Analysis of the rebalancing frequency in log-optimal portfolio selection

Abstract

In a dynamic investment situation, the right timing of portfolio revisions and adjustments is essential to sustain long-term growth. A high rebalancing frequency reduces the portfolio performance in the presence of transaction costs, whereas a low rebalancing frequency entails a static investment strategy that hardly reacts to changing market conditions. This article studies a family of portfolio problems in a Black–Scholes type economy which depend parametrically on the rebalancing frequency. As an objective criterion we use log-utility, which has strong theoretical appeal and represents a natural choice if the primary goal is long-term performance. We argue that continuous rebalancing only slightly outperforms discrete rebalancing if there are no transaction costs and if the rebalancing intervals are shorter than about one year. Our analysis also reveals that diversification has a dual effect on the mean and variance of the portfolio growth rate as well as on their sensitivities with respect to the rebalancing frequency.

Keywords:

• Portfolio selection
• Log utility
• Growth-optimal portfolio
• Rebalancing frequency
• Kelly criterion

Acknowledgement

Daniel Kuhn thanks the Swiss National Science Foundation for financial support.

Notes

†In addition, the use of the ‘max’-operators in the proposition statement is justified.

†Short selling is possible, however, if the rebalancing dates are not predetermined but may depend on the realized asset price paths.

†Notice that the maximization problems 𝒫(τ) and 𝒫′(τ), τ ≥ 0, have the same optimal value and (essentially) the same solution. However, the variance of the optimal portfolio's growth rate over unit time can only be calculated from the objective function of problem 𝒫(τ).

†In fact, the portfolio growth rate does not saturate before τ ≈ 200 years. The saturation regime is outside the range of figure 2 as rebalancing periods longer than a few years are of minor interest.

†For small values of k the ψ _i,k are found by expanding ∂ _{w i} ψ(w*(λ), λ) in powers of λ.