Rebalancing Frequency Considerations for Kelly-Optimal Stock Portfolios in a Control-Theoretic Framework

Abstract

In this paper, motivated by the celebrated work of Kelly, we consider the problem of portfolio weight selection to maximize expected logarithmic growth. Going beyond existing literature, our focal point here is the rebalancing frequency which we include as an additional parameter in our analysis. The problem is first set in a control-theoretic framework, and then, the main question we address is as follows: In the absence of transaction costs, does high-frequency trading always lead to the best performance? Related to this is our prior work on betting, also in the Kelly context, which examines the impact of making a wager and letting it ride. Our results on betting frequency can be interpreted in the context of weight selection for a two-asset portfolio consisting of one risky asset and one riskless asset. With regard to the question above, our prior results indicate that it is often the case that there are no performance benefits associated with high-frequency trading. In the present paper, we generalize the analysis to portfolios with multiple risky assets. We show that if there is an asset satisfying a new condition which we call dominance, then an optimal portfolio consists of this asset alone; i.e., the trader has "all eggs in one basket" and performance becomes a constant function of rebalancing frequency. Said another way, the problem of rebalancing is rendered moot. The paper also includes simulations which address practical considerations associated with real stock prices and the dominant asset condition.

The Impact of Execution Delay on Kelly-Based Stock Trading: High-Frequency Versus Buy and Hold

“…The reader is also referred to [9] for a rather comprehensive exposition covering many aspects of the theory. This paper is most closely related to more recent work such as [10]- [13] which provide results on the effect of rebalancing frequency on optimal trading performance. Specifically, in [12] and [13], it is shown that in a so-called idealized market with a stock satisfying a certain "sufficient attractiveness" condition, the buy and holder can match the performance of the high-frequency trader.…”

On Asymptotic Log-Optimal Buy-and-Hold Strategy

“…Consistent with the existing work in [11,13], for any rebalancing period n ≥ 1, we study the problem of maximizing the expected logarithmic growth…”

“…which is classical in finance; e.g., see [4,5,13,17]. That is, with K ∈ K, we have a guarantee that 100% of the account is invested.…”

“…In this subsection, we review two technical notions called relative attractiveness and dominant asset that are useful for the later development. Indeed, according to [9,13], these two notions enable us to state a result called the Dominate Asset Theorem, which tells us that if a portfolio contains a dominant asset, then a log-optimal trader should invest all available funds on that asset. Definition 2.1 (Relative Attractiveness and Dominant Asset).…”

“…Compared with many of the recent papers that contributed to the ELG maximization problem and its application to stock trading, the effects of rebalancing frequency are arguably not heavily considered into the existing literature. Some initial results along this line regarding rebalancing frequency effects can be found in [6,7,9,11,13,15]. Specifically, in [15], a portfolio optimization with returns following a continuous geometric Brownian motion was considered.…”