Budget-Constrained Optimal Insurance with an Upper Limit on the Retained Loss

Abstract

Unlike sophisticated institutional insurance buyers, individual insurance seekers often use simple heuristic tools for risk management purposes, such as requiring that an insurance arrangement will not result in a retained loss that exceeds a certain predetermined and fixed level. In this paper, we re-examine the problem of budget-constrained demand for insurance indemnification when the insured and the insurer disagree about the likelihoods associated with the realizations of the insurable loss; but we further impose an additional upper-limit constraint on the retained loss and assume that the insurer distorts his subjective probability measure. We do not impose the no sabotage condition on admissible indemnities. Instead, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, which rules out ex post moral hazard issues that could otherwise arise from possible misreporting of the loss by the insured. We characterize the optimal retention function and show that it has a simple two-part structure: zero retention (full insurance) on an event to which the insurer assigns zero probability, and a retention that could be described as a limited variable deductible on the complement of this event. As an illustration, we examine the case of a distorted Esscher premium principle.