Chung-Han Hsieh

A log-optimal portfolio is any portfolio that maximizes the expected logarithmic growth (ELG) of an investor's wealth, which typically assumes prior knowledge of the true return distribution. However, in practice, return distributions are often ambiguous; i.e., the true distribution is unknown, making this problem challenging to solve. This paper proposes a supporting hyperplane approximation approach, reformulating a class of distributional robust log-optimal portfolio problems with polyhedron ambiguity sets into tractable robust linear programs. An efficient algorithm is presented to determine the optimal number of hyperplanes. Additionally, to adapt to the constantly changing market, we propose an online trading algorithm using a sliding window approach to solve a sequence of robust linear programs, offering significant computational advantages. The effectiveness of the proposed approach is supported by empirical studies using historical stock price data.

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.