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

Abstract

In this paper we focus our attention on the study of an excess of loss reinsurance with reinstatements, a problem previously studied by Sundt and, more recently, by Mata and Hurlimann. It is well known that the evaluation of pure premiums requires 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 generalised expected value equation, then the initial premium exhibits some regularity properties as a function of the percentages of reinstatement

Initial premium, aggregate claims and distortion risk measures in XL reinsurance with reinstatements

“…The excess of loss reinsurance model we study in this paper is related to the model that has been originally proposed and analyzed by Sundt [9] and that it has been subsequently generalized (see [1] and [3]). Given the initial premium P , the limit on the payment of each claim m, the number of reinstatements K (such that the aggregate limit M satisfies M = (K + 1)m), the aggregate deductible D, the percentages of reinstatement c i , (i = 1, .…”

“…Recently (see [3]) we studied the initial risk adjusted premium P as solution of the equilibrium condition expressing the fact that the value of the total premium income δ(P ) equals the distorted expected value of the aggregate claims S, that is…”