Dual representations for systemic risk measures
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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.
KeywordsSystemic risk Risk measure Financial network Dual representation Convex duality Penalty function Relative entropy Multivariate risk Shortfall risk
Mathematics Subject Classification91B30 46N10 46A20 26E25 90C46
JEL ClassificationC61 D81 E58 G32
This material is based upon work supported by the National Science Foundation under Grant No. 1321794 and the OeNB anniversary fund, Project Number 17793. Part of the manuscript was written when the first author visited Vienna University of Economics and Business. The authors would like to thank an anonymous referee for useful comments and suggestions that helped improving the manuscript. The first author would like to thank Fabio Bellini, Zachary Feinstein and Daniel Ocone for fruitful discussions. The authors would like to thank Cosimo-Andrea Munari and Maria Arduca for pointing out an issue in an earlier version of the paper, as well as Alexander Smirnow and Jana Hlavinova for pointing out a simplification in the proof of the second part of Theorem 3.2.
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