Systemic risk: Conditional distortion risk measures

In this paper, we introduce the rich classes of conditional distortion (CoD) risk measures and distortion risk contribution (ΔCoD) measures as measures of systemic risk and analyze their properties and representations. The classes unify, and significantly extend, existing systemic risk measures such as the conditional Value-at-Risk, conditional Expected Shortfall, and risk contribution measures in terms of the VaR and ES. We provide sufficient conditions for two random vectors to be ordered by the proposed CoD-risk measures and ΔCoD-measures. These conditions are expressed using the conventional stochastic dominance, increasing convex/concave, dispersive, and excess wealth orders for the marginals and canonical positive/negative stochastic dependence notions.