Whether you hold only stocks or a wide variety of assets, a question you’ll be forced to face sooner or later is whether you should rebalance to a specific preset weight periodically or just let some positions get bigger and others smaller. This article aims to help you make that choice wisely.

Part I: The Math

In 1996, A. J. Wise published an article in the British Actuarial Journal called “The Investment Return from a Portfolio with a Dynamic Rebalancing Policy.” It is far and away the most comprehensive and mathematically sound analysis ever written on the probabilities that periodic rebalancing will outperform leaving your portfolio alone (buy and hold), and by how much. Unfortunately, it wasn’t widely read. Twenty years later, Michael Edesess published an article in Advisor Perspectives called “The Academic Failure to Understand Rebalancing,” which provided a valuable overview of the academic literature on the subject and brought Wise’s article back to the forefront. Without a thorough knowledge of higher mathematics, Wise’s article is difficult to follow, but Edesess’s article is straightforward and compelling reading. I highly recommend the first to anyone who wants a thorough grounding in the mathematics behind rebalancing, and the second to anyone with an interest in the subject. The purpose of my article is to summarize their conclusions and make a few observations of my own.

First, here’s the basic math behind rebalancing and not rebalancing. Let’s express an asset’s return over a set period as its price at the end of the period (adjusted for dividends and splits) divided by its price at the beginning. (Usually we express returns by doing this division and then subtracting one; I want to skip that last part for the sake of this illustration.) Let’s say we have a finite number of these period returns for a finite number of assets. Then the total buy-and-hold return would be the average of the products of each asset’s returns. And the total rebalanced return would be the product of the average of each asset’s returns in each period.

To illustrate, let’s say we were holding four stocks equally with the following returns:

So the basic mathematical problem is:

when is the product of averages greater than the average of products?

This is a very thorny problem indeed, without any easy solution. But we can make some very basic conclusions.

First, however, I must introduce a very fundamental concept. The expected return of an asset or portfolio is the probability-weighted average of all possible outcomes. This is based on unannualized total returns. If you annualize the returns, you end up with very different—and very flawed—results. Averaging annualized returns makes no sense—you’re combining a geometric mean (annualization) with an arithmetic one. (In a cogent 2014 article, “Does Rebalancing Really Pay Off?,” Michael Edesess points out this and other fundamental flaws in William Bernstein’s 1996 article “The Rebalancing Bonus,” an article which has had a significant and unfortunate impact on financial advisors’ advice. Basically, advisors have been using Bernstein’s and others’ academic articles to advocate rebalancing portfolios of all types without regard to the serious flaws in those articles; some of these advisors have a vested interest in giving this advice, since rebalancing costs are not negligible, and brokers make significant profits by implementing the practice.)