Capital allocation with multivariate risk statistics with positive homogeneity and subadditivity

Abstract

In this paper, we aim to discuss an axiom system for capital allocation with multivariate risk statistics. First, we introduce a class of multivariate risk statistics satisfying the properties of positive homogeneity and subadditivity, and provide the representations. Second, the axiom system of capital allocation with multivariate risk statistics are discussed. Specifically, the existence and the uniqueness of the capital allocation principles with positively homogeneous and subadditive risk statistics are studied. Finally, examples are also given to illustrate the proposed axioms for capital allocation with multivariate risk statistics, where the explicit capital allocation principles are derived for two kinds of multivariate risk statistics based on mean and standard deviation. The proposed capital allocations with multivariate risk statistics directly base on the samples of the random vectors, and thus they are more tractable than those with multivariate risk measures.