Conditional risk (co-risk) measures and risk contribution measures are increasingly used in quantitative risk analysis to evaluate the systemic risk that the failure (or loss) of a component spreads to other components or even to the entire system. Co-risk measures are conditional versions of measures usually employed to assess isolated risks, while risk contribution measures evaluate how a stress situation for one component incrementally affects another one. In this article , we introduce the rich classes of conditional distortion (CoD) risk measures and distortion risk contribution (\DeltaCoD) measures , which contain the well-known conditional Value-at-Risk (CoVaR) , conditional Expected Shortfall (CoES) , and risk contribution measures in terms of the VaR and ES (%DIFDELCMD < \DeltaCoVaR %%% and%DIFDELCMD < \DeltaCoES%%% ) as special cases. Sufficient conditions are presented for two random vectors to be ordered by the proposed CoD-risk measures and distortion risk contribution measures. These conditions are stated in terms of the stochastic dominance, increasing convex/concave, dispersive order , and excess wealth orders of the marginals under some assumptions of positive/negative stochastic dependence  . Numerical examples are provided to illustrate our theoretical findings. This paper is the second in a triplet of papers on systemic risk by the same authors. In Dhaene et al. (2018)}\hspace{0pt}%DIFAUXCMD } , we introduce and analyze some new stochastic orders related to systemic risk. In a third (forthcoming) paper, we attribute systemic risk to the different participants in a given risky environment.
In this paper , we introduce the rich classes of conditional distortion (CoD) risk measures and distortion risk contribution (\DeltaCoD) measures as measures of systemic risk and analyze their properties and representations. The classes include the well-known conditional Value-at-Risk , conditional Expected Shortfall , and risk contribution measures in terms of the VaR and ES %DIFDELCMD < \DeltaCoVaR %%% %DIFDELCMD < \DeltaCoES%%% as special cases. Sufficient conditions are presented for two random vectors to be ordered by the proposed CoD-risk measures and distortion risk contribution measures. These conditions are expressed using the conventional stochastic dominance, increasing convex/concave, dispersive , and excess wealth orders of the marginals and canonical positive/negative stochastic dependence notions . Numerical examples are provided to illustrate our theoretical findings. This paper is the second in a triplet of papers on systemic risk by the same authors. In \mbox{%DIFAUXCMD \cite{DLZorder2018a}\hspace{0pt}%DIFAUXCMD } , we introduce and analyze some new stochastic orders related to systemic risk. In a third (forthcoming) paper, we attribute systemic risk to the different participants in a given risky environment.
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