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Optimal Reinsurance From The Perspectives Of Both An Insurer And A Reinsurer

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  • Cai, Jun
  • Lemieux, Christiane
  • Liu, Fangda

Abstract

Optimal reinsurance from an insurer's point of view or from a reinsurer's point of view has been studied extensively in the literature. However, as two parties of a reinsurance contract, an insurer and a reinsurer have conflicting interests. An optimal form of reinsurance from one party's point of view may be not acceptable to the other party. In this paper, we study optimal reinsurance designs from the perspectives of both an insurer and a reinsurer and take into account both an insurer's aims and a reinsurer's goals in reinsurance contract designs. We develop optimal reinsurance contracts that minimize the convex combination of the Value-at-Risk (VaR) risk measures of the insurer's loss and the reinsurer's loss under two types of constraints, respectively. The constraints describe the interests of both the insurer and the reinsurer. With the first type of constraints, the insurer and the reinsurer each have their limit on the VaR of their own loss. With the second type of constraints, the insurer has a limit on the VaR of his loss while the reinsurer has a target on his profit from selling a reinsurance contract. For both types of constraints, we derive the optimal reinsurance forms in a wide class of reinsurance policies and under the expected value reinsurance premium principle. These optimal reinsurance forms are more complicated than the optimal reinsurance contracts from the perspective of one party only. The proposed models can also be reduced to the problems of minimizing the VaR of one party's loss under the constraints on the interests of both the insurer and the reinsurer.

Suggested Citation

  • Cai, Jun & Lemieux, Christiane & Liu, Fangda, 2016. "Optimal Reinsurance From The Perspectives Of Both An Insurer And A Reinsurer," ASTIN Bulletin, Cambridge University Press, vol. 46(3), pages 815-849, September.
    • Handle: RePEc:cup:astinb:v:46:y:2016:i:03:p:815-849_00
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    Cited by:

    1. Guohui Guan & Zongxia Liang & Yilun Song, 2022. "A Stackelberg reinsurance-investment game under $\alpha$-maxmin mean-variance criterion and stochastic volatility," Papers 2212.14327, arXiv.org.
    2. Chen, Yanhong & Cheung, Ka Chun & Zhang, Yiying, 2024. "Bowley solution under the reinsurer's default risk," Insurance: Mathematics and Economics, Elsevier, vol. 115(C), pages 36-61.
    3. Ambrose Lo & Zhaofeng Tang, 2019. "Pareto-optimal reinsurance policies in the presence of individual risk constraints," Annals of Operations Research, Springer, vol. 274(1), pages 395-423, March.
    4. Jiang, Wenjun & Hong, Hanping & Ren, Jiandong, 2021. "Pareto-optimal reinsurance policies with maximal synergy," Insurance: Mathematics and Economics, Elsevier, vol. 96(C), pages 185-198.
    5. Suci Sari & Arief Hakim & Ikha Magdalena & Khreshna Syuhada, 2023. "Modeling the Optimal Combination of Proportional and Stop-Loss Reinsurance with Dependent Claim and Stochastic Insurance Premium," JRFM, MDPI, vol. 16(2), pages 1-20, February.
    6. Bäuerle, Nicole & Glauner, Alexander, 2018. "Optimal risk allocation in reinsurance networks," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 37-47.
    7. Cai, Jun & Liu, Haiyan & Wang, Ruodu, 2017. "Pareto-optimal reinsurance arrangements under general model settings," Insurance: Mathematics and Economics, Elsevier, vol. 77(C), pages 24-37.
    8. Asimit, Alexandru V. & Cheung, Ka Chun & Chong, Wing Fung & Hu, Junlei, 2020. "Pareto-optimal insurance contracts with premium budget and minimum charge constraints," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 17-27.
    9. Chi, Yichun & Liu, Fangda, 2021. "Enhancing an insurer's expected value by reinsurance and external financing," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 466-484.
    10. Cheung, Ka Chun & He, Wanting & Wang, He, 2023. "Multi-constrained optimal reinsurance model from the duality perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 199-214.
    11. Wang, Ruodu & Wei, Yunran, 2020. "Characterizing optimal allocations in quantile-based risk sharing," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 288-300.
    12. Wenhua Lv & Linxiao Wei, 2023. "Distributionally Robust Reinsurance with Glue Value-at-Risk and Expected Value Premium," Mathematics, MDPI, vol. 11(18), pages 1-23, September.
    13. Li, Danping & Young, Virginia R., 2021. "Bowley solution of a mean–variance game in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 98(C), pages 35-43.
    14. Wenjun Jiang & Jiandong Ren & Ričardas Zitikis, 2017. "Optimal Reinsurance Policies under the VaR Risk Measure When the Interests of Both the Cedent and the Reinsurer Are Taken into Account," Risks, MDPI, vol. 5(1), pages 1-22, February.
    15. Chen Li & Xiaohu Li, 2018. "On the Optimal Risk Sharing in Reinsurance with Random Recovery Rate," Risks, MDPI, vol. 6(4), pages 1-16, October.
    16. Chi, Yichun & Tan, Ken Seng & Zhuang, Sheng Chao, 2020. "A Bowley solution with limited ceded risk for a monopolistic reinsurer," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 188-201.
    17. Guan, Guohui & Hu, Xiang, 2022. "Equilibrium mean–variance reinsurance and investment strategies for a general insurance company under smooth ambiguity," The North American Journal of Economics and Finance, Elsevier, vol. 63(C).
    18. Khreshna Syuhada & Arief Hakim & Suci Sari, 2021. "The Combined Stop-Loss and Quota-Share Reinsurance: Conditional Tail Expectation-Based Optimization from the Joint Perspective of Insurer and Reinsurer," Risks, MDPI, vol. 9(7), pages 1-21, July.
    19. Nicole Bauerle & Alexander Glauner, 2017. "Optimal Risk Allocation in Reinsurance Networks," Papers 1711.10210, arXiv.org.
    20. Balbás, Alejandro & Balbás, Beatriz & Balbás, Raquel & Heras, Antonio, 2022. "Risk transference constraints in optimal reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 103(C), pages 27-40.
    21. Tim J. Boonen & Fangda Liu & Ruodu Wang, 2021. "Competitive equilibria in a comonotone market," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 72(4), pages 1217-1255, November.
    22. Cheung, Ka Chun & Yam, Sheung Chi Phillip & Zhang, Yiying, 2019. "Risk-adjusted Bowley reinsurance under distorted probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 64-72.
    23. Balbás, Beatriz & Balbás, Raquel, 2017. "Differential equations connecting VaR and CVaR," IC3JM - Estudios = Working Papers 24017, Instituto Mixto Carlos III - Juan March de Ciencias Sociales (IC3JM).

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