Rebalancing With Shannon’s Demon

Claude Shannon was a prolific individual when it came to mathematics and science. The former Bell Labs researcher and MIT professor helped develop a field of study known as information theory and played a major role in inventing the way that computers compute. He also had an interest in the stock market, and would hold occasional meetings at MIT on the topic of scientific investing. One such method he proposed required no knowledge of future market behavior–a strategy that was designed to profit from a completely random walk.

Shannon’s Demon, as the method is known, is really nothing more than a “diversify and rebalance” strategy. The “demon” in this context is nothing evil or satanic, but instead refers to the action of sorting or rebalancing. Shannon’s hypothetical investment strategy considers a stock whose price moves about in a completely random manner and has no upward or downward trend. It ends the period of study at the same price that it started with. Shannon proposed investment capital be split between two allocations: 50% in this hypothetical stock and 50% in a cash holding. The portfolio is then rebalanced each day back to these original 50/50 allocation. To make things a little more interesting the stock is highly volatile. On any given day it can either double in price or drop by 50%. [1]

Shannon's Demon: Cash and Stock

In this scenario the buy-and-hold investor that simply went all in on the stock holding saw his interest go nowhere. On the other hand the investor that followed Shannon’s strategy, hedging and rebalancing with cash, ended up with a profitable portfolio. This example, of course, is only meant to show what is possible. The behavior of the stock in this setting, with either a +100% or -50% return, is highly unrealistic. An asset such as this has a theoretical average rate of return of 25% and a standard deviation of 75%. [2] Nothing in real financial markets comes close to replicating these types of numbers.

Shannon’s experiment is more than just a mathematical game with random numbers. It provides an excellent framework for thinking about diversification and rebalancing. Below are three different scenarios, but instead of using cash and stock I used two random stocks with the same characteristics as the one used previously.

Scenario 1: Both stocks generate a positive return
Shannon's Demon: both stocks positive

Scenario 2: One stock generates a positive return, the other a negative return
Shannon's Demon: one stock positive, one stock negative

Scenario 3: Both stocks generate a negative return
Shannon's Demon: both stocks negative

In each scenario the overall portfolio was profitable regardless of how the individual stocks performed. A major reason for this was that the stocks behaved completely independently of each other. In other words they were uncorrelated. When assets are perfectly correlated they will, by definition, behave in the exact same manner. As a result there is no opportunity to rebalance, and thus no opportunity to reduce any potential downside. The investor gets whatever the stock gives.

Shannon's Demon: perfect correlation

From a practical standpoint most equity assets exist in a gray zone when it comes to correlation. Correlation coefficients are often somewhere between 0 and 1, and change over time. During periods of high correlation assets will move together and are unlikely to show much of a change with respect to portfolio allocation. In our increasingly globalized world, correlations for the most part are only increasing. As a consequence the opportunities to improve returns, reduce volatiltiy, or reduce drawdowns through rebalancing are likely to be few and far between. The performance of US Large Company stocks (S&P 500) and International Developed Market stocks (MSCI EAFE) is an excellent example of this. From 1970 through 2015 these two assets had a correlation of 0.66 with only seven independent years where annual returns moved in different directions.

Year S&P 500
Total Return
Total Return
1970 3.87% -10.51%
1977 -7.15% 19.42%
1982 21.55% -0.86%
1992 7.65% -11.85%
2011 2.12% -11.73%
2014 13.69% -4.48%
2015 1.41% -0.39%

Looking only at years where returns went in different directions is an oversimplification of correlation as the magnitude of returns also plays a role. It’s entirely realistic to have both indexes move in the same direction but with drastically different magnitudes. An example of this occurred in 1987 when the S&P 500 closed the year up 5.26%, but the MSCI EAFE returned 24.93%.

The real questions here are the following: When does it make sense to rebalance a portfolio? Could a portfolio be rebalanced at a lower frequency (only when assets move in different directions) and produce similar or even better results than one which is rebalanced annually?

Consider two portfolios that both start out with 50/50 allocations to the S&P 500 and MSCI EAFE. The first portfolio is rebalanced annually. The second portfolio is only rebalanced when annual returns move in different directions (the seven years indicated in the table above)–call this the “diverging” portfolio.

Annualized Return Volatility
S&P 500 10.3% 17.3%
MSCI EAFE 9.5% 22.1%
Rebalanced Annually 10.1% 18.0%
Diverging Portfolio 10.0% 18.1%

Between the two rebalancing strategies there wasn’t much difference in returns or volatility, but that’s the point. Put another way, with assets that are closely correlated it didn’t matter how often the portfolio was rebalanced, so long as it was done at some point. However, there may be an advantage to rebalancing with a lower frequency not captured in these simulations. Less rebalancing will inherently spare investors fees, commissions and taxes, thus improving returns compared to the annually rebalanced portfolio.

Having an awareness of these types of situations is perhaps one of the benefits of manually managing assets rather than resorting to a robo-advisor. Taking advantage of those occasional opportunities to rebalance when it makes sense–when one asset is up while the other is down for instance–and eliminating needless transactions when there is less opportunity.

After spending some time building the above simulations and generating random time series there are inherently some odd things that stand out. Most notably is the appearance of “boom and bust cycles” in these completely random data sets. Something to consider…

Random Boom and Bust

1. Poundstone, William. Fortune’s Formula. Hill and Wang. New York, NY. 2005. pp. 15-23, 201-205.
2. Some notes on probabilities, averages and standard deviations: Statistical Eval of Avgs and Std Devs.pdf

What I’m Reading
Personal Finance is Personal (Ben Carlson)
Why The DoL Fiduciary Rule Won’t Kill Annuities, It Will Make Them Stronger! (Michael Kitces)
The Thrill Is Gone From Owning Stocks (Barry Ritholtz)
Learning From Your Mistakes … When You Win (Farnam Street)