Optimal reinsurance from the perspectives of both insurers and reinsurers under the VaR risk measure and Vajda condition

In this article, we revisit the optimal reinsurance problem by minimizing the convex combination of the VaRs of the insurer's loss and the reinsurer's loss. To prevent moral hazard and to reflect the spirit of reinsurance, we assume that the set of admissible ceded loss function is the class of ceded loss functions such that the retained loss functions are increasing and the ceded loss functions satisfy Vajda condition. We analyze the optimal solutions for a wide class of reinsurance premium principles that satisfy the following three properties: law invariance, risk loading property and stop-loss ordering preserving. Meanwhile, we use the expected value premium principle to derive the explicit expressions for the optimal reinsurance treaties. Finally, we construct a numerical example to illustrate our results.