Mining Matrix Data with Bregman Matrix Divergences for Portfolio Selection

Abstract

In the early fifties, Markowitz contributed a theory that considerably simplified portfolio choices. He narrowed down the traditional expected utility model and assumed that investors only care for mean and variance. The mean-variance portfolio theory was born. As it name suggests, mean-variance theory is predicated on simple assumptions that are unfortunately seldomly met in real life. Indeed, it is now a well-established fact that for a host of reasons financial returns do not obey Gaussian distributions. This paper first draws on ideas from econometrics, finance and statistics to derive a rigorous generalization of Markowitz’ mean-variance model to a mean-divergence model, lifted to matrix entries, grounded on exponential families of distributions, that we argue is both more realistic and better suited to further developments in learning. The generalized model turns out to heavily rely on Bregman divergences. There has recently been a burst of attention in on-line learning to learn portfolios having limited risk in Markowitz’ setting. In an on-line framework, we then tackle the problem of finding adaptive portfolio strategies based on our generalized model. We devise a learning algorithm based on new matrix generalizations of p-norms to track non stationary target portfolios with limited risk. Theoretical bounds and preliminary experiments over nearly twelve years of S$\&$P 500 confirm the validity of the generalized model, the capacity it brings to spot important events that would otherwise be dampened in the mean-variance model, and the potential of the algorithm. Finally, we make an in depth analysis of the matrix divergences and risk premia derived in our model that shed some theoretical light on the ways the risk premium may be blown up as the investor’s portfolio shifts away from a so-called natural market allocation which defines the best (unknown) allocation at the market’s scale.