From risk reduction to risk elimination by conditional mean risk sharing of independent losses

Abstract

This paper studies diversification effects resulting from pooling insurance losses according to the risk allocation rule proposed by Denuit and Dhaene (2012). General comparison results are established for conditional expectations given sums of independent random variables. It is shown that these expectations decrease in the number of terms comprised in the conditioning sums. Additional inequalities are obtained under regression dependence in the sum. These general results are used to derive the monotonicity of the respective contributions of the participants with respect to the convex order, showing that increasing the number of participants is always beneficial under conditional mean risk sharing. New convergence results are obtained, showing that the variance of individual contributions tends to zero in many interesting cases. This provides actuaries with conditions ensuring that the risk can be fully eliminated within the pool, at the limit.