The Design of an Optimal Retrospective Rating Plan

A retrospective rating plan, whose insurance premium depends upon an insured's actual loss during the policy period, is a special insurance agreement widely used in liability insurance. In this paper, the design of an optimal retrospective rating plan is analyzed from the perspective of the insured who seeks to minimize its risk exposure in the sense of convex order. In order to reduce the moral hazard, we assume that both the insured and the insurer are obligated to pay more for a larger realization of the loss. Under the further assumptions that the minimum premium is zero, the maximum premium is proportional to the expected indemnity, and the basic premium is the only free parameter in the formula for retrospective premium given by Meyers (2004) and that the basic premium is determined in such a way that the expected retrospective premium equates to the expected indemnity with a positive safety loading, we formally establish the relationship that the insured will suffer more risk for a larger loss conversion factor or a higher maximum premium. These findings suggest that the insured prefers an insurance policy with the expected value premium principle, which is a special retrospective premium principle with zero loss conversion factor. In addition, we show that any admissible retrospective rating plan is dominated by a stop-loss insurance policy. Finally, the optimal retention of a stop-loss insurance is derived numerically under the criterion of minimizing the risk-adjusted value of the insured's liability where the liability valuation is carried out using the cost-of-capital approach based on the conditional value at risk.