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REACHING A BEQUEST GOAL WITH LIFE INSURANCE: AMBIGUITY ABOUT THE RISKY ASSET'S DRIFT AND MORTALITY'S HAZARD RATE

Published online by Cambridge University Press:  05 December 2019

Xiaoqing Liang  and
Virginia R. Young
Xiaoqing Liang
Affiliation:
School of Sciences, Hebei University of Technology, Tianjin 300401, P. R. China, E-Mail: liangxiaoqing115@hotmail.com
Virginia R. Young*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA, E-Mail: vryoung@umich.edu
*
E-Mail: vryoung@umich.edu
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Abstract

We determine the optimal robust strategy of an individual who seeks to maximize the (penalized) probability of reaching a bequest goal when she is uncertain about the drift of the risky asset and her hazard rate of mortality. We assume the individual can invest in a Black–Scholes market. We solve two optimization problems with ambiguity. The first is to maximize the penalized probability of reaching a bequest goal without life insurance in the market. In the second problem, in addition to investing in the financial market, the individual is allowed to purchase term life insurance to help her reach her bequest goal. As the individual becomes more ambiguity averse concerning the drift of the risky asset, she becomes more conservative with her investment strategy. Also, as she becomes more ambiguity averse about her hazard rate of mortality, numerical work indicates she is more likely to buy life insurance when the ambiguity towards the return of the risky asset is not too large.

Keywords

Bequest motiveoptimal investmentterm life insuranceambiguity aversionstochastic controlrobust controlpenalized probabilityC61C73D81G22
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 1 , January 2020 , pp. 187 - 221
Copyright
© Astin Bulletin 2019 

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Footnotes

*

X. Liang thanks the National Natural Science Foundation of China (11701139, 11571189) and the Natural Science Foundation of Hebei Province (A2018202057) for the financial support of her research.

†

V. Young thanks the Cecil J. and Ethel M. Nesbitt Professorship for the partial financial support of her research.

References

Anderson, E.W., Hansen, L.P. and Sargent, T.J. (2003) A quartet of semi-groups for model specification, robustness, prices of risk, and model detection. Journal of the European Economic Association, 1(1), 68–123.CrossRefGoogle Scholar
Angoshtari, B., Bayraktar, E. and Young, V.R. (2015) Minimizing the expected lifetime spent in drawdown under proportional consumption. Finance Research Letters, 15, 106–114.CrossRefGoogle Scholar
Bayraktar, E., David Promislow, S. and Young, V.R. (2014) Purchasing life insurance to reach a bequest goal. Insurance: Mathematics and Economics, 58, 204–216.Google Scholar
Bayraktar, E., David Promislow, S. and Young, V.R. (2016) Purchasing term life insurance to reach a bequest goal while consuming. SIAM Journal on Financial Mathematics, 7(1), 183–214.CrossRefGoogle Scholar
Bayraktar, E. and Young, V.R. (2007) Correspondence between lifetime minimum wealth and utility of consumption. Finance and Stochastics, 11(2), 213–236.CrossRefGoogle Scholar
Bayraktar, E. and Young, V.R. (2016) Optimally investing to reach a bequest goal. Insurance: Mathematics and Economics, 70, 1–10.Google Scholar
Bayraktar, E. and Zhang, Y. (2015) Minimizing the probability of lifetime ruin under ambiguity aversion. SIAM Journal on Control and Optimization, 53(1), 58–90.CrossRefGoogle Scholar
Browne, S. (1997) Survival and growth with a liability: Optimal portfolio strategies in continuous time. Mathematics of Operations Research, 22(2), 468–493.CrossRefGoogle Scholar
Browne, S. (1999a) Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark. Finance and Stochastics, 3(3), 275–294.CrossRefGoogle Scholar
Browne, S. (1999b) Reaching goals by a deadline: Digital options and continuous-time active portfolio management. Advances in Applied Probability, 31(2), 551–577.CrossRefGoogle Scholar
Dubins, L.E. and Savage, L.J. (1965, 1976) How to Gamble if You Must: Inequalities for Stochastic Processes. 1965 edition, New York: McGraw-Hill. 1976 edition, New York: Dover.Google Scholar
Gu, A., Viens, F.G. and Yi, B. (2017) Optimal reinsurance and investment strategies for insurers with mispricing and model ambiguity. Insurance: Mathematics and Economics, 72, 235–249.Google Scholar
Hu, D., Chen, S. and Wang, H. (2018) Robust reinsurance contracts with uncertainty about jump risk. European Journal of Operational Research, 266(3), 1175–1188.CrossRefGoogle Scholar
Jaimungal, S. and Sigloch, G. (2012) Incorporating risk and ambiguity into a hybrid model of default. Mathematical Finance, 22(1), 57–81.CrossRefGoogle Scholar
Li, D., Zeng, Y. and Yang, H. (2018) Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps. Scandinavian Actuarial Journal, 2018(2), 145–171.CrossRefGoogle Scholar
Li, D. and Young, V.R. (2019) Optimal reinsurance to minimize the probability of ruin under ambiguity. Insurance: Mathematics and Economics, 87, 143–152.Google Scholar
Luo, S., Wang, M. and Zhu, W. (2019) Maximizing a robust goal-reaching probability with penalization on ambiguity. Journal of Computational and Applied Mathematics, 348, 261–281.CrossRefGoogle Scholar
Maenhout, P.J. (2004). Robust portfolio rules and asset pricing. Review of Financial Studies, 17(4), 951–983.CrossRefGoogle Scholar
Milevsky, M. A. and Robinson, C. (2000) Self-annuitization and ruin in retirement, with discussion. North American Actuarial Journal, 4(4), 112–129.CrossRefGoogle Scholar
Moore, K.S. and Young, V.R. (2006) Optimal and simple, nearly-optimal rules for minimizing the probability of financial ruin in retirement. North American Actuarial Journal, 10(4), 145–161.CrossRefGoogle Scholar
Pestien, V.C. and Sudderth, W.D. (1985) Continuous-time red and black: How to control a diffusion to a goal. Mathematics of Operations Research, 10(4), 599–611.CrossRefGoogle Scholar
Pirvu, T. A. and Zhang, H. (2012) Optimal investment, consumption and life insurance under mean-reverting returns: The complete market solution. Insurance: Mathematics and Economics, 51(2), 303–309.Google Scholar
Polyanin, A.D. and Zaitsev, V.F. (2003) Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edition. Boca Raton, Florida: Chapman and Hall.Google Scholar
Promislow, S.D. and Young, V.R. (2005) Minimizing the probability of ruin when claims follow Brownian motion with drift. North American Actuarial Journal, 9(3), 109–128.Google Scholar
Rogers, L.C.G. (2013) Optimal Investment. Berlin: Springer.CrossRefGoogle Scholar
Schmidli, H. (2002) On minimizing the ruin probability of investment and reinsurance. Journal of Applied Probability, 12(3), 890–907.Google Scholar
Sudderth, W.D. and Weerasinghe, A. (1989) Controlling a process to a goal in finite time. Mathematics of Operations Research, 14(3), 400–409.CrossRefGoogle Scholar
Yi, B., Li, Z., Viens, F.G. and Zeng, Y. (2013) Robust optimal control for an insurer with reinsurance and investment under Heston’s stochastic volatility model. Insurance: Mathematics and Economics, 53(3), 601–614.Google Scholar
Young, V.R. (2004). Optimal investment strategy to minimize the probability of lifetime ruin. North American Actuarial Journal, 8(4), 105–126.CrossRefGoogle Scholar
Young, V.R. and Zhang, Y. (2016) Lifetime ruin under ambiguous hazard rate. Insurance: Mathematics and Economics, 70, 125–134.Google Scholar
Zeng, Y., Li, D. and Gu, A. (2016). Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps. Insurance: Mathematics and Economics, 66, 138–152.Google Scholar