Abstract
Using a family of modified Weibull distributions encompassing both sub-exponentials and super-exponentials to parametrize the marginal distributions of asset returns and their multivariate generalizations with Gaussian copulas, we offer exact formulae 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 χ function of the asset weights with different functional forms depending on the super- or sub-exponential behaviour of the marginals and on the strength of the dependence between the assets. We then treat in detail the problem of risk minimization using the Value-at-Risk and expected shortfall which are shown to be (asymptotically) equivalent in this framework.