Unexpected Returns: Shannon’s Demon & the Rebalancing Bonus

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One of the core assumptions baked into all of the Portfolio Charts calculations is the idea that the portfolios are rebalanced once a year back to their target percentages. While that simple process seems rather mundane on the surface, there’s actually a bit of mathematical magic going on that often gets lost in broader portfolio discussions. Yes, maintaining your target asset allocation is an important part of risk management, but it goes so much deeper than that. What if I told you that, like a lonely plant in a barren desert, in the right conditions rebalancing can cause profits to seemingly appear out of nowhere?

That peculiar phenomenon can clearly have a major impact on the way people think about diversification in their investments. So let’s unpack the mystery and talk about the elusive rebalancing bonus.

The devil in the details


The prospect of generating something from nothing may sound way too good to be true, but it’s not some crackpot theory. In fact, it was clearly demonstrated back in the 1940’s by one of the greatest geniuses of our time, a man named Claude Shannon. If you’re not familiar with Shannon and have two minutes to spare, I highly recommend this short trailer for a movie about his life.

Shannon was a brilliant electrical engineer who worked at Bell Labs, taught at MIT, and effectively invented digital circuit design. Often referred to as “the father of information theory”, his work was the foundation that allowed the digital device you’re reading this on to exist. His impact on the modern world cannot be overstated.

A mind like Shannon’s was far too curious to be bound to a single subject, and one of his most interesting lesser-known observations involved the field of investing.

Imagine a theoretical investment that perfectly oscillates every year between a 50% gain and a 33.3% loss. For the purposes of this example, let’s also ignore inflation to keep things simple. If you map the compounded returns over time the chart would look something like the one below. By immediately giving back every gain, the investment is extremely volatile but tracks sideways over time with an expected long-term return of zero.

Next, picture an alternative investment of true paper cash that pays no interest at all. Option B also tracks sideways with no expected return.

If one were to invest in both option A and B and let them each do their thing with no interference, the long term return is clearly zero. But what if you allocated 50% of your money in each and rebalanced your portfolio every year? That’s when things start to get interesting.

A visual representation of Shannon's Demon and the rebalancing bonus

Even though the two underlying investments are assets with no expected return, the simple act of rebalancing between them generated significant portfolio growth equivalent to roughly 2% a year!

This mystery source of returns resulting from the act of rebalancing rather than the inherent returns of the assets alone is known as Shannon’s Demon. The name is dual nod to the man who first uncovered it and a similar thought problem in thermodynamics where an unseen force creates unexpected outcomes. But as spooky as Shannon’s Demon may seem, make no mistake — the end result is very real.

How the rebalancing bonus works


Luckily, while the outcome may be unintuitive the source of the unexpected return is mathematically very simple. So even if math isn’t your strong suit, bear with me. You’ll be smarter for the effort!

Let’s turn back to our original Investment A with alternating 50% gains and 33.3% losses. While the compounded expected return is zero, the average return is decidedly positive.

Average return: (50 + -33.3)/2 = 8.4%

So where did all of those gains go? They were eroded away by the volatility. The standard deviation of the returns in the charted series is a very high 42.1%, and due to the influence of volatility drag the actual compound return you experience in real life is always less than the average. The compound annual growth rate (CAGR) can be easily approximated from the average (AVG) and standard deviation (SD) like this:

CAGR ~ AVG – (SD² / 2)

0.084 – (0.421² / 2)

CAGR ~ 0%

Note: I use ~ instead of = to indicate that it’s only an approximation. But it should be pretty close without going overboard with the calculations.

The most important thing to note is that while the arithmetic average return is clearly the starting point of the geometric compound return, it is reduced by the square of the standard deviation. That means that the volatility has a much greater effect than you might normally think, and it always reduces the return you experience in real life. And it’s that volatility reduction that unlocks the mystery of the rebalancing bonus.

When you regularly rebalance 50% of Asset A with 50% cash in the above example, both the average return and standard deviation are cut in half to 4.2% and 21%, respectively. So let’s re-run the numbers for the rebalanced portfolio:

CAGR ~ AVG – (SD² / 2)

0.042- (0.21² / 2)

CAGR ~ 2%

Remember our earlier 2% discrepancy? It looks like we found our demon.

Because of the square in the equation, halving the volatility has a greater positive effect on the compound return than the negative effect of halving the average. The end result is a positive return from two assets that, on their own, have zero expected return. And the outcome is only possible through the act of periodic rebalancing.

You know when people talk about the benefits of buying low and selling high? Shannon’s Demon is just a mathematical proof that it works even in a simple regularly-rebalanced index portfolio. It’s not so much about “locking in profits” in a trading mindset but about harnessing that clever demon to increase compound returns by reducing volatility drag. Or put another way, the rebalancing bonus comes from playing multiple assets off of each other to make money more consistently.

Where’s my free money?


Of course, the classic explanation of Shannon’s Demon is admittedly pretty idealized. For example, no normal investment oscillates so perfectly or avoids the effects of inflation over time. So let’s take a look at a more practical example by studying real-world assets instead of theoretical numbers.

This shows the inflation-adjusted returns of US stocks (large cap blend) and interest-bearing cash (T-bills) from the beginning of 1970 all the way through the end of 2021. The 50/50 line tracks a regularly rebalanced combination of both.

As you can see, the best portfolio over this timeframe was… (checks notes)… the one with 100% stocks. So what happened to our free money? Was it all just smoke and mirrors after all? Actually, the rebalancing bonus is absolutely still there. You just have to look deeper.

From the beginning of 1970 through the end of 2021, the performance of the 3 different portfolios looked like this.

StocksT-Bills50/50
AVG8.1%1.0%4.5%
SD17%2.7%8.7%
CAGR6.7%0.9%4.2%
For bonus points, calculate the theoretical CAGR on your own from the AVG and SD. It should be within about 0.1% due to simple rounding error.

Pay close attention to the CAGR numbers. If you take the weighted average CAGR of stocks and T-Bills, the return we’d expect to see of a 50/50 combination of both is 3.8% (the average of the two numbers). But the actual CAGR was 4.2%. That’s a difference of 0.4% a year, or 11% more than the theoretical compound return for the 50/50 portfolio when you only look at the assets in isolation. That’s Shannon’s Demon at work! It’s just hidden in the shadows of the superior stock returns.

As a reminder, here’s the equation we discussed for the annualized compound returns:

CAGR ~ AVG – (SD² / 2)

Both increasing the average return of the portfolio and decreasing the volatility increases the compound return. However, it’s a balancing act. Sometimes the average dominates (such the above example showing the very long-term track record of stocks), and in that case the rebalancing bonus may get lost in the shuffle.

But other times it’s a little more obvious. Take for example the same assets during the 13-year investing timeframe starting in 2000. In that particularly poor timeframe for US stocks, the real CAGR was a dismal -0.8% per year. The CAGR for T-Bills was a slightly better -0.1%. And the regularly rebalanced 50/50 split of the two had a real CAGR of +0.1%. That’s not huge by any means, but it’s a good example of how rebalancing can serve to generate portfolio returns greater than either asset in isolation.

The lesson here is that while the rebalancing bonus is very real, rebalancing between multiple assets does NOT guarantee a positive premium over investing in the best performing asset on its own. Sometimes it will, and sometimes it won’t. But importantly, it still does have a measurable positive effect on risk-adjusted returns. The compounded returns of your regularly rebalanced portfolio will likely be more than you expect by only looking at the weighted average of each asset, so keep that in mind when thinking about tradeoffs.

When does it matter most?


Lest you come away from the last section with the idea that the rebalancing bonus is overblown, let’s look at one more example that puts it on full display. As smart people like Michael Kitces and William Bernstein have pointed out, the rebalancing bonus is most prominent when:

  1. The overall returns of each asset are similar. Think of our original example when they were both zero. This puts the AVG portion of the CAGR equation on equal footing.

2. Each asset is highly volatile and negatively correlated. When one asset strongly reacts to a poor year for the other and offsets the losses, it greatly reduces the volatility of the portfolio as a whole.

Basically, if you significantly reduce the portfolio volatility without sacrificing returns then good things happen.

For a prime example of this, take a look at the returns for stocks and gold for the 40-year investing period starting in 1970. Over that 4-decade timeframe, the returns for stocks only outpaced gold by less than 1% per year. That’s not exactly the same, but it’s a lot closer than most stock fans expect over the long run and definitely similar enough for our example. Both assets are highly volatile on their own, and were reasonably negatively correlated so that one did well when the other struggled. So that meets all of our criteria for a productive environment for Shannon’s Demon to work its magic.

Check out the regularly rebalanced portfolio of 50% each. It’s not a typo.

StocksGold50/50
40-year real CAGR starting in 19704.9%4.0%5.8%

Regularly rebalancing between stocks and gold generated a return of 5.8%, nearly a 1% annual premium over investing in the highest returning individual asset. A lot of active fund managers spend entire careers trying and failing to accomplish that type of alpha. Who knew that all they had to do was mechanically rebalance between two complementary assets?

Now I’m sure that some of you are quick to notice that ending the analysis in 2009 could rightly be considered cherry-picking. Yes, I chose that period to showcase the greatest effect. So for full disclosure, here are the numbers for the entire period I have on record from 1970 through 2021.

StocksGold50/50
Real CAGR from Jan 1970 to Dec 20216.7%3.5%6.3%

This is actually a great example of a previous point regarding guarantees. No, rebalancing between two assets did not succeed in generating more money than investing in the single best performer. But I bet it got WAY closer than you probably expected by only looking at the stock and gold numbers! It accomplished this feat exactly because it experienced significantly less volatility than either asset in isolation. And going back to the last 40-year example, it even powered through particularly rough timeframes for stocks while generating returns lots of people might think are impossible without even attempting to time the markets.

That’s a pretty appealing tradeoff for a lot of people, and one that they’d never know about if they focus solely on weighted averages rather than digging into the positive effects of portfolio rebalancing.

Conclusions


The topic of the rebalancing bonus is one of the more math-centric issues I’ve touched in a while, and I’ve done my best to distill it down to the most important points without losing people with esoteric things like correlation and covariance calculations. But hopefully you’ve learned something useful and perhaps have a new appreciation for a more sophisticated and nuanced side of portfolio theory. To summarize:

  • Shannon’s Demon illustrates how regular rebalancing can generate a positive portfolio return even if the individual assets have no long-term return on their own.
  • That rebalancing bonus, however, is not guaranteed to generate “free money” above and beyond any one investment. Sometimes it will, and sometimes it won’t. But even when it doesn’t, it can still improve the risk-adjusted returns of the portfolio by generating more money than you might otherwise expect.
  • The benefits of rebalancing are most strongly seen in portfolios with volatile uncorrelated assets with similar returns. And the results can be pretty compelling for investors looking to significantly reduce risk without necessarily sacrificing a proportional amount of reward.

To explore these ideas in more depth on your own, a good first step would be to check out a few tools that will help you run some of the same numbers for yourself. For example, rather than calculating the average and standard deviation of a specific combination of assets, the Annual Returns chart can do it for you automatically. And to find the compound annual growth rate of any investing period since 1970, the Heat Map is particularly useful.

Also, I’d be remiss if I didn’t reference a few great articles that served as inspiration for this one. If you want to learn more about this topic, here are some excellent additional sources:

And for an important investing perspective beyond mere math, perhaps the knowledge of the measurable financial benefits of rebalancing can shed new light on how certain portfolios like the Golden Butterfly and Larry Portfolio operate under the hood to produce unexpected results. There’s often more to investing theory than even highly educated investors truly appreciate.

The next time you look at portfolio ideas, be sure to think beyond the returns of each individual asset and explore how timeless helpers such as Shannon’s Demon can work in your favor via the power of regular rebalancing. There may be more interesting possibilities than you ever realized.


Rebalancing into financial education creates a bonus for everyone.