Using Aumann-Shapley Values to Allocate Insurance Risk

Abstract

The problem of allocating responsibility for risk among members of a portfolio arises in a variety of financial and risk-management contexts. Examples are particularly prominent in the insurance sector, where actuaries have long sought methods for distributing capital (net worth) across a number of distinct exposure units or accounts according to their relative contributions to the total “risk” of an insurer’s portfolio. Although substantial work has been done on this problem, no satisfactory solution has yet been presented for the case of inhomogeneous loss distributions— that is, losses X ∼ F _{X| λ} such that F _X|tλ (X) ≠ F _{tX| λ} (X) for some t > 0. The purpose of this article is to show that the value-assignment method of nonatomic cooperative games proposed in 1974 by Aumann and Shapley may be used to solve risk-allocation problems involving losses of this type. This technique is illustrated by providing analytical solutions for a useful class of multivariatenormal loss distributions.