The Shape of Rebalancing: Why Some Studies Don’t Find a Rebalancing Benefit

The great age of rebalancing is upon us, and yet many people will ignore this superpower. They will point out some studies claim rebalancing doesn’t help. Those studies are flawed, and I’m going to show you why.

By exploring further how rebalancing works, you will see why prior studies on rebalancing may have missed its benefits. By examining more about where and why rebalancing helps, you will also learn how to construct better portfolios yourself in order to capture this benefit.

I prefer to understand these problems through the lens of pure randomness first, so for our exploration, we’re going to take Shannon’s Demon, and “expand” it through time and space.

Rebalancing the Demon at Different Frequencies

Shannon’s Demon is theoretically rebalanced each “round”. But what happens if we expand it through time? What if we don’t rebalance every round and instead rebalance less often?

Using the same values from the prior post modeled as a coin flip we will rebalance back to 50% cash/ 50% coin flip with this payout:

• Heads, up 50%
• Tails, down 33.33%

Here the arithmetic return is 8.33%, and the geometric return is zero. Rebalanced over 100,000 rounds, the following chart shows the compound growth rate per round in terms of rebalancing period frequency.

Shannon’s Demon is the top left point, but you can see there is a lot more here to process than just one point. Less frequent rebalancing (longer periods) leads to lower compound return. Furthermore, it’s not a linear relationship. It’s convex. The shorter the frequency, the greater the additional benefit.

The Shape Matches What Randomness Predicts

The shape of this curve, the way it curves flatter as the rebalancing period decreases isn’t a coincidence. It’s exactly what rebalancing compounded random returns should create.

This is of course not strange, since Shannon’s Demon is nothing but pure randomness rebalanced. The true math for Shannon’s Demon is a bit more complicated than what’s shown in this chart (see notes at the bottom for deeper math and chart¹), but this convex curve from the higher arithmetic return towards the lower geometric return is the common shape of compounded randomness.

I’ve discussed this shape multiple times before. I first brought it up in my initial post on the geometric return, then again in the Market Index post, then expanded on it by discussing how to determine the compound random returns of a simple portfolio through time, and finally discussed calculating the expected return of a single asset through time when short selling.

The expected benefit of rebalancing isn’t random. It’s complicated but calculatable. The natural progression follows this similar shape, a convex curve downward from the arithmetic return to the geometric return.

Same Shape Found With Real Investments

It should be no surprise then that real investments match this shape too.

In my post on Why Market Investing Works, I discussed how frequent rebalancing of the 30 Dow components leads to higher returns. In an effort to streamline that post, I removed an interesting chart, which I will bring back now.

This chart shows the returns from rebalancing the 30 Dow Jones Industrial Average components at different periods over 70 years from 1940-2010.² The returns display the same shape as we saw with Shannon’s Demon, and the same shape projected by geometric randomness.

The rebalancing benefit is real. You can see it appearing clearly in real investments, closely following the pattern randomness prescribes it to follow.

What Can We Learn From This?

There are two very interesting points to notice here³.

#1: The Rebalancing Benefit is Convex to Frequency

The benefit from rebalancing every week vs. every month is higher than the benefit from rebalancing every 11 months vs. every year. You can see that the benefit of moving from monthly rebalancing (21 days) to weekly rebalancing (5 days), is worth more than the benefit of moving from one year (250 days) to one quarter (63 days). This is very important to understand since:

2#: The Data Follows a Pattern, But It’s Still Random

It’s easy to see the data does have a definable shape, which is predictable. But it’s still random especially with less frequent rebalancing. This randomness means while you should find a small benefit from rebalancing semi-annually vs. annually, the randomness in the data may cloud the benefit. But understand, the math says the benefit is still there, it just may take some time to show itself.

So if you see a study that looks at the impact of rebalancing, but only does so over longer term intervals, just ignore it. It’s never going to cut through the noise with enough benefit to show a statistically relevant result.

The real benefit from rebalancing comes at higher frequencies, and there it should cut through the noise to show itself.

Lessons For Your Own Portfolio

What can we take from this to our own portfolio construction?

• The “rebalancing premium” is real, and it can be formulated and predicted.
• If you can do so in a cost effective manner, think about rebalancing to the proper portfolio ratio’s more frequently. However, always keep in mind the trading cost because even though they are much lower now than they used to be, they still exist and can drain your returns if ignored (and taxes too).

The proper ratio part is key, though, to both your portfolio and to understanding the effectiveness of rebalancing. Let’s explore why.

Shannon’s Assumption

Some have characterized Shannon’s Demon as investing with no prior assumptions about the future. This is definitely not true as there is an enormous assumption built into Shannon’s Demon: The 50/50 rebalancing ratio.

50/50 is the correct ratio here since the the risky asset in question has no geometric return. Think back to our post on balancing a portfolio.

When the geometric return matches the return of cash, the risk free rate falls right in the middle of the mixing range. Therefore the proper rebalancing ratio is 50% risky / asset 50% cash.

Shannon perfectly matched up the rebalancing ratio to the maximum return for the game. In the real world we can’t ensure 50/50 is the correct ratio. It might be, it might not be. So what happens if we get it wrong? What happens if we rebalance the same game to different ratios?

We’ve expanded Shannon’s Demon through time, now let’s expand it through “space”.

The Demon at Different Rebalancing Ratios

Here is the same 100,000 rounds of the 50% up, 33.33% down coin flip, but this time rebalanced back to different coin/cash ratios.

You can see that the 50/50 rebalancing ratio does perform the best, as expected. Moving from 50/50 to 75/25 or 25/75 reduces the compound growth rate of the portfolio at all rebalancing frequencies. Moving to 10/90 and 90/10 reduces the returns even further.

The best returns come from rebalancing at the proper ratio (50/50) and as often as possible.

Here are the charts individually, with my formula from the footnotes¹ included to show that this shape is predictable.

Click through to see each of the 5 charts.

Rebalancing Surface

Since we have a formula which shows the relationship between expected compound growth rate, rebalancing ratio, and rebalancing frequency, we can create a 3 dimensional surface of the returns from those variables.

Click the slideshow to rotate the surface

Here we see the full gamut of possibilities. The peak (Kelly peak) at the proper rebalancing ratio, the convexity of the returns through rebalancing frequency, and if you look closely, less frequent rebalancing increasing returns at ratios away from 50/50.

Wait what? Rebalancing helps returns at some ratios, and hurts returns at others?

Yes it does.

Rebalancing Hurts Some Portfolios

I want to zoom in on the 10/90 portfolio specifically here.

Notice how early on rebalancing less frequently improves returns. This isn’t random. The predictive formula says this should happen. The random results nearly exactly track the prediction. It’s expected at a rebalancing ratio this far from optimal to see less frequent rebalancing help returns.

Rebalancing Helps at the Correct Frequency

You only receive the peak return when you rebalance back to the preferred portfolio. This is very easy to see in a geometric return chart (similar to those created in this post).

You can see that investing at 10% of the coin flip only leads to a 0.747% geometric return. This is far less than what’s possible by investing at the peak of 50% / 50%. This portfolio also behaves very differently through time than a portfolio constructed at the peak.

What Time Does to The 10/90 Portfolio

Any change to a portfolio created at the peak always moves it away from the peak. It’s downhill in either direction. Natural volatility in returns always makes the portfolio worse.

But what happens when we start a portfolio at 10/90, and then let it sit? Well, since the coin flip has a positive arithmetic return it’s likely going to grow a bit. After one round the portfolio is either 6.9%/93.1% or 14.3%/85.7%, averaging 10.6%/89.4%. After two rounds the average is 11.2%/88.8%. Because this portfolio is conservative, time on average moves the portfolio toward a better ratio for maximizing the geometric return.

When you let the portfolio sit, it moves up this curve and becomes more optimal through time. So by NOT rebalancing, you are likely letting the portfolio become more optimally balanced for maximizing return. Every time you rebalance early you likely make the portfolio worse for returns.

Now after a long enough time, the many possible portfolios will have either:

• passed far over the peak of the curve
• or fallen backwards down the curve

This is why the benefit rolls over after 27 periods in our example and begins to fall back downward. The original 10/90 portfolio is a better portfolio than most portfolios after 28 rounds, therefore you should rebalance back to it. But before those 28 rounds, the portfolios are probably going to provide a better return than 10/90.

Rebalancing Still Improves Return/Risk

Now while frequent rebalancing of this portfolio doesn’t help returns, it does help to control risk. After 27 rounds, all those portfolios on the right out near 100% are nearly 10 times as volatile as the initial 10/90 portfolio. The average portfolio after 27 rounds is 2.8 times as volatile as the start.

So rebalancing always helps control risk in this example. Interestingly, the highest compound return at each specific volatility level on the surface is found by rebalancing every period. We’ll explore this further in later posts.

Does the Study Rebalance at an Appropriate Ratio?

This example is similar to a 60/40 portfolio of bonds and stocks⁴ and many other traditional portfolios. Many studies take a portfolio of equal stocks and bonds, or 60/40 stocks and rebalance them. They don’t find rebalancing necessarily helps returns. However this makes sense as these portfolios aren’t anywhere near optimal from rebalancing perspective. Rebalancing shouldn’t necessarily help returns.

Mathematically these studies find exactly what they should find.

Now hopefully you can see why.

Rebalancing does work when you rebalance back to an appropriate ratio. If the study doesn’t do this, it’s not really testing the benefit of rebalancing in terms of returns, it’s testing the benefit of rebalancing to a specific portfolio. Rebalancing does not increase returns in every portfolio composition.

Rebalancing Works When You Rebalance to the Proper Portfolio

This is a very simple example of how rebalancing works. Maybe the simplest example you could develop. Real investments don’t have binary heads/tails outcomes. Their distributions are more complicated, and therefore the math of rebalancing is also more complicated.⁵ But the concepts and the relationships are the same.

I hope you see how rebalancing isn’t a mindless exercise. You can’t just rebalance back to any portfolio and expect it to help. However the math says if you rebalance back to the correct portfolio, rebalancing provides a higher compound return with enough repetitions.

It might take time. Returns are random, so there is certainly no guarantee rebalancing provides the best return in the near future (there’s no guarantee it ever will). But rebalancing to the correct portfolio does provide the investor with the highest expected compound return. Furthermore, rebalancing frequently to any portfolio left of the peak provides an investor with a better return per “risk”.

Remember the Shape of Rebalancing

If you see a study that claims to show rebalancing doesn’t work, ask yourself:

• Does the study have enough data to smooth out the randomness?
• Does the study rebalance frequently enough to show a benefit?
• Does the study rebalance to the right ratio, or even acknowledge the ratio is important?

If the answer isn’t yes to all 3, the study might be incomplete.

Finally, when it comes to developing your own rebalancing strategies, keep the entire rebalancing surface in mind. You don’t have to rebalance to the optimal peak portfolio (I don’t), but the rebalancing surface helps you think about what portfolio to rebalance too and how frequently to rebalance that portfolio.

Footnotes:

1-This is my derivation of the formula for the expected compound return from a coin flip random game rebalanced with cash at a certain period “F”. This formula is a “bit” more complicated than other formulas I’ve shown for geometric returns over a period, and I didn’t want anyone’s eyes to gloss over it in the body of this post. Understanding this formula is not key to understanding the shape of rebalancing. As usual, there could be a mistake here, so for the mathematically talented, please let me know if you see anything that looks off, or if there is a way to “compress” this.

Here is that formula applied to the random results created from the “Shannon’s Demon” example above. You can see it tracks nearly perfectly. I had to stop it at a rebalancing frequency of 170 days because the factorials at that level exceeded my computer’s calculation capacity.

Key point, there is a shape of expectations here created by randomness which is not random. Therefore there might be ways to predict, and optimize, rebalancing random returns for your desired goals.

Here’s the same formula converted to use arithmetic return and standard deviation. It’s going to produce the same results for coin flips but I’m not sure it’s correct to apply to other distributions. At a minimum I think all the factorial “pascal’s triangle” part in the top right is only for binomial distributions.

2-Raw data Here. Part of the reason this worked was the Dow stocks have similar expected returns and standard deviations. We obviously know in hindsight this is reasonable, but I think it’s a good assumption for most stocks, especially large cap stocks. This means equal weight (which this example used) is the proper rebalancing ratio.

3-There is also a very clear third takeaway from that chart, but it is ancillary to the point of this post, so I’ll just drop it down here in the footnotes. The fact that the geometric return curves like this is not surprising. The fact that the arithmetic return also curves is VERY surprising. I’ve shared this chart with some people and they all thought I made a calculation mistake. The data is here though, and you can recreate it yourself. Others found the same thing. You can get the Shannon’s Demon example to do this for some starting portfolios. The reason why this happens is extremely interesting, leading to really meaningful discoveries on market behavior. But that’s a topic for another time.

4-An interesting comparison: In the coinflip example, all portfolios between 21% and 79% should be rebalanced at all periods for more returns. This seems to generally hold for different returns as well (holding geometric return at 0). The peak of this surface is full Kelly. So 21% is a little under half Kelly. A 60/40 portfolio could historically be leveraged around 4 times to maximize returns, aka quarter Kelly. The 10/90 portfolio in the example is at fifth Kelly. So a 60/40 portfolio’s rebalancing surface is actually somewhat similar to the 10/90 surface in general shape (although take nothing from the example’s proper holding period of 27 rounds as that’s a function of volatility).

5-This example has high volatility. The volatility and the binary outcome helps the charts to “pop” and makes it easy to see the relationship. Lower volatilities still create the same shape, but the slopes are tamer.