Optimal reinsurance subject to Vajda condition

- We explore optimal reinsurance that minimizes the value of an insurer's liability;
- The insurer's liability is evaluated using a cost-of-capital approach;
- We consider a class of ceded loss functions that are subject to Vajda condition;
- Reinsurance premium principles are assumed to preserve convex order;
- We show that a piecewise linear ceded loss function is always optimal.

In this paper, we study optimal reinsurance design by minimizing the risk-adjusted value of an insurer's liability, where the valuation is carried out by a cost-of-capital approach based either on the value at risk or the conditional value at risk. To prevent moral hazard and to be consistent with the spirit of reinsurance, we follow Vajda (1962) and assume that both the insurer's retained loss and the proportion paid by a reinsurer are increasing in indemnity. We analyze the optimal solutions for a wide class of reinsurance premium principles which satisfy three axioms (law invariance, risk loading and preserving convex order) and encompass ten of the eleven widely used premium principles listed in Young (2004). Our results show that the optimal ceded loss functions are in the form of three interconnected line segments. Further simplified forms of the optimal reinsurance are obtained for the premium principles under an additional mild constraint. Finally, to illustrate the applicability of our results, we derive the optimal reinsurance explicitly for both the expected value principle and Wang's principle.