An Optimal Life Insurance Policy in the Continuous-Time Investment-Consumption Problem

Affiliation(s)

Faculty of Business Administration, Kyoto Sangyo University, Kyoto, Japan.

Faculty of Economics, Osaka Sangyo University, Osaka, Japan.

Faculty of Business Administration, Kyoto Sangyo University, Kyoto, Japan.

Faculty of Economics, Osaka Sangyo University, Osaka, Japan.

ABSTRACT

This paper considers an optimal life insurance for a household subject to mortality risk. The household receives wage income continuously, which could be terminated by unexpected premature loss of earning power. In order to hedge the risk of losing income stream, the household enters a life insurance contract. The household may also invest their wealth into a financial market. Therefore, the problem is to determine an optimal insurance/investment/consumption strategy. To reflect a real-life situation better, we consider an incomplete market where the household cannot trade insurance contracts continuously. We provide explicit solutions in a fairly general setup.

Cite this paper

H. Iwaki and Y. Osaki, "An Optimal Life Insurance Policy in the Continuous-Time Investment-Consumption Problem,"*Journal of Mathematical Finance*, Vol. 3 No. 2, 2013, pp. 291-306. doi: 10.4236/jmf.2013.32029.

H. Iwaki and Y. Osaki, "An Optimal Life Insurance Policy in the Continuous-Time Investment-Consumption Problem,"

References

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[2] S. A. Persson and K. K. Aase, “Valuation of the Minimum Guaranteed Return Embedded in a Life Insurance Products,” Journal of Risk and Insurance, Vol. 64, No. 4, 1997, pp. 599-617. doi:10.2307/253888

[3] M. J. Brennan and E. S. Schwartz, “The Pricing of Equity-linked Life Insurance Policies with an Asset Value Guarantee,” Journal of Financial Economics, Vol. 3, No. 3, 1976, pp. 195-213. doi:10.1016/0304-405X(76)90003-9

[4] J. A. Nielsen and K. Sandman, “Equity-Linked Life Insurance: A Model with Stochastic Interest Rates,” Insurance: Mathematics and Economics, Vol. 16, No. 3, 1995, pp. 225-253. doi:10.1016/0167-6687(95)00007-F

[5] E. Marceau and P. Gaillardetz, “On Life Insurance Reserves in a Stochastic Mortality and Interest Rates Environment,” Insurance: Mathematics and Economics, Vol. 25, 1999, pp. 261-280. doi:10.1016/S0167-6687(99)00019-0

[6] A. R. Bacinello, “Equity Linked Life Insurance,” In: E. Melnick and B. Everitt, Eds., Encyclopedia of Quantitative Risk Analysis and Assessment, John Wiley & Sons, Hoboken, 2008. doi:10.1002/9780470061596.risk0346

[7] H. Iwaki, M. Kijima and Y. Morimoto, “An Economic Premium Principle in a Multiperiod Economy,” Insurance: Mathematics and Insurance, Vol. 28, No. 3, 2001, pp. 325-339. doi:10.1016/S0167-6687(00)00081-0

[8] H. Iwaki, “An Economic Premium Principle in a Continuous-Time Economy,” Journal of the Operations Research Society of Japan, Vol. 45, 2002, pp. 346-361.

[9] R. C. Merton, “Life Time Portfolio Selection under Uncertainty,” Review of Economics and Statistics, Vol. 51, No. 3, 1969, pp. 247-257. doi:10.2307/1926560

[10] R. C. Merton, “Optimum Consumption and Portfolio Rules in a Continuous-time Model,” Journal of Economic Theory, Vol. 3, No. 4, 1971, pp. 373-413. doi:10.1016/0022-0531(71)90038-X

[11] S. F. Richard, “Optimal Consumption, Portfolio and Life Insurance Rules for an Uncertain Lived Individual in a Continuous Time Model,” Journal of Financial Economics, Vol. 2, No. 2, 1975, pp. 187-203. doi:10.1016/0304-405X(75)90004-5

[12] R. A. Campbell, “The Demand for Life Insurance: An Application of the Economics of Uncertainty,” Journal of Finance, Vol. 35, No. 5, 1980, pp. 1155-1172. doi:10.1111/j.1540-6261.1980.tb02201.x

[13] D. F. Babbel and E. Ohtsuka, “Aspects of Optimal Multi-period Life Insurance,” Journal of Risk and Insurance, Vol. 56, No. 3, 1989, pp. 460-481. doi:10.2307/253168

[14] Y. Zhu, “One-period Model of Individual Consumption, Life Insurance, and Investment Decisions,” Journal of Risk and Insurance, Vol. 74, No. 3, 2007, pp. 613-636. doi:10.1111/j.1539-6975.2007.00227.x

[15] Z. Bodie, R. C. Merton and W. Samuelson, “Labor Supply Flexibility and Portfolio Choice in a Life Cycle Model,” Journal of Economic Dynamics and Control, Vol. 16, No. 3-4, 1992, pp. 427-449. doi:10.1016/0165-1889(92)90044-F

[16] H. He and H. F. Pagès, “Labor Income, Borrowing Constraints and Equilibrium Asset Prices; A Duality Approach,” Economic Theory, Vol. 3, No. 4, 1993, pp. 663-696. doi:10.1007/BF01210265

[17] L. E. O. Svensson and I. M. Werner, “Nontradable Assets in Incomplete Markets: Pricing and Portfolio Choice,” European Economic Review, Vol. 37, No. 5, 1993, pp. 1149-1168. doi:10.1016/0014-2921(93)90113-O

[18] I. Karatzas and S. E. Shreve, “Methods of Mathematical Finance,” Springer-Verlag, New York, 1998.

[19] J. Cvitanic, W. Schachermayer and H. Wang, “Utility Maximization in Incomplete Markets with Random Endowment,” Finance and Stochastics, Vol. 5, No. 2, 2001, pp. 259-272. doi:10.1007/PL00013534

[20] J. Grandell, “Double Stochastic Poisson Processes,” Springer-Verlag, New York, 1976.

[21] A. Yashin and E. Arjas, “A Note on Random Intensities and Conditional Survival Functions,” Journal of Applied Probability, Vol. 25, No. 3, 1988, pp. 630-635. doi:10.2307/3213991

[22] D. Kramkov and W. Schachermayer, “The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets,” Annals of Applied Probability, Vol. 9, No. 3, 1999, pp. 904-950. doi:10.1214/aoap/1029962818

[23] N. Bellamy and M. Jeanblanc, “Incompleteness of Markets Driven by a Mixed Diffusion,” Finance and Stochastics, Vol. 4, No. 2, 2000, pp. 209-222. doi:10.1007/s007800050012

[24] S. Wang, V. R. Young and H. Panjier, “Axiomatic Characterization of Insurance Prices,” Insurence: Mathematics and Economics, Vol. 21, No. 2, 1997, pp.173-183. doi:10.1016/S0167-6687(97)00031-0

[25] V. R. Young and T. Zariphopoulou, “Computation of Distorted Probabilities for Diffusion Processes via Stochastic Control Methods,” Insurance: Mathematics and Economics, Vol. 27, No. 1, 2000, pp. 1-18. doi:10.1016/S0167-6687(99)00061-X

[26] D. Cuoco, “Optimal Consumption and Equilibrium Prices wiht Portfolio Constraints and Stochastic Income,” Journal of Economic Theory, Vol. 72, No. 1, 1997, pp. 33-73. doi:10.1006/jeth.1996.2207

[27] J. Cvitanic and I. Karatzas, “Convex Duality in Constrained Portfolio Optimization,” Annals of Applied Probability, Vol. 2, No. 4, 1992, pp. 767-818. doi:10.1214/aoap/1177005576

[28] R. T. Rockafellar, “Convex Analysis,” Princeton University Press, New Jersey, 1970.

[1] M. O. Albizzati and H. Geman, “Interest Rate Risk Management and Valuation of the Surrender Option in Life Insurance Policies,” Journal of Risk and Insurance, Vol. 61, No. 4, 1994, pp. 616-637. doi:10.2307/253641

[2] S. A. Persson and K. K. Aase, “Valuation of the Minimum Guaranteed Return Embedded in a Life Insurance Products,” Journal of Risk and Insurance, Vol. 64, No. 4, 1997, pp. 599-617. doi:10.2307/253888

[3] M. J. Brennan and E. S. Schwartz, “The Pricing of Equity-linked Life Insurance Policies with an Asset Value Guarantee,” Journal of Financial Economics, Vol. 3, No. 3, 1976, pp. 195-213. doi:10.1016/0304-405X(76)90003-9

[4] J. A. Nielsen and K. Sandman, “Equity-Linked Life Insurance: A Model with Stochastic Interest Rates,” Insurance: Mathematics and Economics, Vol. 16, No. 3, 1995, pp. 225-253. doi:10.1016/0167-6687(95)00007-F

[5] E. Marceau and P. Gaillardetz, “On Life Insurance Reserves in a Stochastic Mortality and Interest Rates Environment,” Insurance: Mathematics and Economics, Vol. 25, 1999, pp. 261-280. doi:10.1016/S0167-6687(99)00019-0

[6] A. R. Bacinello, “Equity Linked Life Insurance,” In: E. Melnick and B. Everitt, Eds., Encyclopedia of Quantitative Risk Analysis and Assessment, John Wiley & Sons, Hoboken, 2008. doi:10.1002/9780470061596.risk0346

[7] H. Iwaki, M. Kijima and Y. Morimoto, “An Economic Premium Principle in a Multiperiod Economy,” Insurance: Mathematics and Insurance, Vol. 28, No. 3, 2001, pp. 325-339. doi:10.1016/S0167-6687(00)00081-0

[8] H. Iwaki, “An Economic Premium Principle in a Continuous-Time Economy,” Journal of the Operations Research Society of Japan, Vol. 45, 2002, pp. 346-361.

[9] R. C. Merton, “Life Time Portfolio Selection under Uncertainty,” Review of Economics and Statistics, Vol. 51, No. 3, 1969, pp. 247-257. doi:10.2307/1926560

[10] R. C. Merton, “Optimum Consumption and Portfolio Rules in a Continuous-time Model,” Journal of Economic Theory, Vol. 3, No. 4, 1971, pp. 373-413. doi:10.1016/0022-0531(71)90038-X

[11] S. F. Richard, “Optimal Consumption, Portfolio and Life Insurance Rules for an Uncertain Lived Individual in a Continuous Time Model,” Journal of Financial Economics, Vol. 2, No. 2, 1975, pp. 187-203. doi:10.1016/0304-405X(75)90004-5

[12] R. A. Campbell, “The Demand for Life Insurance: An Application of the Economics of Uncertainty,” Journal of Finance, Vol. 35, No. 5, 1980, pp. 1155-1172. doi:10.1111/j.1540-6261.1980.tb02201.x

[13] D. F. Babbel and E. Ohtsuka, “Aspects of Optimal Multi-period Life Insurance,” Journal of Risk and Insurance, Vol. 56, No. 3, 1989, pp. 460-481. doi:10.2307/253168

[14] Y. Zhu, “One-period Model of Individual Consumption, Life Insurance, and Investment Decisions,” Journal of Risk and Insurance, Vol. 74, No. 3, 2007, pp. 613-636. doi:10.1111/j.1539-6975.2007.00227.x

[15] Z. Bodie, R. C. Merton and W. Samuelson, “Labor Supply Flexibility and Portfolio Choice in a Life Cycle Model,” Journal of Economic Dynamics and Control, Vol. 16, No. 3-4, 1992, pp. 427-449. doi:10.1016/0165-1889(92)90044-F

[16] H. He and H. F. Pagès, “Labor Income, Borrowing Constraints and Equilibrium Asset Prices; A Duality Approach,” Economic Theory, Vol. 3, No. 4, 1993, pp. 663-696. doi:10.1007/BF01210265

[17] L. E. O. Svensson and I. M. Werner, “Nontradable Assets in Incomplete Markets: Pricing and Portfolio Choice,” European Economic Review, Vol. 37, No. 5, 1993, pp. 1149-1168. doi:10.1016/0014-2921(93)90113-O

[18] I. Karatzas and S. E. Shreve, “Methods of Mathematical Finance,” Springer-Verlag, New York, 1998.

[19] J. Cvitanic, W. Schachermayer and H. Wang, “Utility Maximization in Incomplete Markets with Random Endowment,” Finance and Stochastics, Vol. 5, No. 2, 2001, pp. 259-272. doi:10.1007/PL00013534

[20] J. Grandell, “Double Stochastic Poisson Processes,” Springer-Verlag, New York, 1976.

[21] A. Yashin and E. Arjas, “A Note on Random Intensities and Conditional Survival Functions,” Journal of Applied Probability, Vol. 25, No. 3, 1988, pp. 630-635. doi:10.2307/3213991

[22] D. Kramkov and W. Schachermayer, “The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets,” Annals of Applied Probability, Vol. 9, No. 3, 1999, pp. 904-950. doi:10.1214/aoap/1029962818

[23] N. Bellamy and M. Jeanblanc, “Incompleteness of Markets Driven by a Mixed Diffusion,” Finance and Stochastics, Vol. 4, No. 2, 2000, pp. 209-222. doi:10.1007/s007800050012

[24] S. Wang, V. R. Young and H. Panjier, “Axiomatic Characterization of Insurance Prices,” Insurence: Mathematics and Economics, Vol. 21, No. 2, 1997, pp.173-183. doi:10.1016/S0167-6687(97)00031-0

[25] V. R. Young and T. Zariphopoulou, “Computation of Distorted Probabilities for Diffusion Processes via Stochastic Control Methods,” Insurance: Mathematics and Economics, Vol. 27, No. 1, 2000, pp. 1-18. doi:10.1016/S0167-6687(99)00061-X

[26] D. Cuoco, “Optimal Consumption and Equilibrium Prices wiht Portfolio Constraints and Stochastic Income,” Journal of Economic Theory, Vol. 72, No. 1, 1997, pp. 33-73. doi:10.1006/jeth.1996.2207

[27] J. Cvitanic and I. Karatzas, “Convex Duality in Constrained Portfolio Optimization,” Annals of Applied Probability, Vol. 2, No. 4, 1992, pp. 767-818. doi:10.1214/aoap/1177005576

[28] R. T. Rockafellar, “Convex Analysis,” Princeton University Press, New Jersey, 1970.