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April  2017, 13(2): 737-755. doi: 10.3934/jimo.2016044

Optimal reinsurance and investment strategy with two piece utility function

Lv Chen 1,, and  Hailiang Yang 2,

1. 

School of Statistics, East China Normal University, Shanghai, 200241, China
2. 

Departments of Statistics of Actuarial Science, The University of Hong Kong, Hong Kong, China

* Corresponding author: Lv Chen

Received  November 2015 Published  August 2016

Fund Project: The first author is supported by Research Grants Council of the Hong Kong Special Administrative Region (project No. HKU 705313P), National Natural Science Foundation of China (grant number 11231005,11571113), Program of Shanghai Subject Chief Scientist (grant number 14XD1401600).

This paper studies optimal reinsurance and investment strategies that maximize expected utility of the terminal wealth for an insurer in a stochastic market. The insurer's preference is represented by a two-piece utility function which can be regarded as a generalization of traditional concave utility functions. We employ martingale approach and convex optimization method to transform the dynamic maximization problem into an equivalent static optimization problem. By solving the optimization problem, we derive explicit expressions of the optimal reinsurance and investment strategy and the optimal wealth process.

Keywords: Two piece utility function, martingale technique, optimal control, investment, proportional reinsurance.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Lv Chen, Hailiang Yang. Optimal reinsurance and investment strategy with two piece utility function. Journal of Industrial & Management Optimization, 2017, 13 (2) : 737-755. doi: 10.3934/jimo.2016044
References:
[1]

M. Alias, Le comportement de l'homme rationel devant le risque: Critique des postulats et axioms de l'ecole americaine, Econometrica, 21 (1953), 503-546.  doi: 10.2307/1907921.  Google Scholar

[2]

D. E. Bell, Disappointment in decision making under uncertainty, Oper. Res., 33 (1985), 1-27.  doi: 10.1287/opre.33.1.1.  Google Scholar

[3]

S. Benartzi and R. H. Thaler, Myopic loss aversion and the equity premium puzzle, Quart. J. Econ., 110 (1995), 73-92.  doi: 10.2307/2118511.  Google Scholar

[4]

A, B. Berkelaar, R. Kouwenberg and T. Post, Optimal Portfolio Choice under Loss Aversion, Review of Economics and Statistics, 86 (2004), 973-987.   Google Scholar

[5]

C. Bernard and M. Ghossoub, Static portfolio choice under cumulative prospect theory, Financ. Econ., 2 (2010), 277-306.  doi: 10.1007/s11579-009-0021-2.  Google Scholar

[6]

S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Math. of Oper. Res., 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.  Google Scholar

[7]

K. C. Chueng, W. F. Chong and S. C. P. Yam, The optimal insurance under disappointment theories, Insurance Math. Econom., 64 (2015), 77-90.  doi: 10.1016/j.insmatheco.2015.04.004.  Google Scholar

[8]

K. C. Chueng, W. F. Chong, R. J. Elliot and S. C. P. Yam, Disappointment aversion premium principle, Astin Bulletin, 45 (2015), 679-702.  doi: 10.1017/asb.2015.12.  Google Scholar

[9]

W. J. Guo, Optimal portfolio choice for an insurer with loss aversion, Insurance Math. Econom., 58 (2014), 217-222.  doi: 10.1016/j.insmatheco.2014.07.004.  Google Scholar

[10]

C. Hipp and M. Plum, Optimal investment for insurers, Insurance Math. Econom., 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4.  Google Scholar

[11]

C. Hipp and M. Plum, Optimal investment for investors with state dependent income, and for insurers, Finance Stoch., 7 (2003), 299-321.  doi: 10.1007/s007800200095.  Google Scholar

[12]

H. Jin and X. Y. Zhou, Behavior portfolio selection in continuous time, Math. Finance, 18 (2008), 385-426.  doi: 10.1111/j.1467-9965.2008.00339.x.  Google Scholar

[13]

D. Kahneman and A. Tversky, Prospect Theory-Analysis of Decision under risk, Econometrica, 47 (1979), 263-291.   Google Scholar

[14]

I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer-Verlag, New York, 1998. doi: 10.1007/b98840.  Google Scholar

[15]

C. S. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin, N. Am. Actuar. J., 8 (2004), 11-31.  doi: 10.1080/10920277.2004.10596134.  Google Scholar

[16]

G. Loomes and R. Sugden, Disappointment and dynamic consistency in choice under uncertainty, Rev. Econom. Stud., 53 (1986), 271-282.  doi: 10.2307/2297651.  Google Scholar

[17]

L. L. Lopes and G. C. Oden, The role of aspiration level in risky choice: A comparison of cumulative prospect theory and SP/A theory, J. Math. Psych, 43 (1999), 286-313.  doi: 10.1006/jmps.1999.1259.  Google Scholar

[18]

R. Mehra and E. C. Prescott, The equity premium: A puzzle, J. Monetary Econ, 15 (1985), 145-161.  doi: 10.1016/0304-3932(85)90061-3.  Google Scholar

[19]

H. Mi and S. G. Zhang, Continuous time portfolio selection with loss aversion in an incomplete market, Oper. Res Trans, 16 (2012), 1-12.   Google Scholar

[20]

H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J., 1 (2001), 55-68.  doi: 10.1080/034612301750077338.  Google Scholar

[21]

H. Shefrin and M. Statman, Behavioral portfolio theory, J. Financ. Quant. Anal., 35 (2000), 127-151.  doi: 10.2307/2676187.  Google Scholar

[22]

K. C. J. Sung, S. C. P. Yam, S. P. Yung and J. H. Zhou, Behavioral optimal insurance, Insurance Math. Econom., 49 (2011), 418-428.  doi: 10.1016/j.insmatheco.2011.04.008.  Google Scholar

[23]

A. Tsanakas and E. Desli, Risk measures and theories of choice, British Acturial Journal, 9 (2003), 959-991.  doi: 10.1017/S1357321700004414.  Google Scholar

[24]

A. Tversky and D. Kahneman, Advances in prospect theory: Cumulative representation of uncertainty, Chapter: Readings in Formal Epistemology, 1 (2016), 493-519.  doi: 10.1007/978-3-319-20451-2_24.  Google Scholar

[25]

L. Xu, R. Wang and D. Yao, On maximizing the expected terminal utility by investment and reinsurance, J. ind. manag. optim., 4 (2008), 801-815.  doi: 10.3934/jimo.2008.4.801.  Google Scholar

[26]

H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (1995), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[27]

D. Yao, H. Yang and R. Wang, Optimal financing and dividend strategies in a dual model with proportional costs, J. ind. manag. optim., 6 (2010), 761-777.  doi: 10.3934/jimo.2010.6.761.  Google Scholar

[28]

X. Zhang and T. K. Siu, On optimal proportional reinsurance and investment in a markovian regime-switching economy, Acta Mathematica Sinica, English Series, 28 (2012), 67-82.  doi: 10.1007/s10114-012-9761-7.  Google Scholar

show all references

References:
[1]

M. Alias, Le comportement de l'homme rationel devant le risque: Critique des postulats et axioms de l'ecole americaine, Econometrica, 21 (1953), 503-546.  doi: 10.2307/1907921.  Google Scholar

[2]

D. E. Bell, Disappointment in decision making under uncertainty, Oper. Res., 33 (1985), 1-27.  doi: 10.1287/opre.33.1.1.  Google Scholar

[3]

S. Benartzi and R. H. Thaler, Myopic loss aversion and the equity premium puzzle, Quart. J. Econ., 110 (1995), 73-92.  doi: 10.2307/2118511.  Google Scholar

[4]

A, B. Berkelaar, R. Kouwenberg and T. Post, Optimal Portfolio Choice under Loss Aversion, Review of Economics and Statistics, 86 (2004), 973-987.   Google Scholar

[5]

C. Bernard and M. Ghossoub, Static portfolio choice under cumulative prospect theory, Financ. Econ., 2 (2010), 277-306.  doi: 10.1007/s11579-009-0021-2.  Google Scholar

[6]

S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Math. of Oper. Res., 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.  Google Scholar

[7]

K. C. Chueng, W. F. Chong and S. C. P. Yam, The optimal insurance under disappointment theories, Insurance Math. Econom., 64 (2015), 77-90.  doi: 10.1016/j.insmatheco.2015.04.004.  Google Scholar

[8]

K. C. Chueng, W. F. Chong, R. J. Elliot and S. C. P. Yam, Disappointment aversion premium principle, Astin Bulletin, 45 (2015), 679-702.  doi: 10.1017/asb.2015.12.  Google Scholar

[9]

W. J. Guo, Optimal portfolio choice for an insurer with loss aversion, Insurance Math. Econom., 58 (2014), 217-222.  doi: 10.1016/j.insmatheco.2014.07.004.  Google Scholar

[10]

C. Hipp and M. Plum, Optimal investment for insurers, Insurance Math. Econom., 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4.  Google Scholar

[11]

C. Hipp and M. Plum, Optimal investment for investors with state dependent income, and for insurers, Finance Stoch., 7 (2003), 299-321.  doi: 10.1007/s007800200095.  Google Scholar

[12]

H. Jin and X. Y. Zhou, Behavior portfolio selection in continuous time, Math. Finance, 18 (2008), 385-426.  doi: 10.1111/j.1467-9965.2008.00339.x.  Google Scholar

[13]

D. Kahneman and A. Tversky, Prospect Theory-Analysis of Decision under risk, Econometrica, 47 (1979), 263-291.   Google Scholar

[14]

I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer-Verlag, New York, 1998. doi: 10.1007/b98840.  Google Scholar

[15]

C. S. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin, N. Am. Actuar. J., 8 (2004), 11-31.  doi: 10.1080/10920277.2004.10596134.  Google Scholar

[16]

G. Loomes and R. Sugden, Disappointment and dynamic consistency in choice under uncertainty, Rev. Econom. Stud., 53 (1986), 271-282.  doi: 10.2307/2297651.  Google Scholar

[17]

L. L. Lopes and G. C. Oden, The role of aspiration level in risky choice: A comparison of cumulative prospect theory and SP/A theory, J. Math. Psych, 43 (1999), 286-313.  doi: 10.1006/jmps.1999.1259.  Google Scholar

[18]

R. Mehra and E. C. Prescott, The equity premium: A puzzle, J. Monetary Econ, 15 (1985), 145-161.  doi: 10.1016/0304-3932(85)90061-3.  Google Scholar

[19]

H. Mi and S. G. Zhang, Continuous time portfolio selection with loss aversion in an incomplete market, Oper. Res Trans, 16 (2012), 1-12.   Google Scholar

[20]

H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J., 1 (2001), 55-68.  doi: 10.1080/034612301750077338.  Google Scholar

[21]

H. Shefrin and M. Statman, Behavioral portfolio theory, J. Financ. Quant. Anal., 35 (2000), 127-151.  doi: 10.2307/2676187.  Google Scholar

[22]

K. C. J. Sung, S. C. P. Yam, S. P. Yung and J. H. Zhou, Behavioral optimal insurance, Insurance Math. Econom., 49 (2011), 418-428.  doi: 10.1016/j.insmatheco.2011.04.008.  Google Scholar

[23]

A. Tsanakas and E. Desli, Risk measures and theories of choice, British Acturial Journal, 9 (2003), 959-991.  doi: 10.1017/S1357321700004414.  Google Scholar

[24]

A. Tversky and D. Kahneman, Advances in prospect theory: Cumulative representation of uncertainty, Chapter: Readings in Formal Epistemology, 1 (2016), 493-519.  doi: 10.1007/978-3-319-20451-2_24.  Google Scholar

[25]

L. Xu, R. Wang and D. Yao, On maximizing the expected terminal utility by investment and reinsurance, J. ind. manag. optim., 4 (2008), 801-815.  doi: 10.3934/jimo.2008.4.801.  Google Scholar

[26]

H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (1995), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar

[27]

D. Yao, H. Yang and R. Wang, Optimal financing and dividend strategies in a dual model with proportional costs, J. ind. manag. optim., 6 (2010), 761-777.  doi: 10.3934/jimo.2010.6.761.  Google Scholar

[28]

X. Zhang and T. K. Siu, On optimal proportional reinsurance and investment in a markovian regime-switching economy, Acta Mathematica Sinica, English Series, 28 (2012), 67-82.  doi: 10.1007/s10114-012-9761-7.  Google Scholar

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