Capital allocation with multivariate convex risk measures

Abstract

Capital allocation is of fundamental importance in risk management. Capital allocation problems with univariate risk measures have been extensively studied in the finance literature. In contrast, the studies about capital allocations with multivariate risk measures are quite insufficient in the literature. In this paper, we study capital allocations with multivariate coherent and convex risk measures. First, we propose the definitions of capital allocation problem and principle for general multivariate risk measures. Then based on the dual representations proved by [41], we provide the capital allocation principles with multivariate coherent and convex risk measures. Their properties are also investigated. For the case of multivariate convex risk measures, we also suggest a reasonable and effective capital allocation scheme. Finally, examples are given to illustrate the effectiveness of the proposed capital allocation scheme, where the involved convex risk measures are multivariate and univariate entropic risk measures, respectively.