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.”