Real-Valued Systemic Measures

We describe the axiomatic approach to real-valued Systemic Measures, which may be a natural counterpart to the nowadays classical univariate theory initiated by Artzner et al. within the seminal paper “Coherent measures of risk”, Math. Finance, (1999). Especially, 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 permit us to motivate why this particular risk measurement regime is fair for both the only agents and therefore the whole system at an equivalent 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 that the suitable setup when additional information is out there at an initial time.