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Shortfall systemic (multivariate) risk measures $\rho$ defined through an$N$-dimensional multivariate utility function $U$ and random allocations can berepresented as classical (one dimensional) shortfall risk measures associatedto an explicitly determined $1$-dimensional function constructed from $U$. Thisfinding allows for simplifying the study of several properties of $\rho$, suchas dual representations, law invariance and stability.

Are Shortfall Systemic Risk Measures One Dimensional?