Equity Valuation Using Multiples

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Equity Valuation Using Multiples
Preliminary. Comments welcome.
Equity Valuation Using Multiples
Jing Liu
Anderson Graduate School of Management
University of California at Los Angeles
(310) 206-5861
[email protected]
Doron Nissim
Columbia University
Graduate School of Business
(212) 854-4249
[email protected]
and
Jacob Thomas
Columbia University
Graduate School of Business
(212) 854-3492
[email protected]
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, Ohio
State, and UCLA.
Equity Valuation Using Multiples
Abstract
In this study we examine the valuation performance of a comprehensive list of commonly used
price multiples. Our analysis indicates the following ranking: forward earnings multiples perform
the best, followed by historical earnings measures, cash flow measures and book value of equity
are tied for third, and sales performs the worst. Contrary to the popular view that different
industries have different “best” multiples, we find that these overall rankings are observed
consistently for all industries examined. Performance is improved by allowing for an intercept in
the linear relation between price and value drivers, relative to the ratio formulation typically
assumed in practice. Performance is not improved, however, by the use of more complex value
drivers, such as the short cut value measures based on generic patterns for residual income
growth past the forecast horizon.
Equity Valuation Using Multiples
1. 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 riskadjusted 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.
1
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-today 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.
2
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
3
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
4
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 posthorizon 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 ratio
formulation. 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.
5
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 and
recommendations than more rigorous valuation models.
4
Details of those results are available from the authors upon request.
6
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
7
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
8
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 comparable
with those in Tasker [1998].
9
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 eps3 = eps2 *(1 + 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.
10
[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.
11
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.
12
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:
pit = bt xit + eit
(1)
where xit is the value driver of firm i in year t, bt is the multiple on the value driver and eit 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 = bt
xit eit
+ .
pit pit
(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 estimated
equation (2) for comparable firms from the cross-section. (When using comparable firms from the same
industry, the estimated multiples generated substantial pricing errors.) We find that the distributions of percent
pricing errors for all multiples are shifted to the right substantially, relative to the distributions for the restricted
case reported in the paper (our distributions tend to peak around zero pricing error). This shift to the right
indicates 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 of
the distribution, relative to our restricted case.
13
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
æe ö
equation (2), subject to the constraint that those errors be zero on average, i.e., Et ççç it ÷÷÷ = 0 .
è pit ø÷
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 bt is the harmonic mean of the price-value driver ratio.
bt =
1
æx ö
Et çç it ÷÷÷
çè pit ÷ø
(3)
To eliminate in-sample bias, we estimate bt 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
bt estimate by the firm’s value driver, and then calculate the percent pricing error as follows:9
eit
p - bˆt xit
.
= it
pit
pit
(4)
The performance of multiples is evaluated by examining the dispersion of the pooled distribution
of eit / pit (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:
pit = at + bt xit + eit .
9
(5)
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.
14
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.
1 = at
1
x
e
+ bt it + it ,
pit
pit pit
(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 ( eit / pit ), subject to the restriction that
the expected value of eit / pit is zero:
min var (ε it / pit ) = var[( pit − α t − β t ⋅ xit ) / pit ] = var[1 − (α t
α ,β
æe ö
s.t. Et ççç it ÷÷÷ = 0.
è pit ÷ø
x
1
+ β t it )]
pit
pit
(7a)
(7b)
To obtain estimates for αt and βt, we restate restriction (7b) as follows
E(
FG
H
IJ
K
ε it
1
x
) = E 1− αt
− β t it = 0
pit
pit
pit
(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.
15
Solve (8) for αt, and substitute into (7a) to restate the minimization problem in terms of the
following regression with no intercept:
F
I LM
GG1 − 1 JJ = β M x
GH p E ( 1p )JK MM p
N
it
t
it
t
it
FG x IJ OP
H pK P
−
F 1I P
p E G JP
H pK Q
Et
it
(9)
t
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.
16
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 æ
E ( epst+s - kt bvt+s-1 ) ö÷ Et ( epst+5 - kt bvt+4 )
÷÷ +
P1*t = bvt + å çç t
÷ø
ç
kt (1 + kt ) 5
(1 + kt ) s
s=1 è
(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 valueweighted 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
17
forecasts available between years 3 and 5, we generated forecasts by applying the mean longterm growth forecast (g) to the mean forecast for the prior year in the horizon; i.e.,
eps t + s = eps t + s −1 * (1 + g ) .
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.
18
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.
19
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 Pˆ / 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.
20
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 interquartile 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.
21
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 valuerelevant 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.
22
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 is
less precise. This fact is also reflected in the increase in the deviation of the sample mean of the valuation
errors from zero.
23
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 crosssectional 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
24
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.
25
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 æ
E (epst+s - kt bvt+s-1 ) ÷ö
÷÷ , and
P2*t = bvt + å çç t
÷ø
ç
(1 + kt ) s
s=1 è
(2) ROE forecasts from year +3 trends linearly toward the industry median by year +12, and
abnormal earnings exhibit zero growth thereafter,
2
 Ε (epst + s − kt bvt + s−1 )  11 [ Εt ( ROEt + s ) − kt ] bvt + s−1 [ Εt ( ROEt +12 ) − kt ] bvt +11
,
P3*t = bvt + ∑  t
+
+∑
(1 + kt ) s
(1 + kt ) s
kt (1 + kt )11
s =1 
 s =3
where Et ( ROEt +s ) 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].
26
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,
P4*t = å Et ( epst+s ) .22 The second measure attempts to control heuristically for the timing and
5
s=1
risk of the different earnings numbers by discounting the EPS forecasts for the next five years,
5 æ
E (epst+s ) ÷ö
÷.
P5*t = å çç t
s ÷
÷ø
ç
s=1 è (1 + kt )
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
27
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.
28
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
29
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.
30
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 æ
E ( epst+s - kt bvt+s-1 ) ö÷ Et ( epst+5 - kt bvt+4 )
÷÷ +
P1*t = bvt + å çç t
÷ø
ç
(1 + kt ) s
kt (1 + kt ) 5
s=1 è
5 æ
E (epst +s - kt bvt +s-1 ) ÷ö
÷÷
P 2*t = bvt + å çç t
÷ø
ç
(1 + kt ) s
s=1 è
31
2
 Ε (epst + s − kt bvt + s−1 )  11 [ Εt ( ROEt + s ) − kt ] bvt + s−1 [ Εt ( ROEt +12 ) − kt ] bvt +11
P3*t = bvt + ∑  t
+
+∑
kt (1 + kt )11
(1 + kt ) s
(1 + kt ) s
s =1 
 s =3
P4*t = å Et ( epst+s )
5
s=1
5 æ
E (epst+s ) ÷ö
÷
P5*t = å çç t
s ÷
ç
÷ø
s=1 è (1 + kt )
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 valueweighted 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 longterm growth forecast (g) to the mean forecast for the prior year in the horizon; i.e.,
eps t + s = eps t + s −1 * (1 + g ) .
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
32
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 P3*t , Et ( ROEt +s ) 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%.
33
Table 1
Distribution 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 debt
and equity holders; CFO is cash flow from operations; EBITDA is earnings before interest, taxes, depreciation and
amortization; CACT is COMPUSTAT earnings before extraordinary items; IACT is IBES actual earnings; EPS1
and 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 of
the year.
5
 Ε (epst + s − kt bvt + s−1 )  Εt (epst +5 − kt bvt +4 )
,
P1*t = bvt + ∑  t
+
(1 + kt ) s
kt (1 + kt )5
s =1 

5 æ
E (epst +s - kt bvt +s-1 ) ÷ö
÷÷ ,
P 2*t = bvt + å çç t
÷ø
ç
(1 + kt ) s
s=1 è
2
 Ε (epst + s − kt bvt + s−1 )  11 [ Εt ( ROEt + s ) − kt ] bvt + s−1 [ Εt ( ROEt +12 ) − kt ] bvt +11
,
P3*t = bvt + ∑  t
+
+∑
(1 + kt ) s
(1 + kt ) s
kt (1 + kt )11
s =1 
 s =3
where Et ( ROEt +s ) 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. To
eliminate outliers, industry median ROEs are Winsorized at the risk free rate and 20%.
5
5 æ
E (epst+s ) ÷ö
÷.
P4*t = å Et ( epst+s ) , and P5*t = å çç t
s ÷
ç
÷ø
s=1
s=1 è (1 + kt )
Sample is trimmed at 1% and 99% for each ratio (excluding the P* ratio) using the pooled distribution. Years
covered are 1981 through 1996. Sample size is 17,505.
Mean
SD
1%
5%
10%
25%
50%
75%
90%
95%
99%
BV/P
MCF/P
FCF/P
CFO/P
Ebitda/P
CACT/P
0.549
0.034
-0.016
0.102
0.179
0.044
0.354
0.079
0.161
0.094
0.142
0.084
0.014
-0.281
-0.675
-0.151
-0.090
-0.302
0.121
-0.092
-0.274
-0.021
0.015
-0.077
0.178
-0.035
-0.154
0.017
0.044
-0.015
0.300
0.016
-0.041
0.053
0.091
0.029
0.483
0.043
0.019
0.093
0.152
0.055
0.729
0.069
0.054
0.146
0.239
0.080
1.001
0.097
0.092
0.207
0.347
0.110
1.205
0.123
0.127
0.254
0.437
0.133
1.653
0.193
0.252
0.391
0.672
0.182
IACT/P
Sales/P
EPS1/P
EPS2/P
0.051
1.387
0.072
0.090
0.067
1.475
0.041
0.039
-0.212
0.054
-0.055
0.010
-0.041
0.159
0.016
0.036
0.006
0.254
0.030
0.048
0.034
0.509
0.050
0.066
0.057
0.953
0.070
0.086
0.081
1.731
0.093
0.110
0.108
2.920
0.119
0.141
0.131
4.050
0.140
0.162
0.176
7.450
0.180
0.211
EPS3/P
P1*/P
P2*/P
P3*/P
P4*/P
P5*/P
Ebitda/TP
Sales/TP
0.106
0.693
0.607
0.790
0.536
0.354
0.123
0.973
0.043
0.308
0.251
0.466
0.217
0.131
0.075
0.891
0.025
0.176
0.161
0.161
0.124
0.076
-0.081
0.052
0.049
0.299
0.258
0.273
0.248
0.169
0.014
0.141
0.060
0.370
0.318
0.350
0.308
0.210
0.041
0.222
0.078
0.492
0.424
0.494
0.395
0.270
0.081
0.408
0.099
0.647
0.571
0.691
0.500
0.339
0.122
0.728
0.127
0.847
0.762
0.987
0.640
0.421
0.163
1.271
0.164
1.048
0.935
1.305
0.822
0.525
0.207
1.971
0.188
1.207
1.053
1.570
0.942
0.594
0.241
2.547
0.247
1.727
1.308
2.477
1.236
0.746
0.325
4.313
34
Table 2
Pearson (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) using
the pooled distribution. Also, observations for which any of the ratios is negative are deleted. Finally, observations
that do not belong to an industry-year group with at least five members are deleted. Years covered are 1981 through
1996. Sample size is 9,658.
BV/P
BV
/P
1.00
CFO Ebitda CACT IACT Sales EPS1 EPS2 EPS3
/P
/P
/P
/P
/P
/P
/P
/P
0.58 0.65 0.53 0.53 0.52 0.58 0.61 0.56
P1*
/P
0.32
P2*
/P
0.91
P3*
/P
0.41
P4*
/P
0.55
P5* Ebitda Sales
/P
/TP
/TP
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
0.38
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
0.43
35
Table 3
Distribution of Percentage Valuation errors for Simple Multiples
Value and value drivers are assumed to be proportional: pit = bt xit + eit . Multiple is estimated excluding the firm
æ xit ö÷
÷ . Percent pricing error is calculated as follows,
çè p ø÷÷
under valuation, using harmonic means: bt = 1/ Et çç
it
eit
p - bˆt xit
; its pooled distribution is reported.
= it
pit
pit
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 before
extraordinary items; IACT is IBES actual earnings; EPS1, EPS2 are one year out and two year out earnings
forecasts; EPS3=EPS2*(1+g), where g is growth forecast. All totals are deflated by the number of shares
outstanding at the end of the year.
5
 Ε (epst + s − kt bvt + s−1 )  Εt (epst +5 − kt bvt +4 )
P1*t = bvt + ∑  t
+
(1 + kt ) s
kt (1 + kt )5
s =1 

Panel A uses the whole cross-section of firms as comparable firms, Panel B uses comparable firms within each
industry (based on IBES industry classification). Panel C also uses comparable firms within each industry, but the
multiple 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 multiples
Mean
Median
SD
75%-25%
BV
0.000
0.063
0.536
0.697
CFO
-0.001
0.104
0.589
0.689
Ebitda
-0.001
0.129
0.611
0.687
CACT
0.000
0.047
0.477
0.579
IACT
0.000
0.041
0.448
0.549
Sales
-0.001
0.259
0.948
0.761
EPS1
0.000
0.029
0.348
0.440
EPS2
0.000
0.026
0.311
0.368
EPS3
0.000
0.038
0.313
0.347
P1*
0.000
0.056
0.403
0.504
90%-10%
1.253
1.304
1.296
1.120
1.060
1.694
0.835
0.743
0.725
0.918
95%-5%
1.620
1.748
1.674
1.510
1.418
2.394
1.118
1.008
0.992
1.203
99%-1%
2.564
2.998
3.019
2.371
2.242
4.797
1.721
1.570
1.653
1.981
Panel B: Valuation using mean industry multiples
Mean
Median
SD
BV
-0.021
0.064
0.530
CFO
-0.023
0.056
0.554
Ebitda
-0.023
0.064
0.572
CACT
-0.014
0.013
0.448
IACT
-0.012
0.015
0.412
Sales
-0.049
0.157
0.934
EPS1
-0.007
0.018
0.312
EPS2
-0.006
0.021
0.289
EPS3
-0.006
0.026
0.293
P1*
-0.010
0.038
0.377
90%-10%
1.152
1.171
1.084
1.001
0.923
1.610
0.711
0.657
0.658
0.799
95%-5%
1.586
1.665
1.555
1.400
1.288
2.304
0.981
0.919
0.927
1.136
99%-1%
2.668
2.938
2.776
2.322
2.177
4.571
1.658
1.550
1.561
1.991
75%-25%
0.543
0.559
0.502
0.462
0.429
0.729
0.333
0.303
0.301
0.369
36
Panel C: Valuation using median industry multiples
Mean
Median
SD
BV
CFO
Ebitda
CACT
IACT
Sales
EPS1
EPS2
EPS3
P1*
-0.109
-0.102
-0.114
-0.039
-0.038
-0.307
-0.028
-0.032
-0.039
-0.063
0.002
0.000
0.000
-0.001
-0.001
0.001
-0.001
0.000
0.000
0.000
75%-25%
90%-10%
95%-5%
99%-1%
0.571
0.587
0.519
0.465
0.434
0.861
0.339
0.305
0.305
0.377
1.284
1.273
1.160
1.037
0.956
1.936
0.729
0.676
0.685
0.838
1.770
1.816
1.712
1.470
1.353
2.905
1.023
0.958
0.968
1.228
3.054
3.334
3.198
2.469
2.256
6.311
1.774
1.633
1.646
2.199
0.601
0.628
0.659
0.470
0.435
1.312
0.323
0.301
0.307
0.411
37
Table 4
Distribution of Percentage Valuation Errors for Intercept Adjusted Multiples
Value and value drivers are assumed to be linear: pit = at + bt × xit + eit . Multiple is estimated excluding the
firm under valuation, by solving a constraint minimization problem:
min var(eit / pit ) = var (( pit - at - bt × xit ) / pit )
at ,b t
æe ö
s.t. Et ççç it ÷÷÷ = 0
è pit ÷ø
Percentage valuation error is calculated as follows,
eit
p - aˆ t - bˆt xit
; its pooled distribution is reported.
= it
pit
pit
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 before
extraordinary items; IACT is IBES actual earnings; EPS1, EPS2 are one year out and two year out earnings
forecasts; EPS3=EPS2*(1+g), where g is growth forecast. All totals are deflated by the number of shares
outstanding at the end of the year.
5
 Ε (epst + s − kt bvt + s−1 )  Εt (epst +5 − kt bvt +4 )
P1*t = bvt + ∑  t
+
(1 + kt ) s
kt (1 + kt )5
s =1 

Panel A uses the whole cross-section of firms as comparable firms, Panel B uses comparable firms within each
industry (based on IBES industry classification). Years covered are 1981 through 1996. Sample size is 9,658.
Panel A: Valuation using intercept adjusted cross-sectional multiples
Mean
Median
SD
75%-25%
90%-10%
BV
0.026
0.084
0.458
0.541
1.038
CFO
0.032
0.108
0.465
0.520
1.013
Ebitda
0.027
0.112
0.480
0.527
1.011
CACT
0.012
0.053
0.401
0.477
0.931
IACT
0.015
0.054
0.380
0.462
0.893
Sales
-0.031
0.160
0.676
0.615
1.370
EPS1
0.013
0.039
0.321
0.392
0.767
EPS2
0.008
0.035
0.300
0.344
0.705
EPS3
0.000
0.042
0.306
0.333
0.706
P1*
0.016
0.068
0.365
0.435
0.809
95%-5%
1.408
1.386
1.413
1.272
1.209
1.908
1.030
0.966
0.976
1.089
99%-1%
2.264
2.427
2.394
2.028
1.887
3.420
1.610
1.533
1.629
1.830
Panel B: Valuation using intercept adjusted industry multiples
Mean
Median
SD
75%-25%
BV
-0.016
0.076
0.477
0.476
CFO
-0.016
0.072
0.477
0.474
Ebitda
-0.020
0.082
0.496
0.443
CACT
-0.012
0.034
0.396
0.406
IACT
-0.011
0.035
0.372
0.385
Sales
-0.035
0.146
0.668
0.574
EPS1
-0.006
0.026
0.300
0.317
EPS2
-0.004
0.028
0.284
0.296
EPS3
-0.004
0.033
0.289
0.293
P1*
-0.008
0.049
0.357
0.342
95%-5%
1.442
1.420
1.384
1.250
1.175
1.852
0.956
0.900
0.912
1.077
99%-1%
2.403
2.496
2.440
2.068
1.917
3.287
1.594
1.511
1.526
1.884
38
90%-10%
1.032
1.023
0.976
0.884
0.826
1.291
0.689
0.642
0.650
0.754
Table 5
Leverage Adjustments for EBITDA and Sales Multiples
The variables are defined as follows: P is stock price; EBITDA is earnings before interest, taxes, depreciation and
amortization; TP is enterprise value (market value of equity plus book value of debt). Valuations using simple and
intercept adjusted multiples are conducted using cross-sectional and industry comparable firms. When TP multiples
are used, equity value is calculated as the predicted enterprise value minus book value of debt. Years covered are
1981 through 1996. Sample size is 9,658.
.Panel A: Valuation using mean cross-sectional multiples
Mean
Median
SD
75%-25%
Ebitda/P
-0.001
0.129
0.611
0.687
Sales/P
-0.001
0.259
0.948
0.761
Ebitda/TP
-0.045
0.039
0.601
0.630
Sales/TP
0.000
0.269
1.259
1.031
Panel B: Valuation using mean industry multiples
Mean
Median
SD
90%-10%
1.296
1.694
1.231
2.189
95%-5%
1.674
2.394
1.671
3.164
99%-1%
3.019
4.797
2.994
6.602
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
Ebitda/TP
Sales/TP
-0.049
-0.035
-0.075
0.157
0.026
0.150
0.934
0.581
1.113
0.729
0.535
0.859
1.610
1.088
1.853
2.304
1.518
2.698
4.571
2.696
5.501
Panel C: Valuation using intercept adjusted cross-sectional multiples
Mean
Median
SD
75%-25%
90%-10%
Ebitda/P
0.027
0.112
0.480
0.527
1.011
Sales/P
-0.031
0.160
0.676
0.615
1.370
Ebitda/TP
-0.017
0.048
0.513
0.521
1.040
Sales/TP
0.050
0.214
0.978
0.880
1.904
95%-5%
1.413
1.908
1.429
2.721
99%-1%
2.394
3.420
2.602
5.407
Panel D: Valuation using intercept adjusted industry multiples
Mean
Median
SD
75%-25%
90%-10%
95%-5%
99%-1%
0.976
1.291
0.976
1.382
1.384
1.852
1.352
1.985
2.440
3.287
2.432
3.524
Ebitda/P
Sales/P
Ebitda/TP
Sales/TP
-0.020
-0.035
-0.002
-0.003
0.082
0.146
0.078
0.179
0.496
0.668
0.492
0.702
0.443
0.574
0.447
0.620
39
Table 6
Sources 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: P1*t = bvt +
 Εt (epst + s − kt bvt + s−1 )  Εt (epst +5 − kt bvt +4 )
,
+
(1 + kt ) s
kt (1 + kt )5
s =1 

5
∑
5
 Ε (epst + s − kt bvt + s −1 ) 
P2*t = bvt + ∑  t
,
(1 + kt )s
s =1 

2
 Ε (epst + s − kt bvt + s−1 )  11 [ Εt ( ROEt + s ) − kt ] bvt + s−1 [ Εt ( ROEt +12 ) − kt ] bvt +11
,
P3*t = bvt + ∑  t
+
+∑
(1 + kt ) s
(1 + kt ) s
kt (1 + kt )11
s =1 
 s =3
where Et ( ROEt +s ) 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. To
eliminate outliers, industry median ROEs are Winsorized at the risk free rate and 20%.
5
5 æ
E (epst+s ) ÷ö
÷.
P4*t = å Et ( epst+s ) , and P5*t = å çç t
s ÷
ç
÷ø
s=1
s=1 è (1 + kt )
Years covered are 1981 through 1996. Sample size is 9,658.
Panel A: Valuation using mean cross-sectional multiples
Mean
Median
SD
75%-25%
P1* /P
0.000
0.056
0.403
0.504
P2* /P
0.000
0.053
0.395
0.548
P3* /P
0.000
0.108
0.551
0.640
P4* /P
0.000
0.041
0.309
0.338
P5* /P
0.000
0.029
0.309
0.366
90%-10%
0.918
0.957
1.185
0.711
0.733
95%-5%
1.203
1.226
1.562
0.980
0.982
99%-1%
1.981
1.846
2.693
1.631
1.592
Panel B: Valuation using mean industry multiples
Mean
Median
SD
P1* /P
-0.010
0.038
0.377
P2* /P
-0.009
0.036
0.346
P3* /P
-0.016
0.062
0.491
P4* /P
-0.006
0.028
0.291
P5* /P
-0.006
0.024
0.288
90%-10%
0.799
0.803
0.993
0.646
0.655
95%-5%
1.136
1.097
1.400
0.915
0.920
99%-1%
1.991
1.756
2.488
1.528
1.507
Panel C: Valuation using intercept adjusted cross-sectional multiples
Mean
Median
SD
75%-25%
90%-10%
P1* /P
0.016
0.068
0.365
0.435
0.809
P2* /P
-0.001
0.045
0.356
0.463
0.846
P3* /P
-0.011
0.079
0.482
0.532
0.988
P4* /P
-0.002
0.039
0.305
0.326
0.700
P5* /P
0.000
0.031
0.299
0.342
0.698
95%-5%
1.089
1.110
1.371
0.971
0.956
99%-1%
1.830
1.733
2.381
1.611
1.545
Panel D: Valuation using intercept adjusted industry multiples
Mean
Median
SD
75%-25%
P1* /P
-0.008
0.049
0.357
0.342
P2* /P
0.000
0.051
0.336
0.356
P3* /P
-0.013
0.074
0.446
0.397
P4* /P
-0.004
0.034
0.286
0.288
P5* /P
0.000
0.034
0.284
0.287
95%-5%
1.077
1.067
1.287
0.894
0.903
99%-1%
1.884
1.720
2.359
1.509
1.495
75%-25%
0.369
0.375
0.444
0.294
0.297
40
90%-10%
0.754
0.777
0.899
0.636
0.647
Table 7
Industry Rankings of Multiples
Valuations using the simple multiple approach are performed in each industry. Multiples are ranked according to the
variance of percent pricing errors using the pooled distribution. Low rank numbers indicate low variance. Industry
classification is from IBES. Code is the first four digits of the IBES industry classification code. Years covered are
1981 through 1996. Sample size is 9,658.
Sector Name
Industry Name
P1*
P2*
P3*
finance
finance & loan
Code BV CFO Ebitda CACT IACT Sales EPS1 EPS2 EPS3
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 ...
41
Table 7 Continued
Sector Name
Industry Name
P1*
P2*
P3*
technology
computers
801
8
10
9
7
6
11
3
2
technology
electronics
803
9
11
5
7
6
10
3
1
1
4
0
5
2
8
0
technology
software & edp services
804
10
11
6
7
9
8
0
4
3
2
5
1
technology
undesignated tech
805
6
11
9
8
7
10
4
0
1
3
5
4
2
technology
other computers
807
10
7
8
9
6
technology
semiconductors/component
808
10
9
8
7
6
11
1
0
3
5
2
4
11
3
2
1
4
0
technology
electronic syst/devices
810
8
10
9
6
5
7
11
3
2
1
5
0
4
technology
office/comm equip
811
10
9
6
basic industries
building & related
901
8
10
11
8
4
11
3
2
1
7
0
5
7
6
9
4
1
3
2
0
basic industries
chemicals
902
10
9
5
8
7
6
11
4
2
1
3
0
basic industries
containers
903
6
5
9
10
8
7
11
5
3
1
4
0
2
basic industries
metal fabricators & dist
904
basic industries
forest products
906
8
6
9
7
3
11
5
2
1
10
0
4
4
8
7
10
9
11
5
1
3
6
2
basic industries
paper
0
907
9
10
7
6
8
11
3
0
1
5
2
basic industries
4
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
9900
2
11
6
7
5
0
miscellaneous/undesignat unclassified
Code BV CFO Ebitda CACT IACT Sales EPS1 EPS2 EPS3
10
9
8
1
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
2.07
42
0.98 1.62 1.42 1.47 2.36 1.45
Figure 1
Distribution of Percentage Valuation Errors Using Simple Industry Multiples
Value and value drivers are assumed to be proportional: pit =
x +
t it
βt, is estimated using the
it
æx ö
e
p - bˆt xit
, is
bt = 1/ Et çç it ÷÷÷ , and the distribution of percent pricing error, it = it
çè pit ÷ø
pit
pit
plotted below. The variables are defined as follows (all amounts are on a per share basis): P is stock price; BV is
book value of equity; EBITDA is earnings before interest, taxes, depreciation and amortization; IACT is IBES
actual earnings; EPS3=EPS2*(1+g), where EPS2 is two year out earnings forecast and g is growth forecast, and
5
 Ε (epst + s − kt bvt + s−1 )  Εt (epst +5 − kt bvt +4 )
P1*t = bvt + ∑  t
+
(1 + kt ) s
kt (1 + kt )5
s =1
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). For
example, for EPS3, the fraction of the sample with pricing error between 0 and 0.1 is just over 18%.
20
18
EPS3
16
P1*
14
IACT
frequency in %
Ebitda
12
BV
10
8
Sales
6
4
2
0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
43
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
-1.1
-1.2
-1.3
-1.4
-1.5
-1.6
-1.7
-1.8
-1.9
-2
percentage valuation error
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