  This paper offers a precise analytical characterization of the distribution
of returns for a portfolio constituted of assets whose returns are described by
an arbitrary joint multivariate distribution. In this goal, we introduce a
non-linear transformation that maps the returns onto gaussian variables whose
covariance matrix provides a new measure of dependence between the non-normal
returns, generalizing the covariance matrix into a non-linear fractional
covariance matrix. This nonlinear covariance matrix is chiseled to the specific
fat tail structure of the underlying marginal distributions, thus ensuring
stability and good-conditionning. The portfolio distribution is obtained as the
solution of a mapping to a so-called phi-q field theory in particle physics, of
which we offer an extensive treatment using Feynman diagrammatic techniques and
large deviation theory, that we illustrate in details for multivariate Weibull
distributions. The main result of our theory is that minimizing the portfolio
variance (i.e. the relatively ``small'' risks) may often increase the large
risks, as measured by higher normalized cumulants. Extensive empirical tests
are presented on the foreign exchange market that validate satisfactorily the
theory. For ``fat tail'' distributions, we show that an adequete prediction of
the risks of a portfolio relies much more on the correct description of the
tail structure rather than on their correlations.
