Optimal insurance for a prudent decision maker under heterogeneous beliefs

In this paper we extend some of the results in the literature on optimal insurance under heterogeneous beliefs in the presence of the no-sabotage condition, by allowing the likelihood ratio function to be non-monotone. Under the assumption of prudence and a mild smoothness condition on the likelihood ratio function, we first partition the whole domain of loss into disjoint regions and then obtain an explicit parametric form for the optimal indemnity function over each piece, by resorting to the marginal indemnity function formulation. The case where there exists belief singularity between the decision maker and the insurer is also studied. As an illustration, we consider a special case of our setting in which the premium principle is a distortion premium principle. We then obtain a closed-form characterization of the optimal indemnity for the cases where premia are determined by Value-at-Risk and Tail Value-at-Risk. Our study complements the literature and provides new insights into several similar problems.