In a previous article I asked whether rebalancing increases return, as the term “rebalancing bonus” implies. I concluded that it does not. In this article I ask whether it is a tool for reducing risk. The answer depends on whether you believe that the standard deviation of long-term returns is the appropriate measure of risk. This article will show why it often is not.
Rebalancing vs. buy-and-hold
My first question was whether rebalancing a multi-asset portfolio increases expected return as compared with buy-and-hold, when the assets all have the same expected return. In my earlier article I answered this question in the negative.
That answer was confirmed in a recent paper by Edward Qian of PanAgora Asset Management, “To Rebalance or Not to Rebalance: A Statistical Comparison of Terminal Wealth of Fixed-Weight and Buy-and-Hold Portfolios.” Qian’s paper is noteworthy for the exceptionally high quality of its mathematics, mathematical reasoning and exposition, especially for an article in the finance field. (Qian’s PhD is in mathematics, not finance.) Qian develops mathematical expressions for the expected value and standard deviation of future wealth, given both a buy-and-hold strategy and a rebalancing strategy (Qian calls it a “fixed-weight” strategy).
When the expected return is the same for all assets, Qian’s formulas show that the expected value of future wealth is the same whether a rebalancing strategy or a buy-and-hold strategy is used.
Thus, Qian’s paper confirms the conclusion of my earlier article: If the assets’ expected returns are the same, there is no expected return bonus for a rebalancing strategy as compared to a buy-and-hold strategy.
The case of unequal expected returns
When the assets have unequal returns, a buy-and-hold strategy will in general produce a higher expected return and higher expected future wealth than a rebalancing strategy. That is because with a buy-and-hold strategy, allocations to the assets that have higher expected returns will tend to drift upward over time to become larger components of the portfolio. The greater the portfolio allocation to the assets with higher expected returns, the higher will be the expected portfolio return.
At the same time however, the greater the allocation to the assets with higher expected returns, the greater the portfolio’s volatility and uncertainty of return. That is because in general, higher-expected-return assets are also those with higher volatility and uncertainty of return – in short, higher standard deviation of return.
Hence, along with a buy-and-hold portfolio’s higher expected return – as compared with a rebalancing (or fixed-weight) strategy – will typically go a higher standard deviation of return, and thus, we assume, higher risk.
Is the buy-and-hold portfolio’s higher risk compensated for by its higher expected return? Or does the rebalancing strategy produce a higher risk-adjusted return?
Qian’s answer
To address this question, Qian calculated the expected ending wealth and standard deviation of ending wealth after 20 years of investing in an initial 50/50 equity/cash portfolio. The equity portion of the portfolio is assumed to have an 8% expected annual return with a 20% standard deviation, and the cash portion is assumed invested in risk-free assets with an annual return of 1%. Qian derived formulas for expected ending wealth and standard deviation of ending wealth for both a buy-and-hold and a rebalancing strategy.
Using these assumptions, he found that with initial investment of $1, expected ending wealth with a buy-and-hold strategy is $2.90 while expected ending wealth with a rebalancing strategy is $2.40 (rounded to the nearest 10 cents). The respective standard deviations of ending wealth, however, are $2.30 for the buy-and-hold portfolio and $1.20 for the rebalancing strategy.
Thus, it appears that expected ending wealth is a little more than 20% greater with the buy-and-hold strategy, but standard deviation of ending wealth is almost twice as great. This suggests that the extra expected return with buy-and-hold is not worth its extra risk.
This conclusion is confirmed, for Qian, by a calculation of “Sharpe ratios” for the buy-and-hold and rebalancing strategies. I place Sharpe ratio in quotes because it is not the usual Sharpe ratio, which relates annual expected (or realized) returns to the annualized standard deviation of returns. Instead, Qian’s is a ratio of ending wealth after 20 years to the standard deviation of ending wealth after 20 years.
Qian’s Sharpe ratio can be calculated in two ways: first, as the ratio of expected ending wealth to standard deviation of ending wealth, and second, as the ratio of the difference between expected ending wealth and ending wealth with a risk-free portfolio to standard deviation of ending wealth. For brevity, I’ll call the former the “quick Sharpe ratio” and the latter the “conventional Sharpe ratio,” because conventional Sharpe ratios for returns usually subtract the risk-free rate from the expected return before dividing by the standard deviation.
The “quick Sharpe ratio” for the buy-and-hold strategy given Qian’s expected wealth and standard deviation figures is 1.3, while the quick Sharpe ratio for the rebalancing strategy is 2.0. Since ending wealth with an all-risk-free portfolio (annual return 1%) would be approximately $1.20, the conventional Sharpe ratio for the buy-and-hold strategy is 0.7 while the conventional Sharpe ratio for the rebalancing strategy is 1.0.
Hence, clearly, the Sharpe ratios are greater for the rebalancing strategy than for the buy-and-hold strategy. Qian concludes from this, with admirable restraint, that the rebalancing strategy “tends to have a higher risk-adjusted terminal wealth.”
Is that the last word?
A superficial interpretation suggests that the added expected return on a buy-and-hold portfolio at a horizon does not justify its added risk as compared with a rebalancing strategy. Qian posits, reasonably, that the added uncertainty for the buy-and-hold portfolio is because not only are periodic returns random, but asset allocations at any point in time other than the starting time are also random because they depend on the random returns. Fixed-weight (rebalanced) portfolio-asset allocations, by contrast, are fixed at the rebalancing intervals and therefore not as random as those of the buy-and-hold portfolio.
All of this, however, depends on assuming that the standard deviation of ending wealth is a satisfactory proxy for risk, and that dividing expected ending wealth by standard deviation of ending wealth is useful to compare the risk-adjusted returns for various portfolio-allocation strategies over time.
This is not a good assumption. To see why, consider Figure 1.1
Figure 1. Probability Distribution of Ending Wealth (20-year period): Buy & Hold Strategy vs. Rebalancing Strategy
1. The results in Figure 1 are based on simulations using bivariate geometric Brownian motion as the returns-generating process — that is, independent lognormally-distributed random annual returns.
Figure 1 displays the probability distribution of ending wealth after 20 years using the assumptions previously cited: 50/50 initial mix between equities and cash, 8% expected annual return on equities with 20% standard deviation and 1% return on risk-free cash investments with 0% standard deviation. In the rebalancing strategy, the equity/cash mix is rebalanced annually to 50/50.
Notice that all or virtually all of the increased standard deviation for the buy-and-hold strategy is due to the right-end tail of its distribution. Above an ending wealth of $3.60 (annualized return 6.6%), the buy-and-hold strategy becomes much more likely to achieve each level of wealth than the rebalancing strategy. In fact, the buy-and-hold strategy is more than twice as likely to achieve an annualized rate of return greater than 6.6%. It is due almost entirely to this right-end tail of its distribution that the buy-and-hold strategy has a larger standard deviation of ending wealth than the rebalancing strategy.
On the downside, the buy-and-hold strategy has a somewhat higher probability of disappointing results than the rebalancing strategy. However, it also has a very slightly smaller probability of worst-case results. The greater standard deviation of ending wealth for the buy-and-hold strategy is due primarily to its upside opportunity, not its downside risk.
A longer-term example
Now let us take another example that could be applied to a retirement planning scenario. Suppose that a portfolio begins with a 50/50 mix between equities and 30-year Treasury-bond STRIPS (i.e., principal-only securities). The STRIPS can be regarded as risk-free, since they are issued by the U.S. Treasury and pay off only at maturity. Hence, there is no reinvestment risk.2 For a buy-and-hold strategy, any price volatility that the Treasury STRIP may experience is irrelevant to the result. A Treasury STRIP maturing Nov. 15, 2043, has an asked yield of approximately 3.7%. We will use this security as a close proxy for a 30-year risk-free investment. (As of this writing, its term to maturity is 29 years and 7 months.)
Figure 2 shows the probability distributions of ending wealth with a buy-and-hold strategy and with a rebalancing strategy. Each portfolio begins with a 50/50 equity/bond mix. The annual equity-return expectation is 8% with a standard deviation of 20%, while the ending wealth of the bond component is assumed to have an annualized return of 3.7% with a 0% standard deviation.
Figure 2. Probability Distribution of Ending Wealth (30-year period): Buy & Hold Strategy vs. Rebalancing Strategy
2. Reinvestment risk is the risk that interest income will only be able to be invested at a return less than the security’s original 3.7% yield to maturity.
We see the same effects as in Figure 1. While the buy-and-hold strategy has a greater standard deviation of ending wealth than the rebalancing strategy, that greater standard deviation is almost entirely due to an increase in upside opportunity rather than an increase in downside risk. The probability of ending wealth greater than $10 (annualized return greater than 8%) is more than twice as great with the buy-and-hold strategy than with the rebalancing strategy.
Furthermore, the buy-and-hold strategy provides a significantly better cushion against worst-case scenarios than the rebalancing strategy. Because the 50 cents initially invested in the Treasury STRIP must grow at the 3.7% annualized rate to reach a value of $1.49 in 30 years’ time, that downside barrier cannot be breached with the buy-and-hold strategy (barring, of course, default by the U.S. Treasury). By contrast the rebalancing strategy has a non-negligible chance of very disappointing results by the end of the 30-year period, even of “breaking the buck” – that is, returning less than the initial dollar invested. And if the simulation methodology had used returns distributions with “fat tails,” the worst-case scenarios with the rebalancing strategy – though not with the buy-and-hold strategy – would have been even worse. (A fat-tailed probability distribution is one that has a greater probability of extreme values, large or small, than a normal distribution does.)
The danger of a worst-case scenario with a rebalancing strategy
Although rebalancing is held forth as a way to control risk, it can – in a worst-case scenario – lead to genuinely dire results in a way that a buy-and-hold strategy cannot. The following is from William Bernstein’s recent short book, “Deep Risk: How History Informs Portfolio Design (Investing for Adults)” (p. 49):
Consider the following exercise; imagine, for a moment, it is June 30, 1929, and you plan to retire in five years and have a $100,000 portfolio that is 75/25 stocks/ bonds; $75,000 of stocks, represented by the CRSP 1-10 index, and $25,000 in 5-year Treasury notes, your LMP [Liability Matching Portfolio]. The $25,000 of bonds, you figure, is adequate to pay for 10 years of living expenses, which you estimate at $2,500 per year, a more than adequate outlay in 1929. You are not too concerned by the fact that your LMP will last you only 10 years; you figure you will be able raise additional living expenses by selling your stocks, and you resolve to rebalance your portfolio back to its 75/25 composition every year on June 30.
Here’s how this scenario plays out: By June 30, 1930, stocks have fallen by 26%, so you have to sell $6,528 of your Treasuries to buy more stocks to bring the portfolio back to 75/25. Over the next year, stocks fall another 26%, and on June 30, 1931, you’ve got to sell $4,616 more of those precious Treasuries.
The next 12 months are even more of a disaster, with stocks losing more than 64%. On June 30, 1932, your Treasury stash is worth $16,959, and you calculate that to get back to 75/25, you’ll have to sell $8,357 of them – nearly half of the notes – to toss into what now clearly looks like a deep-risk rat hole. This will leave you just $9,324 in liquid Treasuries – less than four years of living expenses.
Bernstein’s purpose in developing this scenario is, in part, to illustrate the difficulty of determining in real-time whether a stock market incident like that of 1929-32 is an example of “shallow risk” or “deep risk.” Bernstein defines shallow risk as “a loss of real capital that recovers relatively quickly” while deep risk is “a permanent loss of real capital.” If an investor is concerned about deep risk, he or she may be better off with a buy-and-hold strategy than with a rebalancing strategy.
Two lessons
There are two lessons to be learned here. One, there is nothing special about rebalancing. Two, standard deviation of long-term returns is a poor proxy for risk. Neither buy-and-hold nor rebalancing is necessarily the best strategy in all cases. In fact, any strategy will need to be adjusted periodically based on the investor’s circumstances and the investment prospects. However, buy-and-hold has many attractive features, not least that it requires the minimal amount of trading and that it can provide a cushion against deep risk that a rebalancing strategy and many other similar strategies cannot provide.
Michael Edesess, a mathematician and economist, is a visiting fellow with the Centre for Systems Informatics Engineering at City University of Hong Kong, a partner and chief investment officer of Denver-based Fair Advisors and a research associate at EDHEC-Risk Institute. In 2007, he authored a book about the investment services industry titled The Big Investment Lie, published by Berrett-Koehler. His new book, The Three Simple Rules of Investing, co-authored with Kwok L. Tsui, Carol Fabbri and George Peacock, will be published by Berrett-Koehler in spring 2014.
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