# American Institute of Mathematical Sciences

September  2020, 16(5): 2195-2211. doi: 10.3934/jimo.2019050

## Optimal investment-reinsurance policy with regime switching and value-at-risk constraint

 1 Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China 2 Department of Mathematical Sciences, University of Nevada, Las Vegas, NV89154, United States

*Corresponding author

Received  April 2018 Revised  October 2018 Published  September 2020 Early access  May 2019

Fund Project: This project was supported by Tianjin philosophy and social science planning project (TJGLQN18-005)

This paper studies an optimal investment-reinsurance problem for an insurance company which is subject to a dynamic Value-at-Risk (VaR) constraint in a Markovian regime-switching environment. Our goal is to minimize its ruin probability and control its market risk simultaneously. We formulate the problem as an infinite horizontal stochastic control problem with the constrained strategies. The dynamic programming technique is applied to derive the coupled Hamilton-Jacobi-Bellman (HJB) equations and the Lagrange multiplier method is used to tackle the dynamic VaR constraint. Furthermore, we propose an efficient numerical method to solve those HJB equations. Finally, we employ a practical example from the Korean market to verify the numerical method and analyze the optimal strategies under different VaR constraints.

Citation: Ming Yan, Hongtao Yang, Lei Zhang, Shuhua Zhang. Optimal investment-reinsurance policy with regime switching and value-at-risk constraint. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2195-2211. doi: 10.3934/jimo.2019050
##### References:
 [1] A. Ang and G. Bekaert, International asset allocation with regime shifts, Review of Financial Studies, 15 (2002), 1137-1187.  doi: 10.1093/rfs/15.4.1137. [2] 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, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937. [3] H. Bühlmann, Mathematical Methods in Risk Theory, Springer, Berlin, 1970. [4] Y. Cao and X. Zeng, Optimal proportional reinsurance and investment with minimum probability of ruin, J. Nanjing Norm. Univ. Nat. Sci. Ed., 36 (2013), 1-9. [5] R. Chen, K. A. Wong and H. C. Lee, Underwriting cycles in Asia, Journal of Risk and Insurance, 66 (1999), 29-47.  doi: 10.2307/253876. [6] P. Chen and S. C. P. Yam, Optimal proportional reinsurance and investment with regime-switching for mean-variance insurers, Insurance: Mathematics & Economics, 53 (2013), 871-883.  doi: 10.1016/j.insmatheco.2013.10.004. [7] S. Chen, Z. Li and K. Li, Optimal investment-reinsurance policy for an insurance company with VaR constraint, Insurance: Mathematics & Economics, 47 (2010), 144-153.  doi: 10.1016/j.insmatheco.2010.06.002. [8] S. Choi and P. D. Thistle, The property/liability insurance cycle: A comparison of alternative models, Southern Economic Journal, 68 (2002), 530-548.  doi: 10.2307/1061716. [9] D. Cuoco, H. He and S. Isaenko, Optimal dynamic trading strategies with risk limits, Operations Research, 56 (2001), 358-368.  doi: 10.1287/opre.1070.0433. [10] R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer, 1995. [11] H. G. Fung and R. C. Witt, Underwriting cycles in property and liability insurance: An empirical analysis of industry and byline data, Journal of Risk and Insurance, 65 (1998), 539-561.  doi: 10.2307/253802. [12] A. Gundel and S. Weber, Utility maximization under a shortfall risk constraint, Journal of Mathematical Economics, 44 (2008), 1126-1151.  doi: 10.1016/j.jmateco.2008.01.002. [13] B. G. Jang and K. T. Kim, Optimal reinsurance and asset allocation under regime switching, Journal of Banking and Finance, 56 (2015), 37-47.  doi: 10.1016/j.jbankfin.2015.03.002. [14] Z. Jin, G. Yin and F. Wu, Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods, Insurance: Mathematics & Economics, 53 (2013), 733-746.  doi: 10.1016/j.insmatheco.2013.09.015. [15] Z. Liang and J. Guo, Optimal proportional reinsurance under two criteria: Maximizing the expected utility and minimizing the value at risk, Anziam Journal, 51 (2010), 449-463.  doi: 10.1017/S1446181110000878. [16] J. Liu, K. F. C. Yiu, R. C. Loxton, K. L. Teo, Optimal investment and proportional reinsurance with risk constraint, Journal of Mathematical Finance 3 (4) (2013) 437–447. doi: 10.4236/jmf.2013.34046. [17] J. Liu, K. F. C. Yiu, T. K. Siu and W. K. Ching, Optimal investment-reinsurance with dynamic risk constraint and regime switching, Scandinavian Actuarial Journal, 3 (2013), Article ID: 38147, 11 pages. doi: 10.1080/03461238.2011.602477. [18] H. Schmidli, Stochastic Control in Insurance, Springer, London, 2008. [19] H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 1 (2001), 55-68.  doi: 10.1080/034612301750077338. [20] M. I. Taksar and C. Markussen, Optimal dynamic reinsurance policies for large insurance portfolios, Finance and Stochastics, 7 (2003), 97-121.  doi: 10.1007/s007800200073. [21] K. F. C. Yiu, Optimal portfolios under a value-at-risk constraint, Journal of Economic Dynamics & Control, 28 (2004), 1317-1334.  doi: 10.1016/S0165-1889(03)00116-7. [22] K. F. C. Yiu, J. Liu, T. K. Siu and W. K. Ching, Optimal portfolios with regime switching and value-at-risk constraint, Automatica, 46 (2010), 979-989.  doi: 10.1016/j.automatica.2010.02.027. [23] C. Zhu, Optimal control of the risk process in a regime-switching environment, Automatica, 47 (2011), 1570-1579.  doi: 10.1016/j.automatica.2011.03.007.

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##### References:
 [1] A. Ang and G. Bekaert, International asset allocation with regime shifts, Review of Financial Studies, 15 (2002), 1137-1187.  doi: 10.1093/rfs/15.4.1137. [2] 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, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937. [3] H. Bühlmann, Mathematical Methods in Risk Theory, Springer, Berlin, 1970. [4] Y. Cao and X. Zeng, Optimal proportional reinsurance and investment with minimum probability of ruin, J. Nanjing Norm. Univ. Nat. Sci. Ed., 36 (2013), 1-9. [5] R. Chen, K. A. Wong and H. C. Lee, Underwriting cycles in Asia, Journal of Risk and Insurance, 66 (1999), 29-47.  doi: 10.2307/253876. [6] P. Chen and S. C. P. Yam, Optimal proportional reinsurance and investment with regime-switching for mean-variance insurers, Insurance: Mathematics & Economics, 53 (2013), 871-883.  doi: 10.1016/j.insmatheco.2013.10.004. [7] S. Chen, Z. Li and K. Li, Optimal investment-reinsurance policy for an insurance company with VaR constraint, Insurance: Mathematics & Economics, 47 (2010), 144-153.  doi: 10.1016/j.insmatheco.2010.06.002. [8] S. Choi and P. D. Thistle, The property/liability insurance cycle: A comparison of alternative models, Southern Economic Journal, 68 (2002), 530-548.  doi: 10.2307/1061716. [9] D. Cuoco, H. He and S. Isaenko, Optimal dynamic trading strategies with risk limits, Operations Research, 56 (2001), 358-368.  doi: 10.1287/opre.1070.0433. [10] R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer, 1995. [11] H. G. Fung and R. C. Witt, Underwriting cycles in property and liability insurance: An empirical analysis of industry and byline data, Journal of Risk and Insurance, 65 (1998), 539-561.  doi: 10.2307/253802. [12] A. Gundel and S. Weber, Utility maximization under a shortfall risk constraint, Journal of Mathematical Economics, 44 (2008), 1126-1151.  doi: 10.1016/j.jmateco.2008.01.002. [13] B. G. Jang and K. T. Kim, Optimal reinsurance and asset allocation under regime switching, Journal of Banking and Finance, 56 (2015), 37-47.  doi: 10.1016/j.jbankfin.2015.03.002. [14] Z. Jin, G. Yin and F. Wu, Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods, Insurance: Mathematics & Economics, 53 (2013), 733-746.  doi: 10.1016/j.insmatheco.2013.09.015. [15] Z. Liang and J. Guo, Optimal proportional reinsurance under two criteria: Maximizing the expected utility and minimizing the value at risk, Anziam Journal, 51 (2010), 449-463.  doi: 10.1017/S1446181110000878. [16] J. Liu, K. F. C. Yiu, R. C. Loxton, K. L. Teo, Optimal investment and proportional reinsurance with risk constraint, Journal of Mathematical Finance 3 (4) (2013) 437–447. doi: 10.4236/jmf.2013.34046. [17] J. Liu, K. F. C. Yiu, T. K. Siu and W. K. Ching, Optimal investment-reinsurance with dynamic risk constraint and regime switching, Scandinavian Actuarial Journal, 3 (2013), Article ID: 38147, 11 pages. doi: 10.1080/03461238.2011.602477. [18] H. Schmidli, Stochastic Control in Insurance, Springer, London, 2008. [19] H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 1 (2001), 55-68.  doi: 10.1080/034612301750077338. [20] M. I. Taksar and C. Markussen, Optimal dynamic reinsurance policies for large insurance portfolios, Finance and Stochastics, 7 (2003), 97-121.  doi: 10.1007/s007800200073. [21] K. F. C. Yiu, Optimal portfolios under a value-at-risk constraint, Journal of Economic Dynamics & Control, 28 (2004), 1317-1334.  doi: 10.1016/S0165-1889(03)00116-7. [22] K. F. C. Yiu, J. Liu, T. K. Siu and W. K. Ching, Optimal portfolios with regime switching and value-at-risk constraint, Automatica, 46 (2010), 979-989.  doi: 10.1016/j.automatica.2010.02.027. [23] C. Zhu, Optimal control of the risk process in a regime-switching environment, Automatica, 47 (2011), 1570-1579.  doi: 10.1016/j.automatica.2011.03.007.
$u_1^*(x)$ with different MVaR levels
$u_2^*(x)$ with different MVaR levels
$\pi_1^*(x)$ with different MVaR levels
$\pi_2^*(x)$ with different MVaR levels
$V_1(x)$ with different MVaR levels
$V_2(x)$ with different MVaR levels
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