A Minimum Pricing Error Approach to EquityValuation and its Implications on the Equity

Premium Puzzle

Jianming Ye⇤

Baruch College

Abstract

Equity valuation procedures depend on subjective assumptions on the cost of eq-uity and perpetual growth rate. Di↵erent assumptions may produce drastically di↵erentvaluations. In this paper, I propose a nonlinear modeling approach which utilizes thecross-sectional information of stock prices in the context of present value relationship.This approach allows for an objective and optimal determination of all the assumptions.Compared to the conventional method, this method leads to equity valuation that hassignificantly lower pricing errors than the best alternatives in the literature. The es-timation shows that the optimal choice of perpetual growth rate of residual earningsshould be negative, and the implied equity premium is merely 0.72%. These findingsare surprisingly di↵erent from the existing empirical literature, but they confirm thetheoretical insights that residual earnings should vanish over time and that the equitypremium should be smaller than previously observed.

JEL Classification: L11, L25, and M41.Key Words: Equity Valuation, Discounted Dividend, Discounted Residual Income, PresentValue Relationship, Conservatism, R&D, Cost of Equity

1 Introduction

The theoretical foundation of equity valuation is the present value relationship (PVR),

which expresses the price of a stock as the discounted sum of its future dividends.

While this is generally agreed upon, its practical implementations have been di�cult and

subjective, requiring assumptions on equity premium, forcast horizon, and perpetual

growth rate. Many of the discussions surrounding valuation, even legal disputes, are

related to these assumptions.

To begin with, when determining the cost of equity, one must know the risk-free rate.

Existing literature uses di↵erent several risk-free rates. In asset pricing literature, it is

usually the short-term rate, such as interest rates from six-month or one-year treasury

bills. For example, see Fama and French (2002). Others use the long-term rate from

five- or even ten-year treasury bonds. These rates may di↵er significantly, with no

clear indication as to which is correct. Brunner et. al. (1998) reviewed seven popular

textbooks in corporate finance, and found that three of them recommended short-term

treasury bills and two of them recommended long-term treasury bonds.

Next, we don’t know the correct equity premium. Fernandez (2008) reviewed 100

textbooks in finance and valuation and found that the equity premium recommended

in these textbooks ranged from 3% to 10%. Di↵erent historical periods give di↵erent

rates, and geometric and arithmetic mean also produce significantly di↵erent values. The

equity premium puzzle indicates a historical premium that is unreasonably high in the

US market. On the other hand, in countries with only a short history of equity trading,

the equity premium is di�cult to estimate and may be even negative. When additional

risk factors are involved, it is usually not clear what the equity premium should be at a

1

specific moment.

Thirdly, to express the infinite sequence of PVR as a finite sequence, one needs

to make an assumption about a firm’s perpetual growth rate. A small change in the

perpetual growth rate can change valuation tremendously. Again, there is no consensus

among researchers as to which is the correct rate. Courteau, Kao, and Richardson (2001)

use 0% and 2%, while Claus and Thomas (2002) assume it to be the ten-year t-bond

rate minus 3%. Penman (2004) considers 0%, 6%, and �100%. Little can be said as to

which is the best rate for a given firm.

The objective of this paper is to provide a general approach with which these as-

sumptions in valuation can be chosen objectively. The methodology employed here is

a nonlinear regression approach which utilizes information in the cross-sectional distri-

bution of stock prices. The rationale behind it is the same as in Kaplan and Ruback

(1995); Penman and Sougiannis (1998); Francis, Olsson, and Oswald (2000); Courteau,

Kao, and Richardson (2001); Liu, Nissim, and Thomas (2002); and Jorgensen, Lee, and

Yoo (2007). These authors use the pricing error to compare di↵erent valuation meth-

ods. In this paper, I extend the criterion further, to estimate parameter values and to

determine other aspects of valuation procedures.

The proposed methodology gives a number of interesting findings. First, the method

shows that using the ten-year t-bond is better than the five-year t-bond for risk-free rate,

which is significantly better than one-year t-bill rate. When estimated independently

each year, the implied cost of equity tracks the long-term bonds much more closely than

do short-term t-bills.

Contrary to existing literature, the analysis indicates that the perpetual growth

rate for residual earnings should be negative. Empirical results show that this rate was

2

negative for each year after 1982; the average was �3.8 for the 1990s, and has been lower

at �6.3 since 2000. While this appears to be counterintuitive at first glance, it reflects

the mean reversion nature of residual earnings: they vanish due to business competition.

The mean reversion of residual earnings is generally accepted in the accounting literature

theoretically (see Ohlson (1995), for example) and tested empirically (Dechow, Hutton,

and Sloan (1999)). The result here simply suggests the use of this well-known intuition

in the valuation contexts.

Using the estimated perpetual growth rate, pricing error is reduced significantly over

that of existing methods. For example, pricing error is 25% less than when using the

assumption in Claus and Thomas (2002); that is, the ten-year t-bond rate minus 3%.

The mean reversion of residual earnings has a strong relationship to accounting

conservatism. When accounting is conservative, the book value of assets is severely un-

derstated, and residual earnings are close to earnings so that they do not mean revert

to zero. Penman (1997) and Claus and Thomas (2002) discuss this theoretically. The

e↵ect, however, has been di�cult to incorporate in valuation procedure, since it is di�-

cult to discern the e↵ect of accounting conservatism from the mean reversion of residual

earnings. Using R&D as an indicator of conservatism, I show that perpetual growth rate

depends on the level of R&D (as a fraction of sales), and that a high level of R&D is

related to a low degree of mean reversion in reresidual earnings and thus a less negative

perpetual growth rate of residual earnings.

The incorporation of R&D in perpetual growth rate has two additional benefits.

In the existing literature, perpetual growth rate is generally assumed to be the same

amoung all firms. The incorporation of R&D introduces cross-sectional di↵erence to

this rate. Secondly, R&D provides a test against a measurement-error interpretation of

3

the negative perpetual growth rate. Under this interpretation, the perpetual growth rate

is low because of the di↵erence between analysts’ forecast and market concensus forecast

of earnings, and the discrepancy leads to mean reversion in residual earnings based on

analysts’ forecasts. Since high R&D firms typically have a more volatile earnings and

diversity in forecasts, and thus higher measurement error, one should expect a faster

mean reversion in residual earnings of high R&D firms and thus a negative coe�cient to

R&D in the perpetual growth rate. A strong positive coe�cient for R&D gives evidence

against such interpretation, and indicates that the negative perpetual rate is due mainly

to the substantive behavior of earnings rather than measurement error.

The mean reversion of residual earnings is also related to earnings persistence. Em-

pirical analysis here confirms that a high earnings persistence is related to a less negative

value in the perpetual growth rate of residual earnings. Its contribution is economically

not substantial.

With a negative perpetual growth rate, the estimated equity premium is even lower

than the 3% observed in Claus and Thomas (2002). Relative to ten-year t-bonds, the

equity premium is only 0.72%, significantly below historical returns, and below the 3%

observed in Claus and Thomas (2002). While this estimate appears to be low given the

historical records, it conforms to the theoretical prediction. Based on risk characteristics

of the US market and the theory of financial economics, Mehra and Prescott (1985)

argued that stocks should provide at most a 0.35% annual risk premium over t-bills.

Even by stretching the parameter estimates, they concluded that the premium should

be no more than 1% (Mehra and Prescott (1988, 2003)). This is the so-called “equity

premium puzzle”, which has grown more puzzling in our sample period from 1981-2007.

However, based on the proposed estimation method, the implied equity premium is close

4

to its theoretical value. Choosing an equity premium of 6%, even with the best choice of

perpetual growth rate, has a pricing error which is 75% higher than that of the optimal

choice.

Using CAPM and Beta estimates obtained from CRSP, I also study the risk premium

on Beta. The average estimated coe�cient of Beta is insignificant, which shows that

there is essentially no loss in pricing error in assuming that Beta=1 for all firms. This

gives the same risk premium for all firms at a given time.

The methodology proposed here allows for determination of all other aspects in an

equity valuation process, such as the forecasting horizon and dividend payout ratio. The

valuation procedure with minimum pricing error is a residual valuation model with a

five year horizon and a perpetual growth rate depending on the R&D. Further analysis

can be performed at the industry level, which will potentially further reduce the pricing

errors.

The valuation procedure obtained here is compared to a number of ad hoc and full-

information alternatives, including valuations by various multiples of forecasted earnings,

combinations of book value of equity and forecasted earnings, the residual income val-

uation model of Gephardt, Lee, and Swaminathan (2001), and the valuation model in

Ohlson and Juettner-Nauroth (2005). As shown in Liu, Nissim, and Thomas (2002),

the simple multiple based on the sum of five-year expected earnings performs the best

of these alternatives. However, the valuation through minimum pricing error proce-

dure performs even better, with a pricing error about 10% lower than that of the best

alternatives.

The methodology employed in this paper relates stock price to analysts’ earnings

and to growth forecasts through a finite-horizon PVR. Therefore, any bias in analysts’

5

estimates (Cheng, 2005) will impact the estimates. All results in this paper need to be

understood in this context.

This paper is also connected to the literature on estimating the implied cost of equity

(Claus and Thomas, 2001; Gebhardt, Lee, and Swaminathan, 2001; Easton, et. al., 2002;

Gode and Mohanram, 2003; Guay, Kothari, and Shu, 2003; Easton, 2004; Botosan and

Plumlee, 2005; and Easton and Monahan, 2005). Estimation of the implied cost of

equity can be considered a dual problem to equity valuation and is subject to the same

subjective assumptions discussed above. This paper is especially motivated by recent

attempts in simultaneous estimation of perpetual growth rate and implied cost of equity,

such as in Easton, et. al. (2002) and Easton (2004), which rely on rather restrictive

assumption of the geometric growth of earnings. The method in this paper eliminates

this assumption, and simultaneously estimates perpetual growth rate and cost of equity

in the context of equity valuation.

The paper is organized as follows: In Section 2, I will review the basic problems in

equity valuation to be addressed in Section 4. Section 3 develops the minimum pricing

error estimation methodology. Sections 5-6 discuss estimation results and comparisons

of di↵erent valuation methods. Section 7 concludes.

2 Issues with Existing Equity Valuation Methods

Equity valuation, the core of security analysis, is an essential tool in the investment

community. Simple valuation methods are usually based on multiples such as price-to-

earnings ratio and price-to-book ratio. Textbooks on equity valuation typically discuss

at least three alternatives to full information valuation: discounted dividend, discounted

free cash flows, and residual income valuation. These three approaches are mathemat-

6

ically equivalent in the sense that, under the clean surplus relationship, each implies

the others. However, in practice we can only predict up to a finite horizon, and some

extrapolation is needed. Each version of the three valuation methods makes a di↵erent

assumption, resulting in di↵erent valuations (Penman and Sougiannis, 1998, Lundholm

and O’Keefe, 2001, Penman, 2001).

Consider the present value relationship,

Pi,t

=d

i,t+1

(1 + r

i

)+

d

i,t+2

(1 + r

i

)2+

d

i,t+3

(1 + r

i

)3+ . . . , (3)

where Pi,t

is the ex dividend market value of firm i at time t, with d

i,t

the expected

dividend and r

i

the cost of equity, which is the risk-free rate of return plus the risk

premium. Using a clean surplus relationship, the PVR can be rewritten as

Pi,t

= B

i,t

+ae

i,t+1

(1 + r

i

)+

ae

i,t+2

(1 + r

i

)2+

ae

i,t+3

(1 + r

i

)3+ . . . , (4)

where ae

i,t

is the abnormal earnings of firm i during year t. The model is referred to as

the residual income valuation (RIV).

The discounted dividends valuation and RIV expressions of the PVR are mathemat-

ically equivalent. Empirical implementation, however, must reduce the infinite sequence

to finite terms. To do this, assumptions must be made, and depending on the speci-

fication used, the assumptions often di↵er. See discussions in Lundholm and O’Keefe

(2001) and Penman (2001). Valuation using discounted dividends often assumes a con-

stant growth rate in dividends, while RIV assumes a constant growth rate in residual

earnings. Penman and Sougiannis (1998) and Francis, Olsson, and Oswald (2000) find

that finite horizon valuation using RIV, which assumes a constant growth rate in resid-

ual earnings, has the lowest overall pricing errors. In the following, I will concentrate on

this specification.

7

2.1 Residual Income Valuation with Perpetual Growth

A typical method of reducing the infinite sequence is to assume a constant growth rate

after a given forecasting horizon. An example of this is in Claus and Thomas (2001).

Assume that residual earnings grow at a constant rate g. This reduces the infinite

sequence (4) into the finite sequence

Pi,t

= B

i,t

+5X

⌧=1

ae

i,t+⌧

(1 + r

i,t

)⌧

+(1 + g

i,t

) ae

i,t+5

(ri,t

� g

i,t

)(1 + r

i,t

)5, (5)

where ae

t+⌧

= eps

t+⌧

�r

i

B

t+⌧�1 is the residual (abnormal) earnings and g is the perpetual

growth rate of ae

t+⌧

for ⌧ > 5. The book value of equity B

t+⌧

is calculated based on

B

t

plus all the undistributed earnings, using a payout ratio equal to that in year t.

Following Claus and Thomas (2001), earnings beyond year t + 2 are extrapolated from

year 2 using analysts’ forecast of five-year growth rate (stg), eps

t+⌧

= (1 + g5) eps

t+⌧�1.

The valuation equation has two explicit parameters: cost of equity r

i,t

and the per-

petual growth rate g

i,t

. Additional parameters may also be introduced; for example, one

can adjust for the potential bias in analysts’ growth forecasts by modifying the five-year

growth rate estimate (g⇤5 = � ⇤ g5) and using it as the predicted five-year growth. This

gives the additional parameter �.

Choosing the Perpetual Growth Rate

The perpetual growth rate of g has a crucial impact on valuation results, especially

when it is di↵erent from the short-term growth rate. However, there is not much basis

for determining what g is. While Penman (2004) is deeply concerned about growth

rate, it gives only a passing note on allowing a perpetual growth rate that is di↵erent

from the short-term growth rate. Claus and Thomas (2001) devote a full appendix to

this, with detailed consideration of the perpetual growth rate. They argue that “the

8

popular assumption of zero-growth in abnormal earnings may be too pessimistic,” and

conclude that in the presence of accounting conservatism, a reasonable approximation

of the rate is the expected inflation rate. Specifically they assume that g = r

f

�3%, and

conduct sensitivity analysis for g = r

f

as the optimistic case and for g = r

f

� 6% as the

pessimistic case. They use the 10-year treasury bond as the risk-free rate (rf

). During

1985-1998 in the U.S., they give the average r

f

at 7.64%, which gives their typical g at

4.64% and their pessimistic g at 1.64%.

Based on the existing literature, there are several values of g that have been used for

RIV:

• Case 1: g=-100%. Discussed in Penman (2004) as Case 1; in Palepu, Healy, and

Bernard (2000); and Liu, Nissim, and Thomas (2002).

• Case 2: g=0%. Discussed in Penman (2004) as Case 2; in Frankel and Lee (1998);

Penman and Sougiannis (1998); Francis, et. al. (2000); Courteau, Kao, and

Richardson (2001); and Liu, Nissim, and Thomas (2002).

• Case 3: g=2%. Discussed in Courteau, Kao, and Richardson (2001), who state that

“2 percent growth approximates the rate of inflation during our sample period”;

and in Penman and Sougiannis (1998).

• Case 4: g=r

f

� 3%. Discussed in Claus and Thomas (2001).

• Case 5: g=4%. Used in Francis, et. al. (2000), citing Kaplan and Reback (1995).

Therefore, based on existing literature, it is generally believed that the perpetual

growth rate is between 0% to 4%. Even through residual earnings are likely to mean-

revert, it is generally believed that accounting conservatism is so strong that the mean

9

reversion is not substantial. However, such belief is not based on empirical evidence. In

addition, the general practice of assuming the same perpetual growth rate for all firms

also disregards the impact of conservatism on the rate of mean reversion of residual

earnings.

Risk-free Interest Rates and Equity Premia

The most important element in equity valuation is probably the cost of equity. The

analysis in this paper is confined to using the Capital Asset Pricing Model (CAPM),

which gives the cost of equity in the form

r = r

f

+ Beta ⇤ (rm

� r

f

)

where r

m

� r

f

is the equity premium. For a recent review of CAPM, see Fama and

French (2004). However, estimation of the cost of equity involves many di↵erent issues

thus cannot be treated fully here.

Even when confined only to CAPM, there are many problems. The first problem is in

the determination of the risk-free rate. There are many risk-free rates one can use, such

as 1- and 6-month treasury bills and 3-, 5-, 10-, and 30-year treasury bonds. Their rates

may di↵er substantially. In the literature of asset pricing in which one-year buy-and-hold

returns are often studied, a short-term t-bill rate is typically used. Fama and French

(2002), for example, use the interest rate of six-month commercial paper. In equity

valuation literature, the interest rates of longer-term bonds are often used. Claus and

Thomas (2001) use 10-year t-bond rate, while Courteau, Kao, and Richardson (2001)

use that of the 5-year t-bond and Penman and Sougiannis (1998) use 3-year t-bond rates.

Brunner et. al. (1998) survey seven popular textbooks in corporate finance and find

that three of them suggest using long-term bonds, two of them short-term bills, and the

10

remaining two are unspecified.

The second question is about the size of the equity premium. Equity premium is

fundamental in asset pricing theory. However, it is not well understood due to the

so-called “equity-premium puzzle” (Mehra and Prescott 1985, 1988, 2003). Based on

the standard theory of financial economics, using the historical characteristics and risk

preference of the US market, Mehra and Prescott (1985) show that the annual equity

premium should be at most 0.35%—not more than 1%, using stretched parameter values.

This contrasts starkly with the historical risk premium of 6.2% in their study. The

phenomenon is even more pronounced in recent years, with the annual equity premium

reaching 8.1% during 1979-2005 (Dimson, Marsh, and Staunton, 2006). Fama and French

(2002), using the dividend yield and earnings growth rate, obtain equity premia of 2.55%

and 4.7% over short-term t-bills. Claus and Thomas (2001), using a residual income

framework similar to the one used in this paper, obtain a premium of 3% over ten-year

bonds.

The inconsistency between theoretical and empirical observation in equity premia

creates considerably diverse estimates of the equity premium in practice. Courteau,

Kao, and Richardson (2001), citing Ibbotson and Sinqucfield (1983), use 6%. Citing

Obbotson (1993), Copeland, Koller, and Murrin (1994) give four di↵erent values: the

risk premium from 1926-1992 is 6.9% based on an arithmetic mean return and 5.0%

based on a geometric mean return; from 1962-1993, the mean returns are 4.2% and 3.6%

respectively. Fernandez (2008) surveys 100 textbooks in finance and valuation and finds

that the recommended value ranges from 3% to 10%, with no convergence over time.

The problem is even more pronounced in developing countries where the historical record

is short and the market highly volatile.

11

The third issue with CAPM is its validity. Extensive empirical testing has shown

that the market Beta does not explain the market return. See, for example, Fama and

French (2004) for a summary. Whether to use CAPM or not, of course, has a direct

impact on the valuation results.

Due to the di�culties in the estimation of cost of equity, Penman (2004)–in a tone

unusual for a textbook–concludes that “Disappointingly, despite a huge e↵ort to build

an empirically valid asset pricing model, research in finance has not delivered a reliable

technology. In short, we really don’t know what cost of equity for most firms is.” (pp.

108, Penman, 2004)

2.2 Valuation Using Multiples and Linear Combinations

A very popular class of valuation tools is valuation by multiples. This method emphasizes

the economic drivers behind firm values. Liu, Nissim, and Thomas (2002) study a

comprehensive list of valuation drivers. In this paper, I will include the best performers

from their study.

Given a value driver, a multiple-based valuation can be written as

P = �X + ✏,

where � is a parameter estimated through data, and the value driver X is positive. One

way they use to estimate � is to minimize the pricing error

|P� �X|/P

subject to an unbiasedness restriction. Such a restriction is necessary because the crite-

rion penalizes over-pricing much more than they do under-pricing. Indeed, the maximum

pricing error is 1 in the case of under-valuation, and infinity for over-valuation.

12

Among the many multiples considered in Liu, Nissim, and Thomas (2002), I will

consider several best performers that are especially relevant here.

• Forward earnings: EPS1 and EPS2, the one-year-out and two-year-out EPS fore-

casts;

• Forward earnings and growth: EG1=EPS2⇤(1 + g) and EG2=EPS2⇤g, the one-

year-out and two-year-out EPS forecasts;

• The sum of forward earnings: EPSS=P5

i=1EPSt+i

, the sum of one-year-out to

five-year-out earnings forecasts.

These measures are especially interesting, because Liu, Nissim, and Thomas (2002)

show that they may outperform full-information valuation that they included.

A slightly more complex model is to combine some of the value drivers. Combining

the book value of equity and earnings (Ohlson, 1995; Penman, 1998; Dechow, Hutton,

and Sloan, 1999), we get the Ohlson model. Here we use the model by Easton, Taylor,

Shro↵, and Sougiannis (2002),

Pt

= B

t

+EPS

t+1 � rB

t

r � g

= � g

r � g

B

t

+1

r � g

EPSt+1,

which fits to the data better than the Ohlson model does (Ohlson, 1995), since the model

here uses analysts’ forecasts of future earnings instead of actual earnings.

2.3 Valuation Using GLS (2001) and Ohlson and Juettner-Nauroth (2005)

Gebhardt, Lee, and Swaminathan (2001) do not use analysts’ growth forecasts like Claus

and Thomas (2001) do. Rather, they use the predicted earnings for year t+1 and t+2,

13

and assume that the ROE fades linearly to the industry median ROE over a 10-year

period,

ROEt+2+k

= (1� k

10) ROE

t+2 +k

10IROE,

for k = 1, . . . , 10 and ROEi,t

= eps

i,t

/B

i,t�1, so that ae

i,t+⌧

= (ROEi,t+⌧

� r

i

)Bi,t�1. The

firm-specific ROE is set equal to industry ROE beyond year t + 12, so that ae

i,t+⌧

=

(IROE� r

i

)Bi,t+11 for ⌧ > 11. This gives the valuation model

P

i,t

= B

i,t

+11X

⌧=1

ae

i,t+⌧

(1 + r

i

)⌧

+ae

i,t+12

r

i

(1 + r

i

)11. (6)

Again, the future book value is calculated based on B

t

plus all the undistributed earnings

using a payout ratio equal to that of year t.

Unlike in Claus and Thomas (2001), GLS method has only one explicit parameter,

the cost of equity r

i

. Additionally, since the historical median IROE is not necessarily

a good estimate for the expected IROE, it is potentially beneficial to consider IROE as

an additional parameter.

Using another specification of the present value relationship, Ohlson and Juettner-

Nauroth (2005) derive a parsimonious representation of the price,

P

i,t

=eps

i,t+1

r

i

+agr

i,t+2

r

i

(ri

� g), (7)

where agr

i,t+2 = eps

i,t+2�r

i

d

i,t+1�(1+r

i

) eps

i,t+1 and g is the growth rate of agr, that is,

agr

t+1 = (1+g) agr

t

. This model will be referred to as the OJ model. The quantity agr

measures ‘abnormal’ growth in one-year-ahead earnings exceeding the normal earnings

growth implied by the cost of equity. Ohlson and Juettner-Nauroth (2005) assume that

the abnormal earnings are positive and will grow further each year with rate 1+g, where

0 < g < r

i

. Based on this assumption, earnings each year equal the normal growth of

14

the previous year’s total earnings at the rate of the cost of capital, plus an abnormal

amount which is 1 + g times of previous years abnormal earnings.

2.4 Uncertainty in Equity Valuation

Valuation methods are inaccurate and subjective, so practitioners often wish to assess

the uncertainty of the valuation estimate. Some attempts have been made to deal with

this issue, especially with the valuation uncertainty caused by fluctuation in cash flows

and earnings. Viebig, Poddig, and Varmaz (2008) discuss Monte Carlo simulation using

the software @Risk to evaluate such uncertainty.

The Monte Carlo method of measuring uncertainty is likely to mismeasure the true

uncertainty. There are two reasons for this. First, it can measure valuation uncertainty

only if a distribution of related quantities is given. Specifying the distribution, however,

is di�cult, especially when the uncertainty comes from many correlated sources. Next,

the method cannot measure the uncertainty of unspecified or unspecifiable aspects of

valuation, such as the approximation error of the finite horizon PVR. Therefore, an

overall measure of valuation uncertainty is needed.

3 Methodology Development

In the literature of equity valuation, many papers have been written to compare the

performance of di↵erent valuation methods. These comparisons are generally based on

a relative pricing error metric, which is defined as the di↵erence between actual price and

predicted price, scaled by actual price. See, for examples, Kaplan and Rubuck (1995),

Penman and Sougiannis (1998), Francis et. al. (2002), Jorgensen, Lee, and Yoo (2007).

The approach to be used here is an extension and modification of the above method.

15

Liu, Nissim, and Thomas (2002) use this method to estimate parameters in a multiple-

based valuation model. Their idea generalizes the comparison of discrete valuation

methods to the evaluation of di↵erent parameter values, allowing an objective choice of

the parameter values. I will refer to this as the minimum pricing error criterion.

The approach here further extends the minimum pricing error criterion to the present

value relationship, treating PVR as a nonlinear model. Estimation of nonlinear models

is a conventional statistical procedure. This approach will essentially allow for the choice

of both short-term and perpetual growth rates, the cost of equity, and any other related

parameters.

To be more specific, the present value relationship (PVR) e↵ectively gives the price

model

P = PVR(X, ↵), (8)

where X represents the variables used (such as dividends or book value of equity and

residual earnings) and ↵ represents specification and parameters (such as growth rates

and the cost of equity). Given an estimate ↵, the estimated price P is obtained. The

aforementioned research selects the form of PVR that minimizes the sum of the absolute

value (or square) of the pricing error (RPE) metric

w =P � P

P

.

For the rest of the paper, I will use the absolute value as in Penman and Sougiannis (1998)

and Francis et. al. (2002). This metric has many advantages compared to a pricing

error scaled by variables such as the book value of equity: the metric is economically

meaningful, since it can be interpreted as the arbitrage gain if the price move to its

predicted value, and it is scaleless, as the measure has the same interpretation for both

16

large and small firms.

One problem of the pricing error is that it treats overpricing and underpricing asym-

metrically. An investor who loses 50% would require an 100% gain to o↵set the loss.

However, the RPE metric gives a value of 0.5 for underpricing with 50% (P = 0.5P ),

but a value of 1 for overpricing by 100% (P = 0.5P ). Due to this problem, minimiz-

ing the RPE often chooses a valuation method that under-values. The bias induced by

this asymmetry is so substantial that the criterion by itself does not lead to meaning-

ful estimates. Liu, Nissim, and Thomas (2002) overcome this problem by imposing an

unbiasedness criterion and by eliminating all firms with a share price less than $2.

One way to overcome the artifact created by this asymmetry is to symmetrize the

pricing error. With a symmetric treatment of gain/loss by both buyers and sellers of a

security, one gets the criterion

w

0 =P � P

min{P, P}. (9)

This measure will be used in the rest of the paper and will continue to be referred to as

“pricing error”.

Example 1: Consider five firms with stock prices of $2, $10, $10, $10, and $20.

Suppose one wishes to estimate the value of a new firm with similar financial measures.

Estimates based on the mean and median would be $10.4 and $10 respectively. The

minimizer of the pricing error is P = 2. The harmonic mean, proposed in Liu, Nissim,

Thomas (2002) is P = 5.88, and the minimizer of the symmetrized pricing error is

P = 10.

Note that, in this example, the harmonic mean (P = 5.88) is the only one such that

17

the mean of the unsymmetrized pricing error is zero,

P � 2

2+ 3 ⇤ P � 10

10+

P � 20

20= 0,

which gives a valuation that appears to be too low. The mean, median, and the minimizer

of the symmetrized pricing error all give a positive mean pricing error, that is,

P � 2

2+ 3 ⇤ P � 10

10+

P � 20

20> 0

should be considered more typical.

In practice, I modify the criterion in two minor ways to increase the robustness of its

parameter estimates. First, to avoid a negative denominator, I set the denominator to

be 1/Shares if the lesser of P and P is less than 1/Shares, so that the minimum market

capitalization is $1 million. Secondly, the pricing error w

0 is winsorized to 10 if greater

than 10. This reduces the impact of certain extreme observations.

The estimation is done in SAS using its nonlinear optimization procedure PROC

NLP. The specified optimization technique is Newton-Raphson with automatic calcula-

tions of gradients, maximum iterations of 1,000, and maximum function calls of 5,000.

Like typical nonlinear optimization procedures, a reasonable initial value for the param-

eters is necessarily for the algorithm to converge.

Alternative estimation criteria can also be used. A popular method is the weighted

least squaresP

i

(Pi

� P

i

)2/S

2i

where S

i

is some measure of size. I find that this method

is ine�cient empirically because it is di�cult to find an appropriate measure of size,

and because the distribution of price is highly non-normal. Alternatively, one can use

Pi

(log(Pi

) � log(Pi

))2. The criterion leads to results similar to symmetrized pricing

errors, but somewhat more di�cult to interpret.

18

4 Data

The stock prices and analysts forecasts are obtained from the I/B/E/S Unadjusted

Summary Database. For each firm-year, I use only the summary forecast at the sixth

month after the end of its fiscal year. The actual earnings and book value of equity are

obtained from COMPUSTAT.

Firms are required to have a nonmissing value for earnings forecasts of years +1 and

+2, and a five year growth rate forecast (g5). I also require nonmissing values for price

and the prior year’s book values, earnings, and dividends. Explicit earnings forecasts

for years +3, +4, and +5 are often not given. When these values are missing, I use

the earnings forecast at year +2 and the growth rate g5 to project their values with

EPS

t+1 = EPS

t

⇤ (1 + g5).

For part of the analysis, I use the Beta from estimating cost of capital. The value of

Beta is obtained from CRSP. CRSP provides annual Betas computed using the methods

developed by Scholes and Williams (1977). This is so that the results will represent what

a practical user would get using the Beta value from an authentic source. The value of

Beta is winsorized to �1 and 3, which a↵ects only a few observations.

The paper concentrates on the analysis of firms with EPS1>0, EPS2>0 and BVE>0,

with a sample size of 46,889. There are relatively few observations in 1981-1983 and 2007.

The sample size is small for 2007 when the data set was obtained.

Table 1 presents the summary statistics and correlation table. Panel A shows that the

median market capitalization is $663m and the median BVE $270m. Generally speaking,

this indicates that the data set contains mainly larger firms. The correlation coe�cients

from Panel B confirm typical intuition: EPS1 and EPS2 are highly correlated, firms

19

with a high five-year growth rate (g5) are associated with low earnings in the near future

(EPS1 and EPS2) and low current dividends, and high Beta values are associated with

high R&D expenditures and high growth rates.

5 Determination of Perpetual Growth Rate and Costof Equity

In this section, I will apply the proposed methodology to discuss the estimation of the

related parameters as mentioned in Section 2. Section 2.1 shows that a typical value

for perpetual growth rate is between 0% to 4%, with multiple choices of risk-free rates

and equity premia. In this part, I will evaluate these assumptions from the perspective

of minimum total (or average) pricing errors. For most of this section, we will focus on

the residual income valuation as specified in equation (5). I will first assume that the

equity premium is constant so that

r

i,t

⌘ R

f

(t) + r

p

,

where R

f

(t) is the risk-free rate, and r

p

is the equity premium.

In Table 2 (A), I consider three di↵erent factors, each with three di↵erent possibilities.

First, I consider the choice of risk-free interest rate. The three choices considered are

the 1-year treasury bill rate (T-1) and five- and ten-year t-bond rates (T-5 and T-10).

Thirty-year t-bonds are not considered because they were discontinued for a period of

time. Very short-term t-bills, such as one- and three-month ones, are not considered—we

will draw clear inferences from the one-year t-bill, so shorter-term t-bills are unnecessary.

For each of the risk-free rates, I consider three values of the perpetual growth rate,

0%, 2%, and 4%, and three values of the equity premium, 1%, 3%, and 6%. I include 1%

20

as an equity premium based on the theoretical upper bound from Mehra and Prescott

(2003), and the other values (3% and 6%) are recommended by some textbook authors

(Fernandez, 2008). Table 2(A) gives the mean pricing error for each case, averaged

across all firm-years.

From Table 2(A), among the di↵erent values of perpetual growth rates, one can see

that g = 4% is generally more associated with higher pricing error than when g = 0%

and 2% for all risk-free rates, all risk premia considered. When comparing g = 0% to

g = 2%, one can see that g = 0 with either similar or lower pricing errors. Therefore,

even through g = 0% is generally considered too conservative, as discussed in Section

2.1, it indeed seems superior to g = 2% and 4%.

For g = 0%, when comparing the three di↵erent values of equity premia, we see that

the average pricing error is always lowest when r = 3%. This appears to support Claus

and Thomas (2001), who state that the equity premium is 3%.

Given g = 0% and r

p

= 3%, one can compare the pricing errors of the three di↵erent

risk-free rates. The pricing errors corresponding to 1-year t-bills and 5- and 10-year

t-bonds are 0.588, 0.486, and 0.474 respectively. Therefore, the pricing error drops

significantly when a long-term t-bond rate is used as opposed to that of a 1-year t-bill.

Using a 10-year t-bond rate leads to lower pricing error. However, the improvement is

smaller, potentially due to the small di↵erence between the two rates.

In summary, Table 2(A) shows that the best combination for minimizing pricing

error is the 10-year t-bond rate, a 3% equity premium, and a perpetual growth rate of

0%. However, this is based on the limited alternatives in Table 2(A). In Table 2(B), I

allow r

o

and g to vary continuously until the lowest pricing error is obtained.

Table 2(B) shows some rather unusual results. When the risk-free rate is the 1-year

21

t-bill rate, the minimum pricing error is obtained at g = �9.81% and the equity premium

at 1.18%. The mean pricing error is 0.467, which is about 20% lower than the best case

in Table 2(A) case (a). For the 5-year t-bond rate, the estimated equity premium is

0.97%, and the perpetual growth rate �7.11%. For 10-year t-bonds, it is 0.82% and

�6.28%. They improve over the corresponding best cases in Table 2(A), although not

as substantially as the 1-year t-bill rate case does. For T-10, the pricing error with an

estimated equity premium and perpetual growth rate is 0.438. If the equity premium is

set to 6%, as many textbooks in corporate finance suggest (Fernandez, 2008), then the

pricing error is 0.77, given in Table 2A case 3, which is 75% greater than the optimal

choice.

I will postpone the discussion of values of the equity premium and perpetual growth

rate to Table 3. In Table 2(B), one observes that the minimum pricing error (0.438)

is reached when the risk-free rate is T-10, although the advantage over T-1 is not as

substantial as in Table 2(A). T-10 is also used in Claus and Thomas (2001). With the

suggested parameter values in their paper—that g = T -10�3% and equity premium is

3%—one gets a pricing error of 0.578, which is 32% higher than the best case.

For the rest of the paper, I will only focus on T-10 as the risk-free interest rate.

Table 3 gives more details of the estimate of the perpetual growth rate and cost of

equity. Unlike in Table 2, in which risk premia and perpetual growth rates stay constant,

they are allowed to vary freely over time in Table 3, but remain the same cross-sectionally

each year: r

i,t

= r

t

and g

i,t

= g

t

. The three di↵erent risk-free rates are given in the table

for comparisons.

First we consider the estimate for g

t

. Table 3 shows that the estimated value of g

t

is negative for all years except for 1982, with a small sample size of 71. The average

22

value of g

t

is �4.45%. The t-ratio for testing the null hypothesis that g

t

⌘ 0 is �10.37,

strongly rejecting the hypothesis. This indicates that a negative value of g is highly

stable.

The result that the value of g is negative seems counterintuitive initially, espe-

cially considering that almost nobody in valuation assumes a negative value. Claus

and Thomas (2001) argue that g = 0 is already overly-conservative. However, a nega-

tive value is not inconceivable; indeed, it is even expected. Economics of competition

suggests that residual earnings are abnormal, should disappear, and therefore have a

negative growth rate. The problem is with accounting conservatism. It understates the

book value of capital, making residual earnings somewhat normal and reducing their

mean reversion. If accounting conservatism is not dominant, then we should still expect

a negative value. One reason why existing literature has not proposed a negative g is

probably due to the di�culty of determining its proper value.

Estimates of the cost of equity (COE, r

t

) in Table 3 are plotted in Figure 1 along

with the three di↵erent risk-free rates. From Figure 1, one can see that r

t

has a pattern

more closely resembling that of T-10 but less closely to that of T-1. Indeed, relative to

r

t

, the short-term rates are too volatile, which is probably why T-10 shows lower pricing

error in Table 2. This analysis shows that the duration impounded in equity is likely 10

years or longer.

The estimated COE (rt

) is almost identical to T-10 from 1981 to 1988, with the equity

premium nearly nonexistent (average 0.26%, standard error 0.24%). An increase of the

equity premium is observed in 1989: the equity premium relative to T-10 from 1989 to

2007 averages 1.25% (standard error 0.14%). For the whole sample period, the average

premium is 0.96% (standard error 0.16%). Relative to T-1, the equity premium is higher.

23

The average is 1.48% (standard error 0.31) from 1981-1988 and 2.54% (standard error

0.33) from 1989 to 2007.

An equity premium of 0.96% (standard error 0.16%) relative to T-10 is very low in

view of historical stock performance. During the sample period of 1981-2007, average

annual stock returns exceed 8%. Fernandez (2008) surveys 100 textbooks in corporate

finance, and the recommended equity premium ranges from 3% to 10%. The estimate

here appears to be unusually low.

While the estimated equity premium appears to be lower than popular perception, it

confirm theoretical results from financial economics. Mehra and Prescott (1985, 2003)

raise the famous “equity premium puzzle” and argue that the historical stock returns

have been unconceivably high considering the risk tolerance characteristics of the US

market. They show that typical equity premium should be 0.35%, and no more than

1% even using stretched parameter values. Since then, multiple theories have been

proposed to explain such irregularity. Fama and French (2002) use a di↵erent approach

to estimate equity premium, and show that the risk premium can be 2.55% over 6-

month commercial paper, which is comparable to the premium from a one-year t-bill

(8.11%�5.88%=2.23%). The equity premium over T-10 of 0.97% is much closer to what

the theory predicts.

From Table 3, we observe that the value of g varies mainly between �3% to �7%,

averaging to �4.45%. In the 1990s, the values are around �3% to �5%, and after 2002

it has been between �6% to �8%. This appears to be a decrease in the estimated cost

of equity between the two periods. Excluding the first three years during which the

sample size is small, the values of g

t

and r

t

form approximately the following empirical

24

relationship:

g

t

= �12.8% + 0.92 ⇤ r

t

+ 1.01 ⇤ (rt

�R

f

(t))

where R

f

is T-10. Note that (rt

� R

f

(t)) is the equity premium. The regression has a

R

2 of 67.8%, and both variables, r

t

and (rt

� R

f

(t)) have significant t-values (6.1 amd

�3.4). Thus, the higher the cost of equity and equity premium, the less negative g

t

will

be.

Since the negative value of g is driven by the mean-reversion of abnormal earnings,

and distorted by accounting conservatism, one should be able to observe a positive

relationship between g and some measure of accounting conservatism. In this paper,

I consider a primary form of accounting conservatism: the expensing of research and

development (R&D) expenditure. With expensing, intangible assets created by R&D

activities are missing from the balance sheet, so that firms with a high R&D tend to

understate their book value of equity. This makes residual earnings behave more like

earnings and less likely to mean revert, that is, g is less negative for high R&D firms.

Based on this, I will consider the model

g

i,t

= g

o,t

+ g1,t

⇤ R&Di,t

.

The discussion above implies a positive value of g1,t

. R&D is measured as a fraction of

the total revenue of a firm. Some firms, however, have nearly no revenue, so to prevent

extreme observations, I compare windsorizing R&D to 0.05, 0.1, and 0.2. The results

are similar, therefore 0.1 is used.

Table 4A presents results of the test of association between the perpetual growth

rate and R&D. The estimated value of g1 is positive for all years except 1981 (too small

sample size) and 1988. The average coe�cient of g1,t

is 0.45, with a t-ratio of 5.40, which

25

rejects the hypothesis that g1,t

= 0. It indicates that a high R&D is associated with lower

mean reversion of residual earnings. The association, however, is not constant. Before

1998, the values of g1 are mostly between 0.2 and 0.3. The value increases to 0.7 and

above after 1997. Such an increase may be driven by increasing accounting conservatism

or by market belief that earnings in high R&D firms have become less likely to mean

revert compared to those of non-R&D firms, as we entered the “New Economy” in the

1990’s.

The last two columns of Table 4A give a comparison in terms of pricing error, with

and without the inclusion of R&D as a mean reversion factor in perpetual growth rate.

The overall average pricing error decreases about 3.2% from 0.401 to 0.388. The im-

provement is much more significant after 1997, reducing pricing error by about 6.2%.

A second factor that reduces mean reversion in earnings is their persistence. If

⇢ is a measure of persistence, then we expect g to be positively related to ⇢. Table

4B presents the results, assuming that g

i,t

= g

o,t

+ g1,t

⇤R&Di,t

+ g2,t

⇢

i

. The persistence

parameter ⇢ is estimated for each firm using all historical data based on the AR(1) model

EPS0t

= ⇢

o

+ ⇢ EPS0t�1. To avoid reducing sample size significantly, I estimate ⇢ using

all data. The estimated ⇢ is approximately the forward-looking earnings persistence for

the beginning of the time period and thus is more meaningful to equity valuation, since

it is the backward-looking persistence toward the end of the sample period.

With R&D included in the perpetual growth rate, Table 4A gives a lower estimate

for the equity premium at 0.72% (standard error 0.14%), even closer to what the theory

predicts.

Table 4B shows that the estimated value of g2 is positive each year, with a mean of

0.0489 and t-ratio of 4.59 for testing g2 = 0. The magnitude of the estimate decreases

26

over time, from 0.0904 in 1985 to 0.0096 in 2005, a reduction of almost 90%. Such a

decrease may be due to the way that we estimate the persistence parameter: in 1985, the

estimate is forward-looking and is more relevant, bue it gradually becomes backward-

looking and not as relevant by 2005. The e↵ect of R&D remains roughly the same as in

Table 4A, with a mean of 0.40. Since the inclusion of the earnings persistence parameter

has only a small e↵ect on the pricing error, reducing it from 0.388 to 0.384–about 0.1%.

This may be due to the di�culties in its estimation, it is not included in the final model.

The cost of equity estimated so far is constant across all firms, only for the sake of

simplicity. More complex models for the cost of equity is can be implemented in the

same way. In Table 6, I will include the capital asset model (CAPM) in the form of

r

i,t

= R

f

+ Beta ⇤ r1,t

where r1,t

is the equity premium. Since the equity premium in this context is often

unknown, it is treated as an unknown parameter. The perpetual growth rate is set to

be g

i,t

= g

o,t

+ g1,t

R&Di,t

as in Table 4A.

Table 6 shows the estimate for r1. The average is 0.11%, with a t-ratio of 0.62,

which does not reject the hypothesis that r1 = 0. On the other hand, with 7 out of 27

estimates negative, one can reject the hypothesis that r1 = 0 with a p-value of 2.7%,

using binomial calculation. Therefore, these tests indicate that there may be a positive

risk premium associated with Beta. However, the premium, if any, is small and cannot

be estimated accurately.

The problem with finding a premium for Beta may be because it is too di�cult

to estimate. The estimation in Table 4A, with a constant equity premium, can be

considered a CAPM with Beta=1 for all firms. In Table 4A, the equity premium is

27

positive and significant, estimated to be 0.72% (t-ratio=5.35). The last two columns

of Table 6 show the pricing error for CAPM with Beta from CRSP, and CAPM with

Beta⌘ 1. The average pricing errors are 0.378 and 0.379 respectively. Indeed, the CAPM

with Beta⌘ 1 has a higher pricing error only in 9 of the 27 years.

In summary, the results show that using Beta=1 is simpler, and do not lead to

increase in pricing error.

Table 6 evaluates the forecasting horizon, based on the assumption that Beta=1 and

g = g

o

+ g1 ⇤R&D. In terms of pricing error, Table 6 shows that using 4, 5, or 6 years as

forecasting horizons gives nearly identical results (RPE=0.399, 0.389, 0.388, and 0.392

respectively). Using 7 years or less than 3 years results in a slightly higher loss.

In summary, for a residual valuation model with a perpetual abnormal earnings

growth rate, we find that

• The optimal forecasting horizon is 4 or 5 years.

• The best risk-free rate comes from ten-year t-bonds.

• The best risk premium using CAPM is to assume a constant Beta=1, with an

equity premium of about 0.72%.

• The perpetual growth rate of abnormal earnings is negative in general. With R&D

expenditure as a fraction of sales, the typical rate is g=�6.6 + 0.45*R&D.

6 A Comparison of Alternative Valuation Methods

In this section, I will compare the valuation strategy obtained in Section 4, which is

described in Table 4A and will be referred to as “RIV5 with R&D,” to a number of ad

28

hoc and full information alternatives, as described in Section 2. The section is in the

same spirit as Liu, Nissim, and Thomas (2002). Di↵erences from their paper include a

di↵erent estimation procedure (minimum RPE) and a number of modifications to the

full-information valuation strategy.

Table 7 presents various summary statistics and distributional characteristics of the

pricing error distribution. Even though the pricing error is symmetrized in the estimation

procedure, the pricing error given in the tables is

w =P � P

P

and is not symmetrized, for the sake of comparability to Liu, Nissim, and Thomas

(2002). In Table 7A, I also present the pricing error in the logarithm scale, defined as

log(P )� log(P ) = log(1+w). The logarithm scale reduces the impact of outliers on the

mean and variance.

The first thing that one may notice is that the mean pricing error is about 0.1 for

most procedures, except for the RIV model in Claus and Thomas (2001) with a mean

of 0.26. In Example 1 of Section 3, a zero mean for an unsymmetrized pricing error

indicates underpricing. Therefore, a positive mean is expected.

The variance of pricing error displays erratic behavior. This is because there is a

small number of large outliers in pricing error, and the standard error is very sensitive

to outliers. For this reason, I will focus on two aspects: (1) the interquartile range

and other percentile di↵erences of the pricing error distribution and (2) the mean and

standard error of log price di↵erence as the summary statistics. These two criteria

usually lead to the same conclusions.

Table 7A shows that the mean of the pricing error in a logarithm scale for most val-

29

uation methods is zero, indicating the pricing is nearly unbiased in the logarithm scale.

The only exception is RIV5-CT (Claus and Thomas, 2001), which tends to overprice.

In a di↵erent light: the median pricing errors in Table 7B are close to zero for all meth-

ods except RIV5-CT. This indicates that the minimum pricing error as an estimation

criterion leads to an estimated price that is underpriced about 50% of the time and

overpriced the other 50% of the time.

Comparison of the standard error in the logarithm scale shows that RIV5-with-R&D

performs best among all the alternatives. A similar conclusion can be obtained from

comparing the interpercentile ranges in the pricing errors. A good valuation strategy

minimizes the di↵erence between predicted and actual price, therefore a tight distri-

bution in pricing error indicates good performance. RIV5-with-R&D has the smallest

standard error (0.404). RIV5-const, which is RIV with annually optimized cost of equity

and perpetual growth rate, has the second smallest standard error (0.417).

Among the remaining alternatives, the sum of the forecasted earnings (EPSS) with

a standard error of 0.444 is the best, and EG2 (which is EPS2*(1+g5)) performs nearly

identically. Indeed, Liu, Nissim, and Thomas (2002) show that these two simple methods

outperform every other alternative. This is still true here with the exception of full

information valuation with optimally chosen parameters. The successes of EPSS and

EG2 indicate that a good valuation strategy must consider both EPS2 and the forecasted

growth rate. Overall, the log pricing error of EPSS and EG2 is about 10% higher than

the best alternative of RIV5-with-R&D (0.444 and 0.449 vs. 0.404). The other simple

alternatives, such as EPS1, EG1, and BVE+EPS1, perform poorly.

A similar result can be seen from comparisons of the inter-quantile range of the

pricing errors. Note that the inter-quantile ranges are based on the pricing error w, not

30

on the logarithm of pricing error. The interquartile range (P75�P25) and other inter-

quantile ranges (P90�P10 and P99�P1) all have their smallest value at RIV5 with

R&D. RIV5 with constant COE and perpetual growth rate ranks second. Valuation

based on EPSS and EG1 performs similarly, and is better than all other alternatives.

These comparisons can also be seen in Figures 2 and 3. Figure 2 shows the com-

parison of RIV5-with-R&D against multiples-based alternatives, and Figure 3 shows the

comparison with more sophisticated valuation methods. One can see that RIV5 with

R&D clearly dominates all the other alternatives in both figures.

As shown in Liu, Nissim, and Thomas (2002), the GLS method performs worse than

EPSS and EG2 do. Its performance is not sensitive to the choice of IROE, as long as the

cost of equity is optimized. Therefore, I set IROE=0.1 for convenience. Its performance

relative to EG2 and EPSS is improved over that in Liu, Nissim, and Thomas (2002) due

to optimization in the cost of equity r

o

used in the valuation. The interquartile range is

0.554, about 20% greater than RIV5 with R&D.

The last procedure in Table 7 is the valuation proposed by Ohlson and Juettner-

Nauroth (2005). With g =T-10�3%, as suggested in their paper, their procedure gives

the worst valuation out of all procedures considered. Investigation into the procedure

shows that it is due to an overvaluation in growth. The OJ valuation equation can be

rewritten as

P

i,t

=EPS

i,t+2 � r

o

d

i,t+1 � (1 + g) EPSi,t+1

r

o

(ro

� g).

A quick analysis of the numerator indicates that firms with EPSi,t+2 < (1 + g)EPS

i,t+1

would be priced negatively, due to overpricing of the growth component. When opti-

mized over the potential value of g, the best choice for g is approximately g = �80%.

Using this optimized value, the OJ model’s performance is close to that of EPS2.

31

Given the percentile, an estimate for valuation uncertainty can be obtained. Assume

that one is to obtain a 95% confidence interval for the price of a stock. Given the

percentiles in Table 7B, with a 95% confidence level we have P2.5< w <P97.5. This

translates into 1 + P2.5 < P/P < 1 + P97.5, which means a 95% confidence interval for

the price P , so

P

1 + P97.5< P <

P

1 + P2.5.

For example, for RIV5 with R&D, the 95% confidence interval is 0.44⇤ P < P < 2.27⇤ P

based on P2.5= �0.559 and P97.5=1.261 in Table 7B. Similarly, a 50% confidence

interval is 0.80 ⇤ P < P < 1.25 ⇤ P .

In summary, in this section, I have shown that RIV5 with R&D outperforms every

alternative. Choosing the cost of equity in GLS and OJ valuation procedures using

minimum pricing error approach improves performance, but they still perform worse

than simple alternatives do. The OJ model overemphasizes the role of growth, and the

optimal choice of growth rate is indeed �80%.

7 Conclusions and Limitations

In this paper, I have attempted to determine the form and parameter values in an

equity valuation process. This procedure leads to a number of surprising but intuitive

findings: The best risk-free rate is from a 10-year t-bond; the best perpetual growth

rate of abnormal earnings should be negative (about �6% in recent years); and equity

premium over 10-year t-bonds is only about 0.72% since 1981.

The procedure adopted here is to choose the best parameter values that match the

market valuation of a cross-section of stocks. This approach extracts the parameters

from the market price through the present value relationship. It is essentially a relative

32

pricing approach. This approach cannot be used to answer the question of whether the

overall level of market prices is right. Rather, it determines the appropriate perpetual

growth rate and cost of equity implied by the cross-sectional market prices, and uses

that to determine the price of individual stocks.

The methodology has its advantages and limitations. Historical stock return can be

too high due to a fortunate outcome in real history, or too low due to an unfortunate

outcome. Implied cost of equity is not subject to that. This is especially important

for developing countries with only a short history of having equity markets. However,

whether or not the implied parameters equal the market expectation depends on two

crucial things: whether the functional form (the finite horizon PVR) is the correct rep-

resentation of stock prices, and more importantly, whether analysts’ forecasts represent

market expectations. If these assumptions do not hold, the estimated cost of equity

need not to represent the conceptual parameter which is the required rate of return.

Therefore, the estimated cost of equity must be interpreted in this context.

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36

Table 1: Descriptive Statistics

Panel A: Summary Statistics

Mean StDev Median Minimum MaximumMVE 4451 17310 662.98 0.01 558882BVE 1612 5891 269.73 0.00 221567EPS0 1.49 6.32 1.16 -56.98 1077EPS1 1.76 1.73 1.39 0.01 100.50EPS2 2.07 1.82 1.68 0.02 88.42DPO 0.21 0.27 0.09 0 1.00g5 16.76 9.96 14.88 -23.10 533.00R&D 0.02 0.03 0 0 1.00⇢ 0.42 0.29 0.42 0 1.00Beta 0.90 0.58 0.84 -1 3.00A total of 46,899 firm-years is obtained, except for Beta.The sample size is reduced to 43,428 when Beta is used.

Panel B: Correlation Coe�cients

(Pearson (above) and Spearman (below))

MVE BVE EPS0 EPS1 EPS2 DPO g5 R&D BetaMVE 1.00 0.80 0.04 0.17 0.17 0.07 -0.07 0.05 0.05BVE 0.90 1.00 0.10 0.21 0.21 0.11 -0.11 -0.01 0.04EPS0 0.41 0.45 1.00 0.26 0.25 0.05 -0.07 -0.06 0.01EPS1 0.49 0.51 0.83 1.00 0.98 0.10 -0.26 -0.17 0.03EPS2 0.49 0.51 0.80 0.97 1.00 0.10 -0.24 -0.15 0.05DPO 0.28 0.36 0.33 0.34 0.33 1.00 -0.37 -0.23 -0.18g5 -0.22 -0.38 -0.39 -0.42 -0.38 -0.56 1.00 0.34 0.23R&D 0.02 -0.07 -0.19 -0.19 -0.16 -0.22 0.28 1.00 0.27Beta 0.25 0.19 -0.03 -0.01 0.02 -0.17 0.24 0.25 1.00

Analysts’ forecasts (EPS1 and EPS2, one-year and two-year ahead earnings per shareforecasts) and stock price are obtained from I/B/E/S Unadjusted Summary Database. Thebook value of equity (BVE) and actual earnings (EPS0) are obtained from COMPUSTAT,and they are the values at the prior year end. For each firm-year, only the summary forecastat the sixth month after the end of a firm’s fiscal year is used to ensure that prior year’sfinancial measures are known. Firms are required to have a nonmissing value for earningsforecasts of years +1 and +2, and a five year growth rate forecast (g5). DPO is the dividendpayout ratio, winsorized to 0 or 1 if less than 0 or greater than 1. R&D is a firm’s researchand development expenditure scaled by total revenue, all from the year before. The variable⇢ is an earnings persistence parameter estimated from the AR(1) model of actual earnings,EPS

t

=⇢o

+⇢EPSt�1. The Beta is obtained from CRSP. The CRSP Data Description Guide

states that “CRSP provides annual betas computed using the methods developed by Scholesand Williams (Myron Scholes and Joseph Williams, “Estimating Betas from NonsynchronousData”, Journal of Financial Economics, vol 5, 1977, 309-327).”

37

Table 2: Valuation with Di↵erent Assumptions

Panel A gives the pricing error for di↵erent assumptions of the risk-free interest rates, equitypremia, and perpetual growth rates of residual earnings, assuming that they are identical acrossfirms. Cases 1-3 of Panel B show the pricing error with short-term and long-term interest rateas the discount rate, with optimized equity premia and perpetual growth rates. Case 4 ofPanel B is based on assumptions in Claus and Thomas (2001), in which the risk-free rate isfrom 10-year t-bonds, the equity premium is 3%, and the perpetual growth rate is the 10-yeart-bond rate minus 3%.

Panel A: Pricing Errors

(a) Rf

= One-year T-Bill Rate (T-1)Risk g =

Premium 0% 2% 4%1% 1.109 2.393 4.4103% 0.588 0.860 2.0066% 0.637 0.651 0.751

(b) Rf

= Five-year T-bond Rate (T-5)0% 2% 4%

1% 0.720 1.221 2.9213% 0.486 0.595 0.9716% 0.727 0.723 0.769

(c) Rf

= Ten-year T-bond Rate (T-10)0% 2% 4%

1% 0.622 0.971 2.2233% 0.474 0.544 0.7986% 0.774 0.770 0.807

Panel B: Minimum Pricing Error Estimation

Risk-free Equity Perpetual PricingRate Premium Growth Rate Error

1 T-1 1.18% �9.81 0.4672 T-5 0.97% �7.11 0.4453 T-10 0.82% �6.28 0.438

CT T-10 3.00% T-10�3% 0.578

38

Table 3: Equity Valuation with Perpetual Growth

The table gives the minimum pricing error estimate for the cost of equity and perpetualabnormal earnings growth rate each year, based on the finite horizon residual income valuation(RIV)

Pi,t

= Bi,t

+5X

⌧=1

aei,t+⌧

(1 + rt

)⌧

+(1 + g

t

) aei,t+5

(rt

� gt

)(1 + rt

)5

along with the corrsponding pricing error. Risk-free interest rates are given for comparisons.

Year #Obs T-1(%) T-5(%) T-10(%) rt

(%) gt

(%) RPE1981 67 13.37 13.78 13.67 15.21 -3.28 0.3771982 71 9.14 10.28 10.60 11.42 0.57 0.3991983 146 11.41 12.68 12.84 12.52 -1.26 0.3431984 707 8.58 10.40 10.89 11.09 -2.04 0.2991985 782 6.70 7.60 7.87 7.95 -5.80 0.3101986 907 6.83 8.03 8.42 8.18 -3.44 0.3701987 947 7.40 8.51 8.99 9.12 -5.92 0.3201988 1065 8.85 8.81 8.78 8.68 -7.02 0.3201989 1139 8.17 8.57 8.63 9.67 -3.30 0.3691990 1189 6.20 7.67 8.06 8.93 -2.87 0.3651991 1354 4.14 6.50 7.26 8.39 -3.51 0.3451992 1529 3.42 5.28 6.08 8.16 -2.38 0.3311993 1931 5.06 6.52 6.95 7.51 -5.98 0.3551994 2189 6.07 6.57 6.78 8.24 -4.60 0.3801995 2561 5.58 6.31 6.58 7.87 -2.96 0.4311996 2905 5.77 6.44 6.59 6.96 -4.65 0.3541997 3016 5.39 5.58 5.61 6.51 -5.83 0.4501998 2850 4.87 5.38 5.48 7.63 -4.18 0.5901999 2699 6.13 6.47 6.30 8.57 -1.19 0.8682000 2177 3.94 4.91 5.32 7.31 -3.48 0.5522001 2313 2.24 4.28 4.98 6.66 -4.88 0.4202002 2492 1.22 2.65 3.69 6.08 -6.72 0.4252003 2830 1.77 3.72 4.59 5.20 -7.34 0.3542004 2956 3.33 3.87 4.17 5.12 -7.05 0.3552005 3020 4.95 4.92 5.01 5.30 -8.31 0.3832006 2916 4.89 4.66 4.74 5.08 -6.44 0.3442007 141 3.44 3.62 4.13 5.67 -6.53 0.398

Mean 1737 5.88 6.81 7.14 8.11 -4.46 0.400StDev 2.80 2.71 2.59 2.37 2.19 0.114

T-ratio 17.44 -10.37

39

1980 1985 1990 1995 2000 2005

24

68

1012

14

Year

Rat

es

Estimate10−yr T−bond Rate5−yr T−bond Rate1−yr T−bill Rate

Figure 1: Estimated Cost of Equity versus Risk-free Interest Rates

40

Table 4: Valuation, Conservatism, and Earnings Persistence

This table gives the minimum pricing error estimate of parameters and corrsponding pricingerrors each year for the finite horizon residual income valuation (RIV)

Pi,t

= Bi,t

+5X

⌧=1

aei,t+⌧

(1 + rt

)⌧

+(1 + g

i,t

) aei,t+5

(rt

� gi,t

)(1 + rt

)5

In Panel A, the perpetual growth rate g depends on the R&D, gi,t

= go

+ g1 ⇤R&Di,t�1. R&D

is calculated as a fraction of total revenues based on prior year’s data. RPE(0) corresponds topricing error when g1 = 0. In Panel B,

gi,t

= go

+ g1 ⇤ R&Di,t�1 + g2 ⇤ ⇢

i

,

where ⇢i

is an earnings persistence parameter estimated from the AR(1) model of actualearnings, EPS

t

=⇢o

+⇢i

EPSt�1.

Year ro

�Rf

go

(%) g1 RPE(g1) RPE(0)1981 1.54 -2.76 -0.43 0.371 0.3711982 -0.05 -4.41 0.88 0.404 0.4191983 -0.15 -1.14 0.00 0.324 0.3241984 0.23 -2.16 0.00 0.293 0.2931985 -0.13 -7.85 0.38 0.314 0.3161986 -0.34 -4.70 0.38 0.369 0.3721987 0.13 -6.66 0.31 0.319 0.3201988 -0.14 -6.69 -0.20 0.318 0.3181989 1.00 -4.32 0.25 0.371 0.3721990 0.89 -3.29 0.22 0.360 0.3611991 1.01 -4.56 0.23 0.344 0.3451992 1.89 -3.40 0.21 0.335 0.3361993 0.39 -7.30 0.28 0.351 0.3521994 1.20 -7.93 0.85 0.357 0.3791995 0.94 -5.32 0.44 0.431 0.4381996 0.26 -5.44 0.21 0.351 0.3531997 0.79 -6.41 0.08 0.453 0.4531998 1.59 -7.97 0.72 0.580 0.5961999 1.49 -8.80 1.43 0.750 0.8692000 1.38 -8.36 1.00 0.514 0.5542001 1.33 -7.61 0.65 0.413 0.4262002 1.65 -12.74 1.29 0.389 0.4272003 0.12 -11.28 0.81 0.333 0.3532004 0.61 -10.26 0.75 0.335 0.3552005 0.00 -11.19 0.73 0.362 0.3782006 0.24 -8.01 0.49 0.334 0.3442007 1.64 -7.36 0.32 0.393 0.397

Mean 0.72 -6.59 0.45 0.388 0.401StDev 0.69 2.89 0.43 0.10 0.12

T-Ratio 5.35 -11.62 5.40

41

Year ro

� rf

go

(%) g1 g2 RPE1981 2.78 -14.76 -0.01 0.2609 0.3481982 0.07 -11.92 0.11 0.1507 0.3971983 -0.24 -2.67 0.03 0.0224 0.3241984 0.34 -4.66 0.04 0.0519 0.2901985 -0.14 -12.98 0.41 0.0904 0.3081986 -0.38 -8.65 0.31 0.0705 0.3621987 -0.06 -12.07 0.31 0.0786 0.3151988 -0.13 -11.30 -0.02 0.0763 0.3151989 0.96 -8.31 0.25 0.0741 0.3651990 0.88 -5.75 0.23 0.0484 0.3561991 0.97 -7.19 0.28 0.0442 0.3401992 1.84 -4.89 0.23 0.0254 0.3331993 0.42 -8.99 0.23 0.0403 0.3481994 1.22 -8.47 0.73 0.0154 0.3581995 0.96 -5.80 0.39 0.0137 0.4301996 0.27 -6.36 0.21 0.0211 0.3501997 0.82 -7.42 0.08 0.0263 0.4511998 1.60 -9.02 0.64 0.0282 0.5771999 1.87 -7.23 1.13 0.0027 0.7492000 1.39 -8.92 0.86 0.0208 0.5082001 1.38 -8.61 0.56 0.0285 0.4082002 1.64 -13.32 1.10 0.0181 0.3882003 0.08 -11.64 0.71 0.0074 0.3332004 0.60 -10.59 0.64 0.0099 0.3352005 -0.01 -11.48 0.62 0.0096 0.3622006 0.26 -8.15 0.43 0.0071 0.3342007 1.63 -11.21 0.30 0.0786 0.381

Mean 0.78 -8.98 0.40 0.0489 0.384StDev 0.80 2.95 0.32 0.0544 0.096

T-Ratio 4.95 -15.54 6.43 4.5889

42

Table 5: Estimation based on CAPM

The table gives the minimum pricing error estimate of parameters and corrsponding pricingerror each year for the finite horizon residual income valuation (RIV)

Pi,t

= Bi,t

+5X

⌧=1

aei,t+⌧

(1 + ri,t

)⌧

+(1 + g

i,t

) aei,t+5

(ri,t

� gi,t

)(1 + ri,t

)5

The perpetual growth rate g depends on R&D, gi,t

= go

+ g1 ⇤R&Di,t�1. The cost of equity is

from CAPM, ri,t

= Rf

+ r1Betai,t

. Beta is obtained from CRSP. The CRSP Data DescriptionGuide states that “CRSP provides annual betas computed using the methods developed byScholes and Williams (Myron Scholes and Joseph Williams, “Estimating Betas from Nonsyn-chronous Data”, Journal of Financial Economics, vol 5, 1977, 309-327).”

Year #Obs r1 go

(%) g1 RPE(�) RPE(1)1981 67 0.0154 -3.43 0.00 0.377 0.3711982 69 -0.0203 -11.55 0.42 0.410 0.4281983 137 0.0002 -0.32 0.00 0.320 0.3201984 672 0.0014 -2.10 0.00 0.291 0.2901985 724 -0.0019 -6.68 0.01 0.312 0.3111986 820 -0.0046 -5.08 0.35 0.348 0.3481987 878 0.0024 -5.49 0.15 0.312 0.3131988 998 0.0004 -6.28 -0.00 0.311 0.3111989 1069 0.0087 -5.16 0.36 0.371 0.3691990 1135 0.0047 -4.68 0.25 0.359 0.3541991 1226 0.0089 -4.91 0.26 0.341 0.3431992 1407 0.0050 -8.39 0.40 0.329 0.3231993 1723 0.0005 -8.12 0.31 0.338 0.3361994 2022 0.0004 -12.32 0.89 0.355 0.3511995 2318 0.0042 -7.53 0.48 0.406 0.4031996 2622 -0.0053 -7.39 0.16 0.339 0.3411997 2757 0.0002 -8.62 0.11 0.427 0.4261998 2722 -0.0005 -15.04 0.99 0.576 0.5691999 2510 -0.0237 -21.12 1.90 0.706 0.7312000 2063 -0.0069 -15.47 1.20 0.519 0.5172001 2202 0.0015 -12.26 0.79 0.411 0.4062002 2363 0.0003 -20.28 1.57 0.381 0.3732003 2638 0.0028 -10.40 0.69 0.326 0.3272004 2696 0.0059 -9.91 0.66 0.320 0.3232005 2774 0.0054 -8.43 0.51 0.328 0.3322006 2677 0.0021 -7.64 0.40 0.320 0.3202007 139 0.0237 -4.58 0.02 0.369 0.397

Mean 1608 0.0011 -8.63 0.47 0.378 0.379StDev 0.0091 5.01 0.49

T-ratio 0.6164 -8.78 4.89

43

Table 6: Comparisons of Forecast Horizon

This table compares di↵erent forecasting horizon in terms of valuation errors usingfinite horizon residual income valuation (RIV)

Pi,t

= B

i,t

+KX

⌧=1

ae

i,t+⌧

(1 + r

o

)⌧

+(1 + g) ae

i,t+K

(ro

� g)(1 + r

o

)K

The perpetual growth rate g depends on R&D, g = g

o

+g1⇤R&Dt�1. The values presented

here are the average of the estimates across years.

Horizon (K) r

o

(%) g

o

(%) g1(%) RPE2 -0.80 -4.02 0.45 0.4193 -0.01 -3.98 0.44 0.3994 0.45 -4.83 0.44 0.3895 0.72 -6.59 0.45 0.3886 0.76 -10.08 0.36 0.3927 0.47 -19.45 0.79 0.399

44

Table 7: Comparisons of Valuation Models

This table compares the pricing error of di↵erent valuation models (multiples):

• Forward earnings: EPS1 and EPS2, the one-year-out and two-year-out EPS forecasts;

• Sum of forward earnings: EPSS=P5

i=1EPSt+i

, the sum of one year out to five year outearnings forecasts.

• Linear combination of BVE and forward earnings (BVE+EPS1) Pt

= � g

r�g

Bt

+ 1r�g

EPSt+1.

• Residual income valuation with five year earnings forecasts,

Pi,t

= Bi,t

+5X

⌧=1

aei,t+⌧

(1 + ro

)⌧

+(1 + g) ae

i,t+5

(ro

� g)(1 + ro

)5

For “RIV5 w/ R&D”, the perpetual growth rate is g = go

+g1⇤R&Dt�1. For RIV5-const,

g1 ⌘ 0. For RIV5-CT, the parameters are obtained from Claus and Thomas (2001), withg =T-10�3% and r

o

=T-10+3%. “RIV5-zero AE” assumes that g = �100%, so thatthe residual earnings are always 0 after the five-year forecasting horizon.

• Residual income valuation using Gephardt, Lee, and Swaminathan (2001)

ROEt+2+k

= (1� k

10) ROE

t+2 +k

10IROE

for k = 1, . . . , 10 and ROEi,t

= EPSi,t

/Bi,t�1, with the assumption that ROE

t+k

=IROEfor k � 12. “RIV GLS 0.1” sets IROE=0.1; “RIV GLS 0.01” sets IROE=0.01; the costof equity r

o

is obtained through the minimization of pricing error.

• Valuation using Ohlson and Juettner-Nauroth (2005), with

Pi,t

=EPS

i,t+1

ro

+EPS

i,t+2 � ro

di,t+1 � (1 + r

o

) EPSi,t+1

ro

(ro

� g).

the cost of equity ro

is obtained through the minimization of pricing error. “OJ g =T10�3%”sets g =T10�3%, as suggested in Ohlson and Juettner-Nauroth (2005) and used in Godeand Mohanram (2004). “OJ g = �0.8” sets g = �80%, approximately the minimumpricing error estimate for g.

Given in tables are the summary statistics of pricing error w = P /P�1. Panel A gives the meanand standard deviation of w, and the mean and standard deviation of log(1 + w) = log(P /P ),and the inter-quantile ranges, where Pn represents the n-th percentile. Panel B gives the actualpercentiles.

45

Panel A: Summary Statistics of the Pricing Error Distribution

Mean STD Mean-l STD-l P75�P25 P90�P10 P99�P1EPS1 0.112 0.692 -0.045 0.620 0.625 1.270 2.217EPS2 0.093 0.673 -0.015 0.469 0.516 1.080 1.916EPSS 0.098 0.835 -0.006 0.444 0.486 1.034 1.920EG1 0.188 1.005 -0.004 0.597 0.671 1.482 2.862EG2 0.094 0.661 -0.008 0.449 0.490 1.040 1.907BVE+EPS1 0.109 0.814 -0.019 0.510 0.601 1.183 2.025RIV5 w/R&D 0.101 1.134 0.003 0.404 0.455 0.966 1.820

RIV5-const 0.101 1.236 -0.001 0.417 0.468 0.992 1.883RIV5-CT 0.263 2.079 0.108 0.502 0.565 1.248 2.510RIV GLS 0.1 0.082 1.012 -0.041 0.480 0.554 1.131 2.000OJ g = T10� 3% 0.111 0.713 -0.046 0.647 0.655 1.337 2.328OJ g = �0.8 0.097 0.693 -0.011 0.464 0.508 1.073 1.928

Panel B: Percentiles of the Pricing Error Distributions

P2.5 P10 P25 Median P75 P90 P97.5EPS1 -0.793 -0.523 -0.248 0.053 0.377 0.747 1.425EPS2 -0.657 -0.433 -0.217 0.022 0.299 0.647 1.259EPSS -0.628 -0.401 -0.206 0.011 0.280 0.633 1.292EG1 -0.737 -0.499 -0.271 0.009 0.400 0.983 2.125EG2 -0.634 -0.406 -0.207 0.012 0.282 0.634 1.273BVE+EPS1 -0.689 -0.472 -0.249 0.032 0.352 0.711 1.336RIV5 w/R&D -0.559 -0.367 -0.198 0.000 0.257 0.600 1.261

RIV5-const -0.578 -0.379 -0.205 0.000 0.262 0.613 1.305RIV5-CT -0.639 -0.335 -0.120 0.118 0.445 0.913 1.871RIV GLS 0.1 -0.647 -0.461 -0.268 -0.021 0.286 0.670 1.353OJ g = T10� 3% -0.856 -0.561 -0.263 0.054 0.392 0.776 1.473OJ g = �0.8 -0.648 -0.423 -0.213 0.017 0.295 0.651 1.280

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−1.0 −0.5 0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Pricing Error

Den

sity

RIV5 with R&DEPSSEPS2EPS1EG2BVE+EPS1

Figure 2: Distribution of Pricing Errors from Simple Valuation Methods

−1.0 −0.5 0.0 0.5 1.0 1.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Pricing Error

Den

sity

RIV5 with R&DRIV5−CTRIV−GLS ROE12=0.1OJ g=T10−3%OJ g=−0.8

Figure 3: Distribution of Pricing Errors from RIV Methods

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