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Analytic solution for ratchet guaranteed minimum death benefit options under a variety of mortality laws

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  • Ulm, Eric R.

Abstract

We derive a number of analytic results for GMDB ratchet options. Closed form solutions are found for De Moivre’s Law, Constant Force of Mortality, Constant Force of Mortality with an endowment age and constant force of mortality with a cutoff age. We find an infinite series solution for a general mortality laws and we derive the conditions under which this series terminates. We sum this series for at-the-money options under the realistic Makeham’s Law of Mortality.

Suggested Citation

  • Ulm, Eric R., 2014. "Analytic solution for ratchet guaranteed minimum death benefit options under a variety of mortality laws," Insurance: Mathematics and Economics, Elsevier, vol. 58(C), pages 14-23.
    • Handle: RePEc:eee:insuma:v:58:y:2014:i:c:p:14-23
    DOI: 10.1016/j.insmatheco.2014.06.003
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    References listed on IDEAS

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    1. Eric R. Ulm, 2006. "The Effect of the Real Option to Transfer on the Value of Guaranteed Minimum Death Benefits," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 73(1), pages 43-69, March.
    2. Milevsky,Moshe A., 2006. "The Calculus of Retirement Income," Cambridge Books, Cambridge University Press, number 9780521842587, September.
    3. Dmitry Davydov & Vadim Linetsky, 2001. "Pricing and Hedging Path-Dependent Options Under the CEV Process," Management Science, INFORMS, vol. 47(7), pages 949-965, July.
    4. Goldman, M Barry & Sosin, Howard B & Gatto, Mary Ann, 1979. "Path Dependent Options: "Buy at the Low, Sell at the High"," Journal of Finance, American Finance Association, vol. 34(5), pages 1111-1127, December.
    5. Ulm, Eric R., 2008. "Analytic Solution for Return of Premium and Rollup Guaranteed Minimum Death Benefit Options Under Some Simple Mortality Laws," ASTIN Bulletin, Cambridge University Press, vol. 38(2), pages 543-563, November.
    6. Gerber, Hans U. & Shiu, Elias S.W. & Yang, Hailiang, 2012. "Valuing equity-linked death benefits and other contingent options: A discounted density approach," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 73-92.
    7. Wilmott,Paul & Howison,Sam & Dewynne,Jeff, 1995. "The Mathematics of Financial Derivatives," Cambridge Books, Cambridge University Press, number 9780521497893, September.
    8. Brennan, Michael J. & Schwartz, Eduardo S., 1976. "The pricing of equity-linked life insurance policies with an asset value guarantee," Journal of Financial Economics, Elsevier, vol. 3(3), pages 195-213, June.
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    Cited by:

    1. Kirkby, J. Lars & Nguyen, Duy, 2021. "Equity-linked Guaranteed Minimum Death Benefits with dollar cost averaging," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 408-428.
    2. Huang, Huaxiong & Milevsky, Moshe A., 2016. "Longevity risk and retirement income tax efficiency: A location spending rate puzzle," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 50-62.
    3. Feng, Runhuan & Shimizu, Yasutaka, 2016. "Applications of central limit theorems for equity-linked insurance," Insurance: Mathematics and Economics, Elsevier, vol. 69(C), pages 138-148.
    4. Ulm, Eric, 2020. "Analytic Valuation of GMDB Options with Utility Based Asset Allocation," Working Paper Series 8566, Victoria University of Wellington, School of Economics and Finance.
    5. Wenguang Yu & Yaodi Yong & Guofeng Guan & Yujuan Huang & Wen Su & Chaoran Cui, 2019. "Valuing Guaranteed Minimum Death Benefits by Cosine Series Expansion," Mathematics, MDPI, vol. 7(9), pages 1-15, September.
    6. Runhuan Feng & Xiaochen Jing & Jan Dhaene, 2015. "Comonotonic Approximations of Risk Measures for Variable Annuity Guaranteed Benefits with Dynamic Policyholder Behavior," Tinbergen Institute Discussion Papers 15-008/IV/DSF85, Tinbergen Institute.
    7. Yaodi Yong & Hailiang Yang, 2021. "Valuation of Cliquet-Style Guarantees with Death Benefits in Jump Diffusion Models," Mathematics, MDPI, vol. 9(16), pages 1-21, August.
    8. Ulm, Eric, 2020. "Analytic Valuation of GMDB Options with Utility Based Asset Allocation," Working Paper Series 21060, Victoria University of Wellington, School of Economics and Finance.
    9. Liang, Xiaoqing & Tsai, Cary Chi-Liang & Lu, Yi, 2016. "Valuing guaranteed equity-linked contracts under piecewise constant forces of mortality," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 150-161.

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    More about this item

    Keywords

    Variable annuities; Laplace transforms; Partial differential equations; Guaranteed minimum death benefits; Closed form solutions;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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