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Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle

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  • Zhou, Ming
  • Yuen, Kam C.

Abstract

This paper considers the optimal dividend problem with proportional reinsurance and capital injection for a large insurance portfolio. In particular, the reinsurance premium is assumed to be calculated via the variance principle instead of the expected value principle. Our objective is to maximize the expectation of the discounted dividend payments minus the discounted costs of capital injection. This optimization problem is studied in four cases depending on whether capital injection is allowed and whether there exist restrictions on dividend policies. In all cases, closed-form expressions for the value function and optimal dividend and reinsurance policies are obtained. From the results, we see that the optimal dividend distribution policy is of threshold type with a constant barrier, and that the optimal ceded proportion of risk exponentially decreases with the initial surplus and remains constant when the initial surplus exceeds the dividend barrier. Furthermore, we show that the optimization problem without capital injection is the limiting case of the one with capital injection when the proportional transaction cost goes to infinity.

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  • Zhou, Ming & Yuen, Kam C., 2012. "Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle," Economic Modelling, Elsevier, vol. 29(2), pages 198-207.
    • Handle: RePEc:eee:ecmode:v:29:y:2012:i:2:p:198-207
    DOI: 10.1016/j.econmod.2011.09.007
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    6. Liu, Bing & Meng, Hui & Zhou, Ming, 2021. "Optimal investment and reinsurance policies for an insurer with ambiguity aversion," The North American Journal of Economics and Finance, Elsevier, vol. 55(C).
    7. Meng, Hui & Zhou, Ming & Siu, Tak Kuen, 2016. "Optimal reinsurance policies with two reinsurers in continuous time," Economic Modelling, Elsevier, vol. 59(C), pages 182-195.
    8. Kristoffer Lindensjo & Filip Lindskog, 2019. "Optimal dividends and capital injection under dividend restrictions," Papers 1902.06294, arXiv.org.
    9. Chen, Lv & Shen, Yang, 2019. "Stochastic Stackelberg differential reinsurance games under time-inconsistent mean–variance framework," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 120-137.
    10. Wang, Ning & Zhang, Nan & Jin, Zhuo & Qian, Linyi, 2021. "Stochastic differential investment and reinsurance games with nonlinear risk processes and VaR constraints," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 168-184.
    11. Tamturk, Muhsin & Utev, Sergey, 2018. "Ruin probability via Quantum Mechanics Approach," Insurance: Mathematics and Economics, Elsevier, vol. 79(C), pages 69-74.
    12. Khaled Masoumifard & Mohammad Zokaei, 2020. "Stochastic optimization of the Dividend strategy with reinsurance in correlated multiple insurance lines of business," Papers 2002.03295, arXiv.org.
    13. Xiaoqing Liang & Zbigniew Palmowski, 2016. "A note on optimal expected utility of dividend payments with proportional reinsurance," Papers 1605.06849, arXiv.org, revised May 2017.
    14. Ernst, Philip A. & Imerman, Michael B. & Shepp, Larry & Zhou, Quan, 2022. "Fiscal stimulus as an optimal control problem," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 1091-1108.
    15. Cheng, Gongpin & Zhao, Yongxia, 2016. "Optimal risk and dividend strategies with transaction costs and terminal value," Economic Modelling, Elsevier, vol. 54(C), pages 522-536.
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    18. Liang, Xiaoqing & Liang, Zhibin & Young, Virginia R., 2020. "Optimal reinsurance under the mean–variance premium principle to minimize the probability of ruin," Insurance: Mathematics and Economics, Elsevier, vol. 92(C), pages 128-146.
    19. Chen, Lv & Qian, Linyi & Shen, Yang & Wang, Wei, 2016. "Constrained investment–reinsurance optimization with regime switching under variance premium principle," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 253-267.
    20. Zhang, Xin & Meng, Hui & Zeng, Yan, 2016. "Optimal investment and reinsurance strategies for insurers with generalized mean–variance premium principle and no-short selling," Insurance: Mathematics and Economics, Elsevier, vol. 67(C), pages 125-132.
    21. Kristoffer Lindensjö & Filip Lindskog, 2020. "Optimal dividends and capital injection under dividend restrictions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(3), pages 461-487, December.
    22. Ramsden, Lewis & Papaioannou, Apostolos D., 2019. "On the time to ruin for a dependent delayed capital injection risk model," Applied Mathematics and Computation, Elsevier, vol. 352(C), pages 119-135.
    23. Yao, Dingjun & Yang, Hailiang & Wang, Rongming, 2014. "Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle," Economic Modelling, Elsevier, vol. 37(C), pages 53-64.
    24. Chen, Mi & Peng, Xiaofan & Guo, Junyi, 2013. "Optimal dividend problem with a nonlinear regular-singular stochastic control," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 448-456.

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