The Calculation of Earnings Per Share and Market Value of ...The

The Calculation of Earnings Per Share and Market Value of Equity: Should

Common Stock Equivalents Be Included?

By

Joshua Livnat and Dan SegalLeonard N. Stern School of Business

New York University40 W. 4th St.

NY NY 10012(212) 998-0022

[email protected]@stern.nyu.edu

First Draft: March 1999Current Draft: April 10, 2023

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

dilutive shares. As we shall show below, if the price per share indeed incorporates the

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.

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.

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

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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.

II. Background and Conceptual Perspective

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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.

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.

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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.

Prior Studies Relating to Calculations of EPS:

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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

method required by APB Opinion No. 15. He proposes that companies with convertible

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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

sample. Deberg and Murdoch [1994] examine whether FDEPS contains more

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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.

Investment managers often categorize their holdings as large-capitalization stocks, mid-

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capitalization 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 capitalizationThe 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 aggregatemethodology, 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)

Similarly, Value Line defines Market Capitalization as:

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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.

Conceptual Analysis:

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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.

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):

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Exhibit 1

Market Price

MV(no dilution)

MV(dilution)

M/B(no dilution)

M/B(dilution)

$5 (correct price)

$50($5x10) $100($5x20)(correct MV)

0.5 1.0 (correct

M/B)$6(almost correct price)

$60 $120 0.6 1.2

$9(almost entirely incorrect price)

$90 $180 0.9 1.8

$10 (incorrect price)

$100($10x10)(correct MV)

$200($10x20) 1.0(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

calculating market values. Note that similar results obtain for the market to book ratio in

the exhibit.

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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

2 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 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 the treasury stock method. Thus, the accounting calculations of CSE may not be identical to those of market participants.

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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)

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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.

We estimate the dilution ratio (DIL) for each firm in a given year as follows:

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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

repeat the tests in this study using 2% and 4% cut-off with no major differences in the

results.

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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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 no-

dilution firms in the same year and industry).

ERND – the difference between the predicted market to book ratio and the

no dilution market to book ratio, ERND = MBND – MBP

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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

boundaries of the confidence interval of MBP. It also assigns each of these cases into two

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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

Better Proxy for MBP (MBND or MBDIL)

1 MBND<LBOUND and MBDIL<LBOUND

MBDIL

2 MBND<LBOUND and LBOUND<MBDIL<UBOUND

MBDIL

3 MBND<LBOUND and MBDIL>UBOUND

MBDIL – if (LBOUND-MBND)>(MBDIL-UBOUND), MBND otherwise

4 LBOUND<MBND<UBOUND and LBOUND<MBDIL<UBOUND

MBDIL – if ABS(MBND-MBP)>ABS(MBDIL-MBP), MBND otherwise

5 LBOUND<MBND<UBOUND and UBOUND<MBDIL

MBND

6 UBOUND<MBND and UBOUND<MBDIL

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.

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

20

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 no-

dilution 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

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

21

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

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

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 point at the same direction as those in Table 3.

22

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

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

23

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)

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

24

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 No. of Industries

No. of No-Dilution Firms

No. of Dilution Firms

Total No. of Firms

86 27 468 56 52487 31 549 71 62088 30 587 82 66989 31 564 88 65290 32 619 80 69991 32 652 85 73792 33 564 122 68693 33 580 137 71794 38 701 171 87295 40 716 174 89096 37 924 260 1184

Total 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 2Financial Ratios And Variables For Dilution

And No-Dilution Firms

Year M/B ratio E/B ratio E/P ratioDilution No-

DilutionDilution No-

DilutionDilution No-

Dilution

86 2.2 1.7 0.150 0.037 0.060 0.02987 2.4 1.5 0.165 0.057 0.070 0.03988 2.4 1.7 0.169 0.063 0.073 0.04289 2.8 1.8 0.159 0.061 0.064 0.03990 2.4 1.3 0.154 0.045 0.078 0.05391 2.7 1.7 0.148 0.031 0.063 0.02892 2.7 2.1 0.153 0.058 0.062 0.03393 3.1 2.3 0.144 0.058 0.053 0.03094 2.9 2.0 0.154 0.068 0.061 0.03895 3.1 2.3 0.152 0.058 0.050 0.03296 2.9 2.1 0.147 0.082 0.056 0.043

Mean 2.7 1.9 0.154 0.056 0.063 0.037M/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 Growth Dilution No-

DilutionDilution No-

Dilution

86 1.152 1.027 22.2 10.487 1.164 1.039 22.5 8.288 1.166 1.035 28.9 14.089 1.143 1.034 28.9 13.390 1.153 1.020 27.6 11.591 1.156 1.004 16.3 6.492 1.149 1.028 14.7 6.993 1.143 1.040 16.2 6.294 1.145 1.030 17.6 11.195 1.168 1.059 19.8 13.696 1.132 1.053 21.3 14.3

Mean 1.152 1.033 21.5 10.5B/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

ERND ERDILDIL N Median Signif. % Negat. Median Signif. % Posit.

3% 1326 -0.033 0.502 52% 0.075 0.001 53%4% 937 -0.067 0.981 54% 0.065 0.001 53%5% 681 -0.060 0.820 54% 0.096 0.001 54%6% 477 -0.171 0.165 56% 0.029 0.045 51%7% 361 -0.222 0.034 58% 0.014 0.210 50%8% 287 -0.264 0.004 61% -0.037 0.656 49%9% 231 -0.384 0.000 63% -0.081 0.714 46%

10% 192 -0.458 0.001 65% -0.126 0.533 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

Closer to MBP

95%Confidence Interval

90%Confidence Interval

1 MBND<LBOUND and MBDIL<LBOUND MBDIL 4.4% 6.3%

2 MBND<LBOUND and

LBOUND<MBDIL<UBOUND

MBDIL 0.7% 0.8%

3a MBND<LBOUND and MBDIL>UBOUND and

(LBOUND-MBND)>(MBDIL-UBOUND)

MBDIL 0.0% 0.0%

3b MBND<LBOUND and MBDIL>UBOUND and

(LBOUND-MBND)<(MBDIL-UBOUND)

MBND 0.0% 0.0%

4a LBOUND<MBND, MBDIL<UBOUND and

ABS(MBND-MBT)>ABS(MBDIL-MBT)

MBDIL 43.5% 41.5%

4b LBOUND<MBND, MBDIL<UBOUND and

ABS(MBND-MBT)<ABS(MBDIL-MBT)

MBND 35.1% 31.8%

5 LBOUND<MBND<UBOUND and

UBOUND<MBDIL

MBND 2.2% 2.6%

6 UBOUND<MBND and UBOUND<MBDIL MBND 14.2% 17.0%

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 N PDIL PNODIL PNODIL_

IN_95%

PDIL_

IN_95%

PDILW_95

%

PDILW_90%

3% 1326 48.6% 51.4% 81% 79% 55.4% 56.6%4% 937 49.3% 50.7% 80% 79% 56.2% 57.6%5% 681 48.6% 51.4% 79% 77% 55.1% 56.5%6% 477 51.4% 48.6% 79% 76% 57.6% 58.5%7% 361 52.4% 47.6% 78% 76% 57.8% 57.9%8% 287 54.4% 45.6% 78% 75% 59.8% 59.8%9% 231 57.6% 42.4% 77% 74% 63.3% 63.1%10% 192 58.9% 41.1% 78% 74% 62.8% 62.5%

Notes:

1. DIL – Dilution Ratio2. N – No. of Observations3. PDIL – Proportion of observation for which the dilution market to book ratio is a better estimator of

the predicted market to book ratio4. PNODIL - Proportion of observation for which the no-dilution market to book ratio is a better

estimator of the predicted market to book ratio5. PNODIL_IN_95% - Percentage of the no-dilution market to book observations within the 95%

confidence interval6. PDIL_IN_95% - Percentage of the dilution market to book observations within the 95% confidence

interval7. 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.8. 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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