A simple approach to dual representations of systemic risk measures. (arXiv:1906.10933v1 [q-fin.MF])

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.