Capital Allocation With Multivariate Risk Measures: An Axiomatic Approach

Abstract

Capital allocation is of central importance in portfolio management and risk-based performance measurement. Capital allocations for univariate risk measures have been extensively studied in the finance literature. In contrast to this situation, few papers dealt with capital allocations for multivariate risk measures. In this paper, we propose an axiom system for capital allocation with multivariate risk measures. We first recall the class of the positively homogeneous and subadditive multivariate risk measures, and provide the corresponding representation results. Then it is shown that for a given positively homogeneous and subadditive multivariate risk measure, there exists a capital allocation principle. Furthermore, the uniqueness of the capital allocation principe is characterized. Finally, examples are also given to derive the explicit capital allocation principles for the multivariate risk measures based on mean and standard deviation, including the multivariate mean-standard-deviation risk measures.

Capital Allocation Rules and the No-Undercut Property

“…For the sake of completeness on the subject, we have also made a short focus on the problem of capital allocation in the set-valued context (see [24,25]) whose definition has been motivated by the recent introduction (due to financial reasons) of the theory of set-valued risk measures (see [26,27]). Additionally, in this context, no-undercut has been defined and studied, and some of the results in the streamline of the gradient allocation have been derived (see [24]).…”

“…It is worth recalling that, more recently, the problem of capital allocation has been defined also in a set-valued context (see [24,25]): indeed, in the latest years, the theory of risk measures has been extended to the more general setting, where they can be setvalued (see [26,27]), motivated by financial considerations. For example, in the case of portfolios of financial positions in different currencies that can not be aggregated due to liquidity constraints and/or transaction costs (see [26,27]), it is reasonable to consider risk measures that associate, to any portfolio in different currencies, a set of hedging deterministic positions.…”