Worst allocations of policy limits and deductibles

In the literature, orderings of optimal allocations of policy limits and deductibles were established with respect to a policyholder's preference. However, from the viewpoint of an insurer, the orderings are not enough for the purpose of pricing. In this paper, by applying the equivalent utility premium principle, we study worst allocations of policy limits and deductibles for an insurer, which give rise to the maximum fair premiums. Closed-form solutions are derived. Then we present a result concerning the optimality in a general risk-sharing scheme, by which we obtain optimal allocations for policyholders directly from worst allocations for an insurer. Several results in Cheung [Cheung, K.C., 2007. Optimal allocation of policy limits and deductibles. Insurance Math. Econom. 41, 382-391] are generalized here.