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Optimal Reinsurance under the Linear Combination of Risk Measures in the Presence of Reinsurance Loss Limit

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  • Qian Xiong

    (School of Mathematics and Statistics, Southwest University, Chongqing 400715, China)

  • Zuoxiang Peng

    (School of Mathematics and Statistics, Southwest University, Chongqing 400715, China)

  • Saralees Nadarajah

    (Department of Mathematics, University of Manchester, Manchester M13 9PL, UK)

Abstract

Optimal reinsurance problems under the risk measures, such as Value-at-Risk ( VaR ) and Tail-Value-at-Risk ( TVaR ), have been studied in recent literature. However, losses based on VaR may be underestimated and TVaR allows us to account better for catastrophic losses. In this paper, we propose a new family of flexible risk measures denoted by LVaR , which is a weighted combination of VaR and TVaR . Based on the new risk measures, we deal with the optimal reinsurance problem by minimizing the LVaR of the total risks of an insurer when two types of constraints for reinsurer’s risk exposure are considered. The results indicate that the two-layer reinsurance is always an optimal reinsurance policy with both types of constraints. Also, we find that the optimal reinsurance policy depends on the confidence level, the weight coefficient, the safety loading, the tolerance level, as well as the relations between them. Finally, we illustrate the results by numerical examples and compare them with the results in Lu et al.

Suggested Citation

  • Qian Xiong & Zuoxiang Peng & Saralees Nadarajah, 2023. "Optimal Reinsurance under the Linear Combination of Risk Measures in the Presence of Reinsurance Loss Limit," Risks, MDPI, vol. 11(7), pages 1-26, July.
    • Handle: RePEc:gam:jrisks:v:11:y:2023:i:7:p:125-:d:1190801
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    References listed on IDEAS

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    2. Cai, Jun & Tan, Ken Seng, 2007. "Optimal Retention for a Stop-loss Reinsurance Under the VaR and CTE Risk Measures," ASTIN Bulletin, Cambridge University Press, vol. 37(1), pages 93-112, May.
    3. 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.
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    5. Guerra, Manuel & Centeno, Maria de Lourdes, 2010. "Optimal Reinsurance for Variance Related Premium Calculation Principles 1," ASTIN Bulletin, Cambridge University Press, vol. 40(1), pages 97-121, May.
    6. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
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