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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 and 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.

[2]

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

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[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. 

[20]

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

[21]

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

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[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. 

[20]

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

[21]

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

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

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