Estimation of the coefficient of variation of the excess of loss by means of Jensen's inequality for convex functions. The connection with the theory of life times

Abstract

This note deals with a way to determine upper and lower bounds for the coefficient of variation of the total claim costs in a year in excess of a certain limit value. In many a publication attention has been paid to the question of calculating, in addition to the mean of the excess costs, the variance of these costs. In this connection we only mention here the study of Vajda: Minimum Variance Reinsurance, ASTIN-Bulletin, September 1962. It is evident that in studying such problems of “statistics of large claims” the coefficient of variation, defined as the ratio of the standard deviation and the mean, may be a useful tool. If we exclude special “dangerous” claim distributions and also distributions with a very “irregular” tail, it appears possible to derive bounds for this coefficient of variation and to indicate its asymptotic behaviour. Hereby good use can be made of Jensen's inequality for convex functions. Jensen's inequality has frequently been applied to problems in the field of life insurance mathematics, but as far as the authors of this note know, not to the question mentioned here.Besides, it may have some interest to point out a connection between the question of estimating the excess of total claim costs and the theory of life times. We may translate “the limit value” as “the age already reached” and “the mean of the excess of loss” as “the expectation of life”. The estimation of the coefficient of variation ot the life time now leads to some well-known biometric formulae on life times and at the same time to an interesting observation as regards human life tables.