, A.R. and Ortu, F., 1994, Single and Periodic Premiums for Guaranteed Equity-linked Life Insurance under Interest Rate Risk: the ＂Lognormal＋Vasicek＂ Case, Financial Modeling, L. Peccati and M. Viren (Eds.), Physica-Verlag, Heidelberg, Germany, 1-25.
6. Black, F. and Scholes, M., 1973, The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81, 637-654.
7. Boyle, P.P., Schwartz, E., 1977. Equilibrium Prices of Guarantees under Equity-linked Contracts. Journal of Risk and Insurance 44(2), 639–680.
8. Boyle, P.P. and Schwartz, E.S., 1997, Equilibrium Prices of Guarantees under Equity-linked Contracts. Journal of Risk and Insurance 44, 639-680.
9. Boyle, P.P. and Hardy, M.R., 1997, Reserving for Maturity Guarantees: Two Approaches. Insurance: Mathematics and Economics 21, 113-127.
10. Boyle, P.P. and Hardy, M.R. 2003, Guaranteed Annuity Options. Astin Bulletin 33 (2), 125-152
11. Brennan, M.J., and Schwartz, E.S., 1976. The Pricing of Equity-linked Life Insurance Policies with an Asset Value Guarantee. Journal of Financial Economics 3 (1), 195–213.
12. Brennan, M.J. and Schwartz, E.S., 1979, Alternative Investment Strategies for the Issuers of Equity-linked Life Insurance Policies with an Asset Value Guarantee, Journal of Business 52, 63-93.
13. Duffy, D.J., 2006, Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, 1st ed., Hoboken, NJ: John Wiley & Sons.
14. Hardy, M.R. 2000, Hedging and Reserving for Single-premium Segregated Fund Contracts. North American Actuarial Journal 4 (2), 63-74.
15. Hardy, M.R., 2003, Investment Guarantees: Modeling and Risk Management for Equity-linked Life Insurance. 1st ed., Hoboken, N.J.: Wiley.
16. Huang, H., Milevsky, M.A., Wang, J., 2004. Ruined Moments in Your Life: How Good are the Approximations? Insurance: Mathematics and Economics 34 (3), 421-448
17. London, J., 2005, Modeling Derivatives in C++, 1st ed., Hoboken, N.J.: J. Wiley.
18. Milevsky, M.A., 1997. The Present Value of a Stochastic Perpetuity and the Gamma Distribution. Insurance: Mathematics and Economics 20, 243–250.
19. Milevsky, M.A., 1998. Optimal Asset Allocation towards the end of the Life Cycle: To Annuitize or not to Annuitize? Journal of Risk and Insurance 65 (3), 401–426.
20. Milevsky, M.A., 1999. Martingales, Scale Functions and Stochastic Life Annuities: A Note. Insurance: Mathematics and Economics 24 (1–2), 149–154.
21. Milevsky, M.A., and Robinson, C., 2000. Self-annuitization and Ruin in Retirement. North American Actuarial Journal 4, 113–129.
22. Milevsky, M.A. and Posner, S., 2001, The Titanic Option: Valuation of Guaranteed Minimum Death Benefits in Variable Annuities and Mutual Funds. Journal of Risk and Insurance 68 (1), 55-79
23. Milevsky, M.A., Salisbury, T.S., 2006. Financial Valuation of Guaranteed Minimum Withdrawal Benefits. Insurance: Mathematics and Economics 38, 21-38
24. Morton, K.W., and Mayers, D.F., 2005, Numerical Solution of Partial Differential Equations: An Introduction, 2nd ed., England: Cambridge University Press.
25. Nielsen, J.A. and Sandmann, K., 1995, Equity-linked Life insurance: A Model with Stochastic Interest Rates. Insurance: Mathematics and Economics 16, 225-253.
26. Nielsen, J.A. and Sandmann, K., 1996, Uniqueness of the Fair Premium for Equity-linked Life Insurance Contracts. The Geneva Papers on Risk and Insurance Theory 21, 65-102.
27. Shreve, S., 2004. Stochastic Calculus for Finance II: Continuous-time Models. 1st ed., New York: Springer.
28. Smith, G.D., 1985, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed., New York: Oxford University Press.
29. Wilmott, P., Dewynne, J.N., Howison, S., 1993, Option Pricin : Mathematical Models and Computation, 1st ed., England: Oxford Financial Press.
30. Wilmott, P., Dewynne, J.N., Howison, S., 1995, The Mathematics of Financial Derivatives: A Student Introduction, 1st ed., New York: Cambridge University Press.
31. Young, V.R., 2004. Optimal Investment Strategy to Minimize the Probability of Lifetime Ruin. North American Actuarial Journal 4, 106–126.		zh_TW