Dual representations for systemic risk measures

TL;DR

This paper develops dual representations for scalar and multivariate systemic risk measures, encompassing different frameworks of capital allocation and aggregation, with applications to financial network models.

Contribution

It introduces dual representations for systemic risk measures in both allocation-before and after-aggregation frameworks, unifying various models.

Findings

Derived dual representations for scalar systemic risk measures

Extended duality results to multivariate risk measures with capital allocations

Applied results to Eisenberg-Noe and network flow models

Abstract

The financial crisis showed the importance of measuring, allocating and regulating systemic risk. Recently, the systemic risk measures that can be decomposed into an aggregation function and a scalar measure of risk, received a lot of attention. In this framework, capital allocations are added after aggregation and can represent bailout costs. More recently, a framework has been introduced, where institutions are supplied with capital allocations before aggregation. This yields an interpretation that is particularly useful for regulatory purposes. In each framework, the set of all feasible capital allocations leads to a multivariate risk measure. In this paper, we present dual representations for scalar systemic risk measures as well as for the corresponding multivariate risk measures concerning capital allocations. Our results cover both frameworks: aggregating after allocating and allocating after aggregation. As examples, we consider the aggregation mechanisms of the Eisenberg-Noe model as well as those of the resource allocation and network flow models.