Financial Analysis Journal Volume 62 • Number 6 fAJ ©2006, CFA Institute The Misuse of Expected Returns 1 Eric Hughson, Michael Stutzer, and Chris Yung 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 retum. opular finance textbooks and other method- namely, compounding the arithmetic average to ological treatises emphasize the relevance of produce cumulative retum forecasts at each future a portfolio's expected retum and the use of horizon between 1 and 30 years. For example, the P forecasted cumulative retum after 1 year is 1.054^ time-averaged historical returns as an esti- mate of it. For example, Bodie, Kane, and Marcus = 1.054, and after 30 years, it is 1.054^0 = 4799. that (2004) state: is, an initial investment of SI.00 is forecasted to ... if our focus is on future performance then grow to about $4.80. But this retum forecast is far the arithmetic average is the statistic of interest higher than the 30-year historical cumulative because it is an unbiased estimate of an asset's retum (3.005) shown at the bottom of the last col- future returns, (p. 865) umn, which suggests that the arithmetic average- A more detailed procedure for using this average based forecast in the fourth column may be too is found in a respected researcher's survey article: high. We will now document that such overblown forecasts are very likely to happen in practice. When returns are serially uncorrelated—that is, when one year's retum is unrelated to the The overoptimism inherent when the arith- next year's retum—the arithmetic average metic average retum is used to forecast is illus- represents the best forecast of future retum in trated in Table 2 and Figure 1, which report the any randonUy selected year. For long holding results from a bootstrap simulation of one million periods, the best return forecast is the arith- possible future cumulative retums derived from metic average compounded up appropriately. the annual gross returns given in Table 1.^ Both (Campbell 2001, p. 3) Tabie 2 and Figure 1 clearly show that the mathe- For an illustration of the quoted concepts, con- matical expected cumulative return is always sider a hypothetical broad-based stock index with higher than the median cumulative retum (i.e., the returns that are corxsistent with the ubiquitous ran- retum that has equal chances of being exceeded or dom walk hypothesis. Annual gross (i.e., 1 plus net) not) and that the gap between the two increases as retums for each of the past 30 years from the hypo- the time horizon lengthens and the cumulative thetical index are given in the second column of retum distribution becomes more highly skewed to Table 1. The arithmetic average gross retum is the right. For example, at the 10-year horizon, the 1.054 (i.e., the net retum averages 5.4 a year). The mathematical expected cumulative retum is 1.72, last column provides the historical cumulative which is 18 percent bugher than the median cumu- retums. The fourth column of Table 1 shows the lative return (1.46). At the 30-year horizon, the cumulative retum forecasts calculated by follow- mathematical expected cumulative retum is 67 per- ing the advice in the Campbell (2001) quotation— cent higher than the median cumulative retum. As a result, the mathematical expected cumulative return is less likely to be realized (i.e., met or Eric Hughson and Chris Yung are professors of finance exceeded by the future cumulative retum) in the in the Leeds School of Business at the University of future than the median retum, and this likelihood Colorado, Boulder. Michael Stutzer is professor of is more pronounced for the long horizons used by finance and director oftheBurridge Center for Securities retirement planners. For example, there is a 38 per- Analysis and Valuation in the Leeds School of Business at the University of Colorado, Boulder. cent probability that the mathematical expected 88 www.cfapubs.org ©2006, CFA Institute The Misuse of Expected Returns Table 1. Forecasts Based on Historical Arithmetic Average Returns Historical Period, T Gross Lx)g Gross T-Year Ctunulative Historical {years) Retum Retum Retum Forecast Cumulative Retum 1 1.014 0.014 1.054 1.014 2 0.876 -0.133 1.110 0.888 3 1.100 0.095 1.170 0.976 4 1084 0.250 1.233 1.254 5 1.269 0.239 1.299 1.592 6 1.375 0.319 1.368 2.189 7 0.764 -0.269 1.442 1.673 8 1.024 0.024 1.519 1.713 9 1.250 0.223 1.601 2.141 10 0.901 -0.104 1.687 1.929 11 0.956 -0.045 1.777 1.845 12 0.823 -0.195 1.873 1.518 13 0.804 -0.218 1.973 1.221 14 0.916 -0.088 2.079 1.118 IS 0.944 -0.057 2.191 1.056 16 0.772 -0.259 2.308 0.815 17 0.974 -0.026 2.432 0.794 18 0.998 -0.002 2.563 0.792 19 1.082 0.079 2.700 0.857 1.004 0.004 2.845 0.861 21 1.010 0.010 2.998 0.869 22 1.003 0.003 3.159 0.872 23 1.297 0.260 3.328 1.131 24 1.047 0.046 3.507 1.184 2S 1.031 0.031 3.695 1.221 2£ 0.982 -0.018 3.893 1.199 27 1.426 0.355 4.102 1.709 28 1.208 0.189 4.323 2.064 29 1.515 0.415 4.555 3.126 30 0.961 -0.039 4.799 3.005 Average 1.054 0.037 Table 2. Forecasting the Mathematical Expected and Median T-Year Cumulative Return Compounded Compotmded Horizon Mathematical Arithmetic Average (Tyears) Expected Retum Average Retum Median Retum Log Retum 5 1.31 1.30'' 1.20 1.203*' 10 1.72 1.69 1.46 1.45 20 3.01 2.86 2.18 2.10 30 5.15 4.84 3.09 3.03 40 9.0] 8.20 4.72 4.39 November/December 2006 www.cfapubs.org 89 Financial Analysts Journal Figure 1. Mathematical Expected Return vs. Median Cumuiative Return: Bootstrap Simuiation of Table 1 Returns A. 5-Year Cumulative Returns B. lO'Year Cumulative Returns Frequency Frequency i 7,000 0 1.0 2.0 3.0 4.0 0 1.0 2.0 3.0 4.0 5.0 Cumulative Return Cumulative Return C. 20-Year Cumulative Returns D. 30-Year Cumulative Returns Frequency Frequency 40,000 80,000 70,000 60,000 50,000 40,000 30,000 20,000 10,000 0 0 2.5 5.0 7.5 10.0 12.5 15.0 5.0 10.0 15.0 20.0 25.0 30.0 Cumulative Return Cumulative Return E. 40-Year Cumulative Returns Frequency 140,000 5.0 10.0 15.0 20.0 25.0 30.0 Cumulative Return Median Cumulative Return Expected Cumulative Return 90 www.cfapubs.org ©2006, CFA Institute The Misuse of Expected Returns cumulative retum will be exceeded at the 10-year Why Use the Arithmetic Average horizon and only a 30 percent probability that it will be exceeded at the 30-year horizon. Return In Forecasting? The third column in Table 1 contains the loga- Two motives are put forth for using the arithmetic rithms of the historical gross retums. The average of average retum, but neither is convincing. The first these logarithms is lower than the arithmetic aver- motive is somewhat complex. Recall that the mathe- age of the gross retums themselves. Table 2 con- matical expectation of something is the probability- trasts forecasts that compound the arithmetic weighted average of its possible values. The average gross retum with those that (continuously) quotations that began this article use this mathemat- compound the average hg gross retum (3.7 percent ical definition of expectation. When portfolio gross from Table 1). It is also common for analysts to call retums K, are independently (I) and identically dis- the number e^-^^'^ - 1 == 0.038 percent the geometric tributed (ID), the mathematical expected cumula- average net retum, in which case the last column of tive retum (denoted by E) is the compounded value Table 2 is equivalently produced by ordinary com- of the expected gross retum per period—that is, pounding of the geometric average gross retum (i.e., 1.038'),^ Table 2 shows that the compounded aver- (0 ?• (ID) age of the log gross retums is far closer to the simu- {) Yl{) lated median future cumulative retum than is the t=] compounded arithmetic average (1.054^), which in where Wj denotes the (random) cumulative return tum, is far closer to the simulated mathematical T periods in the future. expected future cumulative return. At the relatively In addition, with the same IID assumption, the long horizons that characterize retirement planning, arithmetic average of historical gross retums is a the unwarranted optimism inherent in the arith- commonly used estimate of the (unknown) con- metic average-based forecasts will probably lead to stant mathematical expected gross retum E{Ri) per excessively high investment in stocks.