estimating the equity risk premium and equity costs: new ways of looking at old data

Journal of Applied Corporate Finance S P R I N G 1 9 9 9 V O L U M E 1 2 . 1

Estimating the Equity Risk Premium and Equity Costs: New Ways of Looking at Old Data

by Laurence Booth, University of Toronto

100JOURNAL OF APPLIED CORPORATE FINANCE

ESTIMATING THE EQUITYRISK PREMIUM ANDEQUITY COSTS:NEW WAYS OF LOOKINGAT OLD DATA

by Laurence Booth,University of Toronto

100BANK OF AMERICA JOURNAL OF APPLIED CORPORATE FINANCE

2. R. G Ibbotson and R. A. Sinqufeld, “Stocks, Bonds, Bills and Inflation: ThePast (1926-76) and the Future (1977-2000), Financial Analyst’s Research Founda-tion, 1977. This is now updated annually by Ibbotson and Associates.

1. The Wealth of Nations, p. 136.

As a result it is impossible to ignore the impact ofinvestment horizon on the discount rate. It is gener-ally acknowledged that the risk-free rate depends onthe time period over which the cash flows are beingvalued. For example, the 30-day Treasury bill yieldreflects the risk free rate for a 30-day holding period,but it is not a risk-free rate for an investor with aninvestment horizon either longer or shorter than 30days. Similarly, the 20-year Treasury yield reflects therisk-free rate for a 20-year investment horizon. Anadditional complication for long-term bond yields isthe reinvestment rate risk associated with reinvestingthe intermediate coupons. However, to the extent thata similar reinvestment rate risk exists for project cashflows, the long-term Treasury bond yield more closelymatches the investment horizon for typical capitalbudgeting and valuation problems than does theTreasury bill yield. As a result, in practice most equitydiscount rate calculations start out with the long-termTreasury bond yield as the appropriate risk-free rate.

The focus in calculating an equity discount ratethen quickly shifts to adding the appropriate riskpremium to the yield on long treasuries. In practice,much of this effort focuses on the estimation of therelevant risk or “beta” coefficient, with relatively littletime spent on the equity risk premium. Normally theequity risk premium is taken to be the long-runaverage excess return of equities over bonds takenfrom the data compiled by Ibbotson and Sinquefeld.2

For example, this is the common approach taken inregulatory hearings to determine the equity discountrate, with either a CAPM risk premium or “averageutility risk premium” added to the long bond yield.

demand extra profit or return in compensation forbearing it, is a cornerstone of modern finance.Further, it was the major factor in the award of theNobel prize in economics to William Sharpe andHarry Markowitz for the development of modernportfolio theory and the Capital Asset Pricing Model(CAPM). Their contribution was to show that the riskpremium, or the extra “rate of profit” in Smith’sterminology, is related to the contribution of asecurity to the risk of a diversified portfolio. Eversince, required rates of profit or discount rates havepredominantly been determined by a three-stepprocedure: (1) estimate the appropriate risk free rate;(2) estimate the equity market, or average, riskpremium; and (3) adjust the equity risk premium forthe particular risk of the security or firm.

For the CAPM, this procedure amounts toimplementing the following equation:

Kj = RF + (E(Re) – RF) * βj (1)

where Kj is the required return or discount rate forsecurity j, RF the risk-free rate, and the equity riskpremium (ERP) is just the difference between theexpected return on the market (E(Re)), ie., theaverage of all securities, and the risk free rate, andβj the security’s beta coefficient.

While the CAPM is a “single period” model,most problems in finance involve multiple periods.

n The Wealth of Nations Adam Smithstates that “The ordinary rate of profitalways rises more or less with the risk.”1

This idea that investors dislike risk, and

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In this paper, I reverse this sequence of events.Instead of focusing on beta coefficients, I focusdirectly on estimating the equity return and thenconsider, among other things, whether it makessense to estimate this by means of adding a constantrisk premium to the prevailing long Treasury yield.Critical questions raised by this process are whetherthe equity risk premium is stable, and whether thecommon practice of adding a premium to the longTreasury yield is “better” than just using the averagehistoric equity return directly and ignoring theperformance of the bond market entirely. I alsoconsider the question of which equity return shouldbe used: That is, is it better to estimate an averagenominal equity return and use that as a forecast offuture nominal returns, or should a real return beestimated and then the current inflation rate beadded?

This paper shows that all of these approachesare subject to potential bias of one kind or another,and that there is thus no automatic “right” answer.Instead, capital market evidence and knowledge ofeconomic events should guide the user’s choiceamong these different estimation techniques.

ESTIMATION PRINCIPLES

The main source of data on the equity riskpremium comes from the seminal work of Ibbotsonand Sinqufeld, who have calculated holding periodreturn data from December 1925 to the present forcommon equities, long-term government bonds,Treasury bills, and the consumer price index. Ibbotsonand Sinquefeld claim that the equity risk premiumover Treasury bills follows a random walk and that,for this reason, the historic average excess return isthe best forecast of the current equity risk premium.3

To understand their conclusion, let’s adopt theirassumption (an assumption I will question later) thatthe equity risk premium is a random walk, as shownin equation 2, with constant mean and constant,independent and identically distributed error terms:4

ERPt = µ + εt (2)

where ERPt, is the actual excess return of equities(Re,t) over 30-day Treasury bills at time t, which is

equal to the constant equity risk premium (µ) plusa random error term (εt). The estimated averagevalue of equation (2) over T periods can then beexpressed as follows:

AERP = µ + (ΣTt=1εt)/T (3)

If the error terms are indeed independent andidentically distributed each period, then the averagefrom T period observations is simply the true equityrisk premium plus the average error term, which withenough observations will eventually sum to zero. Asa result, with larger and larger time periods we canbe more and more confident that the estimatedaverage risk premium is equal to the constantexpected risk premium.

Ibbotson and Sinquefeld calculate the simpleaverage equity risk premium in the above mannerand suggest that this be added to the current 30-dayyield on a Treasury bill to estimate the current equitydiscount rate. This suggestion is then dutifully fol-lowed in almost every introductory finance textbook.

Leaving aside the problem that the 30-dayTreasury bill yield is not a useful risk-free rate formost problems in finance, the astute reader mightwonder why the average equity market return is nota better measure of the expected return on themarket? That is, if the return on equity follows thesame process as equation (2), why not estimate theaverage equity return directly, rather than by addingthe average excess return to a Treasury bill yield?

The reason given by Ibbotson and Sinquefeld isthat the return from holding Treasury bills does notfollow the above process. Because T-bill yieldsfluctuate with monetary policy, they are seriallycorrelated, and thus have clearly had different aver-age values at different points in time. Further, if stockmarket risk and the market price of risk (riskaversion) have been constant over time, then theequity risk premium will have followed a randomwalk, and equity returns are generated by thefollowing process:

Re,t = R30,t + µ + εt (4)

where R30,t is the return on 30 day Treasury bills. Asimple average of equation (4) is then

4. This is a Random Walk 1 as discussed by J. Campbell, A. Lo and A. C.Mackinlay, The Econometrics of Financial Markets, Princeton, 1997. We will usethe term “random walk” to mean the above process.

3. For example, Stocks, Bonds, Bills and Inflation 1986 Yearbook, Ibbotsonand Associates, Inc. 1986, p. 71.


(ΣTt=1Re,t)/T = (ΣT

t=1R30,t)/T + µ + (ΣTt=1εt)/T (5)

where the average equity return is equal to theaverage T-bill yield plus the equity risk premium plusthe average error term. Even if equation (5) issummed over very long periods, there is no reasonto believe that the average equity return will equalthe expected return; in fact, it will only do so if theaverage T-bill yield is equal to the current T-bill yield.This imparts what I will call a risk free rate bias to theequity return estimate. For example, suppose theaverage equity return is 10%, composed of anaverage T-bill return of 6% and an equity riskpremium of 4%. If during the business cycle T-billyields go from 4% to 8%, then the use of the average10% rate will overestimate equity costs when T-billyields are low and underestimate them when theyare high. The problem, of course, is that short-termT-bill yields are extremely volatile, so that even if theaverage T-bill yield over a business cycle is constant,using equation (5) within a business cycle will causeserious estimation errors.

Ibbotson and Sinquefeld’s justification for esti-mating the equity risk premium and adding it to thecurrent Treasury Bill yield is correct, provided thatthe process driving equity returns is that given inequation (4). At its core the assumption is that equi-librium asset prices are determined based on short-term investment horizons and that the correct risk-free rate is the return on a 30-day Treasury bill. Anequally plausible argument, however, is that thecorrect risk-free rate for the equity market is a one-year Treasury bond yield, since this is closer to theimplicit investment horizon, and thus the opportu-nity cost, of the average investor.5 More to the point,it is the horizon over which most cash flows arediscounted for valuation purposes. However, even ifIbbotson and Sinquefeld’s assumption is correct, theresulting equity discount rate is not very useful forvaluation purposes, since the correct “risk-free rate” forthese purposes is closer to the long Treasury yield.

Assume more realistically that the correct risk-free rate used for valuing equities is unknown—andlet’s call it Rm for a medium-term rate, but the pointis that we simply do not know what it is. In this case,equities, long-term bonds, and Treasury bills are allrisky. However, since we are not interested in Trea-sury bills, let’s just consider the returns for equities,

Re,t = Rm,t + µ + εt (6)

and long bonds,

Rl,t = Rm,t + TPt + ηt (7)

where the long term bond return is equal not onlyto the unknown risk-free rate plus an error term (ηt),but also a term premium (TPt) to capture the fact thatlong-term bonds are riskier than short-term bonds.In this case, the average returns for both equity andthe long bond are subject to the same “risk-free rate”problems as in equation (5), where the “correct” risk-free rate is the Treasury Bill yield. However, if thecorrect risk-free rate is a medium term rate, it is lesssusceptible to the volatility due to monetary policythan is the Treasury Bill rate, and the risk-free ratebias is reduced.

The equity risk premium measured over longbonds can then be expressed as follows:

Re,t – Rl,t = µ – TP + εt – ηt (8)

and the average equity risk premium estimated overT periods is

AERP = µ – ΣTt=1TPt/T + ΣT

t=1(εt – ηt)/T (9)

Equation 9 will produce an estimate of the riskpremium over the true risk-free rate that is biasedlow, since long bonds are themselves risky andhence command a term premium.

The implications from equations (6), (7), and (9)are several. First, if the true risk-free rate is of a longerterm than the 30-day return on a 90-day T-bill, thenthe “risk free rate bias” is reduced. In fact, there maybe no reason for using the risk premium approachat all, depending on the behavior of medium-termyields. Moreover, since the 30-day T-bill is also riskyfor a longer investment horizon due to reinvestmentrate risk, there is nothing special about the statisticalproperties of an equity risk premium estimated overTreasury bills.

Second, an equity risk premium estimated overlong bonds is composed of three distinct compo-nents: the true equity risk premium over the un-known risk-free rate; the average term premiumattached to long term bonds; and the residual

5. The obsession with monthly data in finance is driven more by statisticalneeds for “enough” current observations, than any fundamental economic reasons.


estimation error. This latter component criticallydepends on the relationship between interest raterisk and market risk. If the uncertainty in equityreturns is driven by a simple factor process such as

εt = Inflationt + productiont + default riskt + ηt (10)6

where uncertainty in long bond returns (ηt) arisingfrom interest rate changes is a factor driving theequity market as well, then the error term attachedto the equity risk premium in equation (9) will besmaller than directly estimating the average equityreturn. Anything that lowers the estimation risk isbeneficial, since with normal standard errors theestimated average returns are subject to considerableestimation error.7

Third, the risk premium estimated from equa-tion (9) will only be accurate if the average termpremium is equal to the current term premium,otherwise there will be what I will call a termpremium bias. Equity market returns distill the effectof all risks, so that it might be reasonable to assumethat this overall market risk is constant. However,bond market returns are driven mainly by interestrate changes, and it is questionable to assume thatthis term premium has remained constant over thefull time period.

Finally, although equation (6) assumes thatequity returns are generated as a risk premium overa risk-free rate, an alternative longstanding assump-tion in finance is that nominal returns are generatedas a premium over the expected inflation rate. Inequation (6), this would mean that instead of the risk-free rate bias, we would have an inflation rate bias.That is, using the nominal average return implicitly

assumes that the average inflation rate over theestimation period is equal to the current inflationrate. This bias can be removed by calculating theaverage real return by deflating the nominal returnby the change in the consumer price index. Theassumption would then be that the real equity returnfollows equation (2).

The upshot of the above discussion is that it isby no means certain that the equity return is betterestimated indirectly from a risk premium added toa bond yield rather than directly from the proper-ties of realized equity returns. All approaches in-volve known biases and will be subject to estima-tion error. It is a major theme of this paper thatwhich is the best approach can be determined onlyafter an analysis of the data and an understandingof what economic events have occurred over theestimation period. In contrast, the justification givenfor the “risk premium” approach by Ibbotson andSinquefeld is based largely on assumption, notanalysis of the evidence.

WHAT THE DATA TELL US

Table 1 gives various estimates of the averagerealized returns on different security classes for theoverall period 1926-1997.8 Before discussing thedata it is useful to digress on what is meant by an“average” rate of return. The arithmetic mean (AM)is the simple average of the annual rates of return.The geometric mean (GM) is the compound rate ofreturn earned between December 1925 and Decem-ber 1997. The OLS is the ordinary least squaresestimate of the annual growth rate in wealth investedin each asset class.9

TABLE 1ANNUAL RATE OF RETURNESTIMATES 1926-1997

S&P Long US 90 Day Real ExcessEquities Treasury Treasury Bills CPI Return Return

AM 12.95 5.59 3.80 3.20 9.75 7.36Standard error 2.38 1.07 0.38 0.53 2.41 2.43GM 10.99 5.22 3.75 3.10 7.89 5.77OLS 10.91 4.17 3.75 3.79 7.12 6.74standard error 0.20 0.19 0.18 0.12 N./A N./A

6. See for example, N. Chen, R. Roll and S. Ross, “Economic Forces and theStock Market,” Journal of Business, 59, 1986.

7. The standard error of the estimated mean is calculated in the normal wayas the standard deviation divided by the square root of the number of observations,that is, the time period. With 72 years of data and an average equity volatility of20%, the standard error of the estimate is over 2%, since the time period can notusually be changed anything that lowers the standard errors makes for a moreaccurate estimate of the unknown ‘true” expected equity return.

8. Data for 1926-1995 are the Ibbotson and Sinquefeld data on the CRSP datafiles with 1996 & 1997 data updated manually. The estimates are obtained startingfrom December 1925 from the monthly total return indexes.

9. The OLS estimate is obtained by a log-linear regression of the logarithm ofthe cumulative wealth index against time, with the resulting continuous growth rateconverted to an effective annual rate by the anti-log function eX. Note the standarderror of this estimate reflects the lesser variability of total wealth as compared tothe annual return.

All of the approaches to calculating equity costs are subject to potential bias of onekind or another, and there is thus no automatic “right” answer. Instead, capital

market evidence and knowledge of economic events should guide the user’s choiceamong these different estimation techniques.


All three estimates of the average return arevalid and the choice between them depends on theunderlying reason for deriving the estimate. The GMis the compound rate of return over the entire 72-yearperiod; essentially it is the single arithmetic returnthat treats the whole period as a single period. Incontrast, the AM is the simple average of all theannual rates of return. The AM always exceeds theGM, and the difference increases by approximatelyhalf the variance in the rate of return. As a result, thedifference between the AM and GM is greatest for thecommon stock returns and smaller for the CPI and90-day T-bill series. If the data is needed to assesspotential portfolio performance, and if the investor’sholding period is between one and 72 years, then the“best” estimator10 is a weighted average of the AMand GM with the weights depending on the particu-lar holding period.

However, our concern is not with portfolioperformance, but with the estimation of the annualrate of return required by an investor to discountannual cash flows. For this purpose, the AM esti-mated over annual holding periods is probably moreaccurate, although the actual choice depends on theunknown investment horizon.

The final estimate in Table 1 is the ordinaryleast squares (OLS) estimate of the annual rate ofreturn, which (since it is assumed to be reinvested)is also the growth rate in investors’ wealth. Unlikethe simple AM, which weights each annual returnequally, the OLS estimator minimizes the variancein the forecast error. As a result, it should be a“statistically” better estimator.

The message from the data in Table 1 seems tobe straightforward: common equities have earnedbetween 11-13% and long Treasuries 4-6%, onaverage, depending on the estimation method. Thus,the excess return of common stocks over long-termgovernment bonds has been in the range 6.7-7.4%for annual holding periods, declining to 5.8% as theholding period is lengthened. With long Treasuryyields at 6.0% in April 1998, adding the 6.7-7.4%would yield a range of 12.7-13.4%. This estimatewould be at the top of the range derived from thedifferent estimates of the simple average nominalequity return.

Over the same time period, the real equityreturn has been between 7.1% and 9.8% for annual

holding periods with 7.9% for a longer horizon.Note that the volatility of the inflation rate, particu-larly in the 1930s, causes a significant differencebetween the OLS and AM estimates of the averagereal equity return. If a 2.5% inflation estimate isadded to the average real return estimates, thenominal equity return would be in the 9.6%-12.2%range, which is at the lower end of the range ofestimated simple nominal equity returns.

Clearly, the choice of estimation technique canresult in quite large differences in the estimate ofthe expected equity return. This in turn wouldmean large differences in estimates of the cost ofequity capital for different types of firms. Note alsothat the standard error of the excess return esti-mate is greater than the standard error of thedirect estimate, implying that the estimate of therisk premium has not benefited from any interestrate “factor” risk driving equity returns over thewhole period. The choice between the threeestimating techniques rests, therefore, on therelative importance of the “risk-free rate” and“inflation rate” biases of the direct estimates andthe “term premium bias” of the indirect riskpremium estimate.

One way of examining these potential biases isto calculate annually updated averages from equa-tions (6) and (9). Suppose, for example, that wecalculate the average from the first five observationsand then successively add additional observations.If equation (6) holds for the equity return and therisk-free and inflation rate biases are relativelysmall, the estimated “forward averaging”processwill eventually “zero in” on the true mean.

Figure 1 illustrates this process, where therate of return is assumed to have a mean andstandard deviation of 12% and 20%, respectively.For simplicity there are 72 observations—thesame as for the time period in Table 1. Note thatwhile the average oscillates around 12%, thefluctuation around the mean gets smaller andsmaller.11 How much oscillation there is simplydepends on the initial random returns; large“outliers” early on have a more dramatic effectsimply because they are averaged over fewerobservations. Surrounding the average are the95% confidence bands for the estimate. Note thatthe confidence bands get narrower, but since

10. See Marshall Blume, “Unbiased Estimators of Long Run Expected Rates ofReturn,” Journal of the American Statistical Association, (September 1974).

11. Note that this was the result of one simulation, other simulations wouldget different results.


they decline with the square root of the numberof observations (time), the effect of an increasingtime period decreases.12

If the forward averaging exercise is repeatedwith the actual data, we get the pattern shown inFigure 2. There are three forward averaged series:the nominal equity return, the real equity return,and the excess return of equities over bonds.Overall, all three series seem to exhibit a “hump”in the middle years of 1950-1980, indicating in-creasing returns and then decreasing equity re-turns. This is particularly apparent with the realequity and risk premium returns.

Although visual examination can be suspect,since the variability in the equity return is so large,there is little in the Figure 2 to suggest that one seriesfits the random walk model better than another. Ifanything, the simple average nominal equity returnseems to “zero in” on the average more quickly thaneither the equity risk premium or the real equityreturn average.

Another way to look at the data is to rememberthat, from the random walk model of equation (6),each year’s rate of return or risk premium is assumedto be independent and drawn from the same distri-bution. It follows that there is nothing “special”about calculating an average starting out in 1926 andthen working forward to 1997. Instead, we can justas well start from the average return for 1993-1997and work backwards by adding historic data untilwe again get the average for the full 72-year period.

(In fact, if the assumption that the return each yearis independent is not a valid one then, this processmay make more sense since it implicitly weightscurrent data more heavily.)

The pattern that results from this “backwards”averaging process is shown in Figure 3. The lastobservation at the right is for the period 1993-1997.As one then moves towards the origin, progressivelyolder data is continuously added until the firstobservation, marked 1926, adds the very first obser-vation. This first (1926) observation thus representsthe overall average (and is the same as the last (1997)observation in the forward average return graphshown in Figure 2).

FIGURE 1SIMULATED MEAN ESTIMATION: RANDOM WALK MODEL

FIGURE 2FORWARD AVERAGING: 1926-1930 UNTIL 1926-1997

FIGURE 3BACKWARDS AVERAGING: 1993-7 TO 1926-1997

12. With a 20% standard deviation of annual equity returns, the standard errorof the estimated average return is 6.3% with 10 observations, 2.0% with 100, and

1.4% with 200. We would have to wait 10,000 years to get an estimate accuratewithin 20 basis points! Not many managers can wait this long.

The justification given for the “risk premium” approach by Ibbotson and Sinquefeldis based largely on assumption, not analysis of the evidence.


A partial explanation for the results for the riskpremium average can be gained by looking at thetrend (shown in Figure 4) in fixed income yields overtime. Both the yield on long Treasuries and 90-dayT-bills are available back to 1934 and, when com-bined, they certainly bracket the “true” risk-free rateused in equity pricing. A simple visual examinationreveals that yields were quite constant until the early1950s, when they began their long upward trend thatfinished in the early 1980s; since then, they haveconsistently trended downwards. Since it is thechange in yields that generates the uncertainty in theholding period returns for long bonds, it is clear thatactual returns were lower than anticipated from the1950s through 1980, and higher than anticipated inthe last 15 years or so. It is also clear that “interest raterisk” and the term premium have not been constantover the entire period.

Understanding the behavior of interest ratesover the last 72 years makes it clear that trends inaverage equity returns and average equity riskpremiums have been affected by the significantchanges that have occurred in the bond market. It isa reflection on the volatility in equity returns that theclear trend in bond market yields and returns doesnot generate a clearer pattern in the average equityrisk premium—one that would allow us to conclu-sively state that whether or not the equity riskpremium series has followed a random walk with aconstant mean. However, although the “saucer”shape of the average equity risk premium in Figure3 could have resulted randomly, knowledge of thetrend in monetary policy that caused the temporalchange in market interest rates provides a clearindication that the equity risk premium series has asignificant term bias.

WHAT HAS HAPPENED TO BOND MARKETRETURNS?

The previous averaging data indicated thatneither the averages nor their volatility have beenconstant over time, and that changes in the bondmarket were a likely source of some of the prob-lems. One way of looking at this is to examine“rolling” instead of “updated” averages to see whethereither the average return or the volatility has changedover time.

FIGURE 4FIXED INCOME YIELDS

13. Variance ratio tests and “Q” autocorrelation tests indicate the absence ofsignificant autocorrelation in annual returns over the whole time period.

The interesting point about the backwards av-eraging graph in Figure 3 is that, for all three series,the average decreases as you add older data untilabout 1966 (or 1980, in the case of the risk premiumaverage). At this point, they all begin to increasebefore zeroing in on the long run average. Contrast-ing the forward and backward averages, it is difficultnot to conclude that the latest period has witnessedhigher equity returns than the earlier period. Thisconclusion comes from the slow oscillation in thebackwards averaging graph—particularly for the riskpremium average, which shows a definite “saucer”shape. For the earliest period, there is initially morevolatility (but, as will be discussed later, it was a delib-erate decision to create the Ibbotson and Sinquefelddata starting five years prior to the most “extreme” valueof all—the 1929 stock market crash). As a result, the1925 start date was not chosen randomly and the for-ward averaging from 1925 is inherently biased.

The conclusion drawn from the forward andbackward averaging graphs in Figures 2 and 3 is thatthere is so much volatility in the equity return seriesthat it is difficult to state conclusively whether any ofthe series fits the random walk model with a constantmean and variability. In fact, none of the series seemto be free of bias, with the risk premium series havingthe more obvious problems.13 This in turn impliesthat none of the simple averages can be used as anaive forward-looking estimate for calculating dis-count rates. This conclusion is particularly apt for therisk premium series.


Figure 5 shows the standard deviation (volatil-ity) of actual equity and bond returns over a rollingten-year period starting in 1926-1935 and endingwith the period 1988-1997. The choice of a ten-year“window” is arbitrary, but it should capture a longenough period to include a full business cycle andyet be short enough to capture changes over time.

Some immediate implications are apparent.First, note that equity market risk peaked during the1930s, when as we saw earlier the forward averagereturn suffered the most oscillation, and then seemsto have declined ever since, punctuated by in-creases caused by periodic major market move-ments. Although this is probably the result of choos-ing the 1925 start date, there is little in the data toindicate that equity market risk has increased. Infact, the data suggest the opposite—namely, thatequity market risk has been on a long-run seculardecline since the 1930s.

Also note that, in contrast to the equity market,bond market risk has undoubtedly increased. Bondmarket risk seemed to have been moderate in the1930s, before declining into the 1940s and 1950s, aperiod during which interest rates were tightlycontrolled. As a result, there was relatively little bondmarket uncertainty until 1982, when interest ratesplummeted.14 From 1982 to 1997, the ten-year stan-dard deviation of bond market returns was essen-tially the same as the declining risk in the equity

market (even after the huge gains of 1982 rolled outof the ten-year estimation window).

This change in bond market relative to equitymarket risk raises questions about the assumption(in equations (6) and (7)) that the error termsgenerating equity and bond market uncertainty areconstant. In practice, it seems highly unlikely thatreturns have been generated by processes withconstant error terms. A “better” working assumptionwould be that the error term in equation (7) hasincreased for bond market returns, while possiblydecreasing for the equity market. Moreover, if therelative uncertainty has changed, it also seems likelythat investors have reacted by changing their ex-pected return requirements. Why, for example,would investors expect the same return from equi-ties in the 1990s as in the 1930s, when equities havebeen half as volatile? Moreover, why would investorsin the 1990s expect the same risk premium of equitiesover long bonds as in the 1930s when stocks andbonds had roughly the same level of risk—ascompared to the 1930s when equities were six timesas volatile?

I will return to the equity market later, but thereare also more direct implications for the bondmarket. If investors hold diversified portfolios, whatdoes the increased bond market risk mean for theiroverall portfolio? That is, how much of the increasedinterest rate risk is diversifiable?

Figure 6 shows the rolling bond market “beta.”It is estimated from ten years of annual data (ratherthan the normal five years of monthly data) to cap-ture the effects of the assumed annual holding pe-riod. What is immediately apparent is that the in-crease in bond market volatility has been closelyassociated with the levels of bond market betas. Inthe 1930s, bond betas were of the order of 0.10, andthen they declined when interest rate controls wereimposed. They did not become significant again untilthe 1980s. But, by 1990 (covering the period 1981-1990), bond market betas had climbed to about 0.80!Clearly, bond market uncertainty has had a system-atic as well as an unsystematic component.

Figure 7 shows rolling bond market betasestimated using a five-year window and monthlyrates of return. There is no reason risk should be thesame for a monthly as an annual investment hori-

FIGURE 5UNCERTAINTY IN FINANCIAL MARKETS:STANDARD DEVIATION OF RETURNS OVER ROLLINGTEN YEAR YEAR PERIODS

14. Note that higher interest income was offsetting capital losses on bonds asinterest rates increased. As a result, returns were less than anticipated, but bondmarket volatility did not increase with the general level of interest rates. For

example, in 1979, 1980 and 1981 bond market returns were –1.2%, -3.9% and 1.9%,in 1982 the return was 40.4% as higher interest income was combined with largecapital gains.

The choice of estimation technique can result in quite large differences in theestimate of the expected equity return. Moreover, the standard error of the excessreturn estimate is greater than the standard error of the direct estimate, implying

that the estimate of the risk premium has not benefited from any interest rate“factor” risk driving equity returns over the whole period.


zon, and a comparison of Figures 6 and 7 clearlyshows this. For example, although the beta esti-mates show the same general pattern—low esti-mates initially, followed by a decline and then anincrease, as interest rates became increasingly morevolatile—there are differences. As the investmenthorizon shortens, price volatility dominates incomevolatility. Moreover, with a shorter window, ex-treme “unusual” events pass out of the estimationwindow more quickly. For example, the five-yearbetas drop precipitously starting in October 1987.The reason for this is the stock market crashed by21.5%, and to prevent panic the Federal Reservelowered interest rates, causing a 6.2% bond marketreturn. This one negative correlation is so large thatall monthly estimates that include October 1987show a “break.” As October 1987 passed out of thefive-year window in October 1992, beta estimatesstarted to increase again.15

The above discussion of five-year versus ten-year bond betas is important for two reasons. First,it emphasizes the earlier conceptual problem withinvestment horizons. Overall, 1987 was a good yearfor the stock market, it actually went up by 5.2%,while the bond market went down by 2.7%. As aresult, an investor with a one-year horizon wouldlook at 1987 differently from one with a one-monthhorizon. Second, and more fundamental, risk doesn’tdisappear just because an event has not happenedduring a particular estimation period. Bond marketrisk didn’t increase in October 1992 simply because

there was no crash in the previous five-year periodand, as a result, no opportunity for bonds to demon-strate their attributes as a “safe harbor.”

Even accounting for the subtleties of estimat-ing bond market risk, it is clear from both the five-and ten-year bond market betas, as well as thevolatility estimates, that bond market risk has sub-stantially increased in the last third of the period1926-1997. It is highly unlikely, as a result, that theterm premium demanded by long-term bond inves-tors was the same in this latter period as in theearlier period. As a result, the “term premium bias”in using the equity risk premium over bond yieldmethod is likely to be substantial. For example, inequation (8) the implicit term premium subtractedfrom the estimated average risk premium will beweighted 2/3rds for the period when the termpremium was very low and only 1/3rd for the latterperiod, when it was high. This risk premium willthen be added to a current long-term bond yieldthat fully reflects the current term premium. As aresult, the bond yield plus method will unambigu-ously overestimate equity discount rates.

To put the same thing a little differently, ifinterest rate risk is a factor in market risk (as the largeand significant bond market betas indicate), thenbonds should be priced according to the CAPM aswell as equities. In this case, the equity risk premiumover long term bonds is as follows:

ERP = (E(RM) – RF) * (βe – βl) (11)

FIGURE 6US TREASURY “BETAS”

FIGURE 7BOND MARKET BETAS

15. In daily data, as an extreme, almost all the volatility is from price changes.


In the earlier period, where the bond marketrisk was low (with bond betas of 0.1), the estimatedexcess return of equities over long term bonds wasestimating almost the full equity risk premium. But,in the later period when bond market risk was high(with bond betas of 0.8) it is only estimating part ofthe full risk premium. Adding an average of theseexcess returns to the current risky long term bondyield will produce an upward-biased equity cost. Infact, if current bond beta estimates of 0.5-0.6 arevalid, the risk premium of low risk equities over longterm bonds should be close to zero. This has majorimplications for low-risk stocks, like regulated utili-ties, that have betas in the 0.5-0.6 range, and yetestimate equity costs by the bond yield plus method.

WHAT HAS HAPPENED TO EQUITY MARKETRETURNS

If the evidence from the bond market is that theterm premium bias in the average equity risk pre-mium estimate is significant, what about the equitymarket and the risk-free and inflation rate biasesdiscussed earlier? Here what is important is that, oncewe recognize that the term premium bias invalidatesthe average excess return over bonds (and that theexcess return over T-bills is not a meaningful num-

ber), we should then look objectively at the data onequity market returns in isolation from the bond andbill markets.

Data on the equity market has been pushedback to 1802 by Schwert.16 However, the earlierperiod only includes railroads and financial firms.In an article published in 1987, Wilson and Jones17

“cleaned up” the original Cowles data set thatextends back to 1871. As Wilson and Jones explain,this is data of “comparable quality” to the Ibbotsonand Sinquefeld data that starts in December 1925.The only reason for starting in December 1925 wasFisher and Lorie’s18 desire to capture “at least a fullbusiness cycle before the 1929 stock-market crash,”a practice followed by Ibbotson and Sinquefeld.This in turn explains why the equity return seriesdoes not start at a random point in time, instead itdeliberately starts five years prior to the great crash,which biases both the volatility and average returnestimates. The Cowles data starts in 1871, when itconsisted of 31 railroads, 4 utilities, and 13 industri-als; by 1925 it consisted of 29 railroads, 22 utilities,and 207 industrials. The 1871 start date is random inturns of subsequent return estimates, since it waschosen for other reasons.19

The forward averaging of the returns is re-peated for the overall period in Figure 8. Note thatthe nominal return does not seem to have a con-stant mean. The series oscillates for the first 20 yearsor so, but then increases almost continuously fromthe low around 1890. In contrast, the forwardaverage of the real equity returns starts out higher(since there was deflation almost throughout thelatter part of the 19th century), and finishes lower(because of the more recent inflationary period).The average of the real equity return “looks” morelike a random walk with a constant mean, since itdoes not show a drift over time.

The backwards averaging is shown in Figure 9.Again the last observation on the right is for theperiod 1993-1997 and, as the data moves closer to theorigin, older data is added until the first observationfor 1871, which is the average for the whole period1871-1997. Note that the averages again fall, reflect-

FIGURE 8FORWARD AVERAGING: FROM 1871-1875 TO 1871-1997

16. G. William Schwert, “Indexes of Common Stock Returns from 1802 to1987,” Journal of Business 63-3, 1990.

17. J. W. Wilson and C. P. Jones, “A Comparison of Annual Common Stockreturns: 1871-1925 with 1926-1985,” Journal of Business 60-2, 1987. This is also thesource of the early inflation data.

18. L. Fisher and J. Lorie, “Rates of Return on Investments in Common Stocks,”Journal of Business 37-1, 1964.

19. The original Cowles data apparently started in 1871 for two main reasons:first, prior to that date the market was basically railroad stocks; second there werea large number of changes in both securities regulation and trading rules on theNYSE introduced in the 1860s. Neither reason is cause for concern that the startdate was chosen specifically to include particular return observations.

If current bond beta estimates of 0.5-0.6 are valid, the risk premium of low riskequities over long term bonds should be close to zero. This has major implications

for low-risk stocks, like regulated utilities, that have betas in the 0.5-0.6 range.


ing the recent equity bull market, but that thenominal average continues to fall almost continu-ously through to 1871. In contrast, the average of thereal returns falls through to 1971, increases back to1951, and then “zeros in” on the long-run average.Again, of the two series, only the average of the realequity return seems to be consistent with a randomwalk model with a constant expected (real) return.

If the average real equity return seems to be theonly candidate for a simple estimate of the long-runequity market return, we need to determine if thelong-run average “hides” obvious periods of vary-ing returns. To see this, Figure 10 gives the rollingten-year average of the real equity return. Over thewhole time period, the average real equity returnwas 9.02%, and the standard error of the estimate is1.71%.20 However, for an estimate from ten years ofdata the standard error is 6.1%, so that we canexpect, with a 95% confidence interval, the ten-yearaverage real return estimate to be between about –13% and 21%! The actual mean estimates are justwithin this range, but the significant point is thatthere does not seem to be any trend across time.Current average real equity returns are close to thehigh teens that were experienced prior to the 1929stock market crash and the high inflation period ofthe 1970s, both of which subsequently saw muchlower real equity returns.

In Figure 11, the volatility of real equity marketreturns is measured in the same way as a rolling ten-year standard deviation of real returns. Similar to the

average equity return, the volatility is subject toestimation error and fluctuates quite widely depend-ing on whether or not an “extreme” value occursduring the rolling ten-year period. For example thestandard deviation of real equity returns was about19% until 1889, when the 57% gain of 1879 droppedout of the estimation window. The volatility thenstayed around 12-13% until 1907 when the realequity return was –30.8% followed by +47% in 1908.Overall, between 1871 and 1997, the standard devia-tion of real equity returns was 19.24%.

A final way of looking at the volatility of realequity returns is to graph the absolute value of theirannual returns (see Figure 12). The reason for

FIGURE 9BACKWARDS AVERAGING: FROM 1993-1997 TO 1871-1997

FIGURE 10AVERAGE REAL EQUITY RETURNS: ROLLING 10 YEARAVERAGE

FIGURE 11VOLATILITY IN REAL EQUITY: ROLLING 10 YEARESTIMATE

20. The average nominal return was 11.04% with a standard error of 1.67%


looking at absolute values is that it is the magnitudeof the extreme returns that causes the fluctuation inthe volatility estimate and not their sign. Drawinginferences as to whether the equity market hasbecome more volatile over the last 129 years fromFigure 12 is extremely difficult. Major market move-ments seem to occur every 20 to 30 years and havedone so for the last 129 years. Also the Great Crashof 1929 is an outlier, primarily because it was adownwards correction, though annual returns of asimilar magnitude have occurred at other times inthe past.21

In sum, the volatility figures shown abovesuggest that Ibbotson and Sinquefeld’s choice of1925-1997 data (with its start date so close to theGreat Crash) is largely responsible for the conclu-sion that risk has decreased in the equity market. Ifwe instead use data going back to 1871, it becomesmuch more difficult to justify a “decreasing risk”conclusion. Instead, it is hard not to conclude thatthe equity return is approximately driven by a realreturn process with a mean of about 9.0% and aconstant error term.

CONCLUSION

Estimating discount rates is a critical part offinance. The standard approach based on risk-basedpricing models, such as the CAPM, is to estimateequity returns based on a risk-free rate plus a risk

premium. For conventional valuation and capitalbudgeting purposes it is well accepted that the risk-free rate should be a long Treasury yield. In contrast,Ibbotson and Sinquefeld suggest that the equity riskpremium be estimated over Treasury bills, based onthe assumption that the equity risk premium, and notthe full equity return, follows a random walk with aconstant mean. This idea has also been adopted forestimating equity returns as a premium over longbond yields, with the implicit assumption that theexcess return of equities over long bond returns alsofollows a random walk.

In this paper, the assumptions underlying theseestimation techniques have been investigated. Threespecific models have been examined—namely, that(1) the nominal equity return, (2) the real equityreturn and (3) the equity risk premium over longbonds each follows a random walk with a constantmean. All of these models cannot be true simulta-neously, and a priori none of them is expected to betrue. If nominal returns are determined as a true risk-free rate plus a constant equity risk premium, therewill be a risk-free rate bias to using the nominalequity return as an estimate of future equity returns.Similarly, if the nominal return is determined as aconstant real rate over the expected inflation rate,then there will be an inflation rate bias to the averagenominal return. In this case, the bias can be removedby looking at realized real equity returns. Finally, ifthe riskiness of equity or bonds has changed overtime, the equity risk premium will be distorted by aterm premium bias.

The main conclusions of this paper are asfollows:

(1) Examination of bond market performanceand market interest rates experienced since 1925make it abundantly clear that the term premiumbias is significant. As a result, the long- run realizedexcess equity return over long-term bonds cannotbe used as a risk premium to add to current long-term bond yields.

(2) Total bond market risk (as measured bystandard deviation of returns) has significantly in-creased over the last 20 years, and at times has beenalmost equal to that of the equity market. Thisindicates that the equity risk premium over long termbonds is unlikely to have been constant.

FIGURE 12ABSOLUTE ANNUAL RETURNS

21. Note all the return estimates are simple annual rates of return, they are notthe log relative, which would follow from a continuous time lognormal model ofstock prices.

Ibbotson and Sinquefeld’s choice of 1925-1997 data (with its start dateso close to the Great Crash) is largely responsible for the conclusion that

risk has decreased in the equity market. If we instead use data going back to 1871,the equity return appears to be driven by a real return process with a mean of

about 9.0% and a constant error term.


(3) Bond market betas, whether measured basedon ten-year annual returns or five-year monthlyreturns, have increased from the negligible levelprior to the 1970s to the 0.40-0.80 range by 1990s.As a result, conventional risk premiums over long-term bond yields that may have been valid inearlier periods are excessive in the current interestrate environment.

(4) With bond market betas of 0.40-0.80, riskpremiums for lower risk equity securities, such asutilities, should be close to zero.

(5) When equity market data back to 1871 isexamined, the nominal equity return has clearly notfollowed a random walk with a constant mean. As aresult, it is not reasonable to take an average nominalequity return over even a very long period as a proxyfor expected nominal equity returns. The inflationrate bias indicates that this average nominal returnwill only by coincidence be an accurate estimate offuture nominal equity returns.

(6) The real equity return, in contrast to the nominalequity return, seems to show no major drift over time,whether it is averaged forwards or backwards.

(7) The standard error of the real return estimateof 1.71% means that estimating the average returnover short time periods (say ten years) is subject toconsiderable estimation risk. Indeed, estimating theaverage equity return to within 1.0% of the true mean,if it is in fact constant, will take about 400 years.

(8) The standard deviation of real equity returnshas averaged about 19%, and fluctuated between12% and 30%, based on ten-year estimates. Theestimates are critically dependent on the timing ofperiodic major market movements and whether ornot they happen to fall within the estimation win-dow. Overall, there is little to suggest that over thewhole period 1871-1997 stock market risk has changedsignificantly . There is evidence—focusing only onthe Ibbotson and Sinquefeld period that starts in1925—that equity risk has declined. However, thisconclusion is due to the extreme stock marketvolatility at the time of the Great Crash of 1929, whichdetermined the start point of their time period in thefirst place!

The above conclusions are broad and impor-tant, but the central message is simple: The familiarapproach of adding a constant equity risk premiumto the long-term bond yield is suspect, as long as theequity risk premium is mechanically estimated as asimple average of past excess returns. Instead,examining the broad scope of stock market historywould suggest that a better forecasting method is toadd a current inflation expectation to the averagereal equity return of about 9.00%. At the current pointin time, this would probably indicate an equity returnof just over 11.0%, rather than the 13% plus obtainedfrom adding an historic equity risk premium to acurrent long Treasury yield.

LAURENCE BOOTH

holds the Newcourt Chair in Structured Finance at the Universityof Toronto’s Rotman School of Management.

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estimating the equity risk premium and equity costs: new ways of looking at old data

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