Optimal Dynamic Reinsurance Under Heterogeneous Beliefs and CARA Utility

Abstract

This paper examines the optimal dynamic reinsurance policy for an insurance company under belief heterogeneity. We assume the reinsurance premium is calculated according to the mean-conditional value-at-risk principle and impose the incentive compatibility constraint to rule out moral hazard. Under the objective of maximizing the exponential utility function, we obtain the optimal strategies in closed form via a “relaxation and modification” approach. The optimal contracts have more complicated structures than the standard proportional and excess-of-loss reinsurance widely investigated in the literature. In particular, we demonstrate that the insurer may optimally choose to purchase proportional reinsurance in different layers when the reinsurer is more pessimistic about the underlying loss. Our model lends support to the observation that reinsurance contracts in practice often involve proportional reinsurance in multiple layers. We also demonstrate that belief heterogeneity can explain the inverse relationship between the purchase of reinsurance and the size of the insurer's losses observed in the reinsurance market.