Optimal Reinsurance Policies under the VaR Risk Measure When the Interests of Both the Cedent and the Reinsurer Are Taken into Account

Abstract

Optimal forms of reinsurance policies have been studied for a long time in the actuarial literature. Most existing results are from the insurer's point of view, aiming at maximizing the expected utility or minimizing the risk of the insurer. However, as pointed out by Borch (1969), it is understandable that a reinsurance arrangement that might be very attractive to one party (e.g., the insurer) can be quite unacceptable to the other party (e.g., the reinsurer). In this paper, we follow this point of view and study forms of Pareto-optimal reinsurance policies whereby one party's risk, measured by its value-at-risk (VaR), cannot be reduced without increasing the VaR of the counter-party in the reinsurance transaction. We show that the Pareto-optimal policies can be determined by minimizing linear combinations of the VaRs of the two parties in the reinsurance transaction. Consequently, we succeed in deriving user-friendly, closed-form, optimal reinsurance policies and their parameter values.

Pareto-Optimal Reinsurance Revisited: A Two-Stage Optimisation Procedure Approach

“…A general approach to identify Paretooptimal reinsurance policies is to minimize a convex combination of the TVaRs of the two parties. e result can be found in [18][19][20]. Proposition 1.…”

“…Lo [17] discussed the generalized problems in [16] by using the Neyman-Pearson approach. Based on the optimal reinsurance strategy in [16], Jiang et al [18] proved that the optimal reinsurance strategy is a Pareto-optimal reinsurance policy and gave optimal reinsurance strategies using the geometric method. Cai et al [19] studied the Pareto optimality of reinsurance arrangements under general model settings and obtained the explicit forms of the Pareto-optimal reinsurance contracts under the TVaR measure and the expected value premium principle.…”

Optimal Reinsurance From the Viewpoints of Both an Insurer and a Reinsurer Under the Cvar Risk Measure and Vajda Condition

“…Lo and Tang (2019) further studied Pareto-optimal reinsurance design under distortion risk measures and gave explicit solutions to Pareto-optimal reinsurance treaty under VaR and CVaR risk measures and the expected value premium principle. For more works about optimal reinsurance from the perspectives of both the insurer and the reinsurer, we refer to Cheung and Wang (2017), Jiang et al (2017Jiang et al ( , 2018, Huang and Yin (2019), Asimit et al (2020), and the references therein.…”

The Combined Stop-Loss and Quota-Share Reinsurance: Conditional Tail Expectation-Based Optimization from the Joint Perspective of Insurer and Reinsurer

“…In particular, Liu and Fang (2018) implemented this approach on their study to obtain optimal parameters for the quota-share and stop-loss reinsurance designs. More recently, the above optimization criterion was also adopted by Jiang et al (2017), Fang et al (2019), and Chen and Hu (2020) by using the same VaR risk measure. They found that the combined stop-loss and quota-share reinsurance is one of the optimal solutions to their optimization problems.…”