Optimal allocation of policy limits and deductibles under distortion risk measures

In the literature, orderings of optimal allocations of policy limits and deductibles were established by maximizing the expected utility of wealth of the policyholder. In this paper, by applying the bivariate characterizations of stochastic ordering relations, we reconsider the same model and derive some new refined results on orderings of optimal allocations of policy limits and deductibles with respect to the family of distortion risk measures from the viewpoint of the policyholder. Both loss severities and loss frequencies are considered. Special attention is given to the optimization criteria of the family of distortion risk measures with concave distortions and with only increasing distortions. Most of the results presented in this paper can be applied to some particular distortion risk measures. The results complement and extend the main results in Cheung [Cheung, K.C., 2007. Optimal allocation of policy limits and deductibles. Insurance: Mathematics and Economics 41, 291–382] and Hua and Cheung [Hua, L., Cheung, K.C., 2008a. Stochastic orders of scalar products with applications. Insurance: Mathematics and Economics 42, 865–872].