Information-Theoretic Approaches to Portfolio Selection

Abstract

Ever since modern portfolio theory was introduced by Harry Markowitz in 1952, a plethora of papers have been written on the mean-variance investment problem. However, due to the non-Gaussian nature of asset returns, the mean and variance statistics are insufficient to adequately represent their full distribution, which depends on higher moments too. Higher-moment portfolio selection is however more complex; a smaller literature has been dedicated to this problem and no consensus emerges about how investors should allocate their wealth when higher moments cannot be ignored. Among the proposed alternatives, researchers have recently considered information theory, and entropy in particular, as a new framework to tackle this problem. Entropy provides an appealing criterion as it measures the amount of randomness embedded in a random variable from the shape of its density function, thus accounting for all moments. The application of information theory to portfolio selection is however nascent and much remains to explore. Therefore, in this thesis, we aim to explore the portfolio-selection problem from an information-theoretic angle, accounting for higher moments. We review the relevant literature and mathematical concepts in Chapter 1. Then, we consider in Chapter 2 a natural alternative to the popular minimum-variance portfolio strategy using Renyi entropy as information-theoretic criterion. We show that the exponential Renyi entropy fulfills natural properties as a risk measure. However, although Renyi entropy has some nice features, we show that it can be an undesirable investment criterion because it may lead to portfolios with worse higher moments than minimizing the variance. For this reason, we turn in chapters 3 to 5 to different ways of applying entropy, thereby revisiting two popular frameworks -- risk parity and expected utility -- to account for higher moments. In Chapter 3, we investigate the factor-risk-parity portfolio -- a popular strategy among practitioners -- that aims to diversify the portfolio-return risk across uncorrelated factors underlying the asset returns. We show that although principal component analysis (PCA) is very useful for dimension reduction, its resulting factor-risk-parity portfolio is suboptimal. Indeed, PCA merely provides one choice of uncorrelated factors out of infinitely many others, and one would prefer to be diversified over independent factors rather than merely uncorrelated ones. Instead, thus, we propose to diversify the risk across maximally independent factors, provided by independent component analysis (ICA). We show theoretically that this solves the issues related to principal components and provides a natural way of reducing the kurtosis of portfolio returns. In Chapter 4, we apply ICA in a different way in order to obtain robust estimates of moment-based portfolios, such as those based on expected utility. It is well known that these portfolios are difficult to estimate, particularly in high dimensions, because the number of comoments quickly explodes with the number of assets. We propose to address this curse of dimensionality by projecting the asset returns on a small set of maximally independent factors provided by ICA, and neglecting their remaining dependence. In doing so, we obtain sparse approximations of the comoment tensors of asset returns. This drastically decreases the dimensionality of the problem and leads to well-performing and computationally efficient investment strategies with low turnover. In Chapter 5, we introduce an alternative approach to the utility function to capture investors' preferences. The latter is praised by academics but is difficult to specify when higher moments matter. Because investors ultimately care about the distribution of their portfolio returns, our proposal is to capture their preferences via a target-return distribution. The optimal portfolio is then the one whose distribution minimizes the Kullback-Leibler divergence with respect to the target distribution. Our theoretical exploration shows that Shannon entropy plays a central role as higher-moment criterion in this framework, and our empirical analysis confirms that this strategy outperforms mean-variance portfolios out of sample.