The Misuse of Expected Returns

Much textbook emphasis is placed on the mathematical notion of expected return and its historical estimate via an arithmetic average of past returns. But those wanting to forecast a typical future cumulative return should be more interested in estimating the median future cumulative return than in estimating the mathematical expected cumulative return. For that purpose, continuous compounding of the mathematical expected log gross return is more relevant than ordinary compounding of the mathematical expected gross return. Self-test Pensions, endowments, and other long-term investors often want to forecast the future cumulative returns associated with various asset-class indices or investment strategies. Because no one can foretell the future, the future cumulative return is always a random variable that has a probability distribution. As a point forecast of the future cumulative return, some analysts have chosen to estimate the mathematical expectation of the future cumulative return’s distribution.We argue that this choice is misguided because the distribution of the future cumulative return is often heavily (positively) skewed. As a result, the mathematical expectation of its distribution is not as good a measure of its central tendency (i.e., what is more likely to happen) as is the median future cumulative return. The median future cumulative return has a 50 percent chance of being met or exceeded, but we show that the probability of meeting or exceeding the mathematical expectation approaches zero as the forecast horizon grows to infinity. As a result, even an accurate forecast of the mathematical expected future cumulative return is a badly overoptimistic forecast of what is likely to occur over long horizons. For example, our simulations indicate that there is only about a 30 percent probability of meeting or exceeding the mathematical expected future cumulative return of a large-capitalization stock index at the 30-year horizon that typifies retirement planning forecasts.We use a relatively recent result in the theory of statistics to argue that analysts who want to estimate the median future cumulative return should focus their attention on the mathematical expected logarithm of a single period’s gross return distribution. Continuously compounding the expected log gross return through T periods approximates the median future cumulative return at the T-period horizon.A simple point forecast of the median future cumulative return is made by (1) computing the average of the historical log gross returns (e.g., historical daily or monthly return data) in all past measurement periods and then (2) continuously compounding Step 1’s result up to the T-period forecast horizon. Substituting the average historical ordinary net return for the average historical log gross return in Step 1 is not recommended.Unfortunately, use of any historical average return is somewhat problematic, even in ideal statistical circumstances that may not characterize the real world. Even if the distribution of period log gross returns has been (and will remain) stable over time, the volatility of these log gross returns can make historical averages significantly different from future long-term averages. We show that typical stock index return volatility (15 percent) is enough to cause substantial fluctuation in historical averages. For example, even with 54 years of historical log gross return data, the fluctuation in future historical log gross return averages will be ±400 bps.