In this paper, we introduce a multivariate extension of the classical univariate Value-at-Risk (VaR). This extension may be useful to understand how solvency capital requirement is affected by the presence of risks that cannot be diversify away. This is typically the case for a network of highly interconnected financial institutions in a macro-prudential regulatory system. We also generalize the bivariate Conditional-Tail–Expectation (CTE), previously introduced by Di Bern-ardino et al. (2011), in a multivariate setting and we study its behavior. Several properties have been derived. In particular, we show that these two risk measures both satisfy the positive homogeneity and the translation invariance property. Comparison between univariate risk meas-ures and components of multivariate VaR and CTE are provided. We also analyze how they are impacted by a change in marginal distributions, by a change in dependence structure and by a change in risk level. Interestingly, these results turn to be consistent with existing properties on univariate risk measures. Illustrations are given in the class of Archimedean copulas.

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In this paper, we introduce two alternative extensions of the classical univariate Value-at-Risk (VaR) in a multivariate setting. The two proposed multivariate VaR are vector-valued measures with the same dimension as the underlying risk portfolio. The lower-orthant VaR is constructed fromlevel setsofmultivariatedistributionfunctionswhereastheupper-orthant VaR isconstructed from level sets of multivariate survival functions. Several properties have been derived. In particular, we show that these risk measures both satisfy the positive homogeneity and the translation invariance property. Comparison between univariate risk measures and components of multivariate VaR are provided. We also analyze how these measures are impacted by a change in marginal distributions, by a change in dependence structure and by a change in risk level. Illustrations are given in the class of Archimedean copulas.