PROPERTIES OF FUNCTIONS OF THE EXCESS OF LOSS RETENTION LIMIT WITH APPLICATIONS

We study the excess of loss retention limit of an insurance portfolio both from the reinsurer as well as the ceding company's point of view. We prove that expected values of functions as well as distribution fu nctions of both parties liabilities are differentiable with respect to the retention limit, and we give explicit expressions for these deriv atives. The results are used to analyze excess of loss reinsurance pre miums for a number of premium calculation principles. We also formulat e and discuss the problem facing an insurance company how to choose in vestment and reinsurance policy in an optimal way, using the general r esults of the paper to find gradients of the functions involved in thi s optimization problem.