Properties and comparison of risk capital allocation methods

Abstract

If a financial unit (a bank, an insurance company, a portfolio, the financial system of a country, etc.) consists of subunits (divisions, subportfolios, etc.), then the risk of the main unit should be allocated to the subunits using a risk capital allocation method in a fair way.We analyze seven methods widely discussed in the literature or used in practice (Activity based, Beta, Incremental, Cost gap, Marginal Risk Contribution, Shapley, and Nucleolus) in terms of ten reasonable fairness properties (Full Domain, Core Compatibility, Diversification, Strong Monotonicity, Incentive Compatibility Efficiency, Equal Treatment Property, Riskless Portfolio, Covariance, and Decomposition Invariance). We provide proofs or counterexamples for each method and the ten properties that we consider.We also computed how often on average Core Compatibility is satisfied in randomly generated risk capital allocation situations up to nine subunits in 24 treatments for all methods that do not satisfy Core Compatibility.We believe that through the descriptions of the examined methods our paper can serve as a useful guide for both practitioners and researchers.