Optimal reinsurance with expectile under the Vajda condition

In this paper, we revisit optimal reinsurance problems by minimizing the adjusted value of the liability of an insurer, which encompasses a risk margin. The risk margin is determined by expectile. To reflect the spirit of reinsurance of protecting the insurer, we assume that both the insurer’s retained loss and the proportion paid by a reinsurer are increasing in indemnity. The premium principles are assumed to satisfy the following three properties: law invariance, risk loading and convex order preservation. We show that the optimal ceded loss functions take the form of three interconnected line segments. Further, if the reinsurance premium is translation invariant or follows the expected value principle, simplified forms of the optimal reinsurance treaties are obtained. Finally, when the reinsurance premium is assumed to be the expected value principle or Wang’s premium principle, the explicit expression for the optimal reinsurance treaty is also given.