  Systemic risk is the risk that a company- or industry-level risk could
trigger a huge collapse of another or even the whole institution. Various
systemic risk measures have been proposed in the literature to quantify the
domino and (relative) spillover effects induced by systemic risks such as the
well-known CoVaR, CoES, MES and CoD risk measures, and associated contribution
measures. This paper proposes another new type of systemic risk measure, called
the joint marginal expected shortfall (JMES), to measure whether the MES of one
entity's risk-taking adds to another one or the overall risk conditioned on the
event that the entity is already in some specified distress level. We further
introduce two useful systemic risk contribution measures based on the
difference function or relative ratio function of the JMES and the conventional
ES, respectively. Some basic properties of these proposed measures are studied
such as monotonicity, comonotonic additivity, non-identifiability and
non-elicitability. For both risk measures and two different vectors of
bivariate risks, we establish sufficient conditions imposed on copula
structure, stress levels, and stochastic orders to compare these new measures.
We further provide some numerical examples to illustrate our main findings. A
real application in analyzing the risk contagion among several stock market
indices is implemented to show the performances of our proposed measures
compared with other commonly used measures including CoVaR, CoES, MES, and
their associated contribution measures.
