# American Institute of Mathematical Sciences

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  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 & Management Optimization, 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. Google Scholar [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. Google Scholar [3] H. Bühlmann, Mathematical Methods in Risk Theory, Springer, Berlin, 1970. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer, 1995. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] H. Schmidli, Stochastic Control in Insurance, Springer, London, 2008. Google Scholar [19] H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 1 (2001), 55-68. doi: 10.1080/034612301750077338. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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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. Google Scholar [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. Google Scholar [3] H. Bühlmann, Mathematical Methods in Risk Theory, Springer, Berlin, 1970. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer, 1995. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] H. Schmidli, Stochastic Control in Insurance, Springer, London, 2008. Google Scholar [19] H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 1 (2001), 55-68. doi: 10.1080/034612301750077338. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
$u_1^*(x)$ 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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