  This paper is devoted to the quantification and analysis of marginal risk
contribution of a given single financial institution i to the risk of a
financial system s. Our work expands on the CoVaR concept proposed by Adrian
and Brunnermeier as a tool for the measurement of marginal systemic risk
contribution. We first give a mathematical definition of
CoVaR_{\alpha}^{s|L^i=l}. Our definition improves the CoVaR concept by
expressing CoVaR_{\alpha}^{s|L^i=l} as a function of a state l and of a given
probability level \alpha relative to i and s respectively. Based on Copula
theory we connect CoVaR_{\alpha}^{s|L^i=l} to the partial derivatives of Copula
through their probabilistic interpretation and definitions (Conditional
Probability). Using this we provide a closed formula for the calculation of
CoVaR_{\alpha}^{s|L^i=l} for a large class of (marginal) distributions and
dependence structures (linear and non-linear). Our formula allows a better
analysis of systemic risk using CoVaR in the sense that it allows to define
CoVaR_{\alpha}^{s|L^i=l} depending on the marginal distributions of the losses
of i and s respectively and the copula between L^i and L^s. We discuss the
implications of this in the context of the quantification and analysis of
systemic risk contributions. %some mathematical This makes possible the For
example we will analyse the marginal effects of L^i, L^s and C of the risk
contribution of i.
