Real-Valued Systemic Risk Measures

We describe the axiomatic approach to real-valued Systemic Risk Measures, which is a natural counterpart to the nowadays classical univariate theory initiated by Artzner et al. in the seminal paper “Coherent measures of risk”, Math. Finance, (1999). In particular, we direct our attention towards Systemic Risk Measures of shortfall type with random allocations, which consider as eligible, for securing the system, those positions whose aggregated expected utility is above a given threshold. We present duality results, which allow us to motivate why this particular risk measurement regime is fair for both the single agents and the whole system at the same time. We relate Systemic Risk Measures of shortfall type to an equilibrium concept, namely a Systemic Optimal Risk Transfer Equilibrium, which conjugates Bühlmann’s Risk Exchange Equilibrium with a capital allocation problem at an initial time. We conclude by presenting extensions to the conditional, dynamic framework. The latter is the suitable setup when additional information is available at an initial time.

On the Impossibility of Fair Risk Allocation

Allocating risk properly to subunits is crucial for performance evaluation and internal capital allocation of portfolios held by banks, insurance companies, investment funds and other entities subject to financial risk. We show that by using coherent measures of risk it is impossible to allocate risk satisfying simultaneously the natural game theoretical requirements of Core Compatibility and Strong Monotonicity. To obtain the result we characterize the Shapley value on the class of totally balanced games and also on the class of exact games as being the only risk allocation method satisfying Strong Monotonicity, Equal Treatment Property and Efficiency. Moreover, we clarify and interpret the related game theoretical requirements that have appeared in the literature so far and have been applied to risk allocation.