IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v201y2024i3d10.1007_s10957-024-02429-y.html
   My bibliography  Save this article

Optimal Claim-Dependent Proportional Reinsurance Under a Self-Exciting Claim Model

Author

Listed:
  • Fan Wu

    (Southeast University)

  • Yang Shen

    (University of New South Wales)

  • Xin Zhang

    (Southeast University)

  • Kai Ding

    (Southeast University)

Abstract

This paper investigates an optimal reinsurance problem for an insurance company with self-exciting claims, where the insurer’s historical claims affect the claim intensity itself. We focus on a claim-dependent proportional reinsurance contact, where the term “claim-dependent” signifies that the insurer’s risk retention ratio is allowed to depend on claim size. The insurer aims to maximize the expected utility of terminal wealth. By utilizing the dynamic programming principle and verification theorem, we obtain the optimal reinsurance strategy and corresponding value function in closed-form from the Hamilton–Jacobi–Bellman equation under an exponential utility function. We show that the claim-dependent proportional reinsurance is optimal among all types of reinsurance under the exponential utility maximization criterion. In addition, we present several analytical properties and numerical examples of the derived optimal strategy and provide economic insights through analytical and numerical analyses. In particular, we show the optimal claim-dependent proportional reinsurance can be considered as a continuous approximation of the step-wise risk sharing rule between the insurer and the reinsurer.

Suggested Citation

  • Fan Wu & Yang Shen & Xin Zhang & Kai Ding, 2024. "Optimal Claim-Dependent Proportional Reinsurance Under a Self-Exciting Claim Model," Journal of Optimization Theory and Applications, Springer, vol. 201(3), pages 1229-1255, June.
  • Handle: RePEc:spr:joptap:v:201:y:2024:i:3:d:10.1007_s10957-024-02429-y
    DOI: 10.1007/s10957-024-02429-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-024-02429-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10957-024-02429-y?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. Cao, Jingyi & Landriault, David & Li, Bin, 2020. "Optimal reinsurance-investment strategy for a dynamic contagion claim model," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 206-215.
    2. Lihua Bai & Huayue Zhang, 2008. "Dynamic mean-variance problem with constrained risk control for the insurers," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(1), pages 181-205, August.
    3. Yuen, Kam Chuen & Liang, Zhibin & Zhou, Ming, 2015. "Optimal proportional reinsurance with common shock dependence," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 1-13.
    4. Hanspeter Schmidli, 2001. "Optimal Proportional Reinsurance Policies in a Dynamic Setting," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2001(1), pages 55-68.
    5. 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.
    6. S. David Promislow & Virginia Young, 2005. "Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(3), pages 110-128.
    7. Zhongyang Sun & Xin Zhang & Ya-Nan Li, 2021. "A BSDE approach for bond pricing under interest rate models with self-exciting jumps," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 50(14), pages 3249-3261, July.
    8. 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.
    9. Hansjörg Albrecher & Søren Asmussen c, 2006. "Ruin probabilities and aggregrate claims distributions for shot noise Cox processes," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2006(2), pages 86-110.
    10. Alan G. Hawkes, 2022. "Hawkes jump-diffusions and finance: a brief history and review," The European Journal of Finance, Taylor & Francis Journals, vol. 28(7), pages 627-641, May.
    Full references (including those not matched with items on IDEAS)

    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. 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.
    2. 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.
    3. 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.
    4. Xue, Xiaole & Wei, Pengyu & Weng, Chengguo, 2019. "Derivatives trading for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 84(C), pages 40-53.
    5. Bi, Junna & Cai, Jun, 2019. "Optimal investment–reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 1-14.
    6. Junna Bi & Zhibin Liang & Kam Chuen Yuen, 2019. "Optimal mean–variance investment/reinsurance with common shock in a regime-switching market," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 90(1), pages 109-135, August.
    7. 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.
    8. 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.
    9. 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.
    10. Li, Danping & Rong, Ximin & Zhao, Hui, 2015. "Time-consistent reinsurance–investment strategy for a mean–variance insurer under stochastic interest rate model and inflation risk," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 28-44.
    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. Qiang Zhang & Ping Chen, 2020. "Optimal Reinsurance and Investment Strategy for an Insurer in a Model with Delay and Jumps," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 777-801, June.
    13. Yuchen Li & Zongxia Liang & Shunzhi Pang, 2022. "Continuous-Time Monotone Mean-Variance Portfolio Selection in Jump-Diffusion Model," Papers 2211.12168, arXiv.org, revised May 2024.
    14. 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.
    15. Li, Bin & Li, Danping & Xiong, Dewen, 2016. "Alpha-robust mean-variance reinsurance-investment strategy," Journal of Economic Dynamics and Control, Elsevier, vol. 70(C), pages 101-123.
    16. Liang, Zongxia & Song, Min, 2015. "Time-consistent reinsurance and investment strategies for mean–variance insurer under partial information," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 66-76.
    17. 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.
    18. 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.
    19. Peng, Xingchun & Chen, Fenge & Hu, Yijun, 2014. "Optimal investment, consumption and proportional reinsurance under model uncertainty," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 222-234.
    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.

    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:spr:joptap:v:201:y:2024:i:3:d:10.1007_s10957-024-02429-y. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.