Revisit optimal reinsurance under a new distortion risk measure

Abstract

Distortion risk measures have a significant effect on the fields of finance and risk management. In this article, we consider two optimal reinsurance designs under a new distortion risk measure with mixed methods, which was proposed by Zhu and Yin (Communications in Statistics - Theory and Methods 2023, 4151–4164), one with the reinsurer’s default risk and another one with the context of determining the Pareto-optimal reinsurance policies. The closed-form solutions of optimal reinsurance policies in both setting are obtained. The GlueVaR and generalized GlueVaR are considered in the application of designing optimal reinsurance contracts with reinsurer’s default risk as two special cases. Finally, we give two numerical examples, one with default risk and another one without default risk, to illustrate our results.

KEYWORDS:

• Distortion risk measure
• distortion premium principle
• GlueVaR
• generalized GlueVaR
• default risk
• Pareto-optimal reinsurance

Mathematics Subject Classifications (2000):

• C61
• G22
• G32

Declaration of competing interest

None.

Acknowledgments

The authors are grateful to the Associate Editor and two anonymous referees for their comprehensive reviews of an earlier version of this article.

Notes

1 Specificially, they consider the conditional VaR(CVaR), and TVaR and CVaR will be equal when the random variable is contiunous.

2 Two risks $X,Y\in L^{\infty }$ are comonotonic if there exist a common random variable Z and increasing functions f₁, f₂ such that $(X,Y)\stackrel{d}{=}(f_1(Z),f_2(Z))$.

Additional information

Funding

W. Xia was supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (20KJB110014).