"Nonlinear" covariance matrix and portfolio theory for non-Gaussian multivariate distributions

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.