  Gaussian random vectors exhibit the loss of dimension phenomena, which relate
to their joint survival tail behaviour. Besides, the fact that the components
of such vectors are light-tailed complicates the approximations of various
multivariate risk measures significantly. In this contribution we derive
precise approximations of marginal mean excess, marginal expected shortfall and
multivariate conditional tail expectation of Gaussian random vectors and
highlight links with conditional limit theorems. Our study indicates that
similar results hold for elliptical and Gaussian like multivariate risks.
