Geometric or Arithmetic Mean: A ReconsiderationEric Jacquier, Alex Kane, and Alan J. Marcus

An unbiased forecast of the terminal value of a portfolio requirescompounding of its initial lvalue ut its arithmetic mean return for the lengthof the investment period. Compounding at the arithmetic average historicalreturn, however, results in an upwardly biased forecast. This bias does notnecessarily disappear even if the satnple average return is itself an unbiasedestimator 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 generallybe biased downward. The biases are empirically significant. For investmenthorizons of 40 years, the difference in forecasts of cumulative performancecan easily exceed a factor of 2. And the percentage difference in forecastsgroivs with the investment horizon, as well as with the imprecision in theestimate of the mean return. For typical investment horizons, the propnrcompounding rate is in between the arithmetic and geometric values.

/

ncreascd concern for long-term retirementplanning, tho growth of the defined-contribution investment market, and propos-als for U.S. Social Security reform have all

focused considerable attention on forecasts of long-term portfolio returns. Moreover, recent academicstudies suggest that conventional estimates of long-term performance, such as those guided by histori-cal averages from the database in the IbbotsonAssociates yearbooks. Stocks, Bonds, Bills and Infla-tion (SBBl), may paint far too rosy a picture of likelyfuture performance.

We return to an old controversy in the forecast-ing of long-term portfolio performance—namely,given a historical data series from which one esti-mates the mean and variance of portfolio returns,should one use arithmetic or geometric averages toforecast future performance? Finance textbooksgenerally (and correctly) note that if the arithmeticmean of the portfolio's stochastic rate of return isknown, an unbiaseci estimate of cumulative returnis obtained by compounding at that rate.* Despitethis advice, many in the practitioner community

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

seem to prefer geometric averages, which are nec-essarily lower than arithmetic averages.

We show in this article that the practitionersare onto something. Indeed, compounding at thearithmetic average always produces an upwardlybiased forecast of future portfolio vaiue. The geo-metric average is unbiased, however, only in thespecial case when the sample period and the invest-ment horizon are of equal length. In general, anunbiased forecast may be obtained as a weightedaverage of these two competing methods.

Forecasting Cumulative Returnswith Noisy EstimatesSuppose the rate of return on a stock portfolio islognormally distributed. If the stock price today, attime t, is denoted S,, then ln(Sf+i/Sf) has a normaldistribution with mean |i and variance o . Over aninvestment horizon of H periods, if returns areindependent from one period to another, the cumu-lative return on the portfolio will also be log-normally distributed; ln(S,_ / /S,) has a normal dis-tribution with mean ^H and variance u^H.

For any historical sample of stock returns, thegeometric average rate of return is defined as thecompound growth rate of portfolio value over theinvestment period." Suppose, for example, that wehave observed stock prices over a sample periodstarting T periods ago (i.e., starting at time t - T)and ending today, at time f. If the initial value ofthe portfolio was Sf_j, then the geometric averagerate of return, g, is defined by

46 ©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 eachperiod equals [4, the geometric average return is anunbiased estimator (in fact, the maximum likeli-hood estimator) of j.i.

A well-known feature of the lognormal distri-bution, 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 valueexpressed at a continuously compounded rate is\i + (l/2)a^. This quantity is the arithmetic meanrate of return, which exceeds the geometric meanby (l/2)a^. After an investment horizon of H peri-ods, the unbiased forecast of future portfolio valueis, therefore.

E{S t + hi' (2)

Equation 2 is the basis of the "textbook rule" thatto forecast future value, one should compound for-ward at the mean arithmetic return.

The difference in these approaches can beempirically significant. We estimated the arithmeticmean by computing the growth in portfolio valueeach period (i.e., Sf+^/S() and then calculating thesample period average. This average is the estimateof j,M + (l'2)o^ YJQ estimated the geometric meanfrom Equation l.UsingtheSBB/database from 1926to 2001, we found the geometric average annualreturn for the S&P 500 Index (expressed as a contin-uously compounded rate) to be 10.51 percent andthe arithmetic average return to be 12.49 percent.The standard deviation of the index over this periodwas 20.3 percent, or 0.203, so the difference in thetwo measures' returns is about half the variance(1/2 X 0.203- = 0.0206, or 2.06 percent), which isconsistent with the fact that the annual return of theindex is approximately lognormal. For more vol-atile investments, such as small-capitalizationstocks, the difference in arithmetic and geometricaverages is even larger.

An often-overlooked assumption of tho text-book formula is thnt the forecaster knows the truevalues of the parameters \.i and a. In practice, ofcourse, these parameters will be estimated, andeven when the estimators use the best estimation

techniques, the estimation will be subject to sam-pling error.

One might think that simply substituting unbi-ased estimates of yi and a into Equation 2 wouldprovide unbiased estimates of future portfoliovalue. Indeed, this substitution is common practice.For example, Ibbotson Associates simulates futureportfolio values in SBBl by using the historicalarithmetic average, as in Equation 2, and com-pounding forward. Unfortunately, even if the esti-mate of p is unbiased and a is known, the resultingforecast of future portfolio value is biased, possiblyquite severely. The reason is that c *' ^ " is anonlinear function of p. Symmetrical errors in theestimate of p, therefore, have asymmetrical effectson the forecast of S,t''^'^^''^""'^. Positive estima-tion error has a greater impact than an equal-magnitude negative error. Thus, even if the esti-mate of \i is unbiased, with estimation error cen-tered around zero, the estimation error in S,+^ willbe upwardly biased.

Figure 1 illustrates this property. Suppose thetrue valueof pis 10 percent, the standard deviationof annual returns is 20 percent, and we estimate pfrom Equation 1 using returns over a 30-yearperiod. The standard error of estimate p is then20/^30 ^3.65 percent. Panel A shows that theprobability density of p is symmetrically distrib-uted around 10 percent with a standard deviationof 3.65 percent. In Panels B-D, the shaded verticallines correspond to forecasts of final portfoliovalue, based on an initial investment of $1, obtainedby using estimates equal to p = 10 percent ±3.65percent. The probability densities for forecastedfinal wealth are skewed to the right. Eor shortinvestment horizons, such as two years (Panel B),the effect of skewness is minimal, but for a 10-yearhorizon (Panel C), a 1 -standard-error positive errorin the estimate of p increases the forecast of finalportfolio value by $1.50, from $3.30 to $4,80,whereas the symmetrical 1-standard-error nega-tive error in the estimate of p reduces the forecastof final value by only $1.00, to $2.30. The asymme-try at a 20-year horizon (Panel D) is even moredramatic. In all cases, the uncertainty in finalwealth is considerable.

If the underlying stock price process is log-normal, deriving the exact bias in the forecast isrelatively easy.^ If Equation 1 is used to estimate \i,then the estimate p equals the geometric averagereturn over the sample period of length T. Thestandard error of p is a/Jf. The (noisy) forecastextends for H periods, resulting in a standarddeviation of the forecast equal to G(H/JT) and

November/December 2003 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

Dislribuiioti of Forecasted Portfolio Viiluc:Z'Yenr hnv:^tinenl Horizon

4 hFimil Portfolio V.iluf (Si)

D. Distribution «t I'orccuftfd Portfolio Value:

20-Ycar liwcstmciil HorizonDensity

0.07 r

0.05

0.03

0.01 - $5.00

i $11.00 1 $23.00"

1.2 1.3 1.4

Final Portfolio Value ($)

1.5 20 40

Final Porffdlio V.ilLif($)

60

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 [jincreases the range of possible values one may inferfor final portfolio value: In addition to the "irreduc-ible noise" resulting from economic uncertainty(measured by rr), additional noise has come fromusing an estimate of p to forecast. Equation 2 showsthat adding variance to a lognormal returnincreases the forecast of cumulative portfolio

growth by one-half the variance of cumulativereturn. Hence, the upward bias resulting from theextra volatility associated with sampling error isi/2cy-(HVT) NotL- that the bias increases both in

investment horizon Hand in volatility o (which willmake the statistical estimates less precise). Con-versely, bias declines witb Tbecause longer sampleperiods increase the precision of tbe estimates.

48 ©2003, AIMR®

Geometric or AritJnnetic Mean

Table 1 computes this bias as a function ofinvestment horizon, volatility, and sample estima-tion period. Table 1 demonstrates that when rea-sonable parameters are used, the bias can bedramatic, especially when volatility is high or thesample period is short. In these cases, 30- or 40-yearforecasts can bo biased bv factors of 2 or more.

Table 1. Forecast Bias: Ratio of Forecasted toTrue Expected Portfolio Value

1 lor i / im "iLVir.s

20 30 40

A. Sample

13%20'^

25'! .

B. Sivtiple

13%

20"^

25"/.,

30"/;.

period 75 years

1,0131.027

1.043

1.062

period ^0 i/ears

1.038

1.069

i.no1,162

1,062

1,113

1.181

1.271

1,162

1,306

1.517

1,822

1,143

1.271

1.455

1.716

1,401

1,822

2.554

3.857

1.271

1,532

1.948

2,612

1,822

2,906

5.294

11.023

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

We conclude that, although the expectedfuture value at horizon t H- H of a portfolio currentlyworth SI can be described by Equation 2, one maynot simply substitute an estimate of p, such as thehistorical geometric average, into this formula.Substituting p for p adds extra variability to thedistribution of portfolio values and results in thofollowing bias:

- E{5,_^^)c " ' , (3)

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

(4)

Thus, p* + 1 /2a" is the compound growth rate thatprovides unbiased estimates of future portfoliovalue.

Now, notice that this growth rate is a weightedaverage of the geometric and arithmetic averages,with weights that depend on the ratio of the invest-ment horizon to tho sample estimation.

I * 4- — r r ^^1 2 2(H

-(H(5)

The growth rate that gives an unbiased forecastof final portfolio value will be very close to thearithmetic average ftir short investment horizons(i.e., for which H/T is close to zero). But as thehorizon extends, tho weight on tho goomotric aver-age will incroaso. That is, p* falls as the horizonlengthens. Eor H = T, the unbiased forecast com-pounds initial portfolio \'aluo at the geometric aver-ago return. For oven longer horizons, one wouldapply a weight greater than 1.0 to the geometricaverage and a negative weight to the arithmeticaverage, resulting in a growth rate below both geo-metric and arithmetic means.''

This analysis sheds light on an apparent para-dox. Assume that returns come from a distributionthat is stable over time. In this case, the 75-yearhistorical return from the 5BB! database ending in2001 would be a reasonable (albeit imprecise) esti-mate of cumulative return over the next 75 years.Compounding at tho historical geometric averageovor a 75-year horizon would (by construction)match the proportional growth in wealth realizedovor the past 75 years. In contrast, compounding attho sample arithmetic average for 75 years, as typ-ically proscribed by the literature, would necessar-ily give a forecast of growth in wealth greater thanthe one realized historically. Following standardpractice thus ensures a forecast of future portfoliogrowth that exceeds historical experience. The biascorrection described in Equation 5 shows that thisforecasting exercise (with T-H = 75 years) actuallywould call for compounding at the geometric aver-age, so the forecast of 75-yoar cumulative returnwould match the return experienced historically.

Indicative BiasesIs the potential bias in forecasts of cumulativereturns economically significant? Unfortunately, itseems to be.

Assume that p = 10 percent and o-21) percent.Figure 2 shows the biases resulting from the arith-metic and the geometric methods. Panel A showsthe forecasted growth of funds over investment

November/December 2003 49

Financial Anah/sts journal

Figure 2. Competing Forecasts of Final Portfolio Value

Wealth ($)

120

100

80

60

40

20

10

A. I orecnst of Final Portfolio Value

U: T = 30

10 20 30 40

Relative Wealth

3.0 r

2.5

2.0

1.5

1,0

0.5

Investment Horizon

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

10 20 30

Investment Horizon40

Notes: Annuai returns are assumed to be lognormal with )I of 10 percent and a of 20 percent. ForecastA 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; andtwo unbiased growth rates, ll, computed on thobasis of historical sample periods of differentlengths. The unbiased estimator uses weights H/Tand 1 -H/Ttoweighttho geometric and arithmeticrates, so different sample periods result in differentestimators. We assumed for growth rates U that pwas estimated by using either a 75-year sampleperiod (the SBBJ period for the United States) or a30-year period (a shorter longth more typical for an

emerging market—or even appropriate for a devel-oped capital market, such as that of the UnitedStates, if one believes that the post-Vietnam erarepresents a structural economic break).

Panel A shows that the upward bias of thearithmetic estimator is severe at long horizons. Incontrast, the bias of tho geometric estimatordepends on tho relationship between H and T.When they are close, the estimator is relativelyunbiased. In fact. Panel A shows that, as seenbefore, for H = T ^ 30 years, the geometric and

50 ©2003, AIMR®

Geometric or Arithmetic Mean

unbiased ostimators aro equal. In general, tho goo-metric and unbiased ostimators for T ^ 30 years donot diverge much in Panel A for investment hori-zons less than 35 years. When the discrepancybetween H and T is greater in Panol A, however—for example, when T - 75 years—the downwardbias in tho geometric estimator is profound. In fact,it is roughly equal to tho upward bias in tho arith-metic estimator at the equivalent horizon.

Panol B presents another view of tho relativebiases—the ratios of arithmetic or geometric fore-casts of cumulative return to unbiased forecasts.Eor T = 30 years, the bias in the arithmetic estimatorrisos dramatically with investment horizon: At ahorizon of H ^ 20 years, the bias is about 30 percent,but at a 40-year horizon, the bias rises to almost 200percent, Eor T - 75 yoars, the arithmetic estimatorperforms much better but is still subject to anupward bias of about 50 percent at a horizon of 40yoars. A longer sample period obviously allows thearithmetic forecast to perform better, but tho stabil-ity of the underlying return process at over-longerhorizons becomes increasingly suspect. Symmetri-

cally, for long T, the geometric estimator can bebiased severely downward. With T-75 years ofdata and an investment horizon of H = 40 years, forexample, tho goometric forecast of final wealth isonly about 60 percent of the unbiased forecast.

The trade-off between long sample periods,which incroaso precision whon tho underlyingreturn process is stable, and truncated sample peri-ods, which disregard older, possibly no longer rep-resentative, data, is highlighted in Table 2. Table 2presents 40-year return forecasts for a small sampleof countries and indexes based on historical sampleperiods of different lengths. The longest seriesavailable from DataStream, for Erance, Germany,and the United Kingdom, are well longer than acentury. Eor the United Kingdom, with a 201-yeardata span, the arithmetic average return is almostequal to the unbiased compounding rato ovon for ahorizon as long as 40 years. But notice that theestimates of p for these countries ovor these longperiods are far lower than estimates derived fromusing the past 82 years; they are smaller still thanestimates based on the latest 52 years. Is the proper

Table 2.

Country/Index

(TSE)

France(SBF250)

Germany(DAX)

UK (FTAS)

Japan(Nikkei)

Hong Kong'MSCI

EmergingMarkets(inS)

Estimates of Compounding Rates

T(years)

78

52

145

82

52

145

8252

201

82

52

52

' 27,6

13,6

BeginDate

1414

1950

1857

1920

1950

1857

1920

1950

1801

1920

1950

1950

31/May/73

Jan 1988

FindDate

2001

2001

2001

2001

2001

2001

2(K)1

2(H)1

2001

200!

2001

2001

2/J.in/()]

July 2001

andSample Estimates

'

4.87<,

6.6

5,1

8-5

8.7

1,9

5.5

8.0

2,4

5,5

6,4

8,8

10.7

8.2

n

lb.7%

14,9

19,7

24.7

22,2

32,2

37.0

22,8

15,6

20,0

24,7

24,1

30,7

24.2

Future Portfolio Values:Compound Growtt

A

6.4%

8,0

7,3

12,2

11,8

7.3

13.1

11,2

3,7

7.8

9,9

12-4

16.7

11,8

G

4,9"'!,

6,8

5.2

8,9

9,1

1.9

5,78,3

2,4

5.76,6

9.2

11.3

8,5

1 Kates

U

5.6"/

7,1

6,7

10,6

9,7

5,8

9.4

9,0

3,4

6,7

8.3

9,9

9.0

2.5

Countries and IndexesFuture Portfolio Value

V(A)

;. $11,9

21,8

16,7

101.5

87.0

17,0

139.5

69,4

4.2

20-1

43.8

107.9

475,8

85.7

1 (0)

$6,8

14,0

7.7

30,0

32,5

2,1

9,0

24,5

2.6

9.0

12,9

33.8

72.2

26,6

ViU)

$9,0

15,5

13,5

56,0

40.8

9.6

36,7

31,2

3.9

13,6

24.2

44.2

31.0

2.7

Noti-s: TSE is the Toronto Stock Exchange; SBF250 is a French index of 250 stocks; DAX is a Germain index of ,30 stocks; FTAS i the FinancialTimes Stock Exchange All-Share Index, For unbiased estimates U, the horizon is 40 years and Ll was computed over the specified sample period.Corresponding forecasts of future portfolio values (relative to an initial $1 investment) tor an investment horizon of 40 years are denoted V{A),V(C),nnd V(U),Obser\ations were sampled annually except for the Hong Kong index (daily) and the MSCI Emerging Markets Index (monthly),

Seng Index (in S) without dividend yield; p = 13,5 with dividend yield.

November/December 2003 51

Financial Analysts journal

conclusion (1) that the estimates based on longerdata series are more reliable by virtue of their largersample size or (2) that structural change over thelast century or two makes returns from the 19thcentury of dubious value for predicting 21st cen-tury returns?

We point out that, regardless of how one mightanswer this question, arithmetic averages based onrelatively recent experience will result in com-pounding rates that are far too high, at least forthese countries. If one accepts the long-durationestimates, the resulting values of |i are far lowerthan estimates obtained from more recent periods.In contrast, if one relies on more recent data, ji ishigher but the unbiased estimator gives moreweight to the lower geometric average becauseH/T is larger. In either case, the compound growthrates that give unbiased forecasts of cumulativereturns are substantially below results obtained byusing conventional arithmetic averages.

The impact of these considerations on forecastsof future portfolio value is dramatic. For any ofthese countries, compounding at the arithmeticaverage return calculated from sample periods ofeither the most recent 82 or 52 years resuits inforecasts of future value that are roughly doublethe corresponding unbiased forecasts based on thesame data periods.

Fmerging markets present an even greaterproblem than developed markets. At the bottom ofTable 2 are estimates for Hong Kong and the MSCIFmerging Markets Index (both in dollar-denominated returns), where available data spanonly 17.6 and 13.6years, respectively, and volatilityis high, especially for Hong Kong. With such shorthistorical estimation periods, the unbiased forecastof future value at H = 40 is below both the geometricand arithmetic forecasts—and by enormous mar-gins. For the Fmerging Markets Index, with 7' 13.6years, the unbiased forecast is only 3.1 percent ofthe forecast obtained by compounding at the arith-metic average and is only 10.1 percent of the com-pound geometric average.

ConclusionA long-standing debate on forecasting future port-folio value has focused on the relative merits of thegeometric versus arithmetic average return as acompounding rate. We have shown analyticallythat when these averages must be estimated subject

to sampling error, neither approach yields unbi-ased forecasts. For typical investment horizons, theproper compounding rate is in between these twovalues. Specifically, unbiased estimates of futureportfolio value require that the current \alue becompounded forward at a weighted average of thearithmetic nnd geometric rates. The proper weightfor the geometric rate is the ratio of the investmenthorizon to the sample estimation period. Therefore,for short investment horizons, the arithmetic aver-age is close to the "unbiased compounding rate,"and as the horizon approaches the length of theestimation period, the weight on the geometricaverage approaches 1. For even longer horizons,both the geometric and arithmetic average fore-casts will be upwardly biased.

We demonstrated that these biases can be eco-nomically significant. For investment horizons of40 years, for example, the difference in forecasts ofcumulative performance can easily exceed a factorof 2. The percentage differences in forecasts growas the investment horizon and the imprecision inthe estimate of the mean return grow.

A consensus is already emerging that the 1926-2002 historical average returns on such broad mar-ket indexes as the S&P 500 are probably higher thanlikely future performance. Our results are evenmore sobering: Namely, the best forecasts of futurecompound growth rates are even lower than theestimates emerging from this research.

We have reserved for future research questionsconcerning the robustness of our results to distri-butional assumptions about rates of return. In cur-rent research (Jacquier, Kane, and Marcus 2002),our results appear robust to heteroscedasticity andserial correlation. In that work, we also addressoptimal forecasting criteria other than bias—forexample, minimum mean-squared-error forecasts.Forecasts satisfying these criteria also depend crit-ically on the ratio of the investment horizon to thesample estimation period; so, they present issuesqualitatively similar to those addressed here.

This article benefited from our diseussions with BryanCampbell, Wai/ne Fcrson, Rene Garcia, and Eric Renaultand tlie comments from seminar participants at Concor-dia, HEC Montreal, University of Montreal, and the2003 meetings of tlie French Finance Association. Ericjacquier acknowledges support from 1FM2—the Mont-real Mathematical Finance Institute.

52 ©2003, AIMR®

Geometric or Arithmetic Mean

NotesRecent studies by Fama and French (2002) and by Jagan-nathan, McGrattan, and Scherbina (2000) show that esti-mates of expected rt'turn derived from a dividend discountmodel are substantially lower than historical averagereturns. This finding suggests that U.S. experience hasturned out better than market participants expectt-d, sohistorical averages have been greater than equilibrium riskpremiums. In addition, databases that are more inclusivethan the Ibbotson series result in reduced historical riskpremiums. For example, Dimson, Marsh, and Stauntori(2002) showed that extending U.S- data to pre-1926 histori-cal periods reduces historical average returns.These books include that of two of the authors (Bodie, Kane,and Marcus 2002, pp- BID-Sll) but also Brealey and Myers(2003, pp. 156-157) and Ross, Westerfield, anci Jaffe (2002,pp. 232-233).For simplicity, we assume the stock portfolio pays no divi-dends. If it did pay dividends, we would simply add thedividend yield to obtain the total rate of return.We focus on the estimation error in .i and ignore possiblesampling error in a. The justification for this simplificationis that if the return distribution is stable, one can estimate aarbitrarily accurately by sampling returns more frequently.In contrast, the precision of the estimate of p depends onthe length of the sampling period and cannot be enhancedby sampling more frequently. Merton (1980) rigorouslydemonstrated this result.

A more formal derivation of this formula for bias may befound in Jacquier, Kane, and Marcus (2002), Blume (1974)was the first to discuss the bias in forecasted portfolio value.Because he assumed normally, rather than lognormally,distributed returns, however, he did not obtain exact for-mulas for expected \alues or bias. A lognormal specifica-tion is preferable also in that it rules out annual returns lessthan -100 percent.This result is similar to that of Blume. He showed that finalwealth is approximalch/ a weighted average of wealth rela-tives based on geometric and arithmetic means. In contrastto Blume's demonstration, we have obtained an exact resultby focusing on drift rates of wealth, rather than wealth perse, under lognormality.Cooper (1446) analyzed the bias arising from estimationerror in the context of discount factors. Because discountfactors involve powers of the reciprocal of the rate of return,however, the biases he found differ from those here. Hefound that both arithmetic and geometric averages result indownwardly biased estimators of the appropriate discountfactor but that the arithmetic average is typically quite closeto the unbiased discount rate. In contrast, we found that thearithmetic average always results in upward bias, that thebias can be substantial, and that the geometric average canresult in either upward or downward bias.

ReferencesBlume, M, 1974. "Unbiased Estimates of Long-Run HxpectedRates of Return," journai of the American Statistical Association,vol. 69, no. 347 (September): 634-638.

Bodie, Z., A. Kane, and A,J, Marcus. 2002, Investments. 5th ed.New York: McGraw-Hill Irwin,

Brealey, R., and S,C. Myers. 2003, Principles of Corporate Financc-7th ed. New York: McGraw-Hill Irwin,

Cooper, 1.1996, "Arithmetic versus Geometric Mean Estimators:Setting Discount Rates for Capital Budgeting-" EuropeanFinancial Management, vol, 2, no, 2 (July):!57-167,

Dimson, E,, P,R, Marsh, and M. Staunton, 2002, Triumph of theOptimists: U)l Years of Global Ini'estment Returns. Princeton, NJ:Princeton University Press,

Fama, E., and K. French. 2002, "The Equity Premium,"Finance, vol. 57, no, 2 (April):637-660,

Ibbotson Associates. Annual yearbooks. Stocks, Bonds, Bills, andInflation. Chicago, IL: Ibbotson Associates.

Jacquier, E,, A, Kane, and A.J. Marcus, 2002, "Optimal Forecastsof Long-Term Returns and Asset Allocation; Geometric,Arithmetic, or Other Means?" Working paper: www.hec.ca/pages/eri.jacquier.

Jagannathan, R-, F,R. McGrattan, and A, Scherbina. 2000. "TheDeclining U.S. Equity Premium." Federal Reserve Bank ofMinneapolis Quarterly Rcvieiv, vol. 24, no. 4 (Fall):3-19,

Merton, R, 1981). "On Estimating the Expected Return on theMarket: An Exploratory Investigation." Iournal of FinancialEconomics, vol. 8, no, 4 ([)ecember):323-406.

Ross, S, A,, R,W, Westerfield, and J, Jaffe. 2002. Corporate Finance.6th ed. New York: McGraw-Hill Irwin,

November/December 2003 53