The Calculation of Earnings Per Share
and Market Value of Equity: Should
Common Stock Equivalents Be Included?
By
Joshua Livnat and Dan Segal
Leonard N. Stern School of Business
New York University
40 W. 4th St.
NY NY 10012
(212) 998-0022
jlivnat@stern.nyu.edu
dsegal@stern.nyu.edu
First Draft: March 1999
Current Draft: February 5, 2016
The Calculation of Earnings Per Share and Market Value of Equity:
Should Common Stock Equivalents Be Included?
Abstract
Most stock market participants calculate the market value of equity through a
multiplication of the price per share by the number of outstanding shares. This study
shows that this practice is inconsistent with the assumption that the market price per
share incorporates the potential dilution due to outstanding financial instruments that can
be converted to common stocks (common stock equivalents, CSE). If the market price
incorporates all CSE, the market value of equity (including contingent equity) should be
calculated through the multiplication of price per share by the sum of outstanding shares
and shares that will be issued upon dilution.
Information about CSE can be found by examining all the financial instruments of
a firm that can potentially be converted into common stock. However, one easily
accessible source for the CSE is the accounting calculation of Earnings Per Share (EPS),
where the accounting profession follows a rigid set of rules to calculate the potential
dilution in the number of outstanding shares due to CSE. This study examines how close
are the accounting CSE to those inferred by market participants in their determination of
market prices. Our results indicate that the market and the accounting CSE converge for
firms with high levels of potential dilution due to CSE, but not for low levels of potential
dilution (below 5-6%). Thus, the FASB standard on EPS (FASB, 1997) that requires the
disclosure of Basic EPS (with the assumption of zero CSE) and Diluted EPS (with the
assumption of all CSE), enables market users to select the number of shares they deem
most appropriate for the firm’s level of potential dilution.
The Calculation of Earnings Per Share and
Market Value of Equity: Should Common Stock
Equivalents Be Included?
In a recent pronouncement regarding Earnings Per Share (EPS), the Financial
Accounting Standards Board (FASB) now requires the disclosure of Basic and Diluted
EPS, instead of the previously required Primary and Fully Diluted EPS (FASB, 1997).
While one motive for the change was to conform EPS calculations to the majority of
countries and international standards, the FASB has also wanted to supply users of
financial statements with two extreme EPS figures – one with no dilution and one with
full dilution. This change was intended to help investors to better assess the effect of
potential dilution than that achieved under Primary EPS, which required the inclusion of
Common Stock Equivalents (CSE) in the computation.
The calculation of EPS may be important for market participants who rely on
such measures as the Price/Earnings (P/E) ratio to value and select securities for their
portfolios. It is also important for calculating the market value of equity1, because it
provides a comprehensive and easily accessible source about the potential dilution in the
number of outstanding shares due to conversion of CSE. Currently, the market value of
equity is typically calculated as the price per share times the number of outstanding
shares. When asked whether the market value of equity should be calculated by the total
number of shares that would be issued when CSE are converted into common stock, most
academics respond that the current price per share already takes into account these
1
Throughout this study, the calculation of the market value of equity assumes that the person performing
the calculation is trying to assess the value of the firm’s assets after all liabilities are paid off. Thus, it
includes cash flows to current equity holders as well as cash flows to contingent equity holders.
dilutive shares. As we shall show below, if the price per share indeed incorporates the
expected dilutive shares, the market value of equity should be based not only on
outstanding shares, but also on the expected dilutive shares. As a matter of fact,
calculating the market value of equity by using only outstanding shares is theoretically
inconsistent with the assumption that the market price per share incorporates the potential
dilution.
The number of shares used to calculate Primary (or Diluted) EPS can be
employed to assess the accounting profession’s estimate of the additional shares that
would be issued if all CSE are converted to common stock. If the market assessment of
the potentially dilutive shares corresponds exactly to the CSE calculations of the
accounting profession, the number of shares used to calculate EPS can be utilized to
calculate the correct market value of equity. The calculation of EPS then has implications
for applied finance, as well as security analysis, through the calculation of market value
of equity and various per share amounts.
The purpose of this study is to examine whether market participants seem to use
only the outstanding shares, or also the dilution caused by CSE, in determining security
prices. If the accounting procedures underlying EPS calculations are also used by market
participants to determine security prices, the market values will incorporate the dilution
due to CSE as reported by accountants. On the other hand, if market participants use
other procedures to incorporate the effects of dilution on security prices, the accounting
methods used to calculate EPS will produce market value calculations that should be
different from those determined by market participants.
2
Our results show that, for firms with low levels of dilution (below 5-6%), market
participants seem to ignore the possible effect of dilution caused by accounting CSE on
market prices, and seem to only consider the number of outstanding shares. For firms
with higher dilution levels, investors appear to incorporate the possibility of dilution
when they set the price per share, and the accounting CSE seem to correspond to those
assumed by market participants when prices are set. These results indicate that the
accounting procedures underlying the calculations of dilution are only partially shared by
market participants. Our findings indicate that the market perceptions of the adverse
consequences of dilution increase with the dilution level. Furthermore, our results seem
to support the FASB’s conclusion to replace Primary EPS with Basic EPS, which is
based solely on outstanding shares, essentially providing users the two scenarios of no
accounting CSE and full dilution due to accounting CSE. Users can then choose the
scenario that they deem more appropriate for the specific firm, according to its level of
potential dilution.
The next section provides a conceptual perspective for the pricing of shares in the
market when CSE can dilute outstanding shares. It also describes how we test whether
market prices seem to incorporate the dilution due to accounting CSE. The third section
describes the research design employed in the study, while the fourth section describes
the sample and discusses the results. The last section summarizes our findings and
concludes the paper.
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II. Background and Conceptual Perspective
The accounting profession required the computation and disclosure of EPS by a
rigid set of rules as part of Accounting Principles Board (APB) Opinion No. 15 (APB,
1969). Under this standard, firms were required to report Primary EPS and Fully-Diluted
EPS in their financial statements, where both measures included the potential dilution of
common stock equivalents, which are defined as such securities that are presently not
common stock, but can become common stock upon conversion. CSE include convertible
bonds, convertible preferred stocks, stock options and warrants, contingent shares, and
similar instruments. APB Opinion No. 15 required the inclusion of CSE in computation
of Primary EPS if their effect was dilutive. It also specified the rules that determine when
an instrument is classified as a CSE, sometimes only at inception of the instument,
without further adjustments to reflect changes in the probability of conversion to common
stock. Fully-Diluted EPS was based on assumptions about conversion of all potentially
dilutive financial instruments, and resulted in a lower EPS number than Primary EPS.
In 1997, the FASB issued its Statement No. 128 (FASB, 1997), which requires
firms to calculate and disclose Basic EPS and Diluted EPS, where Basic EPS does not
contain any dilution due to CSE, and Diluted EPS includes the dilutive effects of all CSE.
In its explanation for this change, the FASB stated:
“The Board decided to replace primary EPS with basic EPS for the following
reasons:
a. Presenting undiluted and diluted EPS data would give users the most factually
supportable range of EPS possibilities. The spread between basic and diluted
EPS would provide information about an entity’s capital structure by
disclosing a reasonable estimate of how much potential dilution exists.
b. Use of a common international EPS statistic has become even more important
as a result of database-oriented financial analysis and the internationalization
of business and capital markets.
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c. The notion of common stock equivalents as used in primary EPS is viewed by
many as not operating effectively in practice, and “repairing” it does not
appear to be a feasible option.
d. The primary EPS computation is complex, and there is some evidence that the
current guidance is not well understood and may not be consistently applied.
e. If basic EPS were to replace primary EPS, the criticisms about the arbitrary
methods by which common stock equivalents are determined would no longer
be an issue. If entities were required to disclose the details of their convertible
securities, the subjective determination of the likelihood of conversion would
be left to individual users of financial statements (FASB 1997, Par. 89)”.
In fact, the FASB has explicitly stated that providing investors with both Basic
and Diluted EPS will enable them to assess the two extreme scenarios of no dilution and
that of full dilution, an option not available when only Primary EPS figures were
disclosed. Users can use this information to determine any number in between, which
would reflect their assessments of potential dilution.
In a similar manner, users of financial statements can calculate other per share
data, as well as total market value of equity using their assessments of the total number of
common shares. Since the future cash flows of the firm from its investment projects are
probably unaffected directly by the existence of CSE, the economic market value of
equity (including contingent equity) should not be related to the CSE. However, the price
per common share is affected by the assessment of dilution due to CSE, since the total
market value of equity (including contingent equity) is divided over more shares. Thus,
the practical calculation of market value of equity, i.e., the number of shares times the
price per share, should be affected by the existence (and probability) of dilution due to
CSE. This provides the linchpin between the EPS calculation and the calculation of
market value of equity.
5
Prior Studies Relating to Calculations of EPS:
Most of the research studies concerning EPS calculation and disclosure can be
categorized into two broad areas: (1) criticism of the arbitrary methods used to calculate
Earning Per Share, and (2) the information content of the various earnings per share
measures.
The criticism concerning EPS calculations centered mainly around the rule that
determines when convertible bonds and warrants are deemed common stock equivalents.
The APB postulated that convertible debt should be considered CSE if, on the date of
issuance, the cash yield on the bond was less than 2/3 of the bank prime interest rate at
the time of issuance.
Frank and Weygandt [1970] suggest that the method of determining convertible
debt as CSE has three theoretical weaknesses. The first refers to the term structure of
interest rates. Since interest rates are varying and volatile, a low yield on a convertible
debt as compared to the existing prime rate may be a reflection of a change in the
relationship between short- and long-term rates. The second refers to the credit risk of the
company that issues the debt. Riskier companies issue bonds with higher cash yields.
Since the benchmark of the prime interest rate applies to all companies regardless of their
risk, the likelihood that convertible debt will be treated as CSE may be negatively
correlated with the company’s risk. The third refers to the permanent classification of the
convertible debt as CSE or straight debt. This classification may be irrelevant when later
conditions have changed (e.g., by a dramatic change in the stock price) to such extent that
the probability of conversion has been dramatically altered. Curry [1971] suggests a dual
method of accounting for convertible debt in order to overcome the shortcomings of the
6
method required by APB Opinion No. 15. He proposes that companies with convertible
debt will present two Earnings Per Share (EPS) numbers based on the assumptions of
conversion and non-conversion. Vigeland [1981] argues that, based on financial theory,
one can specify ex-ante when voluntary conversion can occur and can estimate the
probability of occurrence. Thus, it is possible to identify possible dates of dilution and
estimate their likelihood. Gibson and Williams [1973], and Bierman [1986] also contend
that the criteria for determining common stock equivalents are not meaningful.
With respect to the second area of EPS research, most studies examined the
information content of the different measures of EPS by investigating which measure
explains more of the variability in security returns. Rice [1978] computed the cumulative
abnormal return (CAR) around the time of the first reports under Opinion 15 for two
portfolios: one which consists of companies that reported Fully Diluted EPS (FDEPS) in
the first financial report after the commencement date of Opinion 15. The second
portfolio consists of firms that did not report FDEPS. He then compares the pattern of the
two CARs and finds that they started to differ a year before the opinion became
mandatory. Based on his findings, he concludes that the information content of FDEPS is
value relevant to investors. Millar, Nunthirapakorn and Courtenay [1987] find that Basic
Earnings Per Share (BEPS) exhibits stronger correlation with stock returns than either
PEPS or FDEPS. They conclude that among the three measures of earnings, Primary EPS
(PEPS) and FDEPS are the least informative. Jenings, LeClere and Thompson [1997],
and Balsam and Lipka [1998] compare the extent to which BEPS, PEPS and FDEPS
explain the variation in stock prices. They find that FDEPS explains better the variation
in stock prices than both BEPS and PEPS, and that PEPS is superior to BEPS in their
7
sample. Deberg and Murdoch [1994] examine whether FDEPS contains more
information than PEPS. They find that PEPS and FDEPS, as well as price-earnings ratios
computed using PEPS and FDEPS, are highly correlated. They conclude that both figures
contain essentially the same information. Kross, Chapman and Strand [1980] assess
whether FDEPS has any incremental value for security prices beyond PEPS. They also
find that there is no difference between PEPS and FDEPS in the degree of association
with both risk levels, measured as beta, and unexpected security returns.
The above studies seem to point out inconclusive results about the superiority of
any one of the three EPS figures. Most studies indicate that there is no material difference
in the information content of PEPS and FDEPS, but that the FDEPS has more value
relevance to investors than BEPS, since it explains better the variations in stock prices.
This suggests that investors take into account the possibility of dilution when they price
securities.
The above studies attempted to distinguish among the various EPS measures, and
indirectly examine whether stock prices incorporate the dilutive effects of CSE. Our
study will address this question directly through the use of valuation models. Also, our
study will examine the potential value relevance of accounting CSE for different levels of
potential dilution, an issue which most of the prior studies ignored.
The Calculation of Market value of Equity:
The calculation of a firm’s market value is one of the most fundamental tasks in
security analysis and investment management. For example, ratio analysis includes the
market-to-book ratio, which is the ratio of market value of equity to book value of equity.
8
Investment managers often categorize their holdings as large-capitalization stocks, midcapitalization stocks, or small-capitalization stocks. Further, market values are also used
as weights in many indices, such as the S&P 500 Index. In all of the above, the
capitalization is based on the market value of the firm’s equity.
The calculation of a firm’s market value seems to be straightforward -- the
number of outstanding shares times the price per share. Recently, however, there has
been some debate about the correct market value of a firm, and, in particular, whether the
computation should also include the potentially dilutive options, warrants, convertible
preferred stocks, convertible bonds, etc. This debate became more heated with the
frequent practice of granting stock options to employees in high-technology firms, and
with the large increases in stock prices in recent years, which made the conversion of
stock options to common stocks more probable (Scism and Bank, 1998).
In practice, most market participants use the following definition of market value
of equity (capitalization):
Market capitalization
The price of a stock multiplied by the total number of shares outstanding.
Also, the market's total valuation of a public company. (The Nasdaq Stock
Market Glossary, emphasis added)
Standard & Poors uses the market value of a firm’s equity in constructing its famous
value-weighted S&P 500 Index. The market value is determined based only on the
number of shares outstanding:
“How is the S&P 500 Index calculated?
The S&P 500 Index is calculated using a base-weighted aggregate
methodology, meaning the level of the Index reflects the total market value
of all 500 component stocks relative to a particular base period. Total
market value is determined by multiplying the price of its stock by the
number of shares outstanding.”(Standard & Poors, emphasis added)
9
Similarly, Value Line defines Market Capitalization as:
Market Capitalization
“The recent price of a stock multiplied by the number of common shares
outstanding.” (The Value Line Investment Survey, Glossary, emphasis
added)
In addition to market capitalization, Value Line provides detailed analysis on specific
companies. The analysis includes numerous per share ratios, such as, Sales, “Cash Flow”,
Earnings, etc., all of which are based on the number of outstanding shares.
These examples show that the prevalent calculation of market value of equity
does not account for the potential dilution in the number of outstanding shares due to
CSE. However, since the number of shares used to calculate Primary EPS has been
reported in the financial statements of firms, market participants could have calculated a
second market value of equity figure, through the multiplication of price per share by the
number of shares used to calculate Primary EPS.
In fact, for each firm, market participants could have obtained two competing
market values of equity; the first based only on the number of outstanding common
shares (the no-dilution market value), and the second based on the diluted number of
shares (the dilution market value). However, only one of these of competing market
values is “correct”, that is, gives a more accurate estimate of the market value of equity as
perceived by investors when they determine the price per share. To determine which of
these two measures is the “correct” market value of equity, consider the following
analysis.
10
Conceptual Analysis:
In an efficient market, investors should determine the price per share by using
their assessments of future cash flows and the expected number of shares, which includes
outstanding shares and the expected number of additional shares due to CSE. To illustrate
the process and the associated issues, we use the following artificial example.
Example:
Consider a firm with $100 cash and no other assets or liabilities, with a book
value of equity of $100. There are 10 shares outstanding, as well as 10 options with an
exercise price of zero, which are exercisable over a period of ten years, beginning
immediately.
The market value of the firm’s assets is $100, the cash on hand. The correct
market price per common share should be $5 ($100/20), because the options should be
deemed as exercised for valuation purposes (the probability of exercise is one). If the
market completely ignores the options, the price per outstanding common share will be
$10 ($100/10). To the extent that market participants assume that not all options will be
exercised, or that not all options will be exercised immediately, the market price will
range between $5 and $10 per common share. At $6 per share, market participants either
assume a high probability of conversion (but less than one), or are almost correct in
setting the market price. At $9 per share, market participants are either almost entirely
incorrect in setting the price, or attributing a low probability of conversion to the stock
options.
11
The following exhibit shows the market value of the firm (and the market to book
ratio) if one only uses outstanding shares (no dilution), and if one uses also dilutive
shares (dilution):
Exhibit 1
Market
Price
$5
(correct price)
$6
(almost correct
price)
$9
(almost entirely
incorrect price)
$10
(incorrect price)
MV
(no dilution)
MV
(dilution)
M/B
(no dilution)
M/B
(dilution)
$50($5x10)
$100($5x20)
(correct MV)
0.5
$60
$120
0.6
1.0
(correct
M/B)
1.2
$90
$180
0.9
1.8
$100($10x10) $200($10x20)
1.0
(correct MV)
(correct M/B)
2.0
The M/B ratio should be one, because both the market and book value of equity
are $100. Note that the correct market value (of $100) is obtained in one of two cases: (i)
the market has priced the stock (correctly) at $5/share, and we (correctly) use the
potential dilution (i.e. 20 shares) in calculating the market value. Or (ii) the market
(incorrectly) ignored the dilution completely, and we (incorrectly) use only the
outstanding shares in calculating the market value. Obtaining the correct market value in
the first case is intuitively simple –the market priced the stock correctly, and we correctly
used the potential dilution in the calculation of market value. Obtaining the correct
market value in the second case is less intuitive – it results from making two errors which
cancel each other; the market prices the stock incorrectly and we ignore dilution in
12
calculating market values. Note that similar results obtain for the market to book ratio in
the exhibit.
Another feature seen in the exhibit is that if we calculate the market value of the
firm without allowing for dilution, i.e., using only the outstanding shares, the calculated
market value is likely to understate the correct market value. To the extent that stock
prices incorporate some dilution (even a small amount as in the case of a price of
$9/share), the calculated market value is below the correct market value of $100. In
contradistinction, if we calculate the market value by assuming full dilution (20 shares),
we are likely to overstate the market value of the firm. As long as the market does not
adjust the price to reflect full dilution (even with a price of $6/share), the calculated
market value will exceed the correct market value of $100. Similar results obtain for the
M/B ratio, where the ratio will be understated if we calculate the market value with no
dilution, and will be overstated if we allow for dilution in calculating the market value.
In practice, determining the price per share is a complex process, which includes
two stages. In the first stage, investors assess the total market value of the firm’s equity
(and contingent equity) based upon its future cash flows. In the second stage, investors
assess the expected number of shares that are used to calculate the price per share. The
expected number of shares includes the number of currently outstanding shares, plus the
expected additional shares due to conversion of CSE. The expected additional shares due
to CSE may be based on various assumptions about the probability of conversion of CSE
to common shares, as well as on potential applications of the increased cash flows from
conversion.2 The calculations that underlie Primary EPS or Diluted EPS reflect one
For example, the “treasury method” may be used for stock options, assuming that the exercise price will
be used to repurchase shares in the marketplace. Similarly, the saved interest (dividends) in the case of
2
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potential way of assessing the expected number of common shares for determining the
price per share. Whether the accounting methods used to calculate EPS are similar to
those used by the market to assess the expected number of shares (and the price per
share), is an empirical question which we examine below.
III. Research design
The research question we examine is whether the price per common share
incorporates the potential dilution of the number of outstanding shares due to CSE. We
address this question by determining which market value, the dilution or no-dilution
market value, deviates further from the “theoretical” (intrinsic) market value. We
estimate the intrinsic market value using Ohlson’s (1995) valuation model, which is
based on the assumption that the share price equals the present value of future dividends.
The intrinsic market value (to both current equity holders and any contingent equity
holders) is, therefore, independent of the capital structure of the firm, and of the
possibility of dilution. This independence enables us to assess which estimate of market
value of equity is a better proxy for the intrinsic value.
Ohlson (1995) shows that the firm’s market value is a linear function of its book
value (BV) of equity and the present value of expected abnormal accounting earnings. If
abnormal earnings follow a specific generating process, the market value can be written
as:
MVt = BVt + Xta
(1)
convertible bonds (preferred stocks) can increase the market value of the equity. However, in practice, if all
stock options were exercised and the firm would have attempted to repurchase shares, the price per share
would have increased and fewer shares could have been repurchased by the proceeds than is indicated by
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where MVt is the market value of equity at the end of period t, Xta is the abnormal
accounting earnings during period t, and BVt is the book value of equity at the end of
period t. The abnormal accounting earnings is defined as:
Xta = Xt – (Rf-1) BVt-1
(2)
where Xt is reported earnings during period t, and Rf is one plus the risk free rate,
assuming risk neutrality. Substituting (2) into (1) gives:
MVt = BVt + ( Xt – (Rf-1)BVt-1)
Scaling by BVt-1 to control for size gives:
MVt/BVt-1 = -(Rf-1) + BVt/ BVt-1 +  Xt/BVt-1
(3)
This model has been extensively used in the literature for both cross-sectional
analysis and time-series analysis. However, when it is applied in cross-sectional analysis,
a third independent variable is usually added to allow for differences in growth rates of
firms – growth firms should have higher M/B ratios than value firms. To account for the
differences in the growth rates, we estimate the following model:
MB = tj + 1tjBB + 2tjEB + 3tjGROWTH + tj
(4)
where MB is the ratio of market value of equity at the end of period t to book
value of equity at the end of period t-1, BB is book value of equity at the end of year t
divided by book value of equity at the end of year t-1, EB is earnings at the end of year t
divided by book value of equity at the end of year t-1, GROWTH is the average annual
growth in sales over the period from t-2 to t,  is the disturbance term, which is assumed
to be independent, homoscedastic and Normally distributed. The subscript j stands for
industry j, since this equation is estimated separately for each 4-digit SIC industry.
the treasury stock method. Thus, the accounting calculations of CSE may not be identical to those of
15
We estimate the dilution ratio (DIL) for each firm in a given year as follows:
DILt = NSEPSt/(0.5 NSOUTt + 0.5 NSOUTt-1)
Where NSEPS is the number of shares used to calculate Primary earnings per
share, and NSOUT is the number of outstanding common shares. If the firm has no
common stock equivalents, and if the weighted average number of outstanding shares can
be estimated by the simple average of outstanding shares at the beginning and end of year
t, then the dilution ratio is one. The dilution ratio will be greater than one if the firm has
common stocks equivalents. To eliminate the bias in the analysis due to cases where
firms issued or repurchased common stocks during the first or last months of the year, we
restricted our analysis to those firms where the number of outstanding shares did not
change during the year by more than 3%3.
To estimate the intrinsic market value of firms with significant potential dilution,
we first partition the sample based on the dilution ratio DIL into two groups: dilution
firms and no dilution firms. We assume that firms with a dilution ratio below 3% are
firms with no significant dilution (no dilution firms), whereas firms with dilution ratios
above 3% are considered to have significant dilution (dilution firms). The cutoff of 3%
stems from APB Opinion 15 (APB, 1969), which states that “any reduction of less than
3% in the aggregate need not be considered as dilution in the computation and
presentation of earnings per share data as discussed throughout this Opinion”. We
market participants.
3
Our estimate of DIL essentially assumes that any issuance or repurchase of common stock occurred in
mid-year, since we assume a simple average of the number of outstanding common shares. In calculating
EPS, firms use the weighted-average number of outstanding shares, which will be sensitive to the deviation
from the mid-year assumption. For cases where the number of outstanding shares did not change during the
year by more than 3%, our simulation indicated that the bias in DIL from assuming a mid-year issuance or
repurchase of stock, will not exceed 1.2%, even if the issuance or repurchase occurred on the first or last
day of the year.
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repeat the tests in this study using 2% and 4% cut-off with no major differences in the
results.
We then estimate the coefficients of the model (Equation 4) for all the no-dilution
firms in a given year and given industry. We use the estimated coefficients to predict the
intrinsic (predicted from the regression equation (4)) market to book ratio for all the
dilution firms in the same year and industry. The predicted market to book ratio should be
independent of the dilution ratio, and should represent the “theoretical” market to book
ratio given the expected future cash flows of the firm.
Using the dilution ratio (DIL) and the predicted market to book ratio, we compute
the following variables for the sample firms with significant (above 3%) potential
dilution:
MBND (Market to Book, No Dilution) – the market to book ratio, where
the market value is computed based only on the number of outstanding
shares at the end of year t.
MBDIL (Market to Book, Dilution) – market to book ratio, where the
market value is calculated based on the dilutive number of shares. MBDIL
is computed as the price per share (at the end of year t) times the number
of common shares outstanding times the dilution ratio, DIL, divided by
book value of equity.
MBP (Predicted market to book ratio) – is the predicted market to book
ratio of dilution firms (using the coefficients estimated for all the nodilution firms in the same year and industry).
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ERND – the difference between the predicted market to book ratio and the
no dilution market to book ratio, ERND = MBND – MBP
ERDIL - the difference between the predicted market to book ratio and
the dilution market to book ratio, ERDIL = MBDIL – MBP
We expect MBND to be the lower bound of the predicted market to book ratio
(MBP) since the market value MBND is calculated using only the outstanding shares, and
the market price per share is likely to include the expected number of CSE that will be
converted to common stock. Conversely, we expect MBDIL to be the upper bound of the
predicted market to book ratio, since the market value MBDIL incorporates all CSE,
where the price per share may include only a portion of the CSE. Thus, our first test is of
the following hypothesis:
H1:
ERND<=0, ERDIL>=0
To determine which of the two competing market to book ratios, MBND and
MBDIL, is a better proxy for the predicted market to book ratio, MBP, we also estimate a
confidence interval for MBP. We compute the proportion of observations for which each
of the two competing market to book ratios is closer to the MBP. Obviously, when one of
the two market to book ratios is included within the confidence interval while the other is
outside the interval, the former is said to be closer to MBP. In the case where both ratios
are included in the confidence interval, we measure which one is closer (in absolute
value) to the mean of MBP. When both ratios are outside the confidence interval, we
posit that the one closer to one of boundaries is the closer to MBP. Exhibit 2 summarizes
all the possible cases, based on the location of MBND and MBDIL relative to the
18
boundaries of the confidence interval of MBP. It also assigns each of these cases into two
groups for which either MBND or MBDIL is a better surrogate of the predicted ratio
MBP.
Our null hypothesis is that the proportion of cases where MBND is closer to MBP
(denoted P(MBND)) is greater than the proportion of cases where MBDIL is closer to
MBP (denoted P(MBDIL)). To the extent that the accounting CSE does not represent the
expected dilution to market participants, i.e., that market participants essentially ignore
the accounting CSE, we expect:
H2: P(MBND)>P(MBDIL)
Exhibit 2
Case Location of MBDIL and MBND
Relatively to the Confidence
Interval of MBP
MBND<LBOUND and
1
MBDIL<LBOUND
MBND<LBOUND and
2
LBOUND<MBDIL<UBOUND
MBND<LBOUND and
3
MBDIL>UBOUND
LBOUND<MBND<UBOUND and
4
LBOUND<MBDIL<UBOUND
LBOUND<MBND<UBOUND and
5
UBOUND<MBDIL
UBOUND<MBND and
6
UBOUND<MBDIL
Better Proxy for MBP (MBND or MBDIL)
MBDIL
MBDIL
MBDIL – if (LBOUND-MBND)>(MBDILUBOUND), MBND otherwise
MBDIL – if ABS(MBND-MBP)>ABS(MBDILMBP), MBND otherwise
MBND
MBND
Notes:
1. MBP is the predicted M/B ratio of dilution firms from Equation (4), estimated only by no-dilution
firms.
2. LBOUND is the lower bound of the confidence interval of MBP.
3. UBOUND is the upper bound of the confidence interval of MBP.
4. MBND is the M/B ratio where the market value of equity is based only on outstanding shares.
5. MBDIL is the M/B ratio where the market value includes also dilutive shares.
19
IV. Sample, Variables and Results
Sample:
Firms in our sample are extracted from the COMPUSTAT Annual Industrial and
Research Files for the years 1986-1996. The sample consists of all firms that meet the
following criteria:
1. Data are available on market value (data item 24 times data item 25) and book
value of equity (data item 60) for both year (t) and year (t-1), earnings (data item
18), total assets (data item 6), primary earnings per share (data item 58), sales
(data item 12) for year (t-2) to year (t), the number of shares used to calculate
earnings per share (data item 54), and the number of outstanding common shares
(data item 25) at the beginning and at the end of year (t).
2. The change in the number of shares outstanding from beginning to end of the year
is less than 3%.
3. The ratio of the absolute value of earnings to book value of equity is smaller than
0.5, and ratio of the absolute value of book value of equity in year (t) to book
value in year (t-1) is smaller than 5 and greater than 0.5. These selection criteria
are intended to eliminate firms with extreme observations.
4. There exist at least 1 dilution firm, and at least 10 no-dilution firms within the
same industry, as designated by the four-digit SIC code.
20
As is common in these types of cross-sectional studies, we eliminated all firms with
extreme observations on any of the regression variables. This was done by eliminating
the bottom and top 0.5% of all the cross-sectional observations for each of the regression
variables in Equation (4). Table 1 presents the number of industries and the total number
of no-dilution and dilution firms in every year. As can be seen, the number of dilution
firms increases as we move forward in time, possibly due to a greater propensity to award
stock options to employees in later years. In every one of the years, we find many
industries represented in the sample, with an average of about 22 firms per industry. The
number of dilution firms exceeds 10% of the number of no-dilution firms, and is about
18% of the no-dilution firms on the average.
(Insert Table 1 about here)
Table 2 presents various financial ratios and variables for the dilution and nodilution firms. The dilution firms have higher means of M/B, E/B, E/P, B/B ratios, and
Growth. These differences are likely driven by their different business profiles. Common
stock equivalents, and particularly stock options, have proliferated mainly in the high
technology and pharmaceutical industries, which are characterized by high growth rates
and earnings. These differences in growth probably account also for the differences in the
M/B, E/P, B/B and E/B ratios. Investors may be willing to pay higher premiums for
dilution firms because of their faster-growing future cash flows. Firms in high technology
industries along with the pharmaceutical industry invest heavily in R&D projects, which,
when proved successful, provide very high returns to capital. Probit analysis (not reported
here) shows that the probability of a firm to issue common stocks equivalents is
positively and significantly correlated with growth and BB. These results suggest that
21
firms that experience high growth, both in sales and in book value, are more likely to
issue CSE. Because of the high growth rate and high ROE (Earnings to Book value of
equity) investors are willing to pay higher premiums for the stock, as measured by the
market to book (M/B) ratio.
(Insert Table 2 about here)
To test the first hypothesis (H1), we predict the MBP of the dilution firms, using
the estimated coefficients of regression equation (4) from the no-dilution firms within the
same industry. We then compute ERND and ERDIL for the entire sample of dilution
firms (those where DIL>3%). We then also report ERND and ERDIL for firms with a
dilution ratio in excess of 4%. We repeat these reports for firms with DIL greater than
5%, 6%, etc., until 10%. This allows us to test the sensitivity of the hypothesis to various
levels of assumed dilution. The medians of ERND and ERDIL are reported in Table 34.
(Insert Table 3 about here)
Table 3 shows that the median ERND is negative for all levels of dilution, and
statistically different from zero for DIL greater then 6%. These results are consistent with
the first part of the hypothesis that ERND is less or equal to zero. Furthermore, the
median ERND increases in absolute value as the dilution cut-off increases, indicating that
the no-dilution M/B further deviates from MBP the higher is the dilution level. Also, the
percentage of observations that are negative increases monotonically with the dilution
level.
The median of ERDIL is positive and significant for dilution levels below 7%,
positive but insignificant for 7%, and negative but insignificantly different from zero for
22
higher dilution levels. These results are also consistent with the null hypothesis that the
market ignores accounting CSE for low dilution levels. Note further that the percentage
of positive observations declines monotonically with the levels of dilution, possibly
because CSE are ignored by market participants for lower levels of dilution but not for
higher dilution levels.
Overall, we find that for low levels of dilution, ERND is not significantly
different from zero and that ERDIL is positive and significant. For high levels of dilution,
ERND is significantly negative and ERDIL is not different from zero. These results
suggest that market participants do (do not) take into account the possible effect of
dilution due to CSE for low (high) levels of dilution. Hence, for low (high) level of
dilution MBND (MBDIL) may be a better proxy for the intrinsic market to book.
A second test we use is not based singly on the mean of the predicted MBP, but
takes into account its variance as well. This is based on the 95% and the 90% confidence
intervals of MBP. We examine what percentage of the observations on MBND and
MBDIL fall outside the confidence interval, and which measure is closer to the interval
boundary. Also, we examine those observations of MBND and MBDIL that fall within
the confidence interval, and again test which one is closer to the mean MBP (the middle
of the confidence interval). This is carried out in Table 4 for the dilution cut-off level of
3%.
(Insert Table 4 about here)
Table 4 reveals that 79% (73%) of both MBND and MBDIL fall within the 95%
(90%) confidence interval. Since the MBP is estimated from the coefficients obtained for
4
We do not use means for the analysis due to the influence of some high prediction errors. We also
repeated this analysis after deletion of the extreme errors, and used both means and medians. The results
23
the sample of no-dilution firms, the 95% confidence interval is supposed to contain 95%
of the MBND if market participants ignore accounting CSE completely, or 95% of the
MBDIL if market participants incorporate the accounting CSE completely in market
prices. Table 4 reveals that 80.8% of the MBND fall within the 95% confidence interval
(cases 4a, 4b, and 5), whereas only 79.3% of the MBDIL fall within the confidence
interval (cases 2, 4a, and 4b). Thus, for a surprisingly large proportion of the firms, the
confidence interval does not contain either MBND or MBDIL. In 4.4% of the sample
cases, both are below the lower boundary of the 95% confidence interval, and in 14.2%
of the sample cases, both MBND and MBDIL are above the upper boundary. That is, the
market tends to deviate from the confidence interval by assigning a higher market value
to dilution firms more often than a lower value. This is more consistent with market
participants who ignore accounting CSE than a market which incorporates all accounting
CSE.
We can now examine two additional proportions in Table 4 that may indicate
whether the MBND is more consistent with the data, or whether MBDIL is. The first is
the proportion of cases in Table 4 where MBDIL is indicated in Table 4 to be closer to
MBP. In Table 4, this includes cases 1, 2, 3a, and 4a, or a total of 48.6% for the 95%
confidence interval. This proportion is indicated by PDIL in the first row of Table 5. The
second is the proportion of cases where (i) both MBDIL and MBND fall within the
confidence interval, and (ii) MBDIL is closer to the mean MBP, indicated by
PDILW_95% and PDILW_90%, for the 95% and 90% confidence interval, respectively,
in Table 5. These proportions are summarized in Table 5 for various levels of dilution.
(Insert Table 5 about here)
point at the same direction as those in Table 3.
24
As can be seen in the table, the proportion of observations that indicate that the
market ignores accounting CSE (PDIL<50%) applies for levels of dilution below 6%.
Above 6%, market participants seem to set security prices more in accordance with
accounting CSE than without. For the firms where both MBND and MBDIL fall within
the confidence interval, we find that the market price is set more consistently with
accounting CSE at all dilution levels, and that this conclusion is monotonically stronger
with increasing levels of dilution. However, the table also indicates that the proportion of
observations which fall outside the confidence interval actually increases as the dilution
levels increase.
Thus, the cumulative evidence provided in the study indicates that the market
seems to ignore, or at least to not fully incorporate the accounting CSE in determining the
price per share for low levels of dilution. For high levels of dilution, the market seems to
set the security price in a manner that is more consistent with the potential dilution of the
accounting CSE.
Sensitivity Analysis:
1. We repeated the tests in the study assuming that 2% and 4% level of dilutions are the
cut-off points between dilution and no-dilution firms. The results point generally in
the same direction as those reported in the study.
2. One of the selection criteria required that the common shares outstanding will not
change between the beginning and end of the year by more than 3%. We
experimented with 2% and 4%, without any material effects on the results.
25
V. Summary and Conclusions:
This study examines whether stock market participants seem to incorporate into
the price per share the potential dilution due to accounting CSE. The study shows that if
market participants do incorporate the accounting CSE in setting the price per share, the
market value of equity should be calculated through a multiplication of the price per
share by the sum of outstanding shares and accounting CSE. In practice, market
professionals ignore accounting CSE in calculations of the market value of equity.
The study shows that for low levels of dilution (typically below 6-7%), the market
seems to set security prices in a manner that ignores accounting CSE. For higher levels of
dilution, the market seems to set the price in a manner that is consistent with the
assumption that the accounting CSE will be converted. These results provide mixed
signals about the ability of accountants to assess correctly the potential dilution due to
CSE. It seems that for low levels of dilution, market participants ignore the accounting
CSE, or assess a low probability for their conversion. For these low levels of dilution, it
is also possible that market participants do not bother with the precise incorporation of
the accounting CSE, because the effect is not material enough, indicating a minor market
inefficiency. However, stock market participants seem to agree with the accounting
profession’s calculation of CSE for high levels of dilution, either because the probability
of conversion is closer to that assessed by the market, or because ignoring CSE will result
in a market inefficiency that will be too large.
The study points out two implications, one for finance professionals, and the other
for accounting regulators. Finance professionals seem justified in ignoring accounting
26
CSE when they calculate the market value of equity for firms with low levels of potential
dilution of accounting CSE. However, when the firm has high levels of dilution, the
calculation of the market value should incorporate accounting CSE. Similarly, when
earnings or other items are calculated on a per share basis, finance professionals can use
outstanding shares for low levels of dilution, but should use outstanding shares plus CSE
for firms with high levels of dilution.
The study also finds that the FASB’s approach to the calculation and presentation
of Basic and Diluted EPS seems to be justified. As a matter of fact, for low levels of
dilution, market participants are better off using Basic EPS, or just the number of shares
used to calculate it (which incorporates changes in outstanding shares during the year).
However, for firms with high levels of dilution, Diluted EPS and the number of shares
used to calculate it seem to be more appropriate. Allowing users to employ whichever
number they prefer seem to be superior to the Primary EPS computation that required the
assumption of conversion of CSE.
27
Table 1
The Number of Industries, Number of Dilution and
The Number of No-Dilution Firms
Year
86
87
88
89
90
91
92
93
94
95
96
Total
No. of
Industries
27
31
30
31
32
32
33
33
38
40
37
No. of NoDilution
Firms
No. of
Dilution
Firms
Total No. of
Firms
468
549
587
564
619
652
564
580
701
716
924
56
71
82
88
80
85
122
137
171
174
260
524
620
669
652
699
737
686
717
872
890
1184
6924
1326
8250
The table is based on 3% as the cut-off for dilution; firms with dilution ratios
below 3% are classified as no-dilution, whereas firms with dilution ratios above
3% are classified as dilution firms.
28
Table 2
Financial Ratios And Variables For Dilution
And No-Dilution Firms
Year
M/B ratio
Dilution
NoDilution
2.2
2.4
2.4
2.8
2.4
2.7
2.7
3.1
2.9
3.1
2.9
86
87
88
89
90
91
92
93
94
95
96
1.7
1.5
1.7
1.8
1.3
1.7
2.1
2.3
2.0
2.3
2.1
E/B ratio
Dilution
NoDilution
0.150
0.165
0.169
0.159
0.154
0.148
0.153
0.144
0.154
0.152
0.147
0.037
0.057
0.063
0.061
0.045
0.031
0.058
0.058
0.068
0.058
0.082
E/P ratio
Dilution
NoDilution
0.060
0.070
0.073
0.064
0.078
0.063
0.062
0.053
0.061
0.050
0.056
0.029
0.039
0.042
0.039
0.053
0.028
0.033
0.030
0.038
0.032
0.043
Mean
2.7
1.9
0.154
0.056
0.063
0.037
M/B – Market value at the end of year t divided by the book value at the end of year t-1.
E/B- Earnings at the end of year t divided by the book value at the end of year t-1.
E/P- Earnings at the end of year t divided by the stock price at the end of year t.
Year
B/B ratio
Dilution
NoDilution
86
87
88
89
90
91
92
93
94
95
96
1.152
1.164
1.166
1.143
1.153
1.156
1.149
1.143
1.145
1.168
1.132
1.027
1.039
1.035
1.034
1.020
1.004
1.028
1.040
1.030
1.059
1.053
Growth
Dilution
NoDilution
22.2
22.5
28.9
28.9
27.6
16.3
14.7
16.2
17.6
19.8
21.3
10.4
8.2
14.0
13.3
11.5
6.4
6.9
6.2
11.1
13.6
14.3
Mean
1.152
1.033
21.5
10.5
B/B- Book value at the end of year t divided by the book value at the end of year t-1.
Growth –average growth of revenues (in percent) over the period t-2 to t.
29
Table 3
Median Differences between the Calculated and Predicted M/B
Ratios for Dilution Firms at different Dilution Levels
DIL
N
Median
3%
4%
5%
6%
7%
8%
9%
10%
1326
937
681
477
361
287
231
192
-0.033
-0.067
-0.060
-0.171
-0.222
-0.264
-0.384
-0.458
ERND
Signif. % Negat.
0.502
0.981
0.820
0.165
0.034
0.004
0.000
0.001
52%
54%
54%
56%
58%
61%
63%
65%
Median
0.075
0.065
0.096
0.029
0.014
-0.037
-0.081
-0.126
ERDIL
Signif. % Posit.
0.001
0.001
0.001
0.045
0.210
0.656
0.714
0.533
53%
53%
54%
51%
50%
49%
46%
45%
Notes:
1. ERDIL is the difference between the calculated M/B ratio, assuming full dilution of CSE, and
the predicted M/B ratio, estimated from the no-dilution group of firms in the same industry and
year.
2. ERND is the difference between the calculated M/B ratio, using only the outstanding shares,
and the predicted M/B ratio, estimated from the no-dilution group of firms in the same industry
and year.
3. N is the sample size.
4. Signif. represents the significance level of the one-sample Wilcoxon statistic that the median is
equal to zero.
5. % Negative is the percentage of observation with negative ERND.
6. % Positive is the percentage of observation with positive ERDIL.
7. Bold table entries are statistically significant at the 5% level or better.
30
Table 4
Distribution of the Dilution and No-Dilution
Market/Book Ratios Relative to the Confidence Interval of the
Predicted Market/Book Ratio
Case Location of MBDIL and MBND
Relative to the Confidence Interval of
MBP
MBND<LBOUND and MBDIL<LBOUND
1
2
MBND<LBOUND and
Closer
95%
90%
Confidence Confidence
to
Interval
Interval
MBP
MBDIL
4.4%
6.3%
MBDIL
0.7%
0.8%
MBDIL
0.0%
0.0%
MBND
0.0%
0.0%
MBDIL
43.5%
41.5%
MBND
35.1%
31.8%
MBND
2.2%
2.6%
MBND
14.2%
17.0%
LBOUND<MBDIL<UBOUND
3a
MBND<LBOUND and MBDIL>UBOUND and
(LBOUND-MBND)>(MBDIL-UBOUND)
3b
MBND<LBOUND and MBDIL>UBOUND and
(LBOUND-MBND)<(MBDIL-UBOUND)
4a
LBOUND<MBND, MBDIL<UBOUND and
ABS(MBND-MBT)>ABS(MBDIL-MBT)
4b
LBOUND<MBND, MBDIL<UBOUND and
ABS(MBND-MBT)<ABS(MBDIL-MBT)
5
LBOUND<MBND<UBOUND and
UBOUND<MBDIL
6
UBOUND<MBND and UBOUND<MBDIL
Notes:
1. MBP is the predicted M/B ratio of dilution firms from Equation (4), estimated only by no-dilution
firms.
2. LBOUND is the lower bound of the confidence interval of MBP.
3. UBOUND is the upper bound of the confidence interval of MBP.
4. MBND is the M/B ratio where the market value of equity is based only on outstanding shares.
5. MBDIL is the M/B ratio where the market value includes also dilutive shares.
31
Table 5
Proportion of Observations which Indicate the Superiority of
Dilution (No-Dilution) M/B for Different Dilution Ratios
DIL
3%
4%
5%
6%
7%
8%
9%
10%
N
1326
937
681
477
361
287
231
192
PDIL
48.6%
49.3%
48.6%
51.4%
52.4%
54.4%
57.6%
58.9%
PNODIL
PNODIL_
PDIL_
PDILW_95
IN_95%
IN_95%
%
81%
80%
79%
79%
78%
78%
77%
78%
79%
79%
77%
76%
76%
75%
74%
74%
55.4%
56.2%
55.1%
57.6%
57.8%
59.8%
63.3%
62.8%
51.4%
50.7%
51.4%
48.6%
47.6%
45.6%
42.4%
41.1%
PDILW_90%
56.6%
57.6%
56.5%
58.5%
57.9%
59.8%
63.1%
62.5%
Notes:
1.
2.
3.
4.
5.
6.
7.
8.
DIL – Dilution Ratio
N – No. of Observations
PDIL – Proportion of observation for which the dilution market to book ratio is a better estimator of
the predicted market to book ratio
PNODIL - Proportion of observation for which the no-dilution market to book ratio is a better
estimator of the predicted market to book ratio
PNODIL_IN_95% - Percentage of the no-dilution market to book observations within the 95%
confidence interval
PDIL_IN_95% - Percentage of the dilution market to book observations within the 95% confidence
interval
PDILW_95% - percentage of observation within the 95% confidence interval for which the dilution
market to book ratio is closer (in absolute value) to the mean predicted market to book ratio.
PDILW_90% - percentage of observation within the 90% confidence interval for which the dilution
market to book ratio is closer (in absolute value) to the mean predicted market to book ratio.
32
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