On asymptotic log-optimal portfolio optimization

Abstract

In this paper, we consider a frequency-dependent portfolio optimization problem with multiple assets using a control-theoretic approach. The expected logarithmic growth (ELG) rate of wealth is used as the objective performance metric. It is known that if the portfolio contains a special asset, the so-called dominant asset, then the optimal ELG level is achieved by investing all available funds in that asset. However, this “all-in” strategy is arguably too risky to implement. As a result, we study the case where the portfolio weights are chosen in a rather ad-hoc manner, and a linear buy-and-hold strategy is subsequently used. We show that if the underlying portfolio contains a dominant asset, buy and hold on that specific asset is asymptotically log-optimal with a logarithmic convergence rate. This result also extends to the scenario when a trader does not have a probabilistic model for returns or does not trust a model based on historical data. Specifically, we prove a version of the one fund theorem, which states that if a market contains a dominant asset, buying and holding a market portfolio with nonzero weights for each asset is asymptotically log-optimal. Additionally, we extend an existing result regarding the property called high-frequency maximality of an ELG-based portfolio from a single asset to a multi-asset portfolio case. This means that, in the absence of transaction costs, high-frequency rebalancing is unbeatable in terms of ELG. This result enables us to further improve the log-optimality obtained previously. Finally, we provide a result on the issue of how often a portfolio should be rebalanced, if needed. Examples using simulations with high-frequency historical trading data are included throughout to illustrate the theory.