Revisiting the optimal insurance design under adverse selection: Distortion risk measures and tail-risk overestimation

This paper studies the design of optimal insurance from an insurer's perspective when it is subject to adverse selection issue. Different from the literature, the insureds who are exposed to different types of risks are allowed to apply different preference measures. By assuming that the insureds' preferences are dictated by some distortion risk measures that always over-estimate the tail risk, we figure out the optimal policy menu without assuming the parametric form of indemnity functions. We also find that the insureds who deem their losses riskier than those of others will always purchase full insurance, which is consistent with the results in past studies. Furthermore, we show that in the presence of adverse selection the optimal policy menu always outperforms the optimal single policy in the sense that the former can yield a larger expected profit for the insurer. This outcome also echoes some existing results in the literature.