Analysis of Bivariate Excess Losses

The concept of excess losses is widely used in reinsurance and retrospective insurance rating. The mathematics related to it has been studied extensively in the property and casualty actuarial literature. However, it seems that the formulas for higher moments of the excess losses are not readily available. Therefore, in the first part of this paper, we introduce a formula for calculating the higher moments, based on which it is shown that they can be obtained directly from the Table of Insurance Charges (Table M). In the second part of the paper, we introduce the concept of bivariate excess losses. It is shown that the joint moments of bivariate excess losses can be computed through methods similar to the ones used in the univariate case. In addition, we provide examples to illustrate possible applications of bivariate excess loss functions.