We describe a general approach to obtain dual representations for systemic risk measures of the "allocate first, then aggregate"-type, which have recently received significant attention in the literature. Our method is based on the possibility to express this type of multivariate risk measures as special cases of risk measures with multiple eligible assets. This allows us to apply standard Fenchel-Moreau techniques to tackle duality also for systemic risk measures . The same approach can be also successfully employed to obtain an elementary proof of the dual representation of "first aggregate , then allocate" -type systemic risk measures. As a final application, we apply our results to derive a simple proof of the dual representation of univariate utility-based risk measures  .
We establish dual representations for systemic risk measures based on acceptance sets in a general setting. We deal with systemic risk measures of both "first allocate, then aggregate" and "first aggregate , then allocate" type. In both cases, we provide a detailed analysis of the corresponding systemic acceptance sets and their support functions. The same approach delivers a simple and self-contained proof of the dual representation of utility-based risk measures for univariate positions .
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describe a general approach to obtain
establish
R
of the "allocate first, then aggregate"-type, which have recently received significant attention in the literature. Our method is based on the possibility to express this type of multivariate risk measures as special cases of risk measures with multiple eligible assets. This allows us to apply standard Fenchel-Moreau techniques to tackle duality also for
based on acceptance sets in a general setting. We deal with
R
. The same approach can be also successfully employed to obtain an elementary proof of the dual representation of
of both
R
aggregate
allocate, then aggregate" and "first aggregate
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-type systemic risk measures. As a final application, we apply our results to derive a simple
type. In both cases, we provide a detailed analysis of the corresponding systemic acceptance sets and their support functions. The same approach delivers a simple and self-contained
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univariate
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for univariate positions