Dual representations for systemic risk measures
Date
2020Source Title
Mathematics and Financial Economics
Print ISSN
1862-9679
Publisher
Springer
Volume
14
Issue
1
Pages
139 - 174
Language
English
Type
ArticleItem Usage Stats
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166
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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.
Keywords
Systemic riskRisk measure
Financial network
Dual representation
Convex duality
Penalty function
Relative entropy
Multivariate risk
Shortfall risk