IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v74y2017icp7-19.html
   My bibliography  Save this article

Optimal investment and reinsurance for an insurer under Markov-modulated financial market

Author

Listed:
  • Xu, Lin
  • Zhang, Liming
  • Yao, Dingjun

Abstract

This study examines optimal investment and reinsurance policies for an insurer with the classical surplus process. It assumes that the financial market is driven by a drifted Brownian motion with coefficients modulated by an external Markov process specified by the solution to a stochastic differential equation. The goal of the insurer is to maximize the expected terminal utility. This paper derives the Hamilton–Jacobi–Bellman (HJB) equation associated with the control problem using a dynamic programming method. When the insurer admits an exponential utility function, we prove that there exists a unique and smooth solution to the HJB equation. We derive the explicit optimal investment policy by solving the HJB equation. We can also find that the optimal reinsurance policy optimizes a deterministic function. We also obtain the upper bound for ruin probability in finite time for the insurer when the insurer adopts optimal policies.

Suggested Citation

  • Xu, Lin & Zhang, Liming & Yao, Dingjun, 2017. "Optimal investment and reinsurance for an insurer under Markov-modulated financial market," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 7-19.
    • Handle: RePEc:eee:insuma:v:74:y:2017:i:c:p:7-19
    DOI: 10.1016/j.insmatheco.2017.02.005
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668716300555
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2017.02.005?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Anna Frolova & Serguei Pergamenshchikov & Yuri Kabanov, 2002. "In the insurance business risky investments are dangerous," Finance and Stochastics, Springer, vol. 6(2), pages 227-235.
    2. Li, Zhongfei & Zeng, Yan & Lai, Yongzeng, 2012. "Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 191-203.
    3. Sid Browne, 1995. "Optimal Investment Policies for a Firm With a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin," Mathematics of Operations Research, INFORMS, vol. 20(4), pages 937-958, November.
    4. Zhang, Xin & Siu, Tak Kuen, 2009. "Optimal investment and reinsurance of an insurer with model uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 81-88, August.
    5. Hipp, Christian & Plum, Michael, 2000. "Optimal investment for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 27(2), pages 215-228, October.
    6. Begoña Fernández & Daniel Hernández-Hernández & Ana Meda & Patricia Saavedra, 2008. "An optimal investment strategy with maximal risk aversion and its ruin probability," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(1), pages 159-179, August.
    7. Azcue, Pablo & Muler, Nora, 2009. "Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 26-34, February.
    8. Bai, Lihua & Guo, Junyi, 2008. "Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 968-975, June.
    9. Browne, S., 1995. "Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin," Papers 95-08, Columbia - Graduate School of Business.
    10. Zeng, Yan & Li, Zhongfei, 2011. "Optimal time-consistent investment and reinsurance policies for mean-variance insurers," Insurance: Mathematics and Economics, Elsevier, vol. 49(1), pages 145-154, July.
    11. French, Kenneth R. & Schwert, G. William & Stambaugh, Robert F., 1987. "Expected stock returns and volatility," Journal of Financial Economics, Elsevier, vol. 19(1), pages 3-29, September.
    12. W. H. Fleming & S. J. Sheu, 2000. "Risk‐Sensitive Control and an Optimal Investment Model," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 197-213, April.
    13. Luo, Shangzhen & Taksar, Michael & Tsoi, Allanus, 2008. "On reinsurance and investment for large insurance portfolios," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 434-444, February.
    14. Yang, Hailiang & Zhang, Lihong, 2005. "Optimal investment for insurer with jump-diffusion risk process," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 615-634, December.
    15. Hojgaard, Bjarne & Taksar, Michael, 1998. "Optimal proportional reinsurance policies for diffusion models with transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 22(1), pages 41-51, May.
    16. Irgens, Christian & Paulsen, Jostein, 2004. "Optimal control of risk exposure, reinsurance and investments for insurance portfolios," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 21-51, August.
    17. Badaoui, Mohamed & Fernández, Begoña, 2013. "An optimal investment strategy with maximal risk aversion and its ruin probability in the presence of stochastic volatility on investments," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 1-13.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Lesław Gajek & Marcin Rudź, 2020. "Finite-Horizon Ruin Probabilities in a Risk-Switching Sparre Andersen Model," Methodology and Computing in Applied Probability, Springer, vol. 22(4), pages 1493-1506, December.
    2. Junna Bi & Jun Cai & Yan Zeng, 2021. "Equilibrium reinsurance-investment strategies with partial information and common shock dependence," Annals of Operations Research, Springer, vol. 307(1), pages 1-24, December.
    3. Hiroaki Hata & Shuenn-Jyi Sheu & Li-Hsien Sun, 2019. "Expected exponential utility maximization of insurers with a general diffusion factor model : The complete market case," Papers 1903.08957, arXiv.org.
    4. Wang, Ning & Zhang, Nan & Jin, Zhuo & Qian, Linyi, 2019. "Robust non-zero-sum investment and reinsurance game with default risk," Insurance: Mathematics and Economics, Elsevier, vol. 84(C), pages 115-132.
    5. Dimitris Andriosopoulos & Michalis Doumpos & Panos M. Pardalos & Constantin Zopounidis, 2019. "Computational approaches and data analytics in financial services: A literature review," Journal of the Operational Research Society, Taylor & Francis Journals, vol. 70(10), pages 1581-1599, October.
    6. Fu-Wei Huang & Panpan Lin & Jyh-Horng Lin & Ching-Hui Chang, 2023. "The impact of war on insurer safety: a contingent claim model analysis," Palgrave Communications, Palgrave Macmillan, vol. 10(1), pages 1-6, December.
    7. Ning Bin & Huainian Zhu & Chengke Zhang, 2023. "Stochastic Differential Games on Optimal Investment and Reinsurance Strategy with Delay Under the CEV Model," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-27, June.
    8. Matteo Brachetta & Claudia Ceci, 2018. "Optimal proportional reinsurance and investment for stochastic factor models," Papers 1806.01223, arXiv.org.
    9. Shihao Zhu & Jingtao Shi, 2019. "Optimal Reinsurance and Investment Strategies under Mean-Variance Criteria: Partial and Full Information," Papers 1906.08410, arXiv.org, revised Jun 2020.
    10. Zhao, Hui & Shen, Yang & Zeng, Yan & Zhang, Wenjun, 2019. "Robust equilibrium excess-of-loss reinsurance and CDS investment strategies for a mean–variance insurer with ambiguity aversion," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 159-180.
    11. Xue, Xiaole & Wei, Pengyu & Weng, Chengguo, 2019. "Derivatives trading for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 84(C), pages 40-53.
    12. Gajek, Lesław & Rudź, Marcin, 2018. "Banach Contraction Principle and ruin probabilities in regime-switching models," Insurance: Mathematics and Economics, Elsevier, vol. 80(C), pages 45-53.
    13. Lesław Gajek & Marcin Rudź, 2020. "Finite-horizon general insolvency risk measures in a regime-switching Sparre Andersen model," Methodology and Computing in Applied Probability, Springer, vol. 22(4), pages 1507-1528, December.
    14. Sun, Jingyun & Yao, Haixiang & Kang, Zhilin, 2019. "Robust optimal investment–reinsurance strategies for an insurer with multiple dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 89(C), pages 157-170.
    15. Brachetta, M. & Ceci, C., 2019. "Optimal proportional reinsurance and investment for stochastic factor models," Insurance: Mathematics and Economics, Elsevier, vol. 87(C), pages 15-33.
    16. Yan, Tingjin & Wong, Hoi Ying, 2020. "Open-loop equilibrium reinsurance-investment strategy under mean–variance criterion with stochastic volatility," Insurance: Mathematics and Economics, Elsevier, vol. 90(C), pages 105-119.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hiroaki Hata & Shuenn-Jyi Sheu & Li-Hsien Sun, 2019. "Expected exponential utility maximization of insurers with a general diffusion factor model : The complete market case," Papers 1903.08957, arXiv.org.
    2. Yi, Bo & Li, Zhongfei & Viens, Frederi G. & Zeng, Yan, 2013. "Robust optimal control for an insurer with reinsurance and investment under Heston’s stochastic volatility model," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 601-614.
    3. Nian Yao & Zhiming Yang, 2017. "Optimal excess-of-loss reinsurance and investment problem for an insurer with default risk under a stochastic volatility model," Papers 1704.08234, arXiv.org.
    4. Li, Zhongfei & Zeng, Yan & Lai, Yongzeng, 2012. "Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 191-203.
    5. Xue, Xiaole & Wei, Pengyu & Weng, Chengguo, 2019. "Derivatives trading for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 84(C), pages 40-53.
    6. Shen, Yang & Zeng, Yan, 2015. "Optimal investment–reinsurance strategy for mean–variance insurers with square-root factor process," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 118-137.
    7. Gu, Ailing & Guo, Xianping & Li, Zhongfei & Zeng, Yan, 2012. "Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model," Insurance: Mathematics and Economics, Elsevier, vol. 51(3), pages 674-684.
    8. Zhao, Hui & Rong, Ximin & Zhao, Yonggan, 2013. "Optimal excess-of-loss reinsurance and investment problem for an insurer with jump–diffusion risk process under the Heston model," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 504-514.
    9. Liang, Zhibin & Bayraktar, Erhan, 2014. "Optimal reinsurance and investment with unobservable claim size and intensity," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 156-166.
    10. Zilan Liu & Yijun Wang & Ya Huang & Jieming Zhou, 2022. "Optimal Time-Consistent Investment and Premium Control Strategies for Insurers with Constraint under the Heston Model," Mathematics, MDPI, vol. 10(7), pages 1-22, March.
    11. Zhu, Huainian & Cao, Ming & Zhang, Chengke, 2019. "Time-consistent investment and reinsurance strategies for mean-variance insurers with relative performance concerns under the Heston model," Finance Research Letters, Elsevier, vol. 30(C), pages 280-291.
    12. Liang, Zhibin & Yuen, Kam Chuen & Guo, Junyi, 2011. "Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process," Insurance: Mathematics and Economics, Elsevier, vol. 49(2), pages 207-215, September.
    13. Yang Shen & Bin Zou, 2021. "Mean-Variance Investment and Risk Control Strategies -- A Time-Consistent Approach via A Forward Auxiliary Process," Papers 2101.03954, arXiv.org.
    14. Li, Yongwu & Li, Zhongfei, 2013. "Optimal time-consistent investment and reinsurance strategies for mean–variance insurers with state dependent risk aversion," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 86-97.
    15. Zhang, Xin-Li & Zhang, Ke-Cun & Yu, Xing-Jiang, 2009. "Optimal proportional reinsurance and investment with transaction costs, I: Maximizing the terminal wealth," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 473-478, June.
    16. Zhu, Huiming & Deng, Chao & Yue, Shengjie & Deng, Yingchun, 2015. "Optimal reinsurance and investment problem for an insurer with counterparty risk," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 242-254.
    17. Bai, Yanfei & Zhou, Zhongbao & Xiao, Helu & Gao, Rui & Zhong, Feimin, 2022. "A hybrid stochastic differential reinsurance and investment game with bounded memory," European Journal of Operational Research, Elsevier, vol. 296(2), pages 717-737.
    18. Guan, Guohui & Liang, Zongxia, 2014. "Optimal reinsurance and investment strategies for insurer under interest rate and inflation risks," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 105-115.
    19. Jiaqi Zhu & Shenghong Li, 2020. "Time-Consistent Investment and Reinsurance Strategies for Mean-Variance Insurers under Stochastic Interest Rate and Stochastic Volatility," Mathematics, MDPI, vol. 8(12), pages 1-22, December.
    20. Shen, Yang & Zou, Bin, 2021. "Mean–variance investment and risk control strategies — A time-consistent approach via a forward auxiliary process," Insurance: Mathematics and Economics, Elsevier, vol. 97(C), pages 68-80.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:insuma:v:74:y:2017:i:c:p:7-19. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/505554 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.