AUTHORS: Yajie Wang, Ximin Rong, Hui Zhao
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ABSTRACT: In this paper, we consider the optimal investment problem for an insurer who has n dependent lines of business. The surplus process of the insurer is described by a n-dimensional compound Poisson risk process. Moreover, the insurer is allowed to invest in a risk-free asset and a risky asset whose price process follows the constant elasticity of variance (CEV) model. The investment objective is maximizing the expected utility of the insurer’s terminal wealth.Applying dynamic programming approach, we establish the corresponding Hamilton-Jacobi-Bellman (HJB) equation. Optimal investment strategy is obtained explicitly for exponential utility. Finally, we provide a numerical example to analyze the effects of parameters on the optimal strategy.
KEYWORDS: Optimal investment, Dependent risks, Constant elasticity of variance (CEV) model, Utility maximization, Hamilton-Jacobi-Bellman (HJB) equation
REFERENCES:[1] S. Browne, Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, Vol.20, No.4, 1995, pp. 937–958.
[2] C. Hipp, M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, Vol.27, No.2, 2000, pp. 215–228.
[3] C. Liu, H. Yang, Optimal investment for an insurer to minimize its probability of ruin, North American Actuarial Journal, Vol.8, No.3, 2004, pp. 11–31.
[4] H. Yang, L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, Vol.37, No.3, 2005, pp. 615–634.
[5] Z. Wang, J. Xia, L. Zhang, Optimal investment for an insurer: The martingale approach, Insurance: Mathematics and Economics, Vol.40, No.2, 2007, pp. 322–334.
[6] L. Bai, J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, Vol.42, No.3, 1995, pp. 968– 975.
[7] D. Li, X. Rong, H.Zhao, Optimal investment problem for an insurer and a reinsurer under the proportional reinsurance model, WSEAS Transactions on Mathematics, Vol.14, No.3, 2015, pp. 20–35.
[8] H. Chang, J. Lu, Utility portfolio optimization with liability and multiple risky assets under the extended CIR model, WSEAS Transactions on Systems and Control, Vol.9, 2014, pp. 255–268.
[9] H. Chang, X. Rong, H. Zhao, Optimal investment and consumption decisions under the HoLee interest rate model, WSEAS Transactions on Mathematics, Vol.12, No.11, 2013, pp. 1065– 1075.
[10] J. Li, H. Liu, Optimal investment for the insurers in Markov-Modulated jump-diffusion models, Computational Economics , Vol.46, No.1, 2015, pp. 143–156.
[11] M.L. Centeno, Dependent risks and excess of loss reinsurance, Insurance: Mathematics and Economics, Vol.37, No.2, 2005, pp. 229–238.
[12] L. Bai, J. Cai, M. Zhou, Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting, Insurance: Mathematics and Economics, Vol.53, No.3, 2013, pp. 664–670.
[13] Z. Liang, K. Yuen, Optimal dynamic reinsurance with dependent risks: variance premium principle, Scandinavian Actuarial Journal, Vol.2016, No.1, 2016, pp. 18–36.
[14] K. Yuen, Z. Liang, M. Zhou, Optimal proportional reinsurance with common shock dependence, Insurance: Mathematics and Economics, Vol.64, 2015, pp. 1–13.
[15] J.C. Cox, S.A. Ross, The valuation of options for alternative stochastic processes, Journal of Financial Economics, Vol.3, No.2, 1976, pp. 145– 166.
[16] S. Beckers, The constant elasticity of variance model and its implications for option pricing, The Journal of Finance, Vol.35, No.3, 1980, pp. 661–673.
[17] D. Davydov, V. Linetsky, The valuation and hedging of barrier and lookback option under the CEV process, Management Science, Vol.47, No.7, 2001, pp. 949–965.
[18] C. Jones, The dynamics of the stochastic volatility: evidence from underlying and options markets, Journal of Econometrics, Vol.116, No.1, 2003, pp. 181–224.
[19] D. Darius, The constant elasticity of variance model in the framework of optimal investment problems, A Dissertation Presented to Faculty of Princeton University, 2005.Working Paper.
[20] D. Li, X. Rong, H. Zhao, Optimal investment problem with taxes, dividends and transaction costs under the constant elasticity of variance (CEV) model, WSEAS Transactions on Mathematics, Vol.12, No.3, 2013, pp. 243–255.
[21] M. Gu, Y. Yang, S. Li, J. Zhang, Constant elasticity of variance model for proportional reinsurance and investment strategies, Insurance: Mathematics and Economics, Vol.46, No.3, 2010, pp. 580–587.
[22] Z. Liang, K. Yuen, K. Cheung, Optimal reinsurance-investment problem in a constant elasticity of variance stock market for jumpdiffusion risk model, Applied Stochastic Models in Business and Industry, Vol.28, No.6, 2012, pp. 585–597.
[23] X. Lin, Y. Li, Optimal reinsurance and investment for a jump diffusion risk process under the CEV model, North American Actuarial Journal, Vol.15, No.3, 2011, pp. 417–431.
[24] D. Li, X. Rong, H. Zhao, Time-consistent reinsurance-investment strategy for an insurer and a reinsurer with mean-variance criterion under the CEV model, Journal of Computational And Applied Mathematics, Vol.283, No.1, 2015, pp. 142–162.
[25] J. Xiao, Z. Hong, C. Qin, The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts, Insurance: Mathematics and Economics, Vol.40, No.2, 2007, pp. 302–310.
[26] J. Gao, Optimal portfolio for DC pension plans under a CEV model, Insurance: Mathematics and Economics, Vol.44, No.3, 2009, pp. 479– 490.
[27] J. Gao, Optimal investment strategy for annuity contracts under the constant elasticity of variance (CEV) model, Insurance: Mathematics and Economics, Vol.45, No.1, 2009, pp. 9–18.
[28] W. Fleming, H. Soner, Controlled Markov Processes and Viscosity Solutions, 1993, Springer.
[29] H. Zhao, X. Rong, Y. Zhao, Optimal excess-ofloss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance: Mathematics and Economics, Vol.53, No.3, 2013, pp. 504–514.
