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