phi(q)-field theory for portfolio optimization: "fat tails" and nonlinear correlations

Physics and finance are both fundamentally based on the theory of random wa lks (and their generalizations to higher dimensions) and on the collective behavior of large numbers of correlated variables. The archetype examplifyi ng this situation in finance is the portfolio optimization problem in which one desires to diversify on a set of possibly dependent assets to optimize the return and minimize the risks. The standard mean-variance solution int roduced by Markovitz and its subsequent developments is basically a mean-fi eld Gaussian solution. It has severe limitations for practical applications due to the strongly non-Gaussian structure of distributions and the nonlin ear dependence between assets. Here, we present in details a general analyt ical characterization of the distribution of returns for a portfolio consti tuted of assets whose returns are described by an arbitrary joint multivari ate distribution. In this goal, we introduce a nonlinear transformation tha t maps the returns onto Gaussian variables whose covariance matrix provides a new measure of dependence between the non-normal returns, generalizing t he covariance matrix into a nonlinear covariance matrix. This nonlinear cov ariance matrix is chiseled to the specific fat tail structure of the underl ying marginal distributions, thus ensuring stability and good conditioning. The portfolio distribution is then obtained as the solution of a mapping t o a so-called phi(q) field theory in particle physics, of which we offer an extensive treatment using Feynman diagrammatic techniques and large deviat ion theory, that we illustrate in details for multivariate Weibull distribu tions. The interaction (non-mean held) structure in this field theory is a direct consequence of the non-Gaussian nature of the distribution of asset price returns. We find that minimizing the portfolio variance (i.e. the rel atively "small" risks) may often increase the large risks, as measured by h igher normalized cumulants. Extensive empirical tests are presented on the foreign exchange market that validate satisfactorily the theory. For "fat t ail" distributions, we show that an adequate prediction of the risks of a p ortfolio relies much more on the correct description of the tail structure rather than on their correlations. For the case of asymmetric return distri butions, our theory allows us to generalize the return-risk efficient front ier concept to incorporate the dimensions of large risks embedded in the ta il of the asset distributions. We demonstrate that it is often possible to increase the portfolio return while decreasing the large risks as quantifie d by the fourth and higher-order cumulants. Exact theoretical formulas are validated by empirical tests. (C) 2000 Elsevier Science B.V. All rights res erved.