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Optimal reinsurance design under distortion risk measures and reinsurer’s default risk with partial recovery

Published online by Cambridge University Press:  04 October 2024

Yaodi Yong ,
Ka Chun Cheung  and
Yiying Zhang Open the ORCID record for Yiying Zhang [Opens in a new window]
Yaodi Yong
Affiliation:
Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, Guangdong Province, P.R. China
Ka Chun Cheung
Affiliation:
Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, P.R. China
Yiying Zhang*
Affiliation:
Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, Guangdong Province, P.R. China
*
Corresponding author: Yiying Zhang; Email: [email protected]
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Abstract

Reinsurers may default when they have to pay large claims to insurers but are unable to fulfill their obligations due to various reasons such as catastrophic events, underwriting losses, inadequate capitalization, or financial mismanagement. This paper studies the problem of optimal reinsurance design from the perspectives of both the insurer and reinsurer when the insurer faces the potential default risk of the reinsurer. If the insurer aims to minimize the convex distortion risk measure of his retained loss, we prove the optimality of a stop-loss treaty when the promised ceded loss function is charged by the expected value premium principle and the reinsurer offers partial recovery in the event of default. For any fixed premium loading set by the reinsurer, we then derive the explicit expressions of optimal deductible levels for three special distortion functions, including the TVaR, Gini, and PH transform distortion functions. Under these three explicit distortion risk measures adopted by the insurer, we seek the optimal safety loading for the reinsurer by maximizing her net profit where the reserve capital is determined by the TVaR measure and the cost is governed by the expectation. This procedure ultimately leads to the Bowley solution between the insurer and the reinsurer. We provide several numerical examples to illustrate the theoretical findings. Sensitivity analyses demonstrate how different settings of default probability, recovery rate, and safety loading affect the optimal deductible values. Simulation studies are also implemented to analyze the effects induced by the default probability and recovery rate on the Bowley solution.

Keywords

Optimal reinsurancedefault riskpartial recoveryexpected value premium principleBowley solutionC61G22G32
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 54 , Issue 3 , September 2024 , pp. 738 - 766
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The International Actuarial Association

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