On tracking portfolios with certainty equivalents on a generalization of Markowitz model: the Fool, the Wise and the Adaptive

Portfolio allocation theory has been heavily influenced by a major contribution of Harry Markowitz in the early fifties: the mean-variance approach. While there has been a continuous line of works in on-line learning portfolios over the past decades, very few works have really tried to cope with Markowitz model. A major drawback of the mean-variance approach is that it is approximation-free only when stock returns obey a Gaussian distribution, an assumption known not to hold in real data. In this paper, we first alleviate this assumption, and rigorously lift the mean-variance model to a more general mean-divergence model in which stock returns are allowed to obey any exponential family of distributions. We then devise a general on-line learning algorithm in this setting. We prove for this algorithm the first lower bounds on the most relevant quantity to be optimized in the framework of Markowitz model: the certainty equivalents. Experiments on four real-world stock markets display its ability to track portfolios whose cumulated returns exceed those of the best stock by orders of magnitude.