What do distortion risk measures tell us on excess of loss reinsurance with reinstatements?

In this paper we focused our attention to the study of an excess of loss reinsurance with reinstatements, a problem previously studied by Sundt and, more recently, by Mata and H¨urlimann. As it is well-known, the evaluation of pure premiums requires the knowledge of the claim size distribution of the insurance risk: in order to face this question, different approaches have been followed in the actuarial literature. In a situation of incomplete information in which only some characteristics of the involved elements are known, it appears to be particularly interesting to set this problem in the framework of risk adjusted premiums. It is shown that if risk adjusted premiums satisfy a generalized expected value equation, then the initial premium exhibits some regularity properties as a function of the percentages of reinstatement.