Preliminary. Comments welcome.

Equity Valuation Using Multiples

Jing LiuAnderson Graduate School of Management

University of California at Los Angeles(310) 206-5861

jing.liu@anderson.ucla.edu

Doron NissimColumbia University

Graduate School of Business(212) 854-4249

dn75@columbia.edu

and

Jacob ThomasColumbia University

Graduate School of Business(212) 854-3492

jkt1@columbia.edu

January, 2000

We received helpful comments from David Aboody, Jack Hughes, Jim Ohlson, Stephen Penman,Michael Williams, and seminar participants at Columbia, Copenhagen Business School, OhioState, and UCLA.

Equity Valuation Using Multiples

Abstract

In this study we examine the valuation performance of a comprehensive list of commonly usedprice multiples. Our analysis indicates the following ranking: forward earnings multiples performthe best, followed by historical earnings measures, cash flow measures and book value of equityare tied for third, and sales performs the worst. Contrary to the popular view that differentindustries have different “best” multiples, we find that these overall rankings are observedconsistently for all industries examined. Performance is improved by allowing for an intercept inthe linear relation between price and value drivers, relative to the ratio formulation typicallyassumed in practice. Performance is not improved, however, by the use of more complex valuedrivers, such as the short cut value measures based on generic patterns for residual incomegrowth past the forecast horizon.

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Equity Valuation Using Multiples1. Introduction

In this study we examine the valuation performance of a comprehensive list of price

multiples. The multiples we consider include three measures of accrual flows (sales,

COMPUSTAT earnings and IBES earnings), one accrual stock measure (book value), four

measures of cash flows (cash flow from operations, free cash flow, maintenance cash flow, and

earnings before interest, taxes, depreciation, and amortization (EBITDA)), and three measures of

forward earnings (EPS1, EPS2, and EPS3: 1, 2, and 3-year out consensus analysts’ earnings

forecasts). We also consider more complex ways to incorporate value-relevant information,

including variants of the popular short cut value measures based on the residual income model.

The multiple approach assumes firm value is directly proportional to some value driver,

such as earnings or book value. The approach is typically applied as follows: first, identify a set

of comparable firms; next, generate a multiple equal to the mean (or median) ratio of market

price to the value driver for that set; and finally, generate firm value by applying that multiple to

the firm’s value driver.

Comprehensive equity valuations, which require detailed pro forma analyses and present

value calculations, should in theory perform better than simple multiples. In addition to bringing

less information to bear on the valuation process, the multiple approach results in “relative” not

“absolute” valuation, since firm value is estimated relative to the pricing of comparable firms.1

There are, however, some concerns associated with implementing comprehensive valuations.

First, the question of how best to control for risk remains largely unresolved. Although risk-

adjusted discount rates are used heuristically in practice, there are concerns that errors in

1 There is an element of relative pricing even in the case of comprehensive valuation, since stocks are valued

relative to risk-free bonds. If there are concerns about the risk-free rate (the so-called risk-free rate puzzle),those concerns remain in stock valuations based on discount rates derived from risk-free rates.

2

assumed rates distort valuations. Second, comprehensive valuations require projections to

infinity. Rather than make specific projections for all future years, simplifying assumptions (such

as constant growth in free cash flows or a multiple of terminal earnings) are normally adopted to

capture a terminal value, representing value beyond a horizon date. Since a large fraction of total

value typically resides in the terminal value, estimates of firm value hinge substantially on the

simplifying assumptions.

Given these concerns about comprehensive valuations, multiples are used often in day-to-

day valuation, either as a substitute for or as a complement to comprehensive valuations. Analyst

reports, regulatory filings, valuations for estate and gift tax purposes, and the financial press

frequently use multiples to value firms. When complementing comprehensive valuations,

multiples are typically used to obtain terminal values and to calibrate the comprehensive

valuation. The advantages of multiples, relative to comprehensive valuations, include

extraordinary simplicity and the use of contemporaneous market information. While this

simplicity reduces information content, it also reduces potential noise. It is not obvious a priori

whether the benefits of reduced noise exceed the costs of reduced information content.

Although the multiple approach bypasses explicit projections and present value

calculations, it relies on the same principles underlying the more comprehensive approach: value

is an increasing function of future payoffs and a decreasing function of risk. Therefore, the

multiple approach should perform reasonably well if the value driver reflects future firm

profitability, and the comparable group is similar to the firm being valued along various value

attributes, such as growth and risk. To study the impact of selecting comparable firms from the

same industry, we contrast our results obtained by using industry (as defined by IBES)

comparables with results obtained when all firms in the cross-section are used as comparables.

3

Regardless of the role of multiples vis-a-vis comprehensive valuations, there is limited

descriptive evidence on the absolute and relative performance of different multiples, and the

variation across industries in that performance (e.g., Boatsman and Baskin [1981], LeClair

[1990], and Alford [1992]). Recently, a number of studies have examined the role of multiples

for firm valuation in specific contexts, such as tax and bankruptcy court cases and initial public

offerings (e.g., Beatty, Riffe, and Thompson [1999], Gilson, Hutchkiss and Ruback [2000], Kim

and Ritter [1999], and Tasker [1998]). Our study continues in the same vein, but is more

comprehensive. As in most prior research, we evaluate multiples by examining the distribution

of percent pricing errors: actual price less price predicted by the multiple, scaled by actual price.

To eliminate in-sample bias and control for differences in the degrees of freedom across tests, we

evaluate all multiples based on out of sample prediction. That is, when calculating multiples we

always exclude the firm being valued.

Our analysis consists of two stages. In the first stage, we use the conventional ratio

representation (i.e., price doubles when the value driver doubles). In the second stage, we relax

the requirement that value is directly proportional to value drivers, while retaining the

assumption that the relation is linear. In essence, the second stage analysis allows for an

intercept, whereas the first stage does not.

In the first stage, multiples are calculated using the harmonic mean of the ratio of price to

value driver (the reciprocal of the mean of the value driver-to-price ratio) for comparable firms.

Although this estimator is rarely used (see Beatty, Riffe, and Thompson [1999]), it offers the

desirable property that the percent pricing error is zero, on average. It is also recommended by

Baker and Ruback [1999], based on detailed econometric analyses of alternative estimators.

While the harmonic mean estimator results in lower pricing errors than the simple mean or

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median, our ranking of the relative performance of different multiples remains unchanged when

the mean or median is used instead of the harmonic mean.

The following is an overview of the relative performance of different multiples:

• forecasted earnings perform the best, even better than more complex short cut valuations

based on generic residual income growth patterns past the terminal date;

• among drivers derived from historical data, earnings perform better than book value; and

IBES earnings (which exclude some one-time items) perform better than COMPUSTAT

earnings;

• cash flow measures, defined in various forms, perform poorly; and

• sales performs the worst.

When comparable firms are restricted to be from the same industry, performance

improves for all multiples. We also find that the relative performance of the multiples we

consider does not vary much across industries. That is, contrary to general perception, we do not

find that different industries are associated with different “best multiples.” This finding suggests

that our result is driven by the intrinsic information content of the different value drivers, rather

than their ability to capture industry-specific value-relevant factors.

Turning from relative performance to absolute performance, the forward earnings

multiples describe actual stock prices reasonably well. For example, for 3 year out forecasted

earnings or EPS3, the standard deviation of pricing error is about 29%, and approximately half

the firms have absolute pricing errors less than 15%. While there are some firms with very large

pricing errors, stock prices for a substantial majority of the firms are explained relatively well by

simple multiples based on two or three year out forecasted earnings. The dispersion of pricing

errors increases substantially for multiples based on historical drivers, such as earnings and cash

5

flows, and is especially large for sales multiples. For example, approximately half the firms have

absolute pricing errors less than 21%, 25%, and 36% for IBES actual earnings, EBITDA, and

sales, respectively.

For the second stage, we estimate the intercept and slope of the price/value driver relation

by minimizing the sample variance of percent valuation errors, subject to the constraint that the

valuation is on average unbiased. The procedure we follow is related to that proposed by Beatty,

Riffe, and Thompson [1999]. As might be expected, allowing for an intercept reduces the

dispersion of valuation errors for all multiples, and the improvement observed is inversely

related to the performance of that multiple in the first stage (no intercept).2 As in the first stage,

we find that moving from a cross-sectional comparison group to using comparable firms within

each industry further reduces pricing errors. These results suggest that the traditional ratio

formulation should be replaced by a relation that allows for an intercept, especially for multiples

that perform poorly in the traditional ratio formulation. We recognize, however, that if simplicity

is the primary motivation to use multiples, the reduction in pricing errors may not be sufficient to

compensate for the additional complexity introduced by adding an intercept.

To contrast multiples with comprehensive valuations, we construct intrinsic value

measures based on the residual income model, assuming generic patterns for residual income

past the forecast horizon. Surprisingly, these more complex value measures perform worse than

simple multiples based on forecasted earnings. We examine three alternative patterns for post-

horizon residual income: (1) constant abnormal earnings past year 5, (2) zero abnormal earnings

past year 5, and (3) ROE trending toward an industry median (between year 3 and year 12).

2 For example, for multiples based on IBES actual earnings, EBITDA, and Sales, approximately half the firms

have pricing errors less than 19%, 22%, and 29%, relative to 21%, 25%, and 36%, respectively, in the ratioformulation. The improvement is much smaller for multiples that perform well in the first stage; e.g. for EPS3,approximately half the firms have pricing errors less than 14.6%, relative to 15% in the ratio formulation.

6

These intrinsic value measures utilize information about forward earnings at different horizons,

equity book values, firm-specific discount rates, and industry profitability. Further, a structure

derived from valuation theory is imposed to aggregate that information. Despite these advantages

of intrinsic value measures over simple forward earnings multiples, they do not perform better

than simple multiples based on forward earnings.3 Preliminary investigations designed to

uncover possible causes for this result suggest that errors in terminal value proxies and estimated

discount rates are partially responsible. We find that simply aggregating earnings forecasts for

years 1 to 5 produces the lowest valuation errors of all multiples.

We also considered two other extensions to the multiple approach (results not reported in

this version).4 First, we combined two or more value drivers (e.g., Cheng and McNamara

[1996]). Our results, based on a regression approach (e.g., Beatty, Riffe, and Thompson [1999])

indicate only small improvements in performance over that obtained for forward earnings.

Second, we investigated conditional earnings and book value multiples. That is, rather than use

the harmonic mean P/E and P/B values of comparable firms, we use a P/E (P/B) that is

appropriate given the forecast earnings growth (forecast book profitability) for that firm. We first

estimate the relation between forward P/E ratios and forecast earnings growth (P/B ratios and

forecast return on common equity) for each industry-year, and then read off from that relation the

P/E (P/B) corresponding to the earnings growth forecast (forecast ROCE) for the firm being

valued. Despite the intuitive appeal of conditioning the multiple on relevant information, we

were unable to document any improvement in performance. Bradshaw [1999a and 1999b] is able

3 Bradshaw [1999a and 1999b] observes results that are related to ours. He finds that PEG, a construct based on

forward P/E ratios and forecast long-term earnings growth rates (g), explains more variation in target prices andrecommendations than more rigorous valuation models.

4 Details of those results are available from the authors upon request.

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to find, however, that a more restrictive form of conditioning (P/E equals forecast growth)

improves performance for his sample of firms.

Our findings have a number of implications for valuation research. First, we confirm the

validity of two precepts underlying the valuation role of accounting numbers: a) accruals

improve the valuation properties of cash flows, and b) despite the importance of top-line

revenues, its value relevance is limited until it is matched with expenses. Second, we confirm

that forward-looking data (specifically, near-term forecasted earnings) contain considerably more

value-relevant information than historical data. Third, we provide evidence on the signal/noise

tradeoff associated with developing more complex valuation drivers. Finally, our results suggest

that forward earnings multiples should be used as long as earnings forecasts are available, since

they outperform the other multiples in all 68 industries we examine.

The rest of the paper is organized as follows: section 2 contains a literature review;

section 3 describes the methodology; section 4 describes our sample selection process; section 5

reports results and discusses implications; and section 6 concludes the paper.

2. Literature Review

While most of the popular textbooks on valuation (e.g., Copeland, Koller, and Murrin

[1994], Damodaran [1996]) devote considerable space to discussing multiples, there is little

empirical research published on the valuation properties of multiples. Most existing papers that

study multiples use a limited data set and consider only a subset of multiples, such as earnings

and EBITDA. The methodology used also varies from one study to another, making it difficult to

compare results from different studies.

Among commonly used value drivers, earnings and cash flows have received most of the

attention. Boatman and Baskin [1981] compare the valuation accuracy of P/E multiples based on

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two sets of comparable firms from the same industry. They find that valuation errors are smaller

when comparable firms are chosen based on similar historical earnings growth, relative to when

they are chosen randomly. Alford [1992] investigates the effects of choosing comparables based

on industry, size (risk), and earnings growth on the precision of valuation using P/E multiples.

He finds that valuation errors decline when the industry definition used to select comparable

firms is narrowed from a broad, single digit SIC code to classifications based on two and three

digits, but there is no additional improvement when the four-digit classification is considered. He

also finds that controlling for size and earnings growth, over and above industry controls, does

not reduce valuation errors.

Kaplan and Ruback [1995] examine the valuation properties of the discounted cash flow

(DCF) approach in the context of highly leveraged transactions. While they conclude that DCF

performs well in valuation, they find that simple EBITDA multiples result in similar valuation

accuracy. Beatty, Riffe, and Thompson [1999] examine different linear combinations of value

drivers derived from earnings, book value, dividends, and total assets. They derive and document

the benefits of using the harmonic mean, and introduce the price-scaled regressions we use. They

find the best performance is achieved by using a) weights derived from harmonic mean book and

earnings multiples and b) coefficients from price-scaled regressions on earnings and book value.

In a recent study, Baker and Ruback [1999] examine econometric problems in identifying

industry multiples, and compare the relative performance of multiples based on EBITDA, EBIT

and revenue. They provide theoretical and empirical evidence that absolute valuation errors are

proportional to value. They further show that industry multiples estimated using the harmonic

mean are close to minimum-variance estimates based on Monte Carlo simulations. Using the

minimum-variance estimator as a benchmark, they find that the harmonic mean dominates

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alternative simple estimators such as the simple mean, median, and value-weighted mean.

Finally, they use the harmonic mean estimator to calculate multiples based on EBITDA, EBIT

and revenue, and find that industry-adjusted EBITDA performs better than EBIT and revenue.

Instead of focusing only on historical accounting numbers, Kim and Ritter [1999] add

forecasted earnings to the conventional list of value drivers, which includes book value, earnings,

cash flows, and sales. They investigate how initial public offering prices are set using multiples.

Consistent with our results, they find that forward P/E multiples (based on forecasted earnings)

dominate all other multiples in valuation accuracy, and that the next year EPS forecast (EPS2)

dominates the current year EPS forecast (EPS1).

It has been recognized that the use of large data sets could diminish the performance of

multiples, since the researcher selects comparable firms in a mechanical way. In contrast, market

participants may select comparable firms more carefully and take into account situation-specific

factors not considered by researchers. Tasker [1998] examines patterns in the selection of

comparable firms across industries in acquisition transactions by investment bankers and

analysts. She finds the systematic use of industry-specific multiples, which is consistent with

different multiples being more appropriate in different industries.5

3. Methodology

In this section we describe the different value drivers considered, and the methodology

used in the two stages of our analyses: estimating the price/value driver relation without and with

an intercept.

5 Since it is not clear whether the objective of investment bankers/analysts is to achieve the most accurate

valuation in terms of smallest dispersion in percent pricing errors, our results may not be directly comparablewith those in Tasker [1998].

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3.1 Value Drivers

The following is a list of value drivers examined in this paper (details of all variables are

provided in the appendix):6

• Accrual stock: current book value (BV).

• Accrual flows: sales, COMPUSTAT earnings (CACT) and IBES earnings (IACT).

• Cash flows: cash flow from operations (CFO), free cash flow to debt and equity holders

(FCF), maintenance cash flow (MCF, equal to free cash flows for the case when capital

expenditures equal depreciation expense), and earnings before interest, taxes, depreciation

and amortization (EBITDA).

• Forward looking information: consensus one year out, two year out and three year out

earnings forecasts (EPS1, EPS2 and EPS3), where 3 2 *(1 )eps eps g= + , and g is the long

term eps growth forecast provided by analysts.

• Intrinsic pricing measure (P1*): This measure, which is based on the residual income (or

abnormal earnings) valuation approach, is considered since it appears in a number of recent

papers and its pricing properties are relatively better understood.7 In essence, intrinsic value

equals the book value plus the present value of future abnormal earnings. For future years

(beyond year +5) with no available earnings forecasts, abnormal earnings are estimated by

assuming that they do not grow. Details of the implementation of P1* are discussed in the

next section.

All the variables listed above have been linked to value before. Accounting book value

and earnings are used extensively for valuation purposes. Ohlson [1995] and Feltham and Ohlson

6 Some value drivers are not easily classified. For example, Sales, which is categorized as an accrual flow, could

contain less accruals than EBITDA, which is categorized as a cash flow measure.

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[1995] build valuation models in which earnings and book value play instrumental roles. In some

market inefficiency studies (e.g., Basu [1977] and Stattman [1980]), earnings and book value are

assumed to represent “fundamentals,” and are even shown to contain value relevant information

not reflected in market prices.

Accruals distinguish accounting numbers from cash flows. Accounting earnings could be

more value-relevant than current cash flows for at least two reasons: a) cash flows do not reflect

value creation in some cases (e.g., asset purchases), and b) accruals allow managers to reflect

their judgment about future prospects. However, the flexibility allowed within GAAP creates the

potential for accounting numbers to be distorted, thereby reducing their value relevance. This

potential for earnings management, in combination with the truism that price reflects the present

value of future cash flows, has caused some to prefer cash flow multiples over multiples based

on accounting numbers. To provide some empirical evidence on this debate, we consider four

cash flow measures, and contrast their value-relevance with two multiples based on accounting

earnings.

The four cash flow measures considered are the most popular ones used in practice. Each

measure removes the impact of accruals to a different extent. EBITDA adjusts pre-tax earnings

to debt and equity holders for the effects of depreciation and amortization only. CFO deducts

interest and tax expense from EBITDA and also deducts the net investment in working capital.

FCF deducts from CFO net investments in all long-term assets, whereas MCF only deducts from

CFO an investment equal to the depreciation expense for that year.

For earnings-based multiples, we consider reported earnings excluding extraordinary

items and discontinued operation from COMPUSTAT, and actual earnings as defined by IBES.

7 Existing literature gauges valuation properties by comparing R2 from cross-sectional regressions. We use a

different metric, which we believe corrects some biases in the popular method.

12

The second measure is derived from the first earnings measure by deleting some one-time items,

such as write-offs and restructuring charges. To the extent that the IBES measure is a better

proxy for “permanent” or “core” earnings (earnings that are expected to persist in the future), it

will be linked more directly to price. Although the use of sales as a value driver has less

theoretical basis, relative to earnings and cash flows, we consider it because of its wide use in

certain emerging industries where earnings and cash flow are perceived to be uninformative.

The potential mismatch between historical data, such as reported earnings and cash flows,

and the forward-looking information captured by prices has long been recognized in the

literature. Analysts’ forecasts of future period earnings offer a possible solution to this mismatch.

Liu and Thomas [1999] find that revisions in analysts’ earnings forecasts and changes in interest

rates explain a large portion of contemporaneous stock returns. We include EPS1 and EPS2

because these two forecasts are usually available for most firms. To incorporate the information

contained in the long-term EPS growth forecast, we construct EPS3 by adding the amount

implied by that growth rate to EPS2.

The discounted residual income model has been widely used as a way to calculate

“intrinsic values.” Several recent studies provide evidence that the model explains stock prices

(e.g., Frankel and Lee [1998], Abarbanell and Bernard [1997], Claus and Thomas [1999]) and

returns (e.g., Liu and Thomas [1999], Liu [1999]). Consistent with many prior studies, we

assume zero growth in abnormal earnings past a horizon date. Although it incorporates more

information than any of the simple multiples, this approach is not as detailed as a comprehensive

valuation based on pro forma projections that allow for firm-specific growth in abnormal

earnings beyond the horizon date.

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3.2 Traditional Multiple Valuation

In the first stage of our analysis, we follow the traditional ratio representation and require

that the price of firm i in year t (pit) is directly proportional to the value driver:

it t it itp xb e= + (1)

where itx is the value driver of firm i in year t, tb is the multiple on the value driver and ite is the

pricing error. Since our focus is on percent pricing errors (εit/pit), not pricing errors, we divide

equation (1) by price, to obtain the following.

1 it itt

it it

x

p p

eb= + . (2)

Baker and Ruback [1999] and Beatty, Riffe, and Thompson [1999] discuss the problems

associated with estimating the slope using equation (1), because the residual in that equation is

approximately proportional to price.

When estimating βt, we elected to impose the restriction that expected percent pricing

errors (ε/p) be zero, even though an unrestricted OLS estimate for βt from equation (2) offers a

lower value of mean squared percent pricing error.8 Empirically, we find that our approach

generates lower pricing errors for most firms, relative to an unrestricted estimate, but it generates

substantially higher errors in the tails of the distribution. By restricting ourselves to unbiased

pricing errors, we are in effect assigning lower weight to extreme pricing errors, relative to the

8 To investigate the tradeoff between bias and dispersion of pricing errors associated with our choice of a

restricted regression, we investigated the distribution of pricing errors for the unrestricted case. We estimatedequation (2) for comparable firms from the cross-section. (When using comparable firms from the sameindustry, the estimated multiples generated substantial pricing errors.) We find that the distributions of percentpricing errors for all multiples are shifted to the right substantially, relative to the distributions for the restrictedcase reported in the paper (our distributions tend to peak around zero pricing error). This shift to the rightindicates that the multiples and predicted valuations for the unrestricted case are on average lower than ours.We find that the bias created by this shift causes greater pricing errors for the bulk of the firms not in the tails ofthe distribution, relative to our restricted case.

14

unrestricted approach. We are also maintaining consistency with the tradition in econometrics

that appears to exhibit a lexicographic preference for reduced bias over reduced dispersion.

Our approach is to minimize mean squared percent pricing errors when estimating

equation (2), subject to the constraint that those errors be zero on average, i.e., 0itt

itp

e Ε = .

Since βt is the only parameter to be estimated in equation (2), the unbiasedness restriction alone

is sufficient to determine that parameter. Applying the expectation operator to equation (2) and

using the restriction, the estimate for tb is the harmonic mean of the price-value driver ratio.

1t

itt

it

x

p

b = Ε

(3)

To eliminate in-sample bias, we estimate tb for each firm using all relevant comparable

firms excluding the firm that is being valued. We predict the value of the firm by multiplying the

tb estimate by the firm’s value driver, and then calculate the percent pricing error as follows:9

ˆit it t it

it it

p x

p p

e b−= . (4)

The performance of multiples is evaluated by examining the dispersion of the pooled distribution

of /it itpe (lower dispersion indicates better performance).

3.3 Intercept Adjusted Multiples

For the second stage of our analysis, we relax the direct proportionality requirement and

allow for an intercept:

it t t it itp xa b e= + + . (5)

9 Note that some studies measure the pricing error as the difference between the predicted value and price (e.g,

Alford [1992]) while we measure the pricing error as the difference between price and the predicted value.

15

There are many factors, besides the value driver under investigation, that affect price. The

average effect on price of such omitted factors is not likely to be zero. The intercept in equation

(5) captures the average effect of omitted factors and misspecifications and thus its inclusion

may improve the precision of out of sample predictions.

As with the simple multiple approach, we divide equation (5) by price to focus on percent

pricing errors.

11 it it

t tit it it

x

p p p

ea b= + + , (6)

OLS estimation of equation (6), with no restrictions, minimizes the sum of the squares of percent

pricing errors, but the expected value of those errors is non-zero.10 For the reasons mentioned in

section 3.2, and to maintain consistency with our estimates from the no-intercept approach, we

impose the restriction that percent pricing errors be unbiased.11 That is, we seek to estimate the

parameters αt and βt that minimize the mean squared error ( /it itpe ), subject to the restriction that

the expected value of /it itpe is zero:

)]1

(1var[]/)var[()/( varmin,

it

itt

ittititttititit p

x

ppxpp βαβαε

βα+−=⋅−−= (7a)

. . 0.itt

it

s tp

e Ε = (7b)

To obtain estimates for αt and βt, we restate restriction (7b) as follows

Ep

Ep

x

pit

itt

itt

it

it

( )ε

α β= − −FHG

IKJ =1

10 (8)

10 In general, this bias could be removed by allowing for an intercept. That avenue is not available, however, when

the dependent variable is a constant (=1), since the intercept captures all the variation in the dependent variable,thereby making the independent variables redundant.

11 As with equation (2), pricing errors from the unrestricted approach for equation (6) were higher for most firms(in the middle of the distribution) but were smaller in the tails.

16

Solve (8) for αt, and substitute into (7a) to restate the minimization problem in terms of the

following regression with no intercept:

11

1 1−

F

H

GGG

I

K

JJJ= −

FHG

IKJ

FHG

IKJ

L

N

MMMM

O

Q

PPPPp Ep

x

p

Ex

p

p Ep

it t

tit

it

t

it t( )

β (9)

where the different Et(.) represent the mean values of those expressions based on the comparable

group. The estimate for βt is then substituted into equation (8) to obtain an estimate for αt. Those

estimates are then used along with the value driver for the firm being valued to generate a

valuation.

4. Sample and Data

To construct the sample, we merge data from three sources: accounting numbers from

COMPUSTAT; price, analyst forecasts, and actual earnings per share from IBES; and stock

returns from CRSP. As of April of each year, we select a cross-section of firms based on the

following criteria: (1) all COMPUSTAT value drivers for the previous year are available; (2) the

fiscal year ends in December; (3) price, actual EPS, forecasted EPS for years +1 and +2, and a

long term growth forecast are available in the IBES summary file; and (4) none of the price

ratios is an outlier (defined as lying outside the 1% to 99% of the pooled distribution). The

resulting sample includes 17,505 observations between 1981 and 1996. This sample is used for

the descriptive statistics reported in Table 1. For the results reported after Table 1, we impose

four additional requirements: (5) share price on the day IBES publishes summary forecasts in

April is greater than or equal to $2;12 (6) monthly stock returns are available in the CRSP files

12 Since our valuation model has an intercept, valuation error would be abnormally large for stocks with very low

share prices.

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for at least 30 of the 60 months prior to April; (7) all multiples are positive; and (8) each

industry-year set has at least five observations (industry as defined by IBES). These requirements

reduce the sample to 9,658 observations.

We adjust all per share numbers for stock splits and stock dividends using IBES

adjustment factors. If IBES indicates that the majority of forecasts for that firm-year are on a

fully diluted basis, we use IBES dilution factors to convert those numbers to a primary basis. We

summarize all variable definitions in the appendix.

The P1* variable is calculated using the discounted residual income model, assuming

zero growth in abnormal earnings after year five:

5* 1 5 4

51

( ) ( )1

(1 ) (1 )t t s t t s t t t t

t t ss t t t

eps k bv eps k bvP bv

k k k+ + − + +

=

Ε − Ε − = + + + + ∑ (8)

where

bvt = book value per share at time t (the end of year t),

epst = earnings per share in year t,

kt = the discount rate for equity at time t.

The discount rate (kt) is calculated as the risk-free rate plus beta times the equity risk

premium. We use the 10-year Treasury bond yield on April 1 of year t+1 as the risk-free rate

and assume a constant 5% equity risk premium. We measure beta as the median beta of all firms

in the same beta decile in year t. We estimate betas using monthly stock returns and value-

weighted CRSP returns for the five years that end in March of year t+1 (at least 30 observations

are required).13

For a subgroup of firm-years (less than 5 percent), we were able to obtain mean IBES

forecasts for all years in the five-year horizon. For all other firms, with less than complete

18

forecasts available between years 3 and 5, we generated forecasts by applying the mean long-

term growth forecast (g) to the mean forecast for the prior year in the horizon; i.e.,

)1(*1 gepseps stst += −++ .

The book values for future years, corresponding to the earnings forecasts, are determined

by assuming the “ex-ante clean surplus relation” (ending book value in each future period equals

beginning book value plus forecasted earnings less forecasted dividends). Since analyst

forecasts of future dividends are not available on IBES, we assume that the current dividend

payout ratio will be maintained in the future. We measure the current dividend payout as the

ratio of the indicated annual cash dividends to the earnings forecast for year t+1 (both obtained

from the IBES summary file).14 To minimize biases that could be induced by extreme dividend

payout ratios (caused by forecast t+1 earnings that are close to zero), we Winsorize payout ratios

at 10% and 50%.15

We also calculate four variants of P1* (P2* through P5*) that we use to investigate the

information/noise tradeoff among the components of P1*. Definitions for these additional

variables are provided in the appendix and the results are discussed in Section 5.

5. Results

5.1 Descriptive Statistics

Table 1 reports the pooled distribution of ratios of value drivers to price. The table

indicates that cash flow multiples are likely to perform poorly. Free cash flow and maintenance

cash flow are often negative (approximately 30% and 20% of the sample, respectively).

13 We use decile median betas, since firm-specific betas are estimated with considerable error.14 Indicated annual dividends are four times the most recent quarter’s declared dividends. We use EPS1 as the

deflator because it varies less than current year's earnings and is less likely to be close to zero or negative.15 The impact of altering the dividend payout assumptions on the results is negligible, because it has a very small

impact on future book value and an even smaller impact on the computed abnormal earnings.

19

Moreover, the mean of FCF/P is negative, and the mean of MCF/P is close to zero, despite the

deletion of observations with extreme values (top and bottom 1%). Given the difficulty of

mapping negative value drivers to positive share values, we conclude that these two value drivers

are not suitable for multiple valuation purposes and drop them from the remainder of the

analysis.

Table 2 reports the Pearson and Spearman correlations among the ratios of value drivers

to price. Most of the ratios are highly correlated, which suggests that they share a large portion of

common information. The correlations among different earnings forecast ratios are especially

high, generally around 90%. Interestingly, the correlation between earnings forecasts ratios and

P1*/P is only about 50%, which suggests that book value and discount rate adjustments have a

significant impact on the information contained in P1*.

5.2 Traditional Multiples

The results of the first stage analysis, based on the traditional ratio representation (no

intercept), are reported in Table 3. The results reported in Panel A use the entire cross-section of

firms as comparables for computing multiples, and the results in Panel B are based on

comparables selected from the same IBES industry group. Out-of-sample value predictions are

made each year, and percentage valuation errors are pooled across firm-years. We report the

following statistics that describe the distribution of the percent pricing errors: two measures of

central tendency (mean and median) and five measures of dispersion (the standard deviation and

four non-parametric dispersion measures: (i) 75%-25%, (ii) 90%-10%, (iii) 95%-5% and (iv)

99%-1%). Since we restrict the multiples to yield unbiased valuation on average, all the means

are close to zero.16

16 Since the valuations are done out of sample, it is natural to expect some means to deviate from zero by chance.

20

The valuation errors in Panel A exhibit slight negative skewness, suggested by the fact

that medians are higher than means. This implies that the multiple approach undervalues most

firms by a small amount and overvalues some firms by large amounts. This occurs because the

predicted values are bounded from below at zero, while they are not bounded above. A potential

way to make the error distribution symmetrical is to take the log of ˆ /P P (Kaplan and Ruback

[1995]). However, we choose not to follow this approach because the percent pricing errors we

consider are easier to interpret.

Examination of the standard deviation and the four non-parametric dispersion measures

in Panel A suggests the following ranking of multiples. Forecasted earnings, as a group, exhibit

the lowest dispersion of percent pricing errors. This result makes intuitive sense because earnings

forecasts reflect future profitability better than historical measures. Consistent with this

reasoning, performance increases with forecast horizon. The dispersion measures for EPS2 are

lower than those for EPS1 (standard deviation decreases from 0.348 to 0.311, inter-quartile range

decreases from 0.440 to 0.368). The improvement from EPS2 to EPS3 is less dramatic,

consistent with measurement error in the long-term growth forecast used to construct EPS3.

In addition to ranking the relative performance of different multiples, the results in Table

3 can also be used to infer absolute pricing errors. Specifically, halving the four non-parametric

dispersion measures provides an estimate of the range of absolute pricing error within which a

certain fraction of the sample lies. For example, the inter-quartile range of 0.347 for EPS3 in

Panel A, indicates that approximately half the sample has an absolute pricing error less than

17%.17

17 This statement assumes the distribution is symmetric around zero. Because the distribution is not precisely

symmetric around zero, the numbers we provide are approximate.

21

Forecasted earnings are followed by P1*, earnings, book value, cash flows, and sales, in

decreasing order of performance. It is perhaps surprising that P1* does not perform as well as

forecasted earnings, even though the information in each of the forecasted earnings is a subset of

that contained in P1*. The valuation error of P1* has a standard deviation of 0.403 and inter-

quartile range of 0.504. This result suggests that although P1* incorporates additional

information such as firm specific beta, market interest rates, book value, and growth; these

explicit adjustments in combination with the assumption that all firms’ abnormal earnings stop

growing after year 5 result in a noisy valuation measure. In section 5.5, we investigate further the

likely causes for the poor relative performance of P1*.

Comparing the two summary accounting numbers, book value and earnings, we find that

earnings clearly outperforms book value, which is consistent with “street intuition.” The

valuation error for book value (BV) has a standard deviation of 0.536 and inter-quartile range of

0.697, compared to a standard deviation of 0.477 and inter-quartile range of 0.579 for

COMPUSTAT earnings (CACT). The performance of historical earnings is further enhanced by

the removal of one-time transitory components. Consistent with the results in Liu and Thomas

[1999], IBES earnings (IACT) have an even lower standard deviation of 0.448 and inter-quartile

range of 0.549.

Contrary to the belief that “Cash is King” in valuation, our results show cash flows

perform significantly worse than accounting earnings. For example, the valuation error of

EBITDA has a standard deviation of 0.611 (28% higher than earnings) and inter-quartile range

of 0.687 (19% higher than earnings). Between the two cash flow measures, CFO and EBITDA,

there is little difference in performance.18

18 The free cash flow and maintenance free cash flow measures, which are excluded from this analysis because of

the large proportion of negative values, exhibit even worse performance.

22

The sales multiple performs the worst. Its valuation error has a standard deviation of

0.948, and inter-quartile range of 0.761, implying that approximately 50% of the firms have

valuation errors larger than 38%. This result suggests that sales do not reflect profitability until

expenses have been considered. A frequent reason for using sales as a value driver is when

earnings and cash flows are negative. Since we restrict our sample to firms with positive earnings

and cash flows, our sample is less likely to contain firms for which the sales multiple is more

likely to be used in practice. In particular, our sample is unlikely to contain Internet stocks (e.g.

Hand [1999] and Trueman, Wong, and Zhang [2000]), and there are reasons to believe our

results cannot be generalized to that group.

To conduct the analysis using comparable firms from the same industry, we searched for

a reasonable industry classification scheme. Because of the evidence that SIC codes frequently

misclassify firms (Kim and Ritter [1999]), we use the industry classification provided by IBES.

IBES indicate that their classification is based loosely on SIC codes, but it is also subject to

detailed adjustments.19 The IBES industry classification has three levels (in increasing fineness):

sector, industry, and group. We use the intermediate (industry) classification level because

sectors are too broad to allow the selection of homogenous firms, and groups are too narrow to

allow the inclusion of sufficient comparable firms (given the loss of observations due to our data

requirements).

The results reported in Panel B, which are based on comparable firms from the same

IBES industry classification, exhibit improved performance over those reported in Panel A. The

improvement is consistent with the joint hypothesis that (1) increased homogeneity in the value-

relevant factors omitted from the multiples results in better valuation, and (2) IBES industry

19 The IBES classification resembles the industry groupings suggested by Morgan Stanley.

23

classification identifies relatively homogeneous groups of firms.20 Generally, the improvement is

larger at the center of the distributions; that is, small valuation errors became much smaller while

large valuation errors do not change much.

The multiples used in calculating the percent pricing errors in Panels A and B were

estimated using the harmonic mean. To make our results comparable to those in previous studies

(e.g., Alford [1992]) as well as to examine their robustness, we replicate Panel B using the

median instead of the harmonic mean. Those results are reported in Panel C. Consistent with the

evidence in Baker and Ruback [1999] and Beatty, Riffe and Thompson [1999], we find that

median multiples perform worse than harmonic mean multiples. The relative performance (i.e.,

ranking) of the different multiples, however, remains the same.

To offer a visual picture of the relative and absolute performance of different categories

of multiples, we provide in Figure 1 the histograms for percent pricing errors for the following

selected multiples: EPS3, P1*, IACT, EBITDA, BV and Sales. The histograms report the

fraction of the sample that lies within ranges of percent pricing error that are of width equal to

10% (e.g. –0.1 to 0, 0 to 0.1, and so on). To reduce clutter, we simply draw a smooth line

through the middle of the top of each histogram column, rather than provide the histograms for

each of the multiples. A multiple is considered better if it has a more peaked distribution. The

differences in performance across the different categories are clearly visible in Figure 1. The

figure also offers a better view of the shapes of the different distributions and enables readers to

find the fraction of firms within different pricing error ranges for each distribution.

20 Even if these conditions are satisfied, it is not clear that there should be an improvement. Moving from the

cross-section to each industry results in a substantial decrease in sample size, and consequently the estimation isless precise. This fact is also reflected in the increase in the deviation of the sample mean of the valuationerrors from zero.

24

5.3 Intercept Adjusted Multiples

In this subsection, we report results based on the second stage analysis, where we allow

an intercept in the relation between price and value drivers. The optimization problem in

equation (7) is solved out of sample to obtain parameter estimates, and valuation errors are then

calculated using these parameters. Again, the analysis is conducted for comparable firms from

the entire cross-section (Table 4, Panel A) and the same industry (Panel B).

As predicted, relaxing the no-intercept restriction improves the performance of all

multiples. The degree of improvement is not uniform, however. Multiples that perform poorly in

Panel A of Table 3 improve more than those that do well. For example, EPS3’s valuation errors

exhibit a small decrease in standard deviation (inter-quartile range): from 0.313 (0.347) to 0.306

(0.333), while sales’ valuation errors decrease from 0.948 (0.761) to 0.676 (0.615). Although the

improvement in absolute performance of the multiples is not uniform, the rank order of multiples

remains unchanged from Table 3 to Table 4.

The improvement generated by allowing for an intercept can also be seen by comparing

the results in Panel A of Table 4, based on comparable firms from the entire cross-section, with

those in Panel B of Table 3, based on comparable firms from the same industry. Although simple

industry multiples are better than simple cross-sectional multiples, the intercept-adjusted cross-

sectional multiples are better than the simple industry multiples for historical value drivers and

are only slightly worse for forecasted value drivers.

The best performance is achieved when we allow for an intercept and select comparable

firms from the same industry (Table 4, Panel B). Comparing these results with those in Panel A

of Table 3 illustrates the joint benefits of allowing for an intercept and restricting comparable

firms to those in the same IBES industry. For example, the standard deviation (inter-quartile

25

range) for sales, the worst performer, improves from 0.948 (0.761) to 0.668 (0.574); and for

EPS3, the best performer, the improvement is from 0.313 (0.347) to 0.289 (0.293). Consistent

with the results in Table 3, the improvement in valuation by allowing for an intercept (i.e., from

Panel A to Panel B) is relatively uniform across multiples.

5.4 Adjustment for Leverage

Since Sales and EBITDA pertain to the value of the whole firm (enterprise value) rather

than equity alone, multiples computed on the market value of equity are potentially in error. To

correct for this mismatch, we repeat the analyses reported in Tables 3 and 4 for these two value

drivers using enterprise value (market value of equity plus book value of debt) instead of equity

value. To facilitate comparability with the results for other multiples in Tables 3 and 4, we

compute percentage errors in terms of equity value. In effect, we construct multiples based on

enterprise value for the two value drivers, use the comparable firm multiples to estimate each

firm’s enterprise value, and then subtract the book value of debt to estimate equity value.

Table 5 reports the results of this analysis. The first two rows in each panel provide the

results without leverage adjustment and are the same as the corresponding rows in tables 3 and 4.

The next two rows are based on the leverage adjustment. To our surprise, leverage adjustment

does not improve the fit. Leverage-adjusted Sales performs worse in all four panels of Table 5.

For EBITDA, the leverage adjustment reduces slightly the valuation errors in Panel A, increases

the valuation errors in Panels B and C, and has only a marginal effect in Panel D. Although

puzzling at first glance, our results are consistent with those of Alford [1992], who finds that

adjusting P/EBIT multiples for differences in leverage across comparable firms decreases

accuracy.

26

5.5. Potential Errors in P1*

We conduct an investigation into possible reasons why the intrinsic value measure P1*

does not outperform forward earnings multiples, even though it incorporates more information

and imposes a structure on that information that is prescribed by theory. One possibility is that

the assumption that abnormal earnings remain constant past year +5 induces errors in the

terminal value. To understand better this potential source of error, we consider two alternative

assumptions regarding terminal values:

(1) zero abnormal earnings past year 5 (i.e., terminal value equals zero for all firms),

5* 1

1

( )2

(1 )t t s t t s

t t ss t

eps k bvP bv

k+ + −

=

Ε − = + + ∑ , and

(2) ROE forecasts from year +3 trends linearly toward the industry median by year +12, and

abnormal earnings exhibit zero growth thereafter,

[ ] [ ]2 11* 1 12 111

111 3

( ) ( )( )3

(1 ) (1 ) (1 )t t s t t s t t t tt t s t t s

t t s ss st t t t

ROE k bv ROE k bveps k bvP bv

k k k k+ + − + ++ + −

= =

Ε − Ε − Ε −= + + + + + +

∑ ∑ ,

where ( )t t sROE +Ε for s = 4, 5, …, 12 is estimated using a linear interpolation to the industry

median ROE. The industry median ROE is calculated as a moving median of the past ten

years’ ROE of all firms in the industry. To eliminate outliers, industry median ROEs are

Winsorized to lie between the risk free rate and 20%.21

Results are reported in Table 6. As with Table 5, we report the results for all four

combinations: two sets of comparable firms (entire cross section and industry) and two value

relations (with and without an intercept). Because the results in all four Panels of Tale 6 are

similar, we discuss only the results in Panel A. P2* produces essentially the same dispersion in

21 This measure has been proposed by Gebhardt, Lee and Swaminathan [1999].

27

valuation errors as that produced by P1*. This suggests that the terminal value proxy in P1*

contains considerable error, since dropping the terminal value altogether in P2* does not affect

the fit adversely.

We turn next to the improvement offered by the more complex terminal value proxy

incorporated in P3*. This intrinsic value measure allows for firm-specific patterns of profitability

between years +3 and +12, and industry-specific terminal profitability after year +12. Despite the

intuitive appeal of the adjustment proposed in P3*, our results indicate that the percent pricing

errors are actually higher than those observed for P1* and P2*. Again, the additional information

that is incorporated in P3*, regarding the tendency for firms in different industries to revert to

industry means, appears to be negated by increased measurement error.

5.6 Simple Aggregation of Earnings Forecast Information

Since the intrinsic value approach for incorporating the information in the different EPS

forecasts fails to improve on simple multiples based on specific EPS forecasts, we examine two

alternative, simpler, ways of incorporating the information in the different earnings forecasts.

The first measure is based on the sum of the EPS forecasts for the next five years,

5*

1

4 ( )t t t ss

P eps +=

= Ε∑ .22 The second measure attempts to control heuristically for the timing and

risk of the different earnings numbers by discounting the EPS forecasts for the next five years,

5*

1

( )5

(1 )t t s

t ss t

epsP

k+

=

Ε = + ∑ .

The standard deviation (inter-quartile range) of valuation errors for P4* is 0.309 (0.338),

which is lower than that for P1*. In fact, P4* performs better than any of the other multiples

considered so far, including the three forward earnings multiples (EPS1-EPS3). This

28

improvement suggests that a simple aggregation of the earnings forecasts at different horizons

allows us to incorporate the information in those forecasts, whereas the structure imposed by

computing P* adds measurement error. The similarity of the valuation error dispersions in the

fourth row (P4*) and fifth row (P5*) indicates that our simple control for the timing and risk of

future earnings does not improve the valuation.

5.7 Ranking Multiples in Each Industry

Given our focus on understanding the underlying information content of the different

multiples, our focus has been on overall patterns, with firms pooled across industries. It has been

suggested, however, that different multiples work best in different industries. For example,

Tasker [1998] reports that investment bankers and analysts appear to use preferred multiples in

each industry. Therefore, we determine the extent to which the relative rank of different

multiples, based on the dispersion of valuation errors within that industry, varies across different

industries. Although we recognize that our search is unlikely to offer conclusive results, since we

do not pick comparable firms with the same skill and attention as others do in different contexts,

we wish to offer some general results.

Since investment professionals use simple multiples (no intercept) and select comparable

firms from the same industry, we use the same approach here. Then we pool the valuation results

over years for each industry and rank multiples by the standard deviation of valuation errors

within each industry. Table 7 reports the results for the 68 industries we analyze. The ranking

goes from 0 (best) to 11 (worst). We also report summary statistics of the rankings at the bottom

of the table.

22 We thank Jim Ohlson for suggesting this value driver.

29

The overall result shows remarkable consistency across all industries. In almost all

industries, forecasted earnings perform the best, while Sales performs the worst. This result,

which is consistent with the results in Kim and Ritter [1999], suggests that the information

contained in forward looking value drivers captures a considerable fraction of value, and this

feature is common to all industries. Turning to the other value drivers, earnings perform better

than book value and cash flows in most industries. Book value performs well in certain industries

in the finance sector, the energy sector (oil and gas), forest products and gas utilities. Perhaps,

accounting practices in these industries cause book values to be related to market values within

these industries in a more consistent manner.

6. Conclusions

In this study we have examined the valuation properties of a comprehensive list of

multiples. We consider both the commonly used multiple approach, which assumes direct

proportionality between price and value driver, and a less restrictive approach that allows for an

intercept. To identify the importance of selecting comparable firms from the same industry, we

also report results based on the comparable group including all firms in the cross-section. Our

results show the following rank ordering of multiples (from more accurate to less accurate):

forecasted earnings, earnings, cash flows tied with book value, and sales. The ranking is robust

to the use of different statistical methods, and similar results are obtained within individual

industries.

We show that both the industry adjustment (selecting comparables from the same

industry) and the intercept adjustment (allowing for an intercept in the price/value driver

relation) improves the valuation properties of all multiples. While the industry adjustment is

commonly used, the intercept adjustment is not. We speculate that multiples are used primarily

30

because they are simple to comprehend and communicate and the additional complexity

associated with including an intercept exceeds the benefits of improved fit.

Our results are consistent with intuition regarding the information in different value

drivers. For example, future information reflects value better than historical information,

accounting accruals add value-relevant information to cash flows, and profitability can be better

measured when revenue is matched with expenses. Some results in this paper are surprising,

however. For example, a discounted residual income analysis which explicitly forecasts terminal

value and adjusts for risk performs worse than simple multiples based on earnings forecasts. And

adjusting for leverage does not improve the valuation properties of EBITDA and Sales. We

investigate these results further and feel that these results indicate the trade-off that exists

between signal and noise when more complex but theoretically correct structures are imposed.

We recognize that our study is designed to provide an overview of aggregate patterns, and thus

we may have missed more subtle relationships that are only apparent in small sample studies.

We note in conclusion that our analysis assumes that market prices are efficient, and we

evaluate multiples based on their ability to mimic market valuations. If market prices vary

systematically from fundamental or intrinsic value, we may need to revise our conclusions about

the relative and absolute performance of the different multiples considered here. To examine this

possibility, we are currently investigating the ability of these multiples to predict future abnormal

returns. The results in this paper are valid if no relation is observed between future abnormal

returns and pricing errors from different multiples.

31

APPENDIX

This appendix describes how the variables are constructed. All the value drivers are

adjusted for changes in number of shares.

BV: book value of equity, COMPUSTAT item #60

SALES: item #12

CACT: COMPUSTAT earnings (EPS excluding extraordinary items), item #58

IACT: IBES actual earnings

EBITDA: earnings before interest, taxes, depreciation and amortization, item #13

CFO: cash flow from operations, measured as EBITDA minus the total of interest expense

(#15), tax expense (#16) and the net change in working capital (#236)

FCF: free cash flow, measured as CFO minus net investment (#107 - #128 - #113 + #109)

MCF: maintenance cash flow, measured as CFO minus depreciation expense (#125)

EPS1: IBES one year out earnings forecast

EPS2: IBES two year out earnings forecast

EPS3: IBES three year out earnings forecast, measured as EPS2*(1+g), where g is IBES

long term growth forecast

The P* measures:

5* 1 5 4

51

( ) ( )1

(1 ) (1 )t t s t t s t t t t

t t ss t t t

eps k bv eps k bvP bv

k k k+ + − + +

=

Ε − Ε − = + + + + ∑5

* 1

1

( )2

(1 )t t s t t s

t t ss t

eps k bvP bv

k+ + −

=

Ε − = + + ∑

32

[ ] [ ]2 11* 1 12 111

111 3

( ) ( )( )3

(1 ) (1 ) (1 )t t s t t s t t t tt t s t t s

t t s ss st t t t

ROE k bv ROE k bveps k bvP bv

k k k k+ + − + ++ + −

= =

Ε − Ε − Ε −= + + + + + +

∑ ∑

5*

1

4 ( )t t t ss

P eps +=

= Ε∑5

*

1

( )5

(1 )t t s

t ss t

epsP

k+

=

Ε = + ∑The variables used in the P* calculations are obtained in the following way:

The discount rate (kt) is calculated as the risk-free rate plus beta times the equity risk

premium. We use the 10-year Treasury bond yield on April 1 of year t+1 as the risk-free rate

and assume a constant 5% equity risk premium. We measure beta as the median beta of all firms

in the same beta decile in year t. We estimate betas using monthly stock returns and value-

weighted CRSP returns for the five years that end in March of year t+1 (at least 30 observations

are required).

For a subgroup of firm-years (less than 5 percent), we were able to obtain mean IBES

forecasts for all years in the five-year horizon. For all other firms, with less than complete

forecasts available between years 3 and 5, we generated forecasts by applying the mean long-

term growth forecast (g) to the mean forecast for the prior year in the horizon; i.e.,

)1(*1 gepseps stst += −++ .

The book values for future years, corresponding to the earnings forecasts, are determined

by assuming the “ex-ante clean surplus” relation (ending book value in each future period equals

beginning book value plus forecasted earnings less forecasted dividends). Since analyst

forecasts of future dividends are not available on IBES, we assume that the current dividend

payout ratio will be maintained in the future. We measure the current dividend payout as the

ratio of the indicated annual cash dividends to the earnings forecast for year t+1 (both obtained

33

from the IBES summary file). To minimize biases that could be induced by extreme dividend

payout ratios (caused by forecast t+1 earnings that are close to zero), we Winsorize payout ratios

at 10% and 50%.

In the calculation of *3tP , ( )t t sROE +Ε for s = 4, 5, …, 12 are forecasted using a linear

interpolation to the industry median ROE. The industry median ROE is calculated as a moving

median of the past ten years’ ROE of all firms in the industry. To eliminate outliers, industry

median ROEs are Winsorized at the risk free rate and 20%.

34

Table 1Distribution of Value Driver to Price Ratios

The variables are defined as follows: P is stock price; BV is book value of equity; MCF is maintenance cash flow(equivalent to free cash flow when depreciation expense equals capital expenditure); FCF is free cash flow to debtand equity holders; CFO is cash flow from operations; EBITDA is earnings before interest, taxes, depreciation andamortization; CACT is COMPUSTAT earnings before extraordinary items; IACT is IBES actual earnings; EPS1and EPS2 are one year out and two year out earnings forecasts; EPS3=EPS2*(1+g), where g is the growth forecast;and TP is enterprise value (price + debt). All totals are deflated by the number of shares outstanding at the end ofthe year.

5* 1 5 4

51

( ) ( )1

(1 ) (1 )t t s t t s t t t t

t t ss t t t

eps k bv eps k bvP bv

k k k+ + − + +

=

Ε − Ε −= + + + +

∑ ,

5* 1

1

( )2

(1 )t t s t t s

t t ss t

eps k bvP bv

k+ + −

=

Ε − = + + ∑ ,

[ ] [ ]2 11* 1 12 111

111 3

( ) ( )( )3

(1 ) (1 ) (1 )t t s t t s t t t tt t s t t s

t t s ss st t t t

ROE k bv ROE k bveps k bvP bv

k k k k+ + − + ++ + −

= =

Ε − Ε − Ε −= + + + + + +

∑ ∑ ,

where ( )t t sROE +Ε for s = 4, 5, …, 12 is forecasted using a linear interpolation to the industry median ROE. The

industry median ROE is calculated as a moving median of the past ten years’ ROE of all firms in the industry. Toeliminate outliers, industry median ROEs are Winsorized at the risk free rate and 20%.

5*

1

4 ( )t t t ss

P eps +=

= Ε∑ , and 5

*

1

( )5

(1 )t t s

t ss t

epsP

k+

=

Ε = + ∑ .

Sample is trimmed at 1% and 99% for each ratio (excluding the P* ratio) using the pooled distribution. Yearscovered are 1981 through 1996. Sample size is 17,505.

Mean SD 1% 5% 10% 25% 50% 75% 90% 95% 99%

BV/P 0.549 0.354 0.014 0.121 0.178 0.300 0.483 0.729 1.001 1.205 1.653

MCF/P 0.034 0.079 -0.281 -0.092 -0.035 0.016 0.043 0.069 0.097 0.123 0.193

FCF/P -0.016 0.161 -0.675 -0.274 -0.154 -0.041 0.019 0.054 0.092 0.127 0.252

CFO/P 0.102 0.094 -0.151 -0.021 0.017 0.053 0.093 0.146 0.207 0.254 0.391

Ebitda/P 0.179 0.142 -0.090 0.015 0.044 0.091 0.152 0.239 0.347 0.437 0.672

CACT/P 0.044 0.084 -0.302 -0.077 -0.015 0.029 0.055 0.080 0.110 0.133 0.182

IACT/P 0.051 0.067 -0.212 -0.041 0.006 0.034 0.057 0.081 0.108 0.131 0.176

Sales/P 1.387 1.475 0.054 0.159 0.254 0.509 0.953 1.731 2.920 4.050 7.450

EPS1/P 0.072 0.041 -0.055 0.016 0.030 0.050 0.070 0.093 0.119 0.140 0.180

EPS2/P 0.090 0.039 0.010 0.036 0.048 0.066 0.086 0.110 0.141 0.162 0.211

EPS3/P 0.106 0.043 0.025 0.049 0.060 0.078 0.099 0.127 0.164 0.188 0.247

P1*/P 0.693 0.308 0.176 0.299 0.370 0.492 0.647 0.847 1.048 1.207 1.727

P2*/P 0.607 0.251 0.161 0.258 0.318 0.424 0.571 0.762 0.935 1.053 1.308

P3*/P 0.790 0.466 0.161 0.273 0.350 0.494 0.691 0.987 1.305 1.570 2.477

P4*/P 0.536 0.217 0.124 0.248 0.308 0.395 0.500 0.640 0.822 0.942 1.236

P5*/P 0.354 0.131 0.076 0.169 0.210 0.270 0.339 0.421 0.525 0.594 0.746

Ebitda/TP 0.123 0.075 -0.081 0.014 0.041 0.081 0.122 0.163 0.207 0.241 0.325

Sales/TP 0.973 0.891 0.052 0.141 0.222 0.408 0.728 1.271 1.971 2.547 4.313

35

Table 2Pearson (Upper Triangle) and Spearman (Lower Triangle) Correlation Matrices

The variables are defined in table 1. Sample is trimmed at 1% and 99% for each ratio (excluding the P* ratio) usingthe pooled distribution. Also, observations for which any of the ratios is negative are deleted. Finally, observationsthat do not belong to an industry-year group with at least five members are deleted. Years covered are 1981 through1996. Sample size is 9,658.

BV/P

CFO/P

Ebitda/P

CACT/P

IACT/P

Sales/P

EPS1/P

EPS2/P

EPS3/P

P1*/P

P2*/P

P3*/P

P4*/P

P5*/P

Ebitda/TP

Sales/TP

BV/P 1.00 0.58 0.65 0.53 0.53 0.52 0.58 0.61 0.56 0.32 0.91 0.41 0.55 0.58 0.54 0.39

CFO/P 0.63 1.00 0.82 0.50 0.52 0.45 0.53 0.51 0.46 0.31 0.60 0.30 0.45 0.50 0.76 0.31

Ebitda/P 0.70 0.87 1.00 0.59 0.61 0.49 0.62 0.59 0.52 0.30 0.65 0.30 0.51 0.55 0.79 0.27

CACT/P 0.52 0.55 0.65 1.00 0.93 0.26 0.82 0.72 0.65 0.35 0.62 0.33 0.66 0.70 0.63 0.22

IACT/P 0.54 0.59 0.67 0.93 1.00 0.27 0.85 0.75 0.67 0.38 0.64 0.35 0.68 0.73 0.65 0.22

Sales/P 0.60 0.55 0.61 0.37 0.39 1.00 0.37 0.44 0.43 0.17 0.46 0.20 0.42 0.40 0.45 0.91

EPS1/P 0.61 0.59 0.68 0.82 0.85 0.49 1.00 0.93 0.85 0.44 0.71 0.38 0.87 0.90 0.64 0.31

EPS2/P 0.63 0.56 0.65 0.72 0.75 0.55 0.93 1.00 0.96 0.44 0.71 0.35 0.96 0.95 0.61 0.38

EPS3/P 0.58 0.50 0.57 0.65 0.68 0.53 0.86 0.96 1.00 0.46 0.67 0.30 0.99 0.95 0.56 0.38

P1*/P 0.42 0.43 0.45 0.45 0.49 0.27 0.53 0.51 0.49 1.00 0.66 0.80 0.46 0.66 0.28 0.13

P2*/P 0.92 0.66 0.73 0.62 0.65 0.58 0.73 0.74 0.69 0.72 1.00 0.65 0.65 0.77 0.53 0.24

P3*/P 0.47 0.40 0.43 0.44 0.47 0.29 0.48 0.43 0.36 0.81 0.70 1.00 0.30 0.50 0.24 0.08

P4*/P 0.57 0.48 0.56 0.66 0.69 0.52 0.87 0.96 0.99 0.49 0.67 0.35 1.00 0.96 0.54 0.33

P5*/P 0.62 0.55 0.63 0.71 0.75 0.52 0.91 0.95 0.95 0.70 0.79 0.56 0.95 1.00 0.55 0.28

Ebitda/TP 0.60 0.83 0.89 0.68 0.70 0.59 0.69 0.65 0.58 0.39 0.53 0.24 0.54 0.55 1.00 0.43

Sales/TP 0.43 0.38 0.38 0.27 0.28 0.93 0.38 0.44 0.45 0.16 0.24 0.08 0.33 0.28 0.50 1.00

36

Table 3Distribution of Percentage Valuation errors for Simple Multiples

Value and value drivers are assumed to be proportional: it t it itp xb e= + . Multiple is estimated excluding the firm

under valuation, using harmonic means: 1/ itt t

it

x

pb

= Ε . Percent pricing error is calculated as follows,

ˆit it t it

it it

p x

p p

e b−= ; its pooled distribution is reported.

The variables are defined as follows: P is stock price; BV is book value of equity; CFO is cash flow from operations;EBITDA is earnings before interest, taxes, depreciation and amortization; CACT is COMPUSTAT earnings beforeextraordinary items; IACT is IBES actual earnings; EPS1, EPS2 are one year out and two year out earningsforecasts; EPS3=EPS2*(1+g), where g is growth forecast. All totals are deflated by the number of sharesoutstanding at the end of the year.

5* 1 5 4

51

( ) ( )1

(1 ) (1 )t t s t t s t t t t

t t ss t t t

eps k bv eps k bvP bv

k k k+ + − + +

=

Ε − Ε −= + + + +

∑Panel A uses the whole cross-section of firms as comparable firms, Panel B uses comparable firms within eachindustry (based on IBES industry classification). Panel C also uses comparable firms within each industry, but themultiple is calculated using the median instead of the harmonic mean. Years covered are 1981 through 1996.Sample size is 9,658.

Panel A: Valuation using mean cross-sectional multiplesMean Median SD 75%-25% 90%-10% 95%-5% 99%-1%

BV 0.000 0.063 0.536 0.697 1.253 1.620 2.564CFO -0.001 0.104 0.589 0.689 1.304 1.748 2.998Ebitda -0.001 0.129 0.611 0.687 1.296 1.674 3.019CACT 0.000 0.047 0.477 0.579 1.120 1.510 2.371IACT 0.000 0.041 0.448 0.549 1.060 1.418 2.242Sales -0.001 0.259 0.948 0.761 1.694 2.394 4.797EPS1 0.000 0.029 0.348 0.440 0.835 1.118 1.721EPS2 0.000 0.026 0.311 0.368 0.743 1.008 1.570EPS3 0.000 0.038 0.313 0.347 0.725 0.992 1.653P1* 0.000 0.056 0.403 0.504 0.918 1.203 1.981

Panel B: Valuation using mean industry multiplesMean Median SD 75%-25% 90%-10% 95%-5% 99%-1%

BV -0.021 0.064 0.530 0.543 1.152 1.586 2.668CFO -0.023 0.056 0.554 0.559 1.171 1.665 2.938Ebitda -0.023 0.064 0.572 0.502 1.084 1.555 2.776CACT -0.014 0.013 0.448 0.462 1.001 1.400 2.322IACT -0.012 0.015 0.412 0.429 0.923 1.288 2.177Sales -0.049 0.157 0.934 0.729 1.610 2.304 4.571EPS1 -0.007 0.018 0.312 0.333 0.711 0.981 1.658EPS2 -0.006 0.021 0.289 0.303 0.657 0.919 1.550EPS3 -0.006 0.026 0.293 0.301 0.658 0.927 1.561P1* -0.010 0.038 0.377 0.369 0.799 1.136 1.991

37

Panel C: Valuation using median industry multiplesMean Median SD 75%-25% 90%-10% 95%-5% 99%-1%

BV -0.109 0.002 0.601 0.571 1.284 1.770 3.054

CFO -0.102 0.000 0.628 0.587 1.273 1.816 3.334

Ebitda -0.114 0.000 0.659 0.519 1.160 1.712 3.198

CACT -0.039 -0.001 0.470 0.465 1.037 1.470 2.469

IACT -0.038 -0.001 0.435 0.434 0.956 1.353 2.256

Sales -0.307 0.001 1.312 0.861 1.936 2.905 6.311

EPS1 -0.028 -0.001 0.323 0.339 0.729 1.023 1.774

EPS2 -0.032 0.000 0.301 0.305 0.676 0.958 1.633

EPS3 -0.039 0.000 0.307 0.305 0.685 0.968 1.646

P1* -0.063 0.000 0.411 0.377 0.838 1.228 2.199

38

Table 4Distribution of Percentage Valuation Errors for Intercept Adjusted Multiples

Value and value drivers are assumed to be linear: it t t it itp xa b e= + ⋅ + . Multiple is estimated excluding the

firm under valuation, by solving a constraint minimization problem:( )

,min var( / ) var ( ) /

. . 0

t tit it it t t it it

itt

it

p p x p

s tp

a be a b

e

= − − ⋅ Ε =

Percentage valuation error is calculated as follows, ˆˆit it t t it

it it

p x

p p

e a b− −= ; its pooled distribution is reported.

The variables are defined as follows: P is stock price; BV is book value of equity; CFO is cash flow from operations;EBITDA is earnings before interest, taxes, depreciation and amortization; CACT is COMPUSTAT earnings beforeextraordinary items; IACT is IBES actual earnings; EPS1, EPS2 are one year out and two year out earningsforecasts; EPS3=EPS2*(1+g), where g is growth forecast. All totals are deflated by the number of sharesoutstanding at the end of the year.

5* 1 5 4

51

( ) ( )1

(1 ) (1 )t t s t t s t t t t

t t ss t t t

eps k bv eps k bvP bv

k k k+ + − + +

=

Ε − Ε −= + + + +

∑Panel A uses the whole cross-section of firms as comparable firms, Panel B uses comparable firms within eachindustry (based on IBES industry classification). Years covered are 1981 through 1996. Sample size is 9,658.

Panel A: Valuation using intercept adjusted cross-sectional multiplesMean Median SD 75%-25% 90%-10% 95%-5% 99%-1%

BV 0.026 0.084 0.458 0.541 1.038 1.408 2.264CFO 0.032 0.108 0.465 0.520 1.013 1.386 2.427Ebitda 0.027 0.112 0.480 0.527 1.011 1.413 2.394CACT 0.012 0.053 0.401 0.477 0.931 1.272 2.028IACT 0.015 0.054 0.380 0.462 0.893 1.209 1.887Sales -0.031 0.160 0.676 0.615 1.370 1.908 3.420EPS1 0.013 0.039 0.321 0.392 0.767 1.030 1.610EPS2 0.008 0.035 0.300 0.344 0.705 0.966 1.533EPS3 0.000 0.042 0.306 0.333 0.706 0.976 1.629P1* 0.016 0.068 0.365 0.435 0.809 1.089 1.830

Panel B: Valuation using intercept adjusted industry multiplesMean Median SD 75%-25% 90%-10% 95%-5% 99%-1%

BV -0.016 0.076 0.477 0.476 1.032 1.442 2.403CFO -0.016 0.072 0.477 0.474 1.023 1.420 2.496Ebitda -0.020 0.082 0.496 0.443 0.976 1.384 2.440CACT -0.012 0.034 0.396 0.406 0.884 1.250 2.068IACT -0.011 0.035 0.372 0.385 0.826 1.175 1.917Sales -0.035 0.146 0.668 0.574 1.291 1.852 3.287EPS1 -0.006 0.026 0.300 0.317 0.689 0.956 1.594EPS2 -0.004 0.028 0.284 0.296 0.642 0.900 1.511EPS3 -0.004 0.033 0.289 0.293 0.650 0.912 1.526P1* -0.008 0.049 0.357 0.342 0.754 1.077 1.884

39

Table 5Leverage Adjustments for EBITDA and Sales Multiples

The variables are defined as follows: P is stock price; EBITDA is earnings before interest, taxes, depreciation andamortization; TP is enterprise value (market value of equity plus book value of debt). Valuations using simple andintercept adjusted multiples are conducted using cross-sectional and industry comparable firms. When TP multiplesare used, equity value is calculated as the predicted enterprise value minus book value of debt. Years covered are1981 through 1996. Sample size is 9,658.

.Panel A: Valuation using mean cross-sectional multiplesMean Median SD 75%-25% 90%-10% 95%-5% 99%-1%

Ebitda/P -0.001 0.129 0.611 0.687 1.296 1.674 3.019

Sales/P -0.001 0.259 0.948 0.761 1.694 2.394 4.797

Ebitda/TP -0.045 0.039 0.601 0.630 1.231 1.671 2.994

Sales/TP 0.000 0.269 1.259 1.031 2.189 3.164 6.602

Panel B: Valuation using mean industry multiplesMean Median SD 75%-25% 90%-10% 95%-5% 99%-1%

Ebitda/P -0.023 0.064 0.572 0.502 1.084 1.555 2.776

Sales/P -0.049 0.157 0.934 0.729 1.610 2.304 4.571

Ebitda/TP -0.035 0.026 0.581 0.535 1.088 1.518 2.696

Sales/TP -0.075 0.150 1.113 0.859 1.853 2.698 5.501

Panel C: Valuation using intercept adjusted cross-sectional multiplesMean Median SD 75%-25% 90%-10% 95%-5% 99%-1%

Ebitda/P 0.027 0.112 0.480 0.527 1.011 1.413 2.394

Sales/P -0.031 0.160 0.676 0.615 1.370 1.908 3.420

Ebitda/TP -0.017 0.048 0.513 0.521 1.040 1.429 2.602

Sales/TP 0.050 0.214 0.978 0.880 1.904 2.721 5.407

Panel D: Valuation using intercept adjusted industry multiplesMean Median SD 75%-25% 90%-10% 95%-5% 99%-1%

Ebitda/P -0.020 0.082 0.496 0.443 0.976 1.384 2.440

Sales/P -0.035 0.146 0.668 0.574 1.291 1.852 3.287

Ebitda/TP -0.002 0.078 0.492 0.447 0.976 1.352 2.432

Sales/TP -0.003 0.179 0.702 0.620 1.382 1.985 3.524

40

Table 6Sources of Measurement Error in P1* Analyzed

Valuations using simple and intercept adjusted multiples are conducted using cross-sectional and industry

comparable firms. Variables are defined as: 5

* 1 5 45

1

( ) ( )1

(1 ) (1 )t t s t t s t t t t

t t ss t t t

eps k bv eps k bvP bv

k k k+ + − + +

=

Ε − Ε −= + + + +

∑ ,

5* 1

1

( )2

(1 )t t s t t s

t t ss t

eps k bvP bv

k+ + −

=

Ε −= + +

∑ ,

[ ] [ ]2 11* 1 12 111

111 3

( ) ( )( )3

(1 ) (1 ) (1 )t t s t t s t t t tt t s t t s

t t s ss st t t t

ROE k bv ROE k bveps k bvP bv

k k k k+ + − + ++ + −

= =

Ε − Ε − Ε −= + + + + + +

∑ ∑ ,

where ( )t t sROE +Ε for s = 4, 5, …, 12 is forecasted using a linear interpolation to the industry median ROE. The

industry median ROE is calculated as a moving median of the past ten years’ ROE of all firms in the industry. Toeliminate outliers, industry median ROEs are Winsorized at the risk free rate and 20%.

5*

1

4 ( )t t t ss

P eps +=

= Ε∑ , and 5

*

1

( )5

(1 )t t s

t ss t

epsP

k+

=

Ε = + ∑ .

Years covered are 1981 through 1996. Sample size is 9,658.

Panel A: Valuation using mean cross-sectional multiplesMean Median SD 75%-25% 90%-10% 95%-5% 99%-1%

P1* /P 0.000 0.056 0.403 0.504 0.918 1.203 1.981P2* /P 0.000 0.053 0.395 0.548 0.957 1.226 1.846P3* /P 0.000 0.108 0.551 0.640 1.185 1.562 2.693P4* /P 0.000 0.041 0.309 0.338 0.711 0.980 1.631P5* /P 0.000 0.029 0.309 0.366 0.733 0.982 1.592

Panel B: Valuation using mean industry multiplesMean Median SD 75%-25% 90%-10% 95%-5% 99%-1%

P1* /P -0.010 0.038 0.377 0.369 0.799 1.136 1.991P2* /P -0.009 0.036 0.346 0.375 0.803 1.097 1.756P3* /P -0.016 0.062 0.491 0.444 0.993 1.400 2.488P4* /P -0.006 0.028 0.291 0.294 0.646 0.915 1.528P5* /P -0.006 0.024 0.288 0.297 0.655 0.920 1.507

Panel C: Valuation using intercept adjusted cross-sectional multiplesMean Median SD 75%-25% 90%-10% 95%-5% 99%-1%

P1* /P 0.016 0.068 0.365 0.435 0.809 1.089 1.830P2* /P -0.001 0.045 0.356 0.463 0.846 1.110 1.733P3* /P -0.011 0.079 0.482 0.532 0.988 1.371 2.381P4* /P -0.002 0.039 0.305 0.326 0.700 0.971 1.611P5* /P 0.000 0.031 0.299 0.342 0.698 0.956 1.545

Panel D: Valuation using intercept adjusted industry multiplesMean Median SD 75%-25% 90%-10% 95%-5% 99%-1%

P1* /P -0.008 0.049 0.357 0.342 0.754 1.077 1.884P2* /P 0.000 0.051 0.336 0.356 0.777 1.067 1.720P3* /P -0.013 0.074 0.446 0.397 0.899 1.287 2.359P4* /P -0.004 0.034 0.286 0.288 0.636 0.894 1.509P5* /P 0.000 0.034 0.284 0.287 0.647 0.903 1.495

41

Table 7Industry Rankings of Multiples

Valuations using the simple multiple approach are performed in each industry. Multiples are ranked according to thevariance of percent pricing errors using the pooled distribution. Low rank numbers indicate low variance. Industryclassification is from IBES. Code is the first four digits of the IBES industry classification code. Years covered are1981 through 1996. Sample size is 9,658.

Sector Name Industry Name Code BV CFO Ebitda CACT IACT Sales EPS1 EPS2 EPS3 P1* P2* P3*

finance finance & loan 101 7 8 11 5 4 10 3 2 1 9 0 6

finance financial services 102 9 5 11 8 7 10 4 3 2 0 1 6

finance Insurance 105 9 8 10 6 5 11 3 0 2 7 1 4

finance Investments 106 6 8 10 9 5 11 3 0 1 7 2 4

finance undesignated finance 109 2 5 10 9 8 11 7 6 4 0 3 1

health care drugs 201 10 9 8 7 6 11 3 2 1 5 0 4

health care hospital supplies 202 9 10 8 6 4 11 3 2 1 7 0 5

health care hospitals 203 9 10 11 7 6 8 3 1 2 4 0 5

health care biotechnology 204 6 9 10 8 7 11 5 2 0 1 4 3

health care medical supplies 205 8 9 10 7 6 11 3 2 1 5 0 4

health care services to medical prof 206 10 7 6 8 5 11 3 0 1 9 2 4

consumer non-durables clothing 301 10 9 8 4 5 11 0 3 2 7 1 6

consumer non-durables consumer containers 302 7 9 10 8 6 11 1 3 4 0 2 5

consumer non-durables cosmetics 303 11 8 9 7 2 10 1 3 4 0 5 6

consumer non-durables food processors 304 10 9 8 5 6 11 3 1 2 4 0 7

consumer non-durables beverages 305 11 9 7 6 5 10 3 2 1 4 0 8

consumer non-durables home products 306 7 10 9 8 4 11 0 2 3 6 1 5

consumer non-durables leisure times 307 9 7 8 10 5 11 0 3 2 6 1 4

consumer non-durables tobacco 309 11 9 8 7 0 10 1 2 4 5 3 6

consumer services communications 401 9 11 8 7 6 10 3 2 1 5 0 4

consumer services leisure 402 9 11 7 8 6 10 4 1 0 5 3 2

consumer services retailing – foods 403 10 9 8 5 4 11 3 2 0 7 1 6

consumer services retailing – goods 404 10 9 8 5 4 11 2 3 1 7 0 6

consumer services industrial services 405 6 11 10 9 8 5 2 1 4 7 3 0

consumer services undesignated conr svc 407 10 9 8 1 0 11 2 5 4 6 3 7

consumer durables automotive mfg 501 6 11 10 7 8 9 4 1 3 5 2 0

consumer durables auto part mfg 502 9 10 8 6 5 11 0 1 3 7 2 4

consumer durables home furnishings 504 8 10 9 7 5 11 3 0 2 6 1 4

consumer durables leisure products 505 10 4 5 8 6 11 1 0 2 7 3 9

consumer durables recreational vehicles 506 10 5 7 9 6 11 2 0 3 8 4 1

consumer durables rubber 507 10 9 5 7 8 11 3 1 0 4 2 6

energy oil 601 6 8 7 10 9 11 5 1 2 4 3 0

energy coal 602 7 10 9 3 6 11 2 0 8 4 5 1

energy gas 607 2 7 6 10 8 11 3 1 5 9 4 0

transportation airlines 701 7 11 9 8 6 10 3 2 1 5 0 4

transportation railroads 702 8 9 11 7 4 10 3 1 2 5 0 6

transportation trucking 703 8 10 9 4 5 11 1 0 2 7 3 6

transportation maritime 705 7 9 10 8 5 11 0 1 3 4 2 6

Continued ...

42

Table 7 Continued

Sector Name Industry Name Code BV CFO Ebitda CACT IACT Sales EPS1 EPS2 EPS3 P1* P2* P3*

technology computers 801 8 10 9 7 6 11 3 2 1 4 0 5

technology electronics 803 9 11 5 7 6 10 3 1 2 8 0 4

technology software & edp services 804 10 11 6 7 9 8 0 3 2 5 1 4

technology undesignated tech 805 6 11 9 8 7 10 0 1 3 5 4 2

technology other computers 807 10 7 8 9 6 11 1 0 3 5 2 4

technology semiconductors/component 808 10 9 8 7 6 11 3 2 1 4 0 5

technology electronic syst/devices 810 8 10 9 6 7 11 3 2 1 5 0 4

technology office/comm equip 811 10 9 6 8 4 11 3 2 1 7 0 5

basic industries building & related 901 8 10 11 7 6 9 4 1 3 2 0 5

basic industries chemicals 902 10 9 8 7 6 11 4 2 1 3 0 5

basic industries containers 903 6 9 10 8 7 11 5 3 1 4 0 2

basic industries metal fabricators & dist 904 8 6 9 7 3 11 5 2 1 10 0 4

basic industries forest products 906 4 8 7 10 9 11 5 1 3 6 2 0

basic industries paper 907 9 10 7 6 8 11 3 0 1 5 2 4

basic industries steel 908 6 9 5 11 8 10 4 2 3 7 0 1

basic industries nonferrous base metals 910 6 10 7 8 9 11 5 0 4 3 1 2

basic industries precious metals 911 9 8 10 6 7 11 1 2 3 5 4 0

basic industries multi-ind basic 912 11 6 10 8 7 9 0 1 3 5 2 4

capital goods defense 1001 7 10 9 8 6 11 3 0 2 5 1 4

capital goods auto oems 1002 8 11 6 5 7 10 0 1 3 9 2 4

capital goods electrical 1003 8 10 9 7 6 11 1 2 5 4 3 0

capital goods machinery 1004 10 9 8 7 5 11 3 1 0 6 2 4

capital goods building materials 1007 8 10 7 9 6 11 1 0 2 5 3 4

capital goods office products 1008 10 7 9 8 6 11 3 0 2 1 4 5

capital goods multi-ind cap good 1010 10 9 8 7 4 11 3 1 0 5 2 6

public utilities electrical utilities 1101 8 10 9 7 6 11 2 0 3 4 1 5

public utilities gas utilities 1102 5 10 8 9 7 11 3 0 2 6 1 4

public utilities telephone utilities 1103 10 9 6 8 5 11 4 0 2 7 1 3

public utilities water utilities 1105 7 9 8 5 3 11 0 4 2 10 1 6

miscellaneous/undesignat unclassified 9900 10 9 8 1 2 11 6 7 5 0 3 4

Mean 8.26 8.93 8.31 7.09 5.72 10.54 2.59 1.54 2.19 5.19 1.60 4.03

Med 9.00 9.00 8.00 7.00 6.00 11.00 3.00 1.00 2.00 5.00 1.00 4.00

SD 1.98 1.60 1.59 1.87 1.87 0.98 1.62 1.42 1.47 2.36 1.45 2.07

43

Figure 1Distribution of Percentage Valuation Errors Using Simple Industry Multiples

Value and value drivers are assumed to be proportional: it t it itp x= + βt, is estimated using the

1/ itt t

it

x

pb

= Ε , and the distribution of percent pricing error,

ˆit it t it

it it

p x

p p

e b−= , is

plotted below. The variables are defined as follows (all amounts are on a per share basis): P is stock price; BV isbook value of equity; EBITDA is earnings before interest, taxes, depreciation and amortization; IACT is IBESactual earnings; EPS3=EPS2*(1+g), where EPS2 is two year out earnings forecast and g is growth forecast, and

5* 1 5 4

51

( ) ( )1

(1 ) (1 )t t s t t s t t t t

t t ss t t t

eps k bv eps k bvP bv

k k k+ + − + +

=

Ε − Ε −= + + + +∑

All multiples are calculated using comparable firms within each industry (based on IBES industry classification),and the firm being valued is excluded when computing industry multiples. Years covered are 1981 through 1996.Sample size is 9,658. The chart below is derived from a histogram with columns of width=0.1 (or 10% of price). Forexample, for EPS3, the fraction of the sample with pricing error between 0 and 0.1 is just over 18%.

0

2

4

6

8

10

12

14

16

18

20

-2 -1.9

-1.8

-1.7

-1.6

-1.5

-1.4

-1.3

-1.2

-1.1

-1 -0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

percentage valuation error

freq

uen

cy in

%

EPS3

IACT

P1*

BV

Ebitda

Sales

44

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