Real-Valued Systemic Risk Measures

Abstract

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.

Strong Comonotonic Additive Systemic Risk Measures

“…Therefore by Proposition 1(1), we know that Λ(X) and Λ(Y) are comonotonic. Consequently, by (11) and the comonotonic additivity of ρ 0 , we obtain that…”

“…These axiomatic approaches to systemic risk measures share more or less the ideas of Artzner et al [4]. For other axiomatic approaches for systemic risk measures, we refer to [5][6][7][8][9][10][11][12][13][14]. For the dual representation results, we refer to [15][16][17][18][19].…”

A description of the COVID-19 outbreak role in financial risk forecasting

“…For instance, a concurrent approach would be to “measure the risk of each of the system components, and then aggregating the risks”. The difference between these two approaches is discussed thoroughly in Doldi and Frittelli (2021) and Biagini, Fouque, Frittelli, and Meyer-Brandis (2019) . Both approaches can be viewed as extensions of the basic theory developed in Artzner, Delbaen, Eber, and Heath (1999) .…”