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.)

Second, all these conclusions assume that there is an ideal world of uncorrelated assets with equal expected returns. I would argue that such conditions cannot actually exist. So take these conclusions with a grain of salt.

Without further ado, here are the mathematical conclusions:

1. The expected return of a continually rebalanced portfolio is equal to the weighted geometric mean of the expected returns of the individual assets, while the actual return of a buy-and-hold portfolio is by definition the weighted arithmetic mean of the expected returns. Since the arithmetic mean will always be greater than or equal to the geometric mean, the expected return of a buy-and-hold portfolio will always exceed or be equal to the expected return of a continually rebalanced portfolio, but the differences between them will not be large.

2. If the expected return of the various assets is more or less the same, the *median *result of a rebalanced portfolio will usually exceed the median result of a buy-and-hold portfolio.

Taking these two points together, the *median return* of a rebalanced portfolio with equal expected returns is higher but the *expected return* is lower. The reason for the difference is that in the relatively few cases that the buy-and-hold portfolio beats the rebalanced portfolio, it can *crush *it, thereby making its expected return over all cases somewhat higher.

3. If the expected return of the various assets is quite different, a buy-and-hold portfolio will usually beat a rebalanced portfolio no matter what measure you use.

4. For relatively *short* time periods and with similar expected returns, it is quite likely that a rebalanced portfolio will beat a buy-and-hold portfolio. With two assets, the probability is about two-thirds, and decreases as you add more assets, but remains above one-half. (Once again, in the cases that the buy-and-hold portfolio beats the rebalanced portfolio, it is likely to beat it by a large margin.)

5.* Ex-post* (after the fact), rebalancing *always* beats buy-and-hold if the *total returns* of the constituent assets are exactly the same. In other words, if you have five assets that all end up with a 25% total return over four years but have very different sequences of returns, it’s always going to be better to rebalance them than to simply buy and hold. *Ex-ante* (before the fact), however, buy-and-hold can beat rebalancing (sometimes) even if the *expected returns *are exactly the same.

6. As the time period stretches toward infinity, if the difference between the expected returns of the assets is small (less than half of their standard deviation), the likelihood that rebalancing will beat buy and hold approaches a certainty; and if the difference between the expected returns of the assets is larger than that, the likelihood that buy and hold will win approaches a certainty. In other words, if you’re going to hold two or more assets for a very, very long time and you have no idea which will perform better, it may be better to rebalance; if, however, you are reasonably certain that one has better or worse expected returns, it’s better to just buy them and not do anything further. (The first part of this point clearly contradicts point 1 above, and I do not know how A. J. Wise would resolve this contradiction.)

None of these conclusions, which are purely mathematical, take into account *transaction costs*, which do not pertain to a buy-and-hold portfolio, but may be significant in a rebalanced portfolio. None of them take into account *risk* either, which, if you consider variability a good measure, is significantly greater with a buy-and-hold portfolio. And none of them take into account *correlation of returns*: rebalancing provides a significant diversification benefit by putting more money into uncorrelated assets.

### Part II: The Reality

So let’s look at some real-life examples. I have chosen, at random, fifty stocks and fifty ETFs, and have put into a database their monthly returns over the last sixteen years. (For convenience, I am only using stocks and ETFs that have survived over the entire sixteen-year-period with beginning and ending prices greater than $1.00 per share.) In the case of the stocks, I am starting with the assumption that the expected returns of the various stocks are more or less equal (in my own opinion, this is nonsense, but many investors think that stock picking can be done by monkeys); in the case of the ETFs, which include some bond and leveraged and short ETFs along with normal equity ETFs, most people would admit that the expected returns were quite unequal.

I am going to choose random portfolios of five stocks and five ETFs (four equity and one bond-based) and measure their performances over four four-year periods (2002 to 2006, 2006 to 2010, 2010 to 2014, 2014 to 2018), both with monthly rebalancing to equal weight and without. I will repeat this experiment two hundred times so that we have eight hundred pieces of data for each asset class. I think this should be sufficient for a general illustration of the advantages and disadvantages of both approaches.

Here are the results:

For the stocks, there’s a 52% chance that rebalancing will improve returns. The average rebalanced return is 4% better (unannualized, over a four-year period) than the average buy-and-hold return, and the median rebalanced return is 6% better. Rebalancing definitely seems to improve returns, *as long as you can keep its cost under 1% per year*. This differs from the results you would get by using a purely mathematical approach, and I believe that’s because rebalancing provides an additional benefit that the purely mathematical approach doesn’t take into account: when you rebalance, you automatically increase your allocation to the less correlated assets.

The portfolio of five ETFs is chosen as follows: four of the ETFs are chosen randomly from a set of forty stock ETFs, and one is chosen randomly from a set of ten bond ETFs. Because bonds have generally underperformed stocks throughout history, the bond ETFs have a lower expected return than the equity ETFs, and the results reflect that difference. With this portfolio, there’s only a 50% chance that rebalancing will improve returns. The average rebalanced portfolio has a lower return by 1.7% than the average buy-and-hold portfolio, and the median is 0.7% lower. Of course, if you add rebalancing costs, the results will favor the buy-and-hold approach even more strongly. The math shows, and the simulation confirms, that with different expected returns, it’s better not to rebalance if you want maximal returns.

### Part III: Conclusions

Most investment advisers tell their clients to rebalance regularly. Those who work for brokerages have a vested interest in doing so. I take a more complex approach, rebalancing according to a ranking system. But, as Wise puts it, “Which strategy should be preferred by any particular investor will depend upon the utility attached to the various potential outcomes. Some will prefer a better-than-even chance of small performance gains over a long period. Others may prefer the return profile of the passive strategy.”

My recommendation is as follows: If you have no idea which assets are going to outperform, and if you are not going to hold them for very long, rebalance. (That is my own situation.) If the standard deviation of your returns is of great concern to you (for example, if you use leverage or are making regular withdrawals), rebalance. If you intend to hold your assets for a very long time and you’re reasonably confident that their total returns will be very similar, rebalance.

But if you expect some of your assets to outperform or underperform others, or if rebalancing incurs substantial transaction costs, don’t bother rebalancing. By rebalancing, you are likely to lower your expected return.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.