Optimal reinsurance in the epsilon-contaminated loss distribution preserving model

We discuss the role of ambiguity in randomized reinsurance contracts, focusing on one-period stop-loss treaties. Ambiguity is modeled as an 𝜀-contamination of a reference joint probability with respect to the class of all finitely additive joint probabilities with a fixed loss marginal distribution. In a risk management perspective, we consider an insurer that systematically adopts a pessimistic attitude towards ambiguity. This translates in computing a lower expected profit, as a Choquet integral, and in adopting an ambiguous formulation of Value-at-Risk, both determined by the lower envelope of the 𝜀-contamination class. Next, the optimal retention level is obtained by maximizing the insurer’s lower expected profit, that is by applying a maximin criterion of choice. We provide a complete characterization of a randomized stop-loss reinsurance contract under ambiguity, and we study the impact of the ambiguity parameter 𝜀 on the optimal retention level as well as on the resulting expected profit. Finally, we highlight the existence of an ambiguity threshold above which the benefit of randomization vanishes.