American Institute of Mathematical Sciences

Advanced
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

[1]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020
[2]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021012
[3]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[4]

Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021007
[5]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[6]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[7]

Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021040
[8]

John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026
[9]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009
[10]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[11]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014
[12]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709
[13]

Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021004
[14]

Yves Dumont, Frederic Chiroleu. Vector control for the Chikungunya disease. Mathematical Biosciences & Engineering, 2010, 7 (2) : 313-345. doi: 10.3934/mbe.2010.7.313
[15]

Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018
[16]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[17]

M. Mahalingam, Parag Ravindran, U. Saravanan, K. R. Rajagopal. Two boundary value problems involving an inhomogeneous viscoelastic solid. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1351-1373. doi: 10.3934/dcdss.2017072
[18]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881
[19]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009
[20]

Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (87)
  • HTML views (382)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences