  In this paper we estimate the conditional value-at-risk by fitting different
multivariate parametric models capturing some stylized facts about multivariate
financial time series of equity returns: heavy tails, negative skew, asymmetric
dependence, and volatility clustering. While the volatility clustering effect
is got by AR-GARCH dynamics of the GJR type, the other stylized facts are
captured through non-Gaussian multivariate models and copula functions. The
CoVaR$^{\leq}$ is computed on the basis on the multivariate normal model, the
multivariate normal tempered stable (MNTS) model, the multivariate generalized
hyperbolic model (MGH) and four possible copula functions. These risk measure
estimates are compared to the CoVaR$^{=}$ based on the multivariate normal
GARCH model. The comparison is conducted by backtesting the competitor models
over the time span from January 2007 to March 2020. In the empirical study we
consider a sample of listed banks of the euro area belonging to the main or to
the additional global systemically important banks (GSIBs) assessment sample.
