  The financial crisis has dramatically demonstrated that the traditional
approach to apply univariate monetary risk measures to single institutions does
not capture sufficiently the perilous systemic risk that is generated by the
interconnectedness of the system entities and the corresponding contagion
effects. This has brought awareness of the urgent need for novel approaches
that capture systemic riskiness. The purpose of this paper is to specify a
general methodological framework that is flexible enough to cover a wide range
of possibilities to design systemic risk measures via multi-dimensional
acceptance sets and aggregation functions, and to study corresponding examples.
Existing systemic risk measures can usually be interpreted as the minimal
capital needed to secure the system after aggregating individual risks. In
contrast, our approach also includes systemic risk measures that can be
interpreted as the minimal capital funds that secure the aggregated system by
allocating capital to the single institutions before aggregating the individual
risks. This allows for a possible ranking of the institutions in terms of
systemic riskiness measured by the optimal allocations. Moreover, we also allow
for the possibility of allocating the funds according to the future state of
the system (random allocation). We provide conditions which ensure
monotonicity, convexity, or quasi-convexity properties of our systemic risk
measures.
