# The Mythical Rebalancing Bonus-Part 2

In Part 1 I showed that as investment time increased there was, at least historically, a smaller probability of realizing a rebalancing bonus in 60/40 stock/bond portfolios. There was a lot that I left unaddressed at the time and I felt a need to develop a better understanding of the mechanics of the rebalancing bonus. Why does it work in some instances, but not in others? In other words, more was needed to demonstrate what actually drives the rebalancing bonus. A good place to start is with the work of Harry Markowitz, which showed that portfolio performance–both return and volatility–was mathematically related to three characteristics of the constituent assets

1. Correlation with other assets
2. Volatility
3. Rate of return

Correlation, or lack thereof, is a well understood concept. Assets must have low correlation for the rebalancing action to exist. As correlation coefficients rise there is less opportunity to rebalance, thus less opportunity for a rebalancing bonus to occur.

Volatility is number two on the list, but before diving into the math, think about what volatility provides: highly volatile assets will experience higher highs and lower lows compared to assets with low volatility. Thus, it would seem, highly volatile assets should provide greater opportunities to buy low and sell high.

Hypothesis: Assets with high volatility should provide a greater rebalancing bonus compared to assets with low volatility

We can experimentally test this hypothesis using the random nature of a coin flip by assigning a rate of return to each potential outcome. For example, if heads produced a return of +30% and tails a return of -10% then the resulting annualized rate of return would be

Expected Annualized Return = [(1+ReturnHeads)*(1+ReturnTails)]1/2 – 1

Expected Annualized Return = [(1+0.3)*(1-0.1)]1/2 – 1 = 1.0817 – 1 = 8.17%

The arithmetic average return

Average Return = ReturnHeads*0.5 + ReturnTails*0.5 = 0.3*0.5 – 0.1*0.5 = 0.1 or 10%

The corresponding variance and standard deviation

Variance = ReturnHeads2*0.5 + ReturnTails2*0.5 – (Average Return)2

Variance = (0.30)2*0.5 + (-0.10)2*0.5 – (0.10)2 = 0.045 + 0.005 – 0.01 = 0.04

Standard Deviation = (Variance)1/2 = (0.04)1/2 = 0.2 or 20%

Admittedly I borrowed these numbers from William Bernstein’s The Intelligent Asset Allocator. [1] Notice these numbers aren’t that far off from the long-term historic return and volatility of US stocks as measured by the S&P 500. More importantly, the numbers here can be altered to produce assets with different rates of return and volatility. For example, take the following three hypothetical assets

 Asset 1 Asset 2 Asset 3 Heads +30.0% +33.0% +39.3% Tails -10.0% -12.0% -16.0% Expected Annualized Return 8.17% 8.16% 8.17% Standard Deviation/Volatility 20.0% 22.5% 27.7% Table 1

Increasing the return on “heads” and simultaneously decreasing the return on “tails,” in the correct amounts, results in assets that have the same annualized rate of return but higher levels of volatility.

Using these theoretical, stock-like assets I formed portfolios with each asset and a cash holding. Half of the portfolio was allocated to each asset and the other half to cash. A rebalanced and non-rebalanced portfolio were then simulated to calculate the rebalancing bonus (the rebalancing bonus is simply the return of the rebalanced portfolio minus the return of the non-rebalanced portfolio).

 Rebalancing Bonus of 50/50 Portfolios Asset/Cash (6 Rebalancing Periods) Asset 1/Cash Asset 2/Cash Asset 3/Cash No. of Heads 3 3 3 No. of Tails 3 3 3 Asset Std Dev/Volatility 20.0% 22.5% 27.7% Annualized Return (Rebalanced) 4.52% 4.64% 4.92% Annualized Return (Not Rebalanced) 4.48% 4.48% 4.48% Rebalancing Bonus 0.04% 0.16% 0.44% Table 2

Looking at the rebalanced portfolios, rates of return increased as volatility increased. As a consequence the rebalancing bonus also increased. I should point out that in these simulations I limited total investment time to 6 years/periods. This was done on purpose. The cash holding in the above scenarios has a return of exactly 0%. It serves to preserve capital and from that perspective is not all that different from a bond (although somewhat extreme since bonds usually provide some positive return and experience some volatility). As I showed in Part 1, rebalancing between stocks and bonds over long periods of time may actually reduce the occurrence and magnitude of the rebalancing bonus. For this reason I kept the investment period short so that I was able to realize a rebalancing bonus and demonstrate the impact of volatility.

What happens when investment time is increased to, say, 50 years?

 Rebalancing Bonus of 50/50 Portfolios Asset/Cash (50 Rebalancing Periods) Asset 1/Cash Asset 2/Cash Asset 3/Cash No. of Heads 25 25 25 No. of Tails 25 25 25 Asset Std Dev/Volatility 20.0% 22.5% 27.7% Annualized Return (Rebalanced) 4.52% 4.64% 4.92% Annualized Return (Not Rebalanced) 6.72% 6.72% 6.72% Rebalancing Bonus -2.20% -2.08% -1.80% Table 3

Although the rebalancing bonus did increase with higher levels of volatility over each 50 year investment scenario, all of the bonuses were substantially negative. This is a consequence of the compounding action of the stock-like asset. Over long periods of time it will grow to become a larger and larger allocation of the overall portfolio, thus driving the return of the non-rebalanced portfolio higher than that of the rebalanced portfolio.

To summarize so far, the rebalancing bonus realized in a portfolio increases as the volatility of the underlying assets increases. However, the rebalancing bonus between stocks and cash/bonds will be reduced both in magnitude and occurrence over longer periods of time.

Consider an additional investment strategy in which each asset is mixed with itself in a 50/50 portfolio. The two assets behave according to separate coin flips, so the sequence of returns are uncorrelated. However, they have the same overall rate of return and volatility.

 Rebalancing Bonus of 50/50 Portfolios Asset/Asset (50 Rebalancing Periods) Asset 1/Asset 1 Asset 2/Asset 2 Asset 3/Asset 3 No. of Heads 25 25 25 No. of Tails 25 25 25 Asset Std Dev/Volatility 20.0% 22.5% 27.7% Annualized Return (Rebalanced) 9.04% 9.27% 9.83% Annualized Return (Not Rebalanced) 8.17% 8.16% 8.17% Rebalancing Bonus 0.87% 1.11% 1.66% Table 4

Again, as in the previous two setups, the rebalancing bonus increased as the volatility of the underlying assets increased. What stands out in this setup is that the rebalancing bonus was positive in each case over the 50 year/period investment time. So what happened? The big difference here was that the rates of return of the underlying assets were identical. The rebalancing took place between two stock-like assets. In other words, rebalancing among assets that have similar rates of return preserves a positive rebalancing bonus over long periods of time. When rates of return are similar there is no one asset that will dominate the portfolio as was shown with stocks and cash/bonds.

In summary, asset allocators should desire assets with the following three characteristics:

1. Assets that have low correlation to enable rebalancing
2. Assets that have similar rates of return to preserve a positive rebalancing bonus over long periods of time
3. Assets with high volatility, as this will enhance the rebalancing bonus (i.e. buy lower and sell higher)

But let’s get real. Looking at common assets available to retail investors and comparing their behavior to Large Company US stocks results in the following qualitative relationships (roughly speaking)

 Low Correlation High Volatility Similar Rates of Return Bonds & Fixed Income Emerging Market Stocks REITs Precious Metal Equities Precious Metal Equities Emerging Market Stocks Small Company Stocks Small Company Stocks Developed Market Stocks

In other words, real world assets typically exhibit one or two of the three desired characteristics. It’s difficult, if not impossible (at least for smaller retail investors) to find assets that exhibit all three.

Rebalancing Between Stocks and Bonds
The rebalancing bonus, while the primary focus of the work above, is not and should not be the only consideration when examining a rebalancing strategy. Going back to the data presented in Table 4, the rebalancing bonus was negative in each scenario as the non-rebalanced portfolio provided higher returns over a longer investment time. There’s more to this story when portfolio volatility is considered

 Rebalancing Bonus of 50/50 Portfolios Asset/Cash (50 Rebalancing Periods) Asset 1/Cash Asset 2/Cash Asset 3/Cash Rebalanced Portfolio Annualized Return 4.52% 4.64% 4.92% Standard Deviation 10.1% 11.4% 14.0% Return/Risk Ratio 0.45 0.41 0.35 Non-Rebalanced Portfolio Annualized Return 6.72% 6.72% 6.72% Standard Deviation 17.2% 19.4% 24.0% Return/Risk Ratio 0.39 0.35 0.28 Table 5

I added a “Risk/Return Ratio”–simply computed as the ratio of annualized return divided by standard deviation–as a rough approximation of the Sharpe Ratio (since these assets are hypothetical I don’t have a risk-free rate of return), but I think it communicates the idea. The rebalanced portfolios look attractive from a risk-adjusted perspective. This is mainly due to lower portfolio volatility–a byproduct of rebalancing.

A brief side note: There is a popular maxim that states “you can’t eat IRR” (Internal Rate of Return). [2] With the above results in mind I would add that risk adjusted returns aren’t always that tasty either.

This observation, that rebalanced portfolios exhibit lower volatility than an equivalent non-rebalanced portfolio, has occurred with real world assets as well. Below are the results from 60/40 portfolios of US stocks and 5-Year Treasury Notes. The volatility advantage is simply the standard deviation of the rebalanced portfolio minus the standard deviation of the non-rebalanced portfolio. Therefore, negative numbers indicate that the rebalanced portfolio had a lower volatility–a good thing. On average, this number drops (becomes more negative) as investment time increases. Results for 20-year Treasury Bonds and corporate bonds were similar.

 Historic Probabilities of a Volatility Advantage 60/40 Stock/Bond Portfolios 1926-2015 Investment Period 5 Years 10 Years 20 Years 30 Years US Large Co. Stocks/5 Year T-Notes 57/86 = 66% 60/81 = 74% 61/71 = 86% 59/61 = 97%

While rebalancing between stocks and bonds doesn’t always provide a rebalancing bonus via a higher rate of return, chances are that it will help lower volatility compared to a non-rebalanced portfolio. For these reasons bonds are best suited for wealth preservation and reducing portfolio volatility. The question now facing investors: How much do you care about the volatility of your portfolio?

POST SCRIPT
This is not the first major effort undertaken to understand the nature of the rebalancing bonus. An article by William Bernstein titled THE REBALANCING BONUS: Theory and Practice was the inspiration for the work that I put together here.

I was recently made aware of a post by Jake at EconomPic titled The Case For High Volatility Strategies, which digs into the role volatility plays in a rebalancing strategy. While the perspective is a little different the conclusions are the same–correlation and volatility drive the rebalancing bonus. (h/t Jake)

Finally, an article by Michael Edesses titled Does Rebalancing Really Pay Off?? appeared on Advisor Perspectives about two years ago. Edesses correctly points out that the rebalancing bonus quantified by Bernstein (above) is somewhat useless since it compares a rebalanced portfolio to the weighted arithmetic average of returns (what Bernstein terms the “Markowitz Return”). However, the work is still applicable as it relates to explaining the mechanics of rebalancing.

References
1. Bernstein, William. The Intelligent Asset Allocator. McGraw-Hill. 2001. pp. 1-4.
2. Marks, Howard. You Can’t Eat IRR. 2006. https://www.oaktreecapital.com/docs/default-source/memos/2006-07-12-you-cant-eat-irr.pdf?sfvrsn=2.