  Using a family of modified Weibull distributions, encompassing both
sub-exponentials and super-exponentials, to parameterize the marginal
distributions of asset returns and their multivariate generalizations with
Gaussian copulas, we offer exact formulas for the tails of the distribution
$P(S)$ of returns $S$ of a portfolio of arbitrary composition of these assets.
We find that the tail of $P(S)$ is also asymptotically a modified Weibull
distribution with a characteristic scale $\chi$ function of the asset weights
with different functional forms depending on the super- or sub-exponential
behavior of the marginals and on the strength of the dependence between the
assets. We then treat in details the problem of risk minimization using the
Value-at-Risk and Expected-Shortfall which are shown to be (asymptotically)
equivalent in this framework.
