Risk allocation with Shapley value in the risk aggregation framework

Abstract

The topic of risk aggregation arises from the need to incorporate in a single measure the overall exposure to the different risk types. In general, the methodologies adopted for the purposes of risk integration are based on the principle that the overall economic capital is lower than the simple algebraic sum of economic capital measures related to individual risks. This phenomenon, due to the existence of an imperfect correlation between the risks, determines, in line with portfolio theory, a "diversification benefit". The issue of risk allocation subsequently arises when the risk value of the diversified aggregated loss needs to be reassigned to the different risk classes. A similar issue has been solved in the framework of cooperative Game Theory, where the Shapley value provides a player-specific contribution of the total surplus generated by the coalition. The paper proposes a novel application of the Shapley formula in the ICAAP context (Pillar II - economic view). In particular, we show that the Shapley value is the unique solution to the allocation problem of an overall risk value, granting the fundamental requested properties, including the efficiency one. An exemplificative model application is reported, as well as a comparison with a benchmark methodology. The experimental part shows the advantages of the novel approach in terms of precision and reliability of the estimates. Finally, it is important to mention that the presented framework can be applied also in other contexts such as, for instance, in the risk class attribution of the operational risk.