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題名 死亡風險的自然避險與商品設計
Natural Hedging of Mortality Risks and Product Design
作者 黃芳文
貢獻者 蔡政憲
黃芳文
關鍵詞 死亡風險
長壽風險
自然避險
mortality risk
longevity risk
natural hedging
日期 2015
上傳時間 3-Feb-2016 11:18:17 (UTC+8)
摘要 對必須維持長期清償能力的壽險公司來說，如何做好死亡率的風險管理是極為基本的。過去的文獻上提出，利用保險公司銷售的壽險商品(如：終身壽險)，因具有死亡風險，對具有長壽風險的商品(如：年金險)可產生避險效果。這種商品組合的自然避險方式是顯而易見的，但可能因為僵化的銷售市場與誘因導向的銷售方式等因素，商品組合的自然避險是較為難以執行的。我們提出的是，把自然避險策略植入商品設計中，可以把避險效果內含於商品內容裡面。關鍵在於讓這張保單在死亡事件發生的時間點所產生的影響可以被抵銷，我們技巧性地選取給付成長參數，即“應給付多少”因子，可以決定死亡給付的金額現值，而利率因子δ洽可反映給付的時間價值。本論文集提供壽險商品與年金商品的理論推導、圖解說明以及數值分析，一一闡述我們的想法以及如何將自然避險融入於商品設計之中。
在第一篇中，我們利用精算的方法推導，得到利用壽險商品設計範疇內可以達到最佳化化的自然避險策略。在第二篇的論文中，依據第一篇自然避險策略的理論基礎下的商品設計，我們除更進一步探討在死亡率與利率條件不確定的因素下，對於以最佳化避險策略的商品，可產生的影響，我們以實際的數據分析，得到即使對於未來的死亡率與利率不確定的情況下，我們所提出的依理論條件下的最佳化避險策略所設計的商品，仍能使商品所產生的死亡率風險極小化，甚至接近於無死亡率風險。除以壽險商品為主要設計外，在第三篇論文中，我們也呈現以年金商品為架構的商品設計。
How to manage mortality rate risks is essential to the long-term solvency of life insurance companies. The literatures proposed to hedge the products subject to the longevity risk (such as annuities) by using the products subject to the mortality risk (e.g., whole life insurance) sold by an insurer. Such natural hedging is intuitive but may be difficult to implement due to the rigid sales market and incentive issues. We propose to embed natural hedging into product design so that the hedging may occur within a product. The key is to offset the impact of mortality on the timing of death that in turn determines the present value of the death benefit by cleverly choose the growth rate of the death benefit, the factor of “how much to pay” while δ would reflect the time value of payment. This article provides theoretical derivations, graphical illustrations, and numerical analyses of both life insurance products and annuity products to illustrate the idea of embedding natural hedging into product design.
In the first essay, we use actuarial methods to come up with the theoretical derivation of the optimal natural hedging strategy and we can easily embed it into insurance product design. In the second essay, furthermore, we evaluate the impact of uncertainty passed on by future mortality rate and interest rate against the product design based on the optimal natural hedging strategy. We use the numerical illustration and obtain the minimum risk or nearly no risk at all under the optimal strategy. We develop not only on the life insurance product design, in the third essay, we also progress the product design of annuity products.
參考文獻 Blake, D., Burrows, W., (2001). Survivor Bonds: Helping to Hedge Mortality Risk. The Journal of Risk and Insurance 68(2), 339-348.
Blake, D., Cairns, A.J.G., Dowd, K., (2006). Living with Mortality: Longevity Bonds and Other Mortality-Linked Securities. British Actuarial Journal 12(1), 153-197.
Blake, D., Cairns, A.J.G., Dowd, K., MacMinn, R., (2006). Longevity Bonds: Financial Engineering, Valuation, and Hedging. The Journal of Risk and Insurance 73(4), 647-672.
Brown, J. R., & Orszag, P. R., (2006). The Political Economy of Government-Issued Longevity Bonds. The Journal of Risk and Insurance 73(4), 611–631.
Cairns, A.J.G., Blake, D., Dowd, K., (2006a). Pricing death: Frameworks for the valuation and securitization of mortality risk. ASTIN Bulletin 36, 79-120.
Cairns, A.J.G., Blake, D., Dowd, K., (2006b). A Two‐Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration. The Journal of Risk and Insurance 73(4), 687-718.
Cox, S. H., Lin, Y., (2007). Natural Hedging of Life and Annuity Mortality Risks. North American Actuarial Journal 11(3), 1-15.
Cox, S. H., Lin, Y., Wang, S., (2006). Multivariate Exponential Tilting and Pricing Implications for Mortality Securitization. The Journal Risk and insurance 73(4), 719-736.
Denuit, M., Devolder, P., and Goderniaux, A. C., (2007). Securitization of Longevity Risk: Pricing Survivor Bonds With Wang Transform in the Lee-Carter Framework. The Journal of Risk and Insurance 74(1), 87-113.
Dowd, K., (2003). Survivor Bonds: A Comment on Blake and Burrows. The Journal of Risk and Insurance, 70(2), 339-348.
Dowd, K., Blake, D., Cairns, A.J.G., Dawson, P., (2006). Survivor swaps. Journal of Risk and Insurance 73, 1-17.
Human Mortality Database, (2005). Available at: http://www.mortality.org (accessed May 2015).
Lee, R. D., & Carter, L., (1992). Modeling and Forecasting the Time Series of U.S. Mortality. Journal of the American Statistical Association 87(3), 659–671.
Lee, R. D., (2000). The Lee-Carter Method for Forecasting Mortality, With Various Extensions and Applications. North American Actuarial Journal, 4(1): 80-93.
Lin, T., Tsai, C. C., (2013). On the Mortality/Longevity Risk Hedging with Mortality Immunization. Insurance: Mathematics and Economics 53(3), 580-596.
Liu, Xiaoming, (2013). Annuity Uncertainty with Stochastic Mortality and Interest Rates. North American Actuarial Journal, 17(2), 136–152.
Marceau, E. and Gaillardetz, P., (1999). On life insurance reserves in a stochastic mortality and interest rates environment. Insurance: Mathematics and Economics 25(3):261-280.
Melnikov, A., Romaniuk, Y., (2006). Evaluating the performance of Gompertz, Makeham and Lee–Carter Mortality Models for Risk Management with Unit-Linked Contracts. Insurance: Mathematics and Economics 39(3), 310-329.
Pollard, J.H., (1973), Mathematical Models for the Growth of Human Populations, Cambridge University Press, London, Great Britain.
Tsai, C.C., Chung, S., (2013). Actuarial Applications of the Linear Hazard Transform in Mortality Immunization. Insurance: Mathematics and Economics 53(1), 48-63.
Tsai; J. T., Wang, J. L., Tzeng, L.Y., (2010). On the Optimal Product Mix in Life Insurance Companies Using Conditional Value at Risk. Insurance: Mathematics and Economics 46, 235-241.
Wang, C., Huang, H.C., Hong, D., (2013). A Feasible Natural Hedging Strategy for Insurance Companies. Insurance: Mathematics and Economics 52(3), 532-541.
Wang, J.L., Huang, H.C., Yang, S.S., Tsai, J.T., (2010). An Optimal Product Mix for Hedging Longevity Risk in Life Insurance Companies: The Immunization Theory Approach. The Journal of Risk and Insurance 77, 473-497.
Wilkie, A. D., Waters, H. R., and Yang, S. S., (2003). Reserving, Pricing and Hedging For Policies with Guaranteed Annuity Options. British Actuarial Journal 9(2),
描述 博士
國立政治大學
風險管理與保險研究所
100358502
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0100358502
資料類型 thesis
dc.contributor.advisor 蔡政憲zh_TW
dc.contributor.author (Authors) 黃芳文zh_TW
dc.creator (作者) 黃芳文zh_TW
dc.date (日期) 2015en_US
dc.date.accessioned 3-Feb-2016 11:18:17 (UTC+8)-
dc.date.available 3-Feb-2016 11:18:17 (UTC+8)-
dc.date.issued (上傳時間) 3-Feb-2016 11:18:17 (UTC+8)-
dc.identifier (Other Identifiers) G0100358502en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/81120-
dc.description (描述) 博士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) 100358502zh_TW
dc.description.abstract (摘要) 對必須維持長期清償能力的壽險公司來說，如何做好死亡率的風險管理是極為基本的。過去的文獻上提出，利用保險公司銷售的壽險商品(如：終身壽險)，因具有死亡風險，對具有長壽風險的商品(如：年金險)可產生避險效果。這種商品組合的自然避險方式是顯而易見的，但可能因為僵化的銷售市場與誘因導向的銷售方式等因素，商品組合的自然避險是較為難以執行的。我們提出的是，把自然避險策略植入商品設計中，可以把避險效果內含於商品內容裡面。關鍵在於讓這張保單在死亡事件發生的時間點所產生的影響可以被抵銷，我們技巧性地選取給付成長參數，即“應給付多少”因子，可以決定死亡給付的金額現值，而利率因子δ洽可反映給付的時間價值。本論文集提供壽險商品與年金商品的理論推導、圖解說明以及數值分析，一一闡述我們的想法以及如何將自然避險融入於商品設計之中。
在第一篇中，我們利用精算的方法推導，得到利用壽險商品設計範疇內可以達到最佳化化的自然避險策略。在第二篇的論文中，依據第一篇自然避險策略的理論基礎下的商品設計，我們除更進一步探討在死亡率與利率條件不確定的因素下，對於以最佳化避險策略的商品，可產生的影響，我們以實際的數據分析，得到即使對於未來的死亡率與利率不確定的情況下，我們所提出的依理論條件下的最佳化避險策略所設計的商品，仍能使商品所產生的死亡率風險極小化，甚至接近於無死亡率風險。除以壽險商品為主要設計外，在第三篇論文中，我們也呈現以年金商品為架構的商品設計。		zh_TW
dc.description.abstract (摘要) How to manage mortality rate risks is essential to the long-term solvency of life insurance companies. The literatures proposed to hedge the products subject to the longevity risk (such as annuities) by using the products subject to the mortality risk (e.g., whole life insurance) sold by an insurer. Such natural hedging is intuitive but may be difficult to implement due to the rigid sales market and incentive issues. We propose to embed natural hedging into product design so that the hedging may occur within a product. The key is to offset the impact of mortality on the timing of death that in turn determines the present value of the death benefit by cleverly choose the growth rate of the death benefit, the factor of “how much to pay” while δ would reflect the time value of payment. This article provides theoretical derivations, graphical illustrations, and numerical analyses of both life insurance products and annuity products to illustrate the idea of embedding natural hedging into product design.
In the first essay, we use actuarial methods to come up with the theoretical derivation of the optimal natural hedging strategy and we can easily embed it into insurance product design. In the second essay, furthermore, we evaluate the impact of uncertainty passed on by future mortality rate and interest rate against the product design based on the optimal natural hedging strategy. We use the numerical illustration and obtain the minimum risk or nearly no risk at all under the optimal strategy. We develop not only on the life insurance product design, in the third essay, we also progress the product design of annuity products.		en_US
dc.description.tableofcontents Essay I : The Optimal Strategy in Insurance Product Design 11
1. Introduction 12
2. Literature Review 16
3. Theoretical Development 21
3.1 Idea Scratching 21
3.2 Formal Development 22
4. Numerical Illustrations 34
4.1 The optimal strategy with γ=δ 34
4.2 The secondary strategy with 0<γ<δ 41
5. Practice and Conclusion 42
Essay II : The Uncertainty to Optimal Strategy in Life Insurance Product design 45
1. Introduction 46
2. Theoretical Development 49
3. Models 53
3.1 Model for Mortality Rate 53
3.2 Model for Interest Rate 56
4. Numerical Illustrations for Uncertainty 57
4.1 The optimal strategy with γ(t) =δ(t) 57
4.2 The secondary strategy with 0<γ(t)<δ(t) 64
5. Conclusion 67
Essay III: The Natural Hedging Strategy for Annuity Product in Product Design 68
1. Introduction 69
2. Theoretical Development for Annuity Products 73
3. Numerical Illustrations for Annuity Products 78
4. Conclusion 80

References 81
Appendix 84
A1. The deriving process of the equation (15) 84
A2. The close form of differential equation of equation (13) 85		zh_TW
dc.format.extent 2227612 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0100358502en_US
dc.subject (關鍵詞) 死亡風險zh_TW
dc.subject (關鍵詞) 長壽風險zh_TW
dc.subject (關鍵詞) 自然避險zh_TW
dc.subject (關鍵詞) mortality risken_US
dc.subject (關鍵詞) longevity risken_US
dc.subject (關鍵詞) natural hedgingen_US
dc.title (題名) 死亡風險的自然避險與商品設計zh_TW
dc.title (題名) Natural Hedging of Mortality Risks and Product Designen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) Blake, D., Burrows, W., (2001). Survivor Bonds: Helping to Hedge Mortality Risk. The Journal of Risk and Insurance 68(2), 339-348.
Blake, D., Cairns, A.J.G., Dowd, K., (2006). Living with Mortality: Longevity Bonds and Other Mortality-Linked Securities. British Actuarial Journal 12(1), 153-197.
Blake, D., Cairns, A.J.G., Dowd, K., MacMinn, R., (2006). Longevity Bonds: Financial Engineering, Valuation, and Hedging. The Journal of Risk and Insurance 73(4), 647-672.
Brown, J. R., & Orszag, P. R., (2006). The Political Economy of Government-Issued Longevity Bonds. The Journal of Risk and Insurance 73(4), 611–631.
Cairns, A.J.G., Blake, D., Dowd, K., (2006a). Pricing death: Frameworks for the valuation and securitization of mortality risk. ASTIN Bulletin 36, 79-120.
Cairns, A.J.G., Blake, D., Dowd, K., (2006b). A Two‐Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration. The Journal of Risk and Insurance 73(4), 687-718.
Cox, S. H., Lin, Y., (2007). Natural Hedging of Life and Annuity Mortality Risks. North American Actuarial Journal 11(3), 1-15.
Cox, S. H., Lin, Y., Wang, S., (2006). Multivariate Exponential Tilting and Pricing Implications for Mortality Securitization. The Journal Risk and insurance 73(4), 719-736.
Denuit, M., Devolder, P., and Goderniaux, A. C., (2007). Securitization of Longevity Risk: Pricing Survivor Bonds With Wang Transform in the Lee-Carter Framework. The Journal of Risk and Insurance 74(1), 87-113.
Dowd, K., (2003). Survivor Bonds: A Comment on Blake and Burrows. The Journal of Risk and Insurance, 70(2), 339-348.
Dowd, K., Blake, D., Cairns, A.J.G., Dawson, P., (2006). Survivor swaps. Journal of Risk and Insurance 73, 1-17.
Human Mortality Database, (2005). Available at: http://www.mortality.org (accessed May 2015).
Lee, R. D., & Carter, L., (1992). Modeling and Forecasting the Time Series of U.S. Mortality. Journal of the American Statistical Association 87(3), 659–671.
Lee, R. D., (2000). The Lee-Carter Method for Forecasting Mortality, With Various Extensions and Applications. North American Actuarial Journal, 4(1): 80-93.
Lin, T., Tsai, C. C., (2013). On the Mortality/Longevity Risk Hedging with Mortality Immunization. Insurance: Mathematics and Economics 53(3), 580-596.
Liu, Xiaoming, (2013). Annuity Uncertainty with Stochastic Mortality and Interest Rates. North American Actuarial Journal, 17(2), 136–152.
Marceau, E. and Gaillardetz, P., (1999). On life insurance reserves in a stochastic mortality and interest rates environment. Insurance: Mathematics and Economics 25(3):261-280.
Melnikov, A., Romaniuk, Y., (2006). Evaluating the performance of Gompertz, Makeham and Lee–Carter Mortality Models for Risk Management with Unit-Linked Contracts. Insurance: Mathematics and Economics 39(3), 310-329.
Pollard, J.H., (1973), Mathematical Models for the Growth of Human Populations, Cambridge University Press, London, Great Britain.
Tsai, C.C., Chung, S., (2013). Actuarial Applications of the Linear Hazard Transform in Mortality Immunization. Insurance: Mathematics and Economics 53(1), 48-63.
Tsai; J. T., Wang, J. L., Tzeng, L.Y., (2010). On the Optimal Product Mix in Life Insurance Companies Using Conditional Value at Risk. Insurance: Mathematics and Economics 46, 235-241.
Wang, C., Huang, H.C., Hong, D., (2013). A Feasible Natural Hedging Strategy for Insurance Companies. Insurance: Mathematics and Economics 52(3), 532-541.
Wang, J.L., Huang, H.C., Yang, S.S., Tsai, J.T., (2010). An Optimal Product Mix for Hedging Longevity Risk in Life Insurance Companies: The Immunization Theory Approach. The Journal of Risk and Insurance 77, 473-497.
Wilkie, A. D., Waters, H. R., and Yang, S. S., (2003). Reserving, Pricing and Hedging For Policies with Guaranteed Annuity Options. British Actuarial Journal 9(2),		zh_TW