Optimal portfolio diversification using the maximum entropy principle

□ Markowitz's mean-variance (MV) efficient portfolio selection is one of the most widely used approaches in solving portfolio diversification problem. However, contrary to the notion of diversification, MV approach often leads to portfolios highly concentrated on a few assets. Also, this method leads to poor out-of-sample performances. Entropy is a well-known measure of diversity and also has a shrinkage interpretation. In this article, we propose to use cross- entropy measure as the objective function with side conditions coming from the mean and variance-covariance matrix of the resampled asset returns. This automatically captures the degree of imprecision of input estimates. Our approach can be viewed as a shrinkage estimation of portfolio weights (probabilities) which are shrunk towards the predetermined portfolio, for example, equally weighted portfolio or minimum variance portfolio. Our procedure is illustrated with an application to the international equity indexes.