Geometric or Arithmetic Mean: A Reconsideration

Geometric or Arithmetic Mean: A Reconsideration Eric Jacquier, Alex Kane, and Alan J. Marcus An unbiased forecast of the terminal value of a portfolio requires compounding of its initial lvalue ut its arithmetic mean return for the length of the investment period. Compounding at the arithmetic average historical return, however, results in an upwardly biased forecast. This bias does not necessarily disappear even if the satnple average return is itself an unbiased estimator of the true mean, the average is computed from a long data series, and returns are generated according to a stable distribution, lu contrast, forecasts obtained by compounding at the geometric average ivill generally be biased downward. The biases are empirically significant. For investment horizons of 40 years, the difference in forecasts of cumulative performance can easily exceed a factor of 2. And the percentage difference in forecasts groivs with the investment horizon, as well as with the imprecision in the estimate of the mean return. For typical investment horizons, the propnr compounding rate is in between the arithmetic and geometric values.

ncreascd concern for long-term retirement planning, tho growth of the definedcontribution investment market, and propos/ als for U.S. Social Security reform have all focused considerable attention on forecasts of longterm portfolio returns. Moreover, recent academic studies suggest that conventional estimates of longterm performance, such as those guided by historical averages from the database in the Ibbotson Associates yearbooks. Stocks, Bonds, Bills and Infla-

tion (SBBl), may paint far too rosy a picture of likely future performance. We return to an old controversy in the forecasting of long-term portfolio performance—namely, given a historical data series from which one estimates the mean and variance of portfolio returns, should one use arithmetic or geometric averages to forecast future performance? Finance textbooks generally (and correctly) note that if the arithmetic mean of the portfolio's stochastic rate of return is known, an unbiaseci estimate of cumulative return is obtained by compounding at that rate.*^ Despite this advice, many in the practitioner community

Eric jacquier is at CIRANO, CJREQ, and professor of finance at HEC Montreal. Alex Kane is professor of finance and economics at the Craduatc School of International Relations and Pacific Studies, Uiiivcrsitif of California at San Diego. Alan /. Marcus is professor of finance at the Wallace E. Carroll School of Managetnent, Boston College. Updates to this research will be posted at wzuw.hcc.ca/pages/cric.jacquier. 46

seem to prefer geometric averages, which are necessarily lower than arithmetic averages. We show in this article that the practitioners are onto something. Indeed, compounding at the arithmetic average always produces an upwardly biased forecast of future portfolio vaiue. The geometric average is unbiased, however, only in the special case when the sample period and the investment horizon are of equal length. In general, an unbiased forecast may be obtained as a weighted average of these two competing methods.

Forecasting Cumulative Returns with Noisy Estimates Suppose the rate of return on a stock portfolio is lognormally distributed. If the stock price today, at time t, is denoted S,, then ln(Sf+i/Sf) has a normal distribution with mean |i and variance o^. Over an investment horizon of H periods, if returns are independent from one period to another, the cumulative return on the portfolio will also be lognormally distributed; ln(S,_^/^/S,) has a normal distribution with mean ^H and variance u^H. For any historical sample of stock returns, the geometric average rate of return is defined as the compound growth rate of portfolio value over the investment period." Suppose, for example, that we have observed stock prices over a sample period starting T periods ago (i.e., starting at time t - T) and ending today, at time f. If the initial value of the portfolio was Sf_j, then the geometric average rate of return, g, is defined by ©2003, AIMR®

Geometric or AritJnnetic Mean

where e is the standard exponential function, approximately 2.718, or, cquivalently, by (lb) Because the expected value of ln(S,_|.|/S() in each period equals [4, the geometric average return is an unbiased estimator (in fact, the maximum likelihood estimator) of j.i. A well-known feature of the lognormal distribution, however, is that if ln(S| + i/S,) has mean \.i, then the expected value of S, + i equals S^e^'^^ ""\ Thus, the expected rate of growth in portfolio value expressed at a continuously compounded rate is \i + (l/2)a^. This quantity is the arithmetic mean rate of return, which exceeds the geometric mean by (l/2)a^. After an investment horizon of H periods, the unbiased forecast of future portfolio value is, therefore. E{S t + hi'

(2)

Equation 2 is the basis of the "textbook rule" that to forecast future value, one should compound forward at the mean arithmetic return. The difference in these approaches can be empirically significant. We estimated the arithmetic mean by computing the growth in portfolio value each period (i.e., Sf+^/S() and then calculating the sample period average. This average is the estimate of j,M + (l'2)o^ YJQ estimated the geometric mean from Equation l.UsingtheSBB/database from 1926 to 2001, we found the geometric average annual return for the S&P 500 Index (expressed as a continuously compounded rate) to be 10.51 percent and the arithmetic average return to be 12.49 percent. The standard deviation of the index over this period was 20.3 percent, or 0.203, so the difference in the two measures' returns is about half the variance (1/2 X 0.203- = 0.0206, or 2.06 percent), which is consistent with the fact that the annual return of the index is approximately lognormal. For more volatile investments, such as small-capitalization stocks, the difference in arithmetic and geometric averages is even larger. An often-overlooked assumption of tho textbook formula is thnt the forecaster knows the true values of the parameters \.i and a. In practice, of course, these parameters will be estimated, and even when the estimators use the best estimation November/December 2003

techniques, the estimation will be subject to sampling error. One might think that simply substituting unbiased estimates of yi and a into Equation 2 would provide unbiased estimates of future portfolio value. Indeed, this substitution is common practice. For example, Ibbotson Associates simulates future portfolio values in SBBl by using the historical arithmetic average, as in Equation 2, and compounding forward. Unfortunately, even if the estimate of p is unbiased and a is known, the resulting forecast of future portfolio value is biased, possibly quite severely. The reason is that c *^' ^ " is a nonlinear function of p. Symmetrical errors in the estimate of p, therefore, have asymmetrical effects on the forecast of S,t''^'^^''^""'^. Positive estimation error has a greater impact than an equalmagnitude negative error. Thus, even if the estimate of \i is unbiased, with estimation error centered around zero, the estimation error in S,+^ will be upwardly biased. Figure 1 illustrates this property. Suppose the true valueof pis 10 percent, the standard deviation of annual returns is 20 percent, and we estimate p from Equation 1 using returns over a 30-year period. The standard error of estimate p is then 20/^30 ^3.65 percent. Panel A shows that the probability density of p is symmetrically distributed around 10 percent with a standard deviation of 3.65 percent. In Panels B-D, the shaded vertical lines correspond to forecasts of final portfolio value, based on an initial investment of $1, obtained by using estimates equal to p = 10 percent ±3.65 percent. The probability densities for forecasted final wealth are skewed to the right. Eor short investment horizons, such as two years (Panel B), the effect of skewness is minimal, but for a 10-year horizon (Panel C), a 1 -standard-error positive error in the estimate of p increases the forecast of final portfolio value by $1.50, from $3.30 to $4,80, whereas the symmetrical 1-standard-error negative error in the estimate of p reduces the forecast of final value by only $1.00, to $2.30. The asymmetry at a 20-year horizon (Panel D) is even more dramatic. In all cases, the uncertainty in final wealth is considerable. If the underlying stock price process is lognormal, deriving the exact bias in the forecast is relatively easy.^ If Equation 1 is used to estimate \i, then the estimate p equals the geometric average return over the sample period of length T. The standard error of p is a/Jf. The (noisy) forecast extends for H periods, resulting in a standard deviation of the forecast equal to G(H/JT) and 47

Financial Analysts journal

Figure 1. Probability Densities of |x and Forecasted Final Portfolio Value, A. Distribution of\i: .Vl-Vmr Estimalion Period

C. Distribution of Fnnriish-d Portfolio Value: lU-Ycar hurst men I Horizon

4 h Fimil Portfolio V.iluf (Si) Dislribuiioti of Forecasted Portfolio Viiluc:

D. Distribution «t I'orccuftfd Portfolio Value:

Z'Yenr hnv:^tinenl Horizon

20-Ycar liwcstmciil Horizon Density 0.07 r

0.05

0.03

0.01 -

$5.00 i $11.00 1 $23.00"

1.2

1.3

1.4

1.5

Final Portfolio Value ($)

20

40

60

Final Porffdlio V.ilLif($)

Notes: True mean, |.i, is 10 percent; standard de\i.ition is 20 percent; standard deviation of M is 20/^30 —that is, 3.65 percent.

variance of o (H /T). Thus, estimation error in [j increases the range of possible values one may infer for final portfolio value: In addition to the "irreducible noise" resulting from economic uncertainty (measured by rr), additional noise has come from using an estimate of p to forecast. Equation 2 shows that adding variance to a lognormal return increases the forecast of cumulative portfolio 48

growth by one-half the variance of cumulative return. Hence, the upward bias resulting from the extra volatility associated with sampling error is ^i/2cy-(HVT) NotL- that the bias increases both in investment horizon Hand in volatility o (which will make the statistical estimates less precise). Conversely, bias declines witb Tbecause longer sample periods increase the precision of tbe estimates. ©2003, AIMR®

Geometric or AritJnnetic Mean

Table 1 computes this bias as a function of investment horizon, volatility, and sample estimation period. Table 1 demonstrates that when reasonable parameters are used, the bias can be dramatic, especially when volatility is high or the sample period is short. In these cases, 30- or 40-year forecasts can bo biased bv factors of 2 or more.

Thus, p* + 1 /2a" is the compound growth rate that provides unbiased estimates of future portfolio value. Now, notice that this growth rate is a weighted average of the geometric and arithmetic averages, with weights that depend on the ratio of the investment horizon to tho sample estimation. I * 4- — r r

^^

1 2

Table 1. Forecast Bias: Ratio of Forecasted to True Expected Portfolio Value 1 l o r i / i m "iLVir.s

20

30

40

A. Sample period 75 years 13% 1,013 20'^ 1.027 25'!^. 1.043 1.062

1,062 1,113 1.181 1.271

1,143 1.271 1.455 1.716

1.271 1,532 1.948 2,612

B. Sivtipleperiod ^0 i/ears 1.038 13% 20"^ 1.069 25"/., i.no 30"/;. 1,162

1,162 1,306 1.517 1,822

1,401 1,822 2.554 3.857

1,822 2,906 5.294 11.023

Note: Biases induced by using arithmetic average return of portfolio over a sample period to forec.ist Fin.il portfolio \alue.

We conclude that, although the expected future value at horizon t H- H of a portfolio currently worth SI can be described by Equation 2, one may not simply substitute an estimate of p, such as the historical geometric average, into this formula. Substituting p for p adds extra variability to the distribution of portfolio values and results in tho following bias: - E{5,_^^)c ^"

' ,

(3)

Equation 3 doos indicate, however, how one can adjust the estimate of the compound growth rate of the portfolio to render tho forecast of portfolio value unbiased. Suppose one starts with the sample ostimate of the continuously compounded arithmetic average rate of roturn (i.e., p+l/2a'^) but then reduces this estimate by the amount 1 /2CT^(H/T). We call this modified estimator p* -I- \l7xP-. This reduction is just sufficient to undo the bias associated with the use of p:

(4)

November/December 2003

2(H

-(H

(5)

The growth rate that gives an unbiased forecast of final portfolio value will be very close to the arithmetic average ftir short investment horizons (i.e., for which H/T is close to zero). But as the horizon extends, tho weight on tho goomotric average will incroaso. That is, p* falls as the horizon lengthens. Eor H = T, the unbiased forecast compounds initial portfolio \'aluo at the geometric averago return. For oven longer horizons, one would apply a weight greater than 1.0 to the geometric average and a negative weight to the arithmetic average, resulting in a growth rate below both geometric and arithmetic means.'' This analysis sheds light on an apparent paradox. Assume that returns come from a distribution that is stable over time. In this case, the 75-year historical return from the 5BB! database ending in 2001 would be a reasonable (albeit imprecise) estimate of cumulative return over the next 75 years. Compounding at tho historical geometric average ovor a 75-year horizon would (by construction) match the proportional growth in wealth realized ovor the past 75 years. In contrast, compounding at tho sample arithmetic average for 75 years, as typically proscribed by the literature, would necessarily give a forecast of growth in wealth greater than the one realized historically. Following standard practice thus ensures a forecast of future portfolio growth that exceeds historical experience. The bias correction described in Equation 5 shows that this forecasting exercise (with T-H = 75 years) actually would call for compounding at the geometric average, so the forecast of 75-yoar cumulative return would match the return experienced historically.

Indicative Biases Is the potential bias in forecasts of cumulative returns economically significant? Unfortunately, it seems to be. Assume that p = 10 percent and o-21) percent. Figure 2 shows the biases resulting from the arithmetic and the geometric methods. Panel A shows the forecasted growth of funds over investment 49

Financial Anah/sts journal

Figure 2. Competing Forecasts of Final Portfolio Value A. I orecnst of Final Portfolio Value

Wealth ($) 120 100 80 60 40 U: T = 30

20

10 10

20

30

40

Investment Horizon Relative Wealth 3.0 r

B. Ratio of Arithmetic or Geometric ForL'cast to Unbiased forecast

2.5

2.0

1.5

1,0

0.5 10

20 Investment Horizon

30

40

Notes: Annuai returns are assumed to be lognormal with )I of 10 percent and a of 20 percent. Forecast A is based on c ^' "^ " ; forecast G is based on t'^ ; forecasts t/are based on t''*'"'^' ~" '" .

horizons ranging up to 40 years for four forocasts— arithmetic average. A; geometric average, G; and two unbiased growth rates, ll, computed on tho basis of historical sample periods of different lengths. The unbiased estimator uses weights H/T and 1 -H/Ttoweighttho geometric and arithmetic rates, so different sample periods result in different estimators. We assumed for growth rates U that p was estimated by using either a 75-year sample period (the SBBJ period for the United States) or a 30-year period (a shorter longth more typical for an

50

emerging market—or even appropriate for a developed capital market, such as that of the United States, if one believes that the post-Vietnam era represents a structural economic break). Panel A shows that the upward bias of the arithmetic estimator is severe at long horizons. In contrast, the bias of tho geometric estimator depends on tho relationship between H and T. When they are close, the estimator is relatively unbiased. In fact. Panel A shows that, as seen before, for H = T ^ 30 years, the geometric and

©2003, AIMR®

Geometric or Arithmetic Mean

unbiased ostimators aro equal. In general, tho goometric and unbiased ostimators for T ^ 30 years do not diverge much in Panel A for investment horizons less than 35 years. When the discrepancy between H and T is greater in Panol A, however— for example, when T - 75 years—the downward bias in tho geometric estimator is profound. In fact, it is roughly equal to tho upward bias in tho arithmetic estimator at the equivalent horizon. Panol B presents another view of tho relative biases—the ratios of arithmetic or geometric forecasts of cumulative return to unbiased forecasts. Eor T = 30 years, the bias in the arithmetic estimator risos dramatically with investment horizon: At a horizon of H ^ 20 years, the bias is about 30 percent, but at a 40-year horizon, the bias rises to almost 200 percent, Eor T - 75 yoars, the arithmetic estimator performs much better but is still subject to an upward bias of about 50 percent at a horizon of 40 yoars. A longer sample period obviously allows the arithmetic forecast to perform better, but tho stability of the underlying return process at over-longer horizons becomes increasingly suspect. Symmetri-

cally, for long T, the geometric estimator can be biased severely downward. With T - 7 5 years of data and an investment horizon of H = 40 years, for example, tho goometric forecast of final wealth is only about 60 percent of the unbiased forecast. The trade-off between long sample periods, which incroaso precision whon tho underlying return process is stable, and truncated sample periods, which disregard older, possibly no longer representative, data, is highlighted in Table 2. Table 2 presents 40-year return forecasts for a small sample of countries and indexes based on historical sample periods of different lengths. The longest series available from DataStream, for Erance, Germany, and the United Kingdom, are well longer than a century. Eor the United Kingdom, with a 201-year data span, the arithmetic average return is almost equal to the unbiased compounding rato ovon for a horizon as long as 40 years. But notice that the estimates of p for these countries ovor these long periods are far lower than estimates derived from using the past 82 years; they are smaller still than estimates based on the latest 52 years. Is the proper

Table 2. Estimates of Compounding Rates and Future Portfolio Values: Countries and Indexes Sample Estimates

Country/ Index (TSE)

France (SBF250)

Germany (DAX)

UK (FTAS)

Japan (Nikkei)

T (years)

Begin Date

Find Date

^'

n

Compound Growtt1 Kates A

G

U

Future Portfolio Value 1^(0)

ViU)

$11,9

$6,8

$9,0

21,8

14,0

15,5

V(A)

78

1414

2001

4.87