The Misuse of Expected Returns 1 Eric Hughson, Michael Stutzer, and Chris Yung

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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 estimate 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

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

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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. period, which becomes a more accurate estimate as To confirm that these problems also occur the calendar history of gross retums lengthens. when actual monthly historical returns are used, we This argument is the typical motivation (as used in applied the same bootstrap simulation technique to the opening quotation) for substituting the arith- the widely used 1926-2004 large-capitalization metic average gross retum (i.e., 1.054 percent) for stock monthly retums produced by CRSP. The the unobserved expected gross retum per period, results, depicted in Figure 2 and Table 3, confirm EiRx).^ But as Figures 1 and 2 and Tables 2 and 3 the previous problems. Moreover, a proof in show, the mathematical expected cumulative Appendix A shows that these phenomena are return, E{Wj), for a stock index is less likely to be generic, not simply the result of the specific data or equaled or exceeded than the median cumulative accuracy of the simulations. retum is. The situation becomes extreme in the long These findings are important because some run because, as proven in Appendix A, investors do use the overly optimistic forecast procedure based on the historical arithmetic average. For example, in 2001, the chief actuary of the U.S. pwb[Wj->E{Wj-)] = 0. Social Security Admirustration described forecast So, perhaps some long-term investors want to procedures used in the organization's study of indi- forecast the expected cumulative retum because vidual retirement account options that had been they use the word "expected" in its dictionary proposed but notyet enacted. The actuary noted that sense, rather than its mathematical sense. Accord- for individual account proposals, analysis of ing to the Merriam-Webster Online Dictionary, expected benefit levels and money's worth "expect" means "to anticipate or look forward to was aiso provided using a higher annual equity yield assumption of about 9.6 percent. the coming or occtirrence of" or "to consider prob- This higher average yield reflected the arith- able or certain." Long-term investors using this metic mean, rather than the geometric mean sense of the word will want to forecast the median (which was 7 percent), of historical data for cumulative retum rather tlian the mathematical annual yields. (Campbell 2001, pp. 55-56) expected cumulative return because the unknown In other words, the actuary made separate forecasts future cumulative retum is more likely to equal or by compounding equity accounts at both the 9.6 exceed the median. Tables 2 and 3 suggest that such percent historical arithmetic average retum and the investors should continuously compound the aver- 7 percent geometric average retum. We now exam- age log gross retum (or, equivalently, simply com- ine possible reasons for compounding at the histor- pound the geometric average retum) when making ical arithmetic average retum rate. forecasts based solely on historical data.

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Figure 2. Mathematical Expected vs. Median Cumulative Return: Bootstrap Simulation of (1926-2004) Large-Cap Stock Returns

A. 60-Month Cumulative Retums B. 120-Month Cumulative Retums Frequency Frequency

8,000 r 16,000 7,000 14,000 \ 6,000 - \ 12,000 10,000 5,000 \ A \ 4,000 ]\ 8,000 6,000 3,000 \ \ 2,000 4,000 1,000 2,000 0 Jj . 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 2.0 4.0 6.0 8.0 10.0 12.0 Cumulative Return Cumulative Retum

C 240-Month Cumulative Retums D. 360-Month Cumulative Retums Frequency Frequency I^ii f)(¥l 4U,UUU 35,000 r i\ 1 100,000 30,000 l\ \ 25,000 •I I 75,000 20,000 -I \ 50,000 15,000 > \ 10,000 W 25,000 5,000 vi \^ 0 1 0 30.0 40.0 50.0 0 25.0 .JO.O 75.0 lro.O 125.0 150.0 e Retum Cumulative Retum

E. 480-Month Cumulative Retums Frequency 1 tn nnn loU/UUU A 140,000 120,000 A 100,000 80,000 60,000 \

40,000 • \

20,000 - ^_ 0 1. 1 .1. ^^^^ 100.0 200.0 300,0 400.0 Cumulative Retum

Median Cumulative Retum Expected Cumulative Retum

Table 3. Forecasting Based on Monthly (1926-2004) Large-Cap Returns Compounded Compounded Horizon Mathematical Arithmetic Average (T years) Expected Return Average Return Median Return Log Return 5 1.79 1.82^ 1.64 1.62^ 10 3.22 3.30 2.68 2.64 20 10.35 10.89 7.16 6.96 ^' 33.26 35.95 19.08 18.38 40 106.69 118.65 50.83 48.51

A second possible motive for interest in fore- Hence, they act "as if" the forecast loss is the forecasting the mathematical expected cumulative cast error itself, rather than the squared error. rehim arises from the statistical theory of best forecasts. This theory requires that the forecaster choose a loss function that quantifies the loss Limitations of Historical Returns incurred by missing the forecast. Of course, the Unfortunately, using a historical average to estimate forecast error is random, so the best forecast is the either the unknown expected gross return per year one that minimizes a misforecasting "cost," or expected log gross return per year requires, even defined to be the mathematical expected loss. under the ideal statistical circumstances embodied When the loss is proportional to the squared forecast in the IID assumption, a very long calendar history error, it is well known (see, for example, Zellner of returns. Under the IID assumption, the estimate 1990) that the best forecast, m this sense ofthe word becomes progressively more accurate as the number "best," is the mathematical expected cumulative of available past years' returns gets larger. But the return, despite the fact that it will be higher than convergence to the unknown true number is typi- the median cumulative return. To understand why, cally slow. Measuring returns more frequently (e.g., consider the loss associated with underforecasting monthly or daily instead of annually) does abso- an unusually high cumulative return. The loss is lutely no good."* extremely high because it is found by squaring the For an illustration of this point, note that the error between the high cumulative return and the hypothetical annual stock returns in Table 1 (and (lower) forecast. For example, suppose the forecast used to produce Figure 1 and Table 2) were ran- error is +3. Then, the loss is 9, which is three times domly sampled from a lognormal distribution the size of the forecast error itself. Now, because with a volatility of 15 percent. Suppose we would cumulative returns are inherently positively like to be 95 percent confident that the historical skewed, the chance of underforecasting by an average log gross annual return is within 400 bps unusually large amount is greater than the chance of the (unknown) expected log gross return per of overforecasting. As a result, to minimize the year. Appendix A shows that we would need more chance of underforecasting by an unusually large than 54 years of past log gross returns to ensure this amount, the forecast that minimizes the mathemat- confidence level. And ±400 bps per year is probably ical expected squared forecast error will be higher too wide an uncertainty band for many financial than the median. But this mathematical result is planning purposes. merely an alternative way of characterizing the Moreover, even if we did have a long calendar behavior of someone who uses the expected cumu- history of returns, how likely is it that those returns lative return as a forecast. It is not a recommenda- would continue being generated by the same IID tion that investors use the squared forecast error to process? If the probability distribution of the mea- measure the loss from misforecasting. In fact, stat- sured returns changes over time, the compounding isticians have also shown that if investors use the of historical averages will be very misleading, espe- forecast error itself to measure the loss from mis- cially for short- or medium-term forecasts. Asness forecasting, the median is the statistically best fore- (2005) noted: cast. After seeing the results in this article, we believe that most long-term investors would con- When it comes to forecasting the future, especially when valuations (and thus histori- sider the median cumulative return to be a better cai returns) are at extremes, the answers we forecast than the expected cumulative return. get from looking at simple historical averages are bunk. (p. 37)

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This opinion may be excessively harsh, but we do (I) T (ID) feel that using historical returns to implement non- l (A2) parametric forecast procedures (such as all the ones mentioned in this article) does not solve all prob- which shows that compounding expected portfolio lems inherent in the difficult task of forecasting gross returns produces the portfolio expected cumuiative returns. cumulative return, a fact underlying the Campbell (2001) quotation at the beginning of this article. Conclusion Representing Equation Al and Equation A2 a bit differently will soon prove useful. To do so, we Textbooks and other methodological sources may take the logarithm of both sides of Equation A2 and discuss the mathematical differences between his- then reexponentiate to show that torical arithmetic average and geometric average returns but may not adequately advise practitioners ,^r^iog£(fl) about the proper use of these concepts when fore- (A3) casting future cumulative returns. Under ideal sta- Taking the logarithm of both sides of Equation Al, tistical assumptions, the historical arithmetic dividing and multiplying by T, and then reexpo- average gross return is an unbiased estimator of the nentiating shows that mathematical expected gross return per period. As others have noted, compounding the mathematical expected gross return (but not the historical arith- (A4) metic average return) produces the mathematical expected cumulative return. But because cumulative We see from Equation A4 that it is the time- returns are positively skewed, the mathematical averaged log gross returns that determine the evo- expected cumulative return substantially overstates lution of a portfolio's cumulative return, regardless the future cumulative return that investors are of whether or not the returns are IID. Because the likely to realize, and the problem grows worse as (nonlog) gross returns are never negative for stock the horizon increases. Those seeking a more realistic and/or bond investments, and raising something forecast procedure can approximate the median normegative to a fixed power greater than 1 is a cumulative return by continuously compounding monotone nondecreasing function. Equation A3 the mathematical expected log gross return per and Equation A4 imply that the probability of doing period, using the historical average log gross return at least as well as the expected cumulative return is to estimate the expected log gross return. Without a hundred years or more of accurate returns to average, however, that procedure may still provide a = proh (A5) T highly inaccurate estimate—even if the return distribution does not change over time. But by the law of large numbers for HD pro- cesses, j Vie authors wish to acknowledge Gitlt Gur-Gershgorin I r r^- for assistance loith the simulations and Garland — ^log/?, - E{\ogR). (A6) Durimm for comments on the mathematics. ^f=i So, from Equation A5 and Equation A6, we see This article qualifies for 1 PD credit. that the long-run behavior of the probability (Equa- tion A5) is governed by the relationship between £(log R) and log £(R). Because Jensen's inequality Appendix A. Derivation of implies that Mathematical Claims E(\QgR)< \ogE[R), (A7) Using Wj to denote the cumulative return in time Equations A5-A7 imply the distribution-free result period T from a dollar invested initially and using in the text; that is, Rf to denote the gross return at time t {i.e., 1 plus the net return), we express Wj- as pToh[WT>E(Wjj\ = 0. (A8) Erom Equation A8, we can clearly see that we should not expect to earn a long-run cumulative return that is greater than or equal to the expected When the gross return process is IID, the ma th- cumulative return! When a variable is not symmet- ematical expectation (denoted by £) is rically distributed, its mathematical expectation is

not generally a good indicator of the variable's central tendency. Figures 1 and 2 show that the 1 (A13) cumulative return distribution is sharply skewed for all horizons T (i.e., there would be no tendency to the right, so the expected cumulative return, for the investment value to drift either up or down, E{Wj), is higher than what will likely occur. despite the seemingly high expected cumulative The horizon-dependent probabilities (Equa- return e^-^^^). An investment with a smaller ii tion A5) are easily calculated when the returns are would result in negative drift (i.e., a tendency to lognormally distributed, as the hypothetical returns lose money). However, a more diversified portfolio used in Table 1 are. Substituting our notation for with a smaller return (|a < 8 percent) would tend to that used by Hull (1993, p. 211), log Wj is normaUy make money if its volatility were low enough to distributed with mean equal to (|i - a^/2)T and variance equal to crT, where |i and CT are, respec- But what about when log R is not normal? We tively, the annualized mean and volatility parame- will now see why compovmding the average log ters. Hull also showed that return produces a reasonable estimate of the median cumulative return whether the IID distribution of log R is normal or not. First, we rewrite (A9) Equation A4 by canceling T to obtain (that is, the compound value of the expected gross return). Similarly, ^ (A14) Because the exponential function is a mono- E(\ogWT) = (\i-o-^/2)T. (A]0) tone (increasing) function of T ^ Because the logarithm is a monotone increasing transformation. Median (AIS) When Tis suitably large, Ethier (2004) used the following approximation:

= prob Z> (All) Median] , = E{\ogR)T ) = prob

where Z denotes the standard normal density func- 6 var (log R) tion. We see that prob[W7-> EiWj-)] approaches zero In practice, the second term in Equation A16 is as the horizon, T, approaches infinity, as proven for quite small compared with the first term. So, sub- the arbitrary distributions. stituting Equation A16 into Equation A15 yields Fortunately, prob[W7- > Median{Wj)] = 1/2, instead of approaching 0 for large T. When the Hull Median (A17) (1993, p. 211) lognormal example is used, the It is easy to show the equivalence between median cumulative return is (?(M-CT-/2)r. ^^i^^ jg^ it compounding the historical average log gross is produced by compounding the expected log return and compounding the historical geometric gross return per year. The percentage difference of average net return. Denote the actual historical the expected and median cumulative returns is rehirns by R\ R'I^ and the historical cumula-

-17" tive return by W'. Then, the historical geometric average, R{g), is defined by (A12) (A18)

But W can also be computed by which is an increasing function of a and T. A particularly stark example of the difference (A19) between the expected and median cumulative returns can be seen by considering a volatile invest- Equation A18 and Equation A19 show that ment (e.g., fi = 8 percent and a = 40 percent). Then, the median cumulative return is i = e (A20)

November/December 2006 www.cfapubs.org 95 Financial Analysts Journat so raising either side of Equation A20 to the power measure log retums I/At times per year {e.g.. At = T produces the same T-period cumulative retum 1/12 when retums are measured monthly). Then, forecast—a "plug-in" estimator of Equation A17. If the log gross retum per measurement period has an R, is used to denote a generic random historical expected value of (^ - a^/2)Af, so a historical aver- gross retum, a desirable property of this plug-in age of N s T/At log gross retums (i.e., T years of estimator is that history) will also be normally distributed with an expected value equal to (fj - o^/2)A(. Hence, an Median (A21) unbiased estimator of ^ - CT^/2 is the historical aver- Median = e age log gross retum divided by Af. Because the log which we dub "median unbiasedness." A more gross retum per measurement period has a variance complex procedure might provide a better estimator of a^Af, the variance of the unbiased estimator is of the median cumulative retum, but simplicity of

Notes T random draws from the annual gross retums in Table 1 ematical expected cumulative return, They added tbe were multiplied to produce a possible T-year cumulative assumption that returns are lognormally distributed and retum. Tbe procedure was repeated one million times to proposed better estimators of the unknown mathematical produce eacb of the smootbed histograms in Figure 1, expected cumulative retum. Our goal is different, however, whose means and medians are reported in Table 2. for we are highlighting flaws in the arguments used to justify See Appendix A for a derivation of this equivalence and the the relevance of estimating tbe mathematical expected other mathematical claims made later in the text. cumulative retum in the first place. We argue that estimat- Even this typical motivation is flawed. An interesting article ing the median cumulative retum is a much more relevant by Jacquier, Kane, and Marcus (2003) highlighted problems objective, regardless of whether the returns are lognormal. resulting from compounding the historical arithmetic aver- See Luenberger (1998) for a simple exposition of this prob- age to produce a data-based estimate of tbe unknown math- lem and Appendix A for a specific calculation.

References Asness, Clifford S. 2005. "Rubble Logic: Wbat Did We Leam Hull, Jobn. 1993. Options, Futures, and Other Derivative Securities. from the Great Stock Market Bubble?" Financial Analysts joiirmi, Upper Saddle River, NJ: Prentice Hall. vol. 61, no. 6 (November/December) ;36-54. Jacquier, Eric, Alex Kane, and Alan Marcus. 2003. "Geometric Bodie, Zvi, Alex Kane, and Alan Marcus. 2004. Essentials of Mean or Arithmetic Mean: A Reconsideration." Financial Invesiments. 6th ed. New York: McGraw-Hill. Analysts journal, vol. 59, no. 6 (November/December):46-53. Campbell,JobnY. 2001. "Forecasting U.S. Equity Returns in the Luenberger, David G. 1998. Investment Science. New York: 21st Century." In Estimating lhe Real Rate of Return on Stocks over Oxford University Press. the long Term. Edited by Jobn Y. Campbell, Peter A, Diamond, Zellner, Arnold. 1990. "Bayesian Inference." In The New Palgrave: and Jobn B. Shoven. Social Security Advisory Board Time Series and Statistics. Edited by John Eatwell, Murray (www.ssab.gov/publications.htm). Milgate, and Peter Newman. New York: Norton. Ethier, S.N. 2004. "The Kelly System Maximizes Median Fortune." Journal of Applied Probability, vol. 41, no. 4 (December): 1230-1236.