earning, adaptation and equity valui - burgstahler, dichev

Earnings, Adaptation and Equity ValueAuthor(s): David C. Burgstahler and Ilia D. DichevSource: The Accounting Review, Vol. 72, No. 2 (Apr., 1997), pp. 187-215Published by: American Accounting AssociationStable URL: http://www.jstor.org/stable/248552Accessed: 01/12/2009 23:31

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THE ACCOUNTING REVIEW Vol. 72, No. 2 April 1997 pp. 187-215

Earnings, Adaptation

and Equity Value

David C. Burgstahler University of Washington

Ilia D. Dichev University of Michigan

ABSTRACT: This paper develops and tests an option-style valuation model, whose main prediction is that equity value is a convex function of both earnings and book value, where the function depends on the relative values of earnings and book value. Earnings provides a measure of how the firm's resources are currently used. Book value provides a measure of the value of the firm's resources, independent of how the resources are currently used. When the ratio earnings/book value is high, the firm is likely to continue its current way of using resources, and earnings is the more important determinant of equity value. When earnings/book value is low, the firm is more likely to exercise the option to adapt its resources to a superior alternative use, and book value becomes the more important determinant of equity value. Evidence from a variety of empirical specifications is consistent with the convexity prediction.

Key Words: Equity valuation, Earnings, Book value, Convexity.

Data Availability: Data used in this study are collected or purchased from public sources identified in the study.

I. INTRODUCTION E vidence from a variety of studies shows that equity value is related to accounting earnings

(e.g., Ball and Brown 1968; Barth et al. 1992; Collins and Kothari 1989). These studies usually rely on some form of the Miller and Modigliani (1961)discounted dividend

valuation model, imbedding an (implicit) assumption that current earnings are an adequate

Financial support has been provided by the Deloitte & Touche Foundation and the Accounting Development Fund at the University of Washington. The comments of Gary Biddle, Fischer Black, Robert Bowen, Dan Collins, Lane Daley, Robert Freeman, S. P. Kothari, Steve Matsunaga, Eric Noreen, Jane Ou, Edward Rice, D. Shores, Richard Sloan, Ross Watts, workshop participants at the Universities of Washington, Alberta, Iowa, and the 1994 Fifth Annual Conference on Financial Economics and Accounting at the University of Michigan and, especially, Thomas Hemmer and Terry Shevlin are gratefully acknowledged.

Submitted February 1995. Accepted October 1996.

187

188 The Accounting Review, April 1997

characterization of expected future earnings and dividends. At the same time, a number of studies show that equity value is related to balance sheet measures of assets and liabilities (e.g., Landsman 1986; Barth 1991; Shevlin 1991). These studies use the book values of the firm's assets and liabilities to draw inferences about the value of the firm, imbedding an assumption that measures of assets and liabilities impound the expected results of future activities.

Valuation models based on earnings and models based on book value are typically viewed as alternative approaches to valuation (e.g., Solomons 1995; Barth and Landsman 1995). Additionally, in theoretical models that assume complete and perfect markets, measures of book value and measures of earnings are redundant alternatives for valuation (see, for example, Beaver and Demski 1979; Barth and Landsman 1995). However, in more realistic settings with market imperfections, accounting systems can provide information about book value and earnings as complementary, rather than redundant, components of equity value. Book values from the balance sheet provide information about the net value of the firm' s resources. This information is based on (primarily historical) market prices and is therefore largely independent of the success with which the firm currently employs its resources.1 In contrast, earnings from the income statement provide a measure of value which reflects the current firm-specific results of employing firm resources. Accordingly, earnings are a relatively more important determinant of value when the firm' s current activities are successful (earnings are high relative to book value) and are to be continued, while book value is more important when the firm's resources are likely to be adapted to some superior alternative use. However, in general, the value of the firm is a function of both earnings and book value because the firm has the ongoing option to either continue its present activities or adapt its resources to alternative uses.

This paper develops a model of firm value which explicitly recognizes that the potential to adapt firm resources to an alternative use represents an option available to the firm and the ex ante value of the option should be reflected in market value.2 The firm is viewed as a combination of a set of resources and a firm-specific business technology, where business technology is defined broadly as the way a firm uses its resources to produce a stream of earnings. The two complementary elements of value are "recursion value" and "adaptation value." Recursion value is the value which results from discounting the stream of future earnings under the assumption that the firm continues to apply its current business technology to its resources. Adaptation value is the value of the firm's resources independent of the firm's current business technology; this value exists whenever resources can be adapted to alternative uses. The alternative uses may involve external adaptations (where the resources are sold to another entity) or internal adaptations (where the firm retains the resources and adapts them to an alternative use).3 Because the firm has the option to adapt firm resources to uses other than their current use, both adaptation value and recursion value are determinants of equity value.

1 Mostbalance sheet measures are based on historical market transactions but some are based on contemporaneous market transactions which do not directly involve the firm (e.g., market valuation of investment securities or the market aspect of lower-of-cost-or-market valuation of inventory).

2 The potential ex ante importance of the adaptation option for equity valuation is supported by a broad literature in finance and accounting which shows that a wide variety of adaptations of firms' activities take place (including liquidations, sell-offs, spin-offs, divestitures, CEO changes, mergers, takeovers, bankruptcies, restructurings, and new capital investments) and these adaptations have substantial ex post effects on firm value (e.g., see Jensen and Ruback 1983).

3 Concepts closely related to our theoretical constructs are found throughout the accounting literature. Our concept of business technology is similar to what Barth and Landsman (1995) label "managerial skill." Our concept of recursion value corresponds roughly to Sterling's (1968) "stationary state" for a firm, in which the firm "will continue in much the same manner as in the past." Sterling (1968, 489) contends that the stationary state is what many accountants have in mind when they refer to a going concern in valuation arguments. Our concept of adaptation value is similar to Wright's (1967) discussion of the role of balance sheet information. Wright contends that balance sheet information is informative about adaptation value, and adaptation value could be either resale value or replacement cost depending on what type of adaptation (external or internal) is more likely.

Burgstahler and Dichev-Earnings, Adaptation and Equity Value 189

We combine earnings and net assets in an option-style model of equity value, where the likelihood of exercising the adaptation option depends on the relative values of earnings and book value and, therefore, the value-relevance of the two components depends on their relative levels. The main prediction of the model is convexity in the valuation function. Equity value is a convex function of earnings holding book value constant and, at the same time, equity value is a convex function of book value holding earnings constant. Empirical evidence on the relation between equity value, expected earnings and net book value is consistent with the convex form predicted by the model. Results on the relation between changes in equity value, expected earnings and net book value also provide evidence of the predicted convexity.

The remainder of the paper is organized as follows. Section II presents the model of equity value and section III provides a discussion of the relation to previous research. Section IV formulates empirical predictions and presents results of empirical tests. Section V presents conclusions and implications for future research.

II. MODEL In the model below, the market value of equity reflects an option-style combination of

recursion value (capitalized expected earnings when the firm recursively applies its current business technology to its resources) and adaptation value (the value of the firm's resources adapted to an alternative use). The relative weights on the two components of value reflect the probability that the firm will exercise the option to adapt the resources to an alternative use. Specifically, the firm will opt out of recursion value in favor of adaptation value when recursion value is low relative to the adaptation value. The shape of the valuation function in each argument (holding the other argument constant) leads to two testable propositions.

The model incorporates four basic terms: MV = Market value of equity, E = Expected future earnings using the firm's current business technology, c = Capitalization factor for earnings, AV = Adaptation value.

E and AV are random variables, whose joint distribution is described by the multivariate normal density parameterized by a vector of means and a variance-covariance matrix, f(E, AV) = f( { RE,

9AV } GE'EAV'GEAVEI}). All available information relevant to the evaluation of expected future earnings and adaptation value is assumed to be captured by the parameters of the multivariate distribution. The covariance between E and AV will generally be positive, i.e., firms with higher levels of earnings typically employ more resources (AV), and firms with more resources typically generate higher levels of earnings.

The recursion value component of equity value is the discounted present value of expected future earnings, which is assumed to be the product of an earnings capitalization factor c and expected future earnings E. The capitalization factorc implicitly reflects risk-adjustment, the schedule of future interest rates, and other features which are sometimes explicitly included in valuation models.

The market value of equity reflects the option to choose either the recursion value or the adaptation value, whichever is larger, at some point in the future.4

MV(E, AV) = Efmax(cE, AV] = JJ max(cEAV)f(E,AV)dAVdE. (1)

4 The derivation assumes that adaptation value can be exercised at only one point in time. Note that because exercise is restricted to a single point in time, it follows that the value derived is a conservative estimate of the value which would result if investors have more opportunities to exercise the adaptation option, as investors cannot be worse off having more frequent opportunities to exercise. Also note that investors are presumed risk-neutral, so that expected wealth maximization is sufficient to characterize their preferences.


Equation (1) provides the structure to examine the form of the relation between market value of equity, recursion value and adaptation value.

Market Value as a Function of Recursion Value for Fixed Adaptation Value In this section, we use equation (1) to examine the value-relevance of recursion value (capital-

ized earnings), holding adaptation value constant. Denoting the conditional density function for E by f(E I AV), and expanding equation (1) for a fixed AV, the market value of equity is given by:5

MV(E I AV) = J AVf(E I AV)dE + J cEf(E I AV)dE. (2)

Note that the limit of integration, AV/c, is the point of investor indifference where recursion value (capitalized earnings) equals adaptation value. For E > AV/c, investors prefer to continue to operate the company using the current technology, while for E < AV/c, they prefer to adapt resources to an alternative use.

Rearranging equation (2) yields:

MV(E I AV) = AV + / (cE -AV)f(E I AV)dE. (3)

Thus, the market value of equity conditional on a fixed adaptation value can be interpreted as the sum of the adaptation value plus the value of an option on the recursion value. Specifically, the integral term in equation (3) is effectively a call option, whose value is increasing in the conditional mean of expected earnings.

The following proposition describes a testable implication of the conditional valuation function (see proof in appendix):

Proposition 1: Market value is an increasing, convex function of expected earnings, for a given adaptation value.

Intuitively, Proposition 1 follows because as the conditional mean of future earnings increases, the value of the call option on earnings increases. Furthermore, the revision in market value due to a given improvement in expected earnings is increasing in the level of expected earnings.

Figure 1 depicts market value of equity as a function of expected earnings for a fixed adaptation value. The horizontal line in the figure represents a fixed adaptation value and the upward sloping line represents recursion value which is linear and increasing in the level of expected earnings. The two lines intersect at the indifference point E = AV/c. The market value of equity curve approaches the fixed adaptation value on the left and approaches capitalized expected earnings on the right. For low expected earnings, most of market value is attributable to adaptation value and a revision in expected earnings matters little for valuation. For high expected earnings, most of the market value is attributable to recursion value and a revision of expected earnings has a large effect on market value. For intermediate levels of earnings, the market value of equity depends on a more balanced combination of adaptation and recursion value. Thus, holding adaptation value constant, the market value revision in response to a change in gE depends on the level of gE. The slope of the relation between market value and expected

5 Because the exercise point is in the future, the expected future adaptation and recursion value should be discounted to the present. However, the discount factor is constant for a constant interest rate and constant time to exercise, and therefore, to simplify the exposition, we omit the explicit discount factor. In the appendix, the discount factor is again omitted because the constant discount factor has no effect on the proofs.


FIGURE 1 Market Value of Equity as a Function of Expected Earnings, Holding

Adaptation Value Constant

Market value I

--- Adaptation value

- Recursion value /

>~~~~~~~~~~~~~~~~~~~~~~~~~~ /8

caiaizto facor.

a / /~~~~~~~~~~~~~~~~~~~~~~~~~~~~

;3 / ,,"~~~~~~~~~~~~~~~~~

IS

Expected earnings

earnings increases over the range of expected earnings toward its limiting value, the earnings capitalization factor.6

6 Though we do not explore the issue in this paper, there could also be an eventual decrease in the slope of the relation at the uppermost levels of expected earnings because high levels of expected earnings for a given resource base are likely to attract competition and consequently may not be sustainable. Miller (1994, 33-34) describes this phenomenon as follows: "Above normal profits always carry with themthe seeds of theirown decay. They attractcompetitors, both from within a country and from abroad, driving profits and share prices relentlessly back toward the competitive norm." Mean-reversion in returns on equity is consistent with this conjecture (see Penman (1991) and Bernard (1994) for evidence and further discussion).

Note also that a different type of non-linearity related to earnings is considered in Freeman and Tse (1992). This non- linearity is motivated by the conjecture that larger unexpected earnings contain more non-recurring components and leads to the prediction of an S-shaped relation between unexpected returns and unexpected earnings. Thus, the Freeman and Tse (1992) non-linearity is different from our predicted convex relation in motivation, type of non-linearity, and predicted effects.


Market Value as a Function of Adaptation Value for Fixed Recursion Value The analysis of market value as a function of adaptation value, holding recursion value

constant is parallel to the analysis above. Rewriting and expanding equation (1) as a function of AV, for a fixed E, yields:

MV(AV I E) = cE + J(AV - cD)f(AV I E)dAV. (4)

In this form, the market value of equity conditional on a fixed level of expected earnings can be interpreted as the sum of capitalized earnings plus the value of an option on "downside protection," i.e., the value of the firm's option to elect the adaptation value if it proves superior to the value of the firm, based on the earnings stream generated with the current business technology. Equation (4) provides the basis for a second proposition (proof is analogous to proof of Proposition 1 in appendix but with roles of AV and E interchanged):

Proposition 2: Market value is an increasing, convex function of adaptation value, for given expected earnings.

Intuitively, Proposition 2 holds because, for a fixed level of expected earnings, an increase in adaptation value provides additional downside protection. Further, the higher the level of adaptation value relative to the fixed level of expected earnings, the larger the revision in market value associated with a revision in adaptation value. If the adaptation value is large relative to capitalized expected earnings, the current business technology of the firm is inferior to alternatives and adaptation value becomes the primary determinant of market value; recursion value is only a secondary determinant of market value.

III. RELATION TO OTHER RESEARCH There is a long stream of research, dating back to Ball and Brown (1968), that documents the

relation between earnings and security prices. The underlying model typically posits that firm value is a function of expected future dividends, which in turn are assumed to be functions of expected future earnings and current earnings. Much of this research focuses on empirical specifications relating returns and changes in expected earnings.

More recent papers including Berger et al. (1996), Hayn (1995) and Subramanyam and Wild (1996) have begun to consider the valuation implications of specific types of adaptation in an option-like framework. Berger et al. (1996) posit that the value of the firm reflects the value of the option to abandon the firm for its liquidation value. For a sample of firms selected because the abandonment option is especially relevant, they find that equity value increases in liquidation value after controlling for expected going-concern cash flows. However, they do not test for convexity as predicted here. Hayn (1995) considers situations where earnings are considered to be sufficiently low as to make liquidation of the firm preferable to continued operation. She focuses on the effect of inclusion of negative earnings (losses) on measures of the information content of earnings and presents evidence that stock price movements are much more strongly linked to positive earnings than negative earnings. Subramanyam and Wild (1996) investigate how the informativeness of earnings is affected by the validity of the going concern assumption and find that the informativeness of earnings is negatively related to the risk of failure (measured using the Altman (1968) Z-score).

The model developed here is more general in that it considers the broader notion of adaptation, which subsumes liquidation of the firm. Actual liquidations are relatively rare, at least


among the population of firms which is most widely studied in academic research.7 Other forms of adaptation (e.g., targeted sales of assets and subsidiaries, spin-offs, and various internal changes and reorganizations) are much more common and their ex ante effect is therefore likely to be relevant for a much larger segment of the firm population.

Our model also differs from other accounting-based valuation models which specify equity value as a function of expected earnings and book value. For example, Ohlson (1990, 1995) expresses price as a linear function of book value and abnormal earnings. Empirical specifications derived from this model (e.g., Easton and Harris 1991; Sougiannis 1994) typically also assume a linear form where equity value is an additive function of earnings and book value. In contrast, our model implies that equity value is a convex non-additive function of expected earnings and book value where the roles of earnings and book value depend on their relative values. When earnings are low relative to book value, the firm is likely to adapt assets to alternative, more productive uses; for these firms, book values are especially relevant. When earnings are high relative to book value, the firm is likely to continue the current, highly productive uses; for these firms, earnings are especially relevant in valuation.

IV. EMPIRICAL TESTS Overview and Choice of Proxies

This section defines empirical proxies for the theoretical variables and specifies the form of tests of the predicted relations. The main focus of the tests is to determine if the form of the relation between equity value, recursion value and adaptation value displays the convexity predicted by the model. Convexity implies that the marginal effects of the two determinants of equity value, adaptation value and recursion value, depend on their magnitudes relative to each other. Thus, tests for convexity require variation in the relative values of recursion value and adaptation value. In the language of our paper, it is variation in business technology (i.e., variation in the way that firms use resources to produce earnings) that leads to variation in the relative values of recursion value and adaptation value. Because the cross-sectional variation in business technology is likely to be substantially larger than the variation in business technology of a particular firm over time, our primary tests are cross-sectional. Additionally, we examine a second specification in changes.

There are three main variables for which empirical proxies are required: market value of equity, recursion value and adaptation value. Market value of equity is directly observable for traded firms, but we need proxies for recursion value and adaptation value.

Recall that recursion value is defined as the value of capitalized future earnings, assuming that the firm continues to apply its current business technology. We use current earnings as a proxy for recursion value for two reasons, the first based on a priori reasoning and the second based on previous empirical research. First, current earnings reflect, by definition, the results of applying the firm's current technology. Although prior research indicates that analyst forecasts or other proxies can provide marginal improvements over current earnings as a proxy for observed future earnings in general (including future earnings which are the result of various adaptations), current earnings is arguably a better proxy for the future earnings expected if the firm continues to apply the current business technology. Second, most univariate time-series research shows that earnings, on average, follow a random walk. In fact, Bernard (1994) finds that it is difficult to improve on current earnings as a predictor of future earnings. For these two reasons, current earnings is the main proxy for recursion value used in the empirical tests. Later, in the results section, two additional earnings specifications are examined in order to test the robustness of the

I For example, for firms with both earnings and book value data on Compustat, the rate of actual liquidations is on the order of approximately one percent or less. Thus, it appears that few firms would have more than a trivial ex ante value of the real liquidation option.


results to alternative measures of recursion value. We also explore the sensitivity of the results to the assumption of a constant earnings capitalization factor.

We use book value as our proxy for adaptation value because adaptation value is closely related to book value by design. The concept of adaptation value is defined as the value of the firm's net assets, independent of the firm's current business technology, while book value is a cost-based measure of the value of the firm's resources, where cost is independent of how the firm will use the resources. Thus, at acquisition, book value is similar or identical to adaptation value. Subsequent to acquisition, book value is also likely to be closely related to adaptation value, although the two may diverge if the various adjustments of the accounting system (e.g., depreciation) do not accurately reflect the changes in adaptation value (e.g., unforeseen obsoles- cence of the assets due to market conditions). However, alternative empirical proxies which might be used (e.g., proxies for replacement cost and net realizable value) are not readily available and must be estimated.8 Further, there is little reason to expect substantively different results using alternative proxies for adaptation value because these proxies typically begin with book value and incorporate relatively minor adjustments. In fact, research in other settings documents that results based on proxies using adjusted book values are strongly correlated with results based on unadjusted book values.9

Design of Cross-Sectional Tests in Levels of MV, BV and E All of the tests are derived from an empirical version of equation (1) expressing the market

value of equity (MV) as a function of both book value (BV) and earnings (E), where BV is a proxy for adaptation value and E is a proxy for recursion value:

MVt= l yBVt+y2Et+e (5)

where e is a normally distributed error term with mean zero and unspecified variance (see footnote 12 for further discussion.) The empirical version of equation (1) approximates market value of equity for a given firm at time t as a linear combination of book value and expected earnings at time t. As E becomes extremely low relative to BV, BV becomes the sole determinant of MV and y2 approaches zero while y, approaches unity.'0 On the other hand, as E becomes extremely high relative to BV, E becomes the sole determinant of MV and y1 approaches zero while y2 approaches the earnings capitalization factor. However, the model in section II implies that market value in general will be a function of both BV and E. Thus, equation (5) must be operationalized in a form which provides for the convexity predicted by the theory so that the coefficients on BV and E are allowed to vary with the level of E relative to BV.

8 Adaptation value is either the value of external or internal adaptation, depending on which provides the higher value. The net amount realizable from sale of the assets is the value for external adaptation. The amount the firm would be willing to pay to acquire the assets and put them into an alternative use is the value for internal adaptation. Wright (1967, 78) draws a similar conclusion from a slightly different perspective, contending that "resale price measures capacity for adaptation if existing activities are to be contracted, whereas replacement cost measures capacity for adaptation if existing activities are to be expanded."

9 For example, results in Beaver and Landsman (1983) suggest that measures of current cost-adjusted book values perform no better than unadjusted book values for purposes of equity valuation. In another setting, Perfect and Wiles (1994) examine a variety of measures of Tobin's q, which is the market value of the firm as a whole divided by the replacement value of the assets, and find substantial agreement across various measures of replacement cost- correlations between Tobin' s q based on book value and based on more refined estimates of replacement cost are high, ranging from .9159 to .9584.

?0More precisely, the coefficient on BV should approach the factor relating BV and adaptation value. If BV were a perfect proxy for adaptation value, this factor would be unity. However, if BV is a biased measure of adaptation value, the bias will be reflected in the estimated coefficient, e.g., if conservatism causes BV to be a downward-biased measure of adaptation value, the coefficient on BV would be expected to exceed one.


Tests of Proposition 1 require that we hold book value constant while testing for the incremental effects of earnings. Tests of Proposition 2 require that we hold earnings constant while testing for the incremental effects of book value. Thus, in tests of Proposition 1, we hold book value constant by dividing market value and earnings by book value. In this form, the results can be interpreted as a test of the incremental value relevance of earnings by examining market value per dollar of net assets as a function of earnings per dollar of net assets. Similarly, in tests of Proposition 2, we hold earnings constant by dividing market value and book value by earnings. In this form, the results provide a test of the incremental relevance of book value by examining market value per dollar of earnings as a function of net assets per dollar of earnings.

Equation (5) expresses MV at time t as a function of BV, (as a proxy for adaptation value at time t) and Et (as a proxy for expected future earnings at time t using the current business technology). We, however, choose to use BVt 1 as the measure of adaptation value for year t. By definition, BVt includes Et as a component. Therefore, empirical tests using BVt-1 will more clearly separate the effects of E and BV.1I

To test for the convex form implied by Proposition 1 where market value is viewed as a function of expected earnings for a fixed level of book value, we divide through by BVt-1 as the measure of adaptation value to obtain:

MVt/BVt-l =y1 BVt/IBVt-l + y2 Et/BVt1 + c* (6) = y1 + y2 Et/BVt_l + a;*

where e* = e/BVt l. In this transformed relation, the estimated intercept is an estimate of the coefficient on BV in the original relation and the estimated slope is an estimate of the coefficient on E. The proposition implies that the values of the coefficients depend on the level of E/BV. At the lowest levels of E/BV, y, should be approximately one and y2 should be close to zero while at higher levels of E/BV, y1 should move toward zero and y2 should move toward the earnings capitalization factor. Results of these tests are reported in tables 2 and 3.

To test for the convex form implied by Proposition 2 where market value is viewed as a function of book value for a fixed level of expected earnings, we divide through by E to obtain:

MVt/Et = y1 BVt- JEt + Y2 EtEt + E* (7) = y2 + Y1 BVt-I/E t+ ;**

where ,** = E/Et. In this transformed relation, the estimated intercept is an estimate of the coefficient on E in the original relation and the estimated slope is an estimate of the coefficient on BV.12 As with equation (6), the analysis implies that the values of the coefficients depend on the relative levels of BV and E. Again, at the lowest levels of E/BV (or, equivalently, the highest levels of BV/E), y1 should be approximately one and y2 should be close to zero while at higher levels of E/BV, yj should move toward zero and y2 should move toward the earnings capitalization factor. Results of these tests are reported in tables 4 and 5.

I'In any event, the operational choice of BVt 1 versus BVt has little influence on the results. The results reported in tables 2-5 below were recalculated using BVt and the results remain qualitatively identical. Note that equations (6) and (7) can also be interpreted as the weighted least squares specifications which follow from the alternative assumptions that the standard deviations of the error terms in equation (5) are either proportional to book value, which leads to equation (6), or proportional to earnings, which leads to equation (7).


In the tests reported below, the coefficients are estimated for three ranges of E/BV (table 3) or BV/E (table 5) to provide a piece-wise linear approximation to the convex valuation function. The cutoff points separating the three ranges are defined to yield an equal number of observations in each of the three ranges. Within each range, the estimated coefficients are restricted to be constant but the coefficients are allowed to vary across the three ranges. The number of ranges in this piece-wise approximation to the continuous function is an empirical choice. A larger number of ranges would provide a larger expected difference between the lowest and highest range but would also reduce the number of observations available to estimate the coefficients for each range. The choice of three ranges seems to be a reasonable compromise.

Results of Cross-Sectional Tests in Levels The sample consists of all firms available on Compustat for the years 1976-1994. Firms in

regulated industries (e.g., utilities) were deleted because the valuation model does not apply to firms with regulated levels of profitability. Similar to other research (e.g., Fama and French 1992), we also exclude banks and financial services firms and firms with negative book value.

The following annual financial statement variables are used in the cross-sectional tests. Earnings is measured as "Earnings Before Extraordinary Items" Available For Common adjusted for Special Items net of tax (Compustat item 237 less Compustat item 17 adjusted by the statutory tax rate). 13 Market value is number of shares outstanding multiplied by the market value per share at fiscal year-end (item 25 by item 199). Book value is the beginning of the year book value (item 60).

Market Value as a Function of Levels of Expected Earnings for a Fixed Adaptation Value Plots of the raw data revealed extreme outliers. To abstract from the influence of extreme

observations, the extreme top and bottom three percent of the observations on E/BV and the top three percent on MV/BV were deleted.14 Descriptive statistics for the trimmed sample are reported in table 1 panel A. The number of observations for each year increases steadily over time from about 1,200 for 1976 to more than 4,100 for 1994. The early years had comparatively few firms with negative earnings but the proportion increased significantly beginning in about 1981 and grew to approximately 30 percent for years after 1986. Average earnings as a percentage of book value declined fairly steadily over the sample period. The average ratio of market to book value hovered around a value slightly above one during the first five years of the sample period and then fluctuated more widely around a range of two to three for the remainder of the sample years.

We begin the analysis with a graphical look at the relation between market value and earnings. Figure 2 plots the empirical relation between market value and earnings, both scaled by book value, for all of the individual observations in the study. The plot is consistent with the predicted shape from figure 1. At low levels of scaled earnings, the relation is fairly flat. However, as the

'At the suggestion of a reviewer, we adjusted for special items because these items are unlikely to recur in earnings if the firm continues its current mode of operations. However, the results are substantially the same for unadjusted earnings.

'4The three percent trimming resulted in an overall deletion of six percent of the observations in the original sample and a MV/BV variable that ranged from zero to 37.46 and an E/BV variable from-i .24 to .83. Deleting the top and the bottom one percent of the observations produced a sample ranging from zero to 105.61 for MV/BV and -3.58 to 2.01 for E/BV. These values and plots of the data convinced us that one percent trimming would leave extreme observations with undue influence in the regression analysis. (Using one percent trimming, the estimated coefficients for the models discussed below were generally consistent with, but more significant than, the reported results for three percent trimming.) The three percent trimming rule used here eliminates fewer observations than the trimming rules used in related previous studies. For example, Collins and Kothari (1989) trimmed all observations with market value to book value in excess of five.

Burgstahler and Dichev-Eamings, Adaptation and Equity Value 197

TABLE 1

Descriptive Statistics

Panel A: Market Value (MV,) and Earnings (E,) Scaled by Book Value (BV,_,)

NegativeEIBt M BV Total Earnings E_/BV, MV7BV1J

Year N N Mean Std. Dev. Median Mean Std. Dev. Median

76 1,208 62 .15 .09 .15 1.39 1.01 1.05 77 1,249 61 .15 .09 .15 1.34 .90 1.08 78 1,335 32 .17 .10 .17 1.53 1.17 1.16 79 1,423 63 .18 .11 .18 1.77 1.59 1.25 80 1,512 108 .17 .12 .16 2.41 2.78 1.43 81 1,694 186 .14 .14 .15 2.17 2.45 1.36 82 1,783 349 .09 .15 .11 2.12 2.09 1.39 83 1,907 364 .11 .18 .12 2.99 3.16 1.91 84 2,124 436 .09 .19 .12 2.24 2.04 1.63 85 2,219 558 .07 .19 .10 2.47 2.29 1.73 86 2,450 674 .05 .22 .09 2.97 3.30 1.93 87 2,758 750 .06 .22 .09 2.60 2.72 1.75 88 2,850 800 .06 .22 .10 2.23 2.01 1.67 89 2,941 875 .05 .23 .09 2.44 2.48 1.65 90 3,049 942 .03 .24 .08 2.08 2.30 1.30 91 3,294 1,045 .02 .25 .07 3.01 4.08 1.63 92 3,681 1,125 .03 .27 .07 3.19 3.96 1.86 93 4,127 1,231 .04 .28 .08 3.73 4.41 2.30 94 4,144 1,103 .05 .27 .10 3.02 3.07 2.07

Pooled 45,748 10,764 .07 .22 .11 2.60 3.03 1.64

Panel B: Market Value (MV,) and Book Value (BV, I) Scaled by Earnings (E,) for Positive Earnings Observations

BVt-l/Et MVt/Et

Year N Mean Std. Dev. Median Mean Std. Dev. Median

76 1,118 7.28 4.24 6.24 8.76 4.21 7.73 77 1,155 7.05 4.04 6.01 8.25 3.61 7.41 78 1,260 6.61 3.96 5.62 8.21 3.98 7.30 79 1,314 6.18 3.87 5.20 8.65 4.96 7.32 80 1,354 6.74 4.51 5.39 12.06 8.63 9.44 81 1,455 7.27 5.05 5.70 11.38 7.72 9.09 82 1,395 9.10 7.48 6.85 15.29 11.19 11.98 83 1,501 7.83 6.09 6.01 18.35 13.01 14.36 84 1,638 7.98 6.21 6.11 14.23 8.77 11.81 85 1,627 9.27 7.90 6.80 18.91 13.75 14.98 86 1,740 9.62 8.71 6.78 20.93 14.87 16.75 87 1,964 9.28 8.81 6.54 17.62 12.25 14.17 88 2,005 8.85 8.05 6.13 15.84 10.59 13.13 89 2,019 8.99 7.83 6.56 17.03 11.35 13.90 90 2,070 9.94 9.06 7.09 15.44 10.36 12.57 91 2,203 11.24 11.30 7.36 21.95 16.05 17.76 92 2,508 10.78 11.28 7.15 21.73 14.56 17.97 93 2,826 9.17 8.46 6.62 23.18 15.72 18.50 94 2,986 8.40 7.17 6.33 19.03 11.76 15.74

Pooled 34,138 8.77 7.96 6.35 16.73 12.64 13.40

(Continued on next page)


TABLE 1 (Continued)

Panel C: Scaled Changes in Market Value (AMV), Scaled Changes in Earnings (AE), and Scaled Changes in Book Value (ABV)

Change in Market Value Change in Earnings Change in Book Value

Year N Mean Std. Dev. Median Mean Std. Dev. Median Mean Std. Dev. Median

76 1,135 .27 .36 .22 .04 .07 .03 .12 .12 .11 77 1,166 .12 .40 .06 .03 .06 .02 .12 .12 .11 78 1,206 .27 .49 .13 .04 .07 .03 .15 .13 .13 79 1,266 .37 .70 .16 .04 .08 .03 .16 .16 .14 80 1,361 .69 1.20 .27 .02 .08 .02 .18 .24 .13 81 1,441 .03 1.01 .01 .02 .09 .02 .17 .28 .11 82 1,607 .34 1.02 .14 -.02 .10 -.01 .09 .20 .07 83 1,629 .71 1.27 .38 .03 .11 .03 .19 .36 .10 84 1,822 -.24 .90 -.11 .03 .11 .03 .10 .24 .09 85 1,955 .42 1.03 .21 .00 .12 .00 .10 .27 .08 86 2,057 .36 1.14 .14 .01 .15 .01 .12 .34 .08 87 2,280 -.05 1.09 -.08 .04 .17 .02 .13 .34 .09 88 2,450 .10 .87 .06 .03 .14 .02 .08 .26 .08 89 2,501 .31 1.06 .07 .02 .15 .01 .08 .29 .07 90 2,578 -.20 1.00 -.22 .00 .15 .01 .07 .30 .05 91 2,704 .86 2.01 .23 .01 .17 .00 .12 .40 .05 92 3,014 .32 1.32 .08 .02 .17 .02 .12 .44 .05 93 3,371 .60 1.62 .21 .02 .17 .02 .15 .46 .07 94 3,525 -.05 1.19 -.05 .03 .16 .03 .15 .39 .10

Pooled 39,068 .27 1.23 .09 .02 .14 .02 .12 .33 .09

Variable Definitions:

MVt = number of shares outstanding at the end of year t (Compustat item 25) times the market value per share at fiscal year- end (item 199).

BVt = book value at the end of year t (item 60).

Et = earnings before extraordinary items available for common adjusted for special items net of tax (item 237 less item 17 adjusted by the statutory tax rate) for year t.

Scaled Change in Market Value (AMV) = [MV, - MV, I]/ BV l.

Scaled Change in Earnings (AE) = [Et- Et,]/BVt_,. Scaled Change in Book Value (ABV) = [BV, - BVt1]/ BV,1.

level of earnings increases, the slope of the market value function clearly increases.15 The predicted convexity is apparent even at the individual firm level for this comprehensive sample drawn from a potentially heterogeneous set of industries and time periods.

To provide a formal test of the predicted shape, we estimate two forms for the regression of scaled market value on scaled earnings specified in equation (6) above. The first regression assumes a simple linear relation over the entire domain of E/BV while the second regression assumes piece-wise linearity as an approximation to the theoretical form derived above. Thus, the

15A similar shape can be inferred from the data reported in Bernard (1994) table 2 panel A. Plotting the mean Price/Book ratios against the mean ROEs shows the convex shape predicted here, especially for the decile 2-9 means. The lack of fit for the first decile in Bernard's (1994) data which includes negative earnings firms is consistent with our empirical results (discussed further below).

Burgstahler and Dichev-Earmings, Adaptation and Equity Value 199

FIGURE 2 Market Value/Book Value versus Earnings/Book Value

1976-1994

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Table 2 contains estimates of the coefficients for the simple linear form relating MVt/BVt_ to Et/BVt 1 in equation (6). Results are presented for individual year regressions, followed by the average of the individual year regression results. Results are also presented for a pooled regression with all firm/years pooled together. The coefficient on earnings is highly significant for all regressions, consistent with value relevance of earnings after controlling for book value. However, the coefficients on earnings averagejust 4.1 1 across all individual year regressions (and range from a high of 8.60 to a low of 1.66), and thus are lower than would be expected for an earnings capitalization factor, especially in later years. The decline of the coefficient estimates and the adjusted R2s over time suggests that either the constraint imposed by the simple linear form becomes less appropriate over time or the importance of earnings decreases in later years. This is discussed in more detail below.

Table 3 contains the results for the piece-wise linear form. The piece-wise regression provides a test of the predicted convex shape by allowing the slope and the intercept of the

TABLE 2 Coefficient Estimates for Market Value as a Function of Earnings

Controlling for the Level of Book Value

MVt/BV,1= b1 + b2E,/BV,1I + E

Year bi tI b2 t2 Adj. R-sq.

76 .43 9.62 6.49 19.84 .35 77 .43 9.62 6.00 19.78 .35 78 .26 4.11 7.33 19.59 .37 79 .50 5.59 7.05 14.39 .24 80 .99 6.53 8.60 10.13 .15 81 1.15 11.50 7.13 10.65 .16 82 1.75 31.95 4.11 9.31 .09 83 2.29 27.37 6.53 10.74 .13 84 2.01 36.70 2.57 7.13 .06 85 2.29 41.81 2.85 6.91 .05 86 2.83 40.02 2.78 5.45 .03 87 2.45 44.73 2.48 6.40 .04 88 2.14 47.85 1.66 5.87 .03 89 2.33 52.10 2.27 7.57 .04 90 2.03 45.39 1.80 6.36 .03 91 2.96 41.86 2.11 3.92 .02 92 3.15 44.55 1.76 3.59 .01 93 3.65 51.62 2.24 4.89 .02 94 2.90 52.95 2.41 8.03 .05

Average 1.92 31.89 4.11 9.50 .12

Pooled Regression 2.45 141.45 2.10 16.60 .02

All t-statistics are based on the heteroskedasticity-consistent covariance matrix (White 1980). Variable Definitions: see table 1.


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Et/BV, 1 into three parts with equal numbers of observations. Dummy variables, DM and DH'

indicate the medium (M) and high (H) ranges of Et/BVt-,. Note that here (and in similar piece-wise linear regression forms in tables 5 and 6), the middle

and high group intercept (b2 and b3) and slope (b5 and b6) coefficients are estimated incremental to the intercept, by, and slope, b4, for the low group. The tests of significance are tests of whether the incremental coefficients are equal to zero and the t-statistics shown in the tables for the middle and high groups are for tests of incremental significance relative to the low group. However, coefficients reported in the tables are the total intercept and slope coefficients for the middle group (i.e., bl+b2 for the intercept and b4+b5 for the slope) and the total coefficients for the high group (i.e., bI+b3 for the intercept and b4+b6 for the slope). Thus, for example, t5 shown in the table is the relevant t-statistic for a test of the difference between the slope coefficients for the middle group and the low group (i.e., a test of the hypothesis that b5=0) while t6 is the relevant t-statistic for a test of the difference between the slope coefficients for the high group and the low group.16 In addition to the t-tests of incremental significance for the middle and high groups relative to the low group, we also report results for t-tests of incremental significance for the high group relative to the middle group using a system of asterisks in table 3 (as well as in tables 5 and 6 to follow). For example, one (two) asterisk(s) on the t-statistic shown for b6 indicates that the difference between the slope coefficients for the high group and the middle group is significant at the .05 (.01) level.

The results in table 3 are consistent with the predictions of the valuation model. As E/BV increases across groups, the intercepts which correspond to the coefficient on BV in equation (5) decrease from a value near 1 to a value of 0 (or less). The slope coefficients on E/BV generally increase as E/BV increases-the t-statistics reveal that b5 and b6 are uniformly significantly positive. Additionally, comparison of the estimates shows that b6 is greater than b5 in every year beyond the first two years and the asterisks indicate that b6 is significantly greater than b5 in every year beyond the first four years. The magnitudes of the estimated slope coefficients for the high earnings group are generally in the range from 10 to 20, consistent with the magnitude expected for an earnings capitalization factor. The piece-wise regression also fits the data much better than the simple linear form. The adjusted R2 increases from .02 to .27 for the pooled regressions. The increases in R2 for the individual year regressions are also large in the latter half of the sample period, though the increases are less dramatic for the first half of the sample period. Also, in contrast to the simple linear form, the R2 is relatively constant over time for the piece-wise form.

One puzzling aspect of the results in table 3 is that the estimated slope coefficients for E/BV for the low earnings interval are significantly negative-the model predicts that the coefficients should be small, but not negative.17 The negative slope of the relation is also apparent from inspection of more extensive versions of the plot in figure 2 which include all of the trimmed data (not presented here).18 To further investigate this phenomenon, we ran a corresponding set of regressions where the E/BV domain was partitioned more finely into five regions rather than

'Note that in this system of presentation, the t-statistics shown adjacent to the estimated coefficients for the middle and high ranges are not for a test of whether the coefficient is equal to zero. For example, in table 3, the overall intercept for the high range in 1976 is .45 and the adjacent t-statistic is -.87. The overall intercept is the sum of the intercept (b =.67) and the incremental coefficient for the high range (b3=-.22), while the t-statistic indicates that the incremental coefficient for the high range is negative (though the magnitude of the t-statistic in this case suggests that the incremental difference is not significant).

17Similar puzzling results associated with negative earnings observations have been reported in other research, e.g., Jan and Ou (1994) and Collins et al. (1996).

"Similar findings carry over to the portfolio level analysis discussed in the next section and presented in figure 3.


three. The first region contained only the negative earnings and the range of positive earnings was divided equally into four regions. The regression using this finer partition showed the same overall shape-the negative region coefficient remained significantly negative, the first positive region coefficient was positive but marginally significant or insignificant in some years and the coefficients for the other three positive regions were significantly positive and increasing. Although we are not able to explore possible explanations for the negative earnings phenomenon further in this paper, one explanation might be that negative values of E/BV are more likely to reflect costs of adaptation. Consequently, more negative values of E/BV may be empirically associated with actions which have been taken to improve E/BV in subsequent years and which are already reflected in relatively higher market values.

Sensitivity Analysis We performed two additional groups of tests to evaluate the sensitivity of our results to

alternative specifications. The first group of tests deals with the effects of measurement error in our proxy for expected future earnings. The second group of tests considers the effect of the constant capitalization factor assumption on our results.

Measurement Error Sensitivity Analysis Since the proxy for expected earnings is subject to measurement error, some of the results

reported in table 3 could be affected by measurement error. As the portion of the variance of the proxy attributable to measurement error (rather than to variation in the level of expected earnings) increases, the estimated coefficient relating market value and earnings will generally be biased toward zero. Thus, the effect of measurement error on the results is a potentially important concern. At the same time, there are reasons to believe that measurement error cannot fully explain the results in table 3. First, in order to explain the large differences across intervals in estimated slope coefficients, large differences in measurement error would be required. Second, the measurement error explanation would also predict smaller estimated coefficients in the highest earnings interval, yet the estimated coefficients in the highest earnings interval are generally larger, rather than smaller.

To evaluate the effects of measurement error, we form portfolios based on lagged scaled earnings, i.e., Et1/BV t2. Transient events which induce measurement error in Et/BVt_1 as a measure of future expected earnings are, by definition, those which do not persist into the future so the measurement error in Et-1/BV t2 should be relatively uncorrelated with the measurement error in E,/BVt-l. Therefore, the relation between the mean (or median) of the dependent variable

(MVt/BVt-) and the independent variable (E,/BVt-1) for portfolios formed on Et- /BVt-2 should be relatively unaffected by measurement error in the individual observations. For each year, observations are sorted on EtJI/BVt-2 to form 30 portfolios. The median values of MVt/BVt-I and

Et/BVtI are then computed for each portfolio.'9 A plot of the portfolio medians, which is analogous to the plot of the individual observations in figure 2, is presented in figure 3.20

The convex shape of the valuation function, discernible in figure 2, is even more apparent for the portfolio medians in figure 3. We also conducted portfolio-level regressions corresponding

'9Results are reported here for portfolio medians, because the use of medians also provides some assurance that the results are not attributable to the effects of outliers. However, this same procedure applied to the portfolio means yielded virtually identical results.

20The horizontal scale in figure 3 is trimmed to correspond to the trimmed scale in figure 2. A complete plot of all the data shows that the observations omitted at the left side of the plot, as a resultof the trimming, continue the slight trend upward to the left which can be seen in the plot in figure 2, and which is also apparent from the negative estimated coefficients for the lowest E/BV intervals in table 3.


to the individual-observation-level regressions reported in tables 2 and 3 using portfolio medians for the dependent and independent variables. The estimated coefficients show the same patterns and approximate magnitudes as those reported in tables 2 and 3. The R2s are much higher for the three-interval portfolio-level regressions than for the single-interval portfolio-level regressions. Also, the R2s are much higher (typically from 85 percent to 95 percent for the three-interval portfolio-level regressions) than for the corresponding individual-observation-level regressions, which is typical in comparisons of portfolio-level regressions versus individual-observation- level regressions.21 In summary, this analysis provides no evidence that measurement error is an important determinant of the magnitude of the coefficient estimates reported in tables 2 and 3.22

Capitalization Factor Sensitivity Analysis The analysis assumes that the earnings capitalization factor c is constant. However, in a large

cross-section of firms, earnings capitalization factors are likely to vary for a number of reasons including risk and size. Thus, it is important to obtain some assurance that the observed convexity results are not attributable to an interaction with factors related to cross-sectional differences in earnings capitalization factors. We partition the sample into more homogeneous leverage and size (measured by total assets) subsamples within which earnings capitalization factors are expected to be relatively constant. If the results observed in the subsamples are similar to those observed in the overall sample, we obtain some assurance that the overall results are not attributable to variation in earnings capitalization factors related to either leverage or size.

We rank the firms on each partitioning variable within each year, split the sample in half, and rerun the regressions from table 2 and table 3 within each subsample. For the two leverage subsamples, results are qualitatively consistent between the two subsamples and consistent with tables 2 and 3. The predicted convexity is also observed in both size subsamples. However, the single-interval regressions for the large firms demonstrate appreciably larger slope coefficients than those in table 2 (the magnitudes range from approximately 5 to 8) and R2 (20-30 percent) while small firms have lower slope coefficients (magnitudes range from -1 to 2) and lower R2 (0- 4 percent). For the three-interval large-firm regressions, evidence of the predicted convexity remains highly significant, accompanied by an R2 on the order of 40-50 percent. The small-firm three-interval regressions show an even more pronounced improvement in fit over the single- interval regressions, demonstrated by a dramatic increase in R2 (in the range of 10-20 percent) and clear evidence of convexity in the slope coefficients.

Summary of Sensitivity Analysis Overall, the convexity in the relation between market value and earnings, controlling for net

asset value, does not seem to be explained by measurement error and is robust to alternative variable specifications. The evidence of convexity remains significant for more homogeneous subsamples of large and small firms and for subsamples with high and low leverage.

2'We also repeated these procedures for portfolios formed on two-year lagged and three-year lagged E/BV. The convex shape was still clearly discernible with these longer lags, although it predictably became less clear as the time lag increases.

22We also used the average of the actual earnings realizations for the following three years as a proxy for ex ante earnings expectations. This proxy for expected earnings has been used in other accounting research (e.g., Bernard 1994), but there are reasons to believe that the proxy is less appropriate here. Our model requires a measure of expected future earnings assuming that the firm continues to apply its current business technology to its resources, but actual future earnings realizations may reflect either the results from continuation of current activities or the results of adaptation. Thus, the proxy may be particularly inappropriate for low-earnings firms where we expect future earnings realizations of low- earnings firms to reflect significant adaptations. Results using this alternative proxy for expected future earnings were qualitatively the same as the primary results based on current earnings.

Burgstahler and Dichev-Earnings, Adaptation and Eciuity Value 205

Market Value as a Function of Levels of Adaptation Value for Fixed Expected Earnings Following Proposition 2, we next examine the relation between market value and adaptation

value for fixed recursion value. Under the assumption that the earnings capitalization factor is constant across firms, we test the significance of the slope coefficients relating market value scaled by earnings and book value scaled by earnings. This approach provides a different perspective on the valuation relation in equation (5) though the results are clearly not statistically independent of the evidence presented in the previous section. Thus, the results here are to be interpreted as alternative specifications of tests of the theory, rather than as independent tests.

Market value and book value in our sample are always positive and, therefore, all observations of earnings-scaled market value versus earnings-scaled book value fall in the first quadrant (for variables scaled by positive earnings) or third quadrant (for variables scaled by negative

FIGURE 3 Median Market Value/Book Value versus Median Earnings/Book Value for

Portfolios Formed on Lagged Earnings/Book Value 6 - 1977-1994

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earnings). However, zero is a discontinuity point for the relation. Holding earnings constant and moving from left to right, scaled book value increases in the first quadrant and decreases in the third quadrant. Similarly, holding earnings constant and moving up, market value increases in the first quadrant and decreases in the third quadrant. Thus, it is clearly inappropriate to examine the predicted relation over the combined domain of positive and negative earnings. As explained below, the relation for the set of observations scaled by positive earnings has a straightforward interpretation. In contrast, the interpretation of the relation for negative earnings is complicated by the anomalous relation between equity value and negative earnings documented in table 3 and other research. Therefore, we restrict the analysis to observations with positive earnings scalars.

For positive earnings scalars, a positive slope of the relation between scaled market value and scaled book value indicates the value relevance of book value. Within the positive region, small values of scaled book value represent a high level of earnings relative to the book value. These are expected to be firms where the largest component of equity value is recursion value-the adaptation value should be relatively unimportant. Thus, the relation between scaled market value and scaled book value should be relatively weak for small values of scaled book value. As the level of scaled adaptation value increases, the slope of the relation should gradually increase toward some constant. If book value were a perfect proxy for adaptation value, this constant would be 1.

We first estimate a constrained version of equation (7) with a single intercept and slope, and then a piece-wise linear version where the observations are sorted on BV/E into three equal-sized groups and separate intercept and slope coefficients are estimated for each of the three groups.

Small values of earnings sometimes result in extremely large values of scaled market value and scaled book value. Therefore, the upper five percent of the observations for both variables are trimmed.23 Descriptive statistics for the scaled and trimmed market value and book value variables are presented in panel B of table 1.

Table 4 contains the coefficient estimates for the simple linear form of equation (7). The slope coefficients in the constrained regression are significantly positive for all yearly regressions and for the pooled regression, indicating the value relevance of book value for a given amount of earnings. As one might expect when earnings and book value are complementary determinants of market value, changes in the incremental explanatory power of earnings in table 2 are mirrored by changes in the incremental explanatory power of book value in table 4. Specifically, the pattern of adjusted R2 over time in table 4 is the opposite of what was observed in table 2.

Table 5 contains the results for the piece-wise linear form of equation (7). The coefficient estimates from the pooled and individual year regressions are consistent with the convex form predicted by the theory. The estimated coefficients on earnings (the intercepts in (7)) and on book value (the slopes in (7)), are similar to those reported in table 3. For the pooled regression, the intercept is 18.97 for the low BV/E region, 6.39 for the middle region, and 6.31 for the high region. The slope coefficient for the region of low BV/E values is -1.01, but then increases to 1.18 for the middle region of BV/E, and .85 for the high region. Thus, the estimated coefficients on earnings and book value are generally consistent with their theoretical values (and consistent with the estimates from the specification reported in table 3). The results for the individual yearly regressions show substantial variability. Nonetheless, the individual year results provide significant evidence of the predicted convexity in comparing the coefficients on book value for the low

23The five percent trimming results in an MV/E variable ranging from 0 to 115 and a BV/E variable ranging from 0 to 71. Deleting the top and the bottom three percent (one percent) of the observations produced a range from 0 to 215 (627) for MV/E and 0 to 122 (357) for BV/E.


TABLE 4 Coefficient Estimates for Market Value as a Function of Book Value

Controlling for the Level of Earnings (For Observations with Positive Earnings Only)

MVt7Et =b + b2BVt-/Et + E

Year bi tj b2 t2 Adj. R2

76 8.03 28.39 .10 3.16 .01 77 7.43 29.14 .12 3.79 .02 78 7.70 31.44 .08 2.53 .01 79 7.91 25.01 .12 2.68 .01 80 11.19 21.95 .13 1.84 .00 81 10.06 25.15 .18 4.02 .01 82 10.97 22.39 .47 8.58 .10 83 14.37 23.31 .51 7.21 .06 84 9.87 28.49 .55 12.30 .15 85 12.00 22.68 .75 11.86 .18 86 13.16 26.32 .81 14.79 .22 87 11.50 29.69 .66 14.76 .22 88 9.41 31.37 .73 23.08 .30 89 11.06 30.67 .66 14.76 .21 90 11.15 33.62 .43 13.60 .14 91 16.48 35.14 .49 13.10 .11 92 16.15 41.70 .52 16.44 .16 93 17.27 39.62 .64 14.31 .12 94 14.04 40.53 .59 15.23 .13

Average 11.57 29.82 .45 10.42 .11

Pooled 11.29 103.06 .62 50.62 .15

All t-statistics are based on the heteroskedasticity-consistent covariance matrix (White 1980). Variable Definitions: see table 1.

and high groups (where the tests are expected to be most powerful) in all but one year. The less powerful comparisons predictably show less consistent results. Comparisons of the slope coefficients of the low and middle groups provide significant evidence of convexity in eight of 19 years, and comparisons of the slope coefficients of the middle and high groups show significant evidence of convexity in 12 of 19 years.

Note that the empirical results in tables 2-5 are not simply a reflection of results from previous papers which suggest that extreme earnings may represent transitory or value-irrelevant items (e.g., Ali and Zarowin (1992)). First, extreme observations are unlikely to account for the convexity results because the lowest and highest three percent of earnings have been eliminated for the tests in tables 2 and 3. Second, a large transitory component in extreme values of earnings would suggest a weak relation with equity value for both low and high earnings. However, here we observe a weak relation for low earnings but a strong relation for high earnings. Similarly, the empirical results in tables 2-5 are not simply a reflection of results from previous papers which


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suggest that negative earnings are less value-relevant than positive earnings because there is widespread evidence of convexity throughout the entire range of earnings. In table 3, there is a clear difference in the coefficients for the medium and high earnings intervals, which include only positive earnings. Additionally, in the auxiliary analysis where the range of earnings was segmented into five intervals (one negative and four positive), there is significant evidence of convexity for the positive earnings intervals. Similar evidence of convexity is found in tables 4 and 5, where all of the negative earnings observations have been eliminated. Finally, the visual evidence in both figures 2 and 3 shows clear evidence of convexity throughout the entire range of earnings.

Results for Changes Over Time Equation (1) states equity value as a function of both expected earnings and adaptation value

at a point in time and equation (5) is the corresponding operational form. An obvious alternative to the levels specifications used in the previous section is to difference equation (5) over time to derive a relation between changes in market value, changes in earnings and changes in book value.

The form of the relation between changes in equity value, recursion value and adaptation value is more complicated than the levels relation. A change in net resources affects both the adaptation value and recursion value components of equity value. For a firm with a low level of expected earnings, since a change in resources will generally translate directly into a change in adaptation value, the change in equity value will be approximately equal to the change in resources. On the other hand, a change in resources will also change recursion value through a change in the level of expected earnings, where the change in recursion value is the capitalized marginal expected earnings provided by additional resources. For a firm with a high level of expected earnings, the change in recursion value will again be approximately equal to the change in resources, as long as the firm has previously invested in resources up to the point where the marginal return is equal to the marginal cost of capital. Consequently, across all levels of expected earnings, we expect the change in equity value to be approximately equal to the change in resources.

In contrast, a change in expected earnings affects recursion value by definition but does not necessarily affect adaptation value. Thus, the effect of a change in expected earnings on equity value will vary with the level of expected earnings. For a firm with a low level of expected earnings where equity value is primarily determined by adaptation value, a change in expected earnings will have a small effect on equity value. For a firm with a high level of expected earnings, the effect on equity value should be approximately the earnings capitalization factor times the change in expected earnings. Thus, the relation between changes in equity value and changes in expected earnings should reflect the convexity observed in the previous section related to the level of expected earnings.

We adopt a specification for the change in equity value which allows the effect of either the change in book value or the change in expected earnings to vary with the level of expected earnings (though variation in the relation is predicted only for the change in expected earnings). This form is derived by simply differencing equation (5) over time and then scaling by book value:

MVt - MVt_1 = yj (BVt- BVt_1) + y2 (Et - Et_,) (8)

Dividing through by BVt_1 yields

MVt - MVt=1 BVt - BVt-j + Y Et - Et-l (9)

B1 1l BVtj BVt


To allow the relation between changes in market value, changes in book value and changes in earnings to vary with the level of E/BV, we fit a piece-wise linear version of equation (9):

AMV = bo +bl AE +b2 DM AE +b3 DH AE +b4ABV +b5 DM ABV +b6 DH ABV + ? (10)

where

AMV= MVt - MVt- BVt-

BVtV

AE Et - Et_1

BVt_1

The piece-wise form in equation (10) allows for separate coefficients on the change in earnings and the change in book value for three groups formed by sorting the observations on Et-1/BVt_1. Observations with a higher ratio of earnings to book value are predicted to be firms for which recursion value is a larger component of value, i.e., the magnitude of the earnings to book ratio should be directly related to the magnitude of the coefficient relating changes in market value to changes in earnings.

Table 6 contains the results of estimating equation (10). The coefficient on the change in book value is significant. Estimates of b4 are significantly different from zero for all but the first year and are generally close to one. The estimates for the middle and high groups, b4 + b5 and b4 + b69

are also close to one but slightly smaller on average. In most cases, differences among the groups are not significant so there is little evidence that the coefficient depends on the level of expected earnings relative to book value. Thus, the estimated coefficients on the change in book value are significant and close to our expectations.

In contrast, there is clear evidence that the coefficient on the change in earnings depends on the level of expected earnings relative to book value. The coefficient on earnings in the pooled regression increases across the three groups from .50 to 2.09 to 3.50 and the incremental estimates are statistically significant. The individual year regressions show the same overall pattern, with the average coefficient increasing across groups from .58 to 2.46 to 3.68 and the incremental estimates for individual years are statistically significant in most, but not all, cases.24

Summary of Empirical Results The observed empirical results are consistent with the predictions of the theory. The data

plotted in figures 2 and 3 and the statistical tests reported in tables 2 and 3 relating market value to earnings conditional on the level of book value show clear evidence of the valuation function

24Note that the estimated coefficients on earnings in table 6 are substantially lower than the corresponding estimates in table 3. Because the change in expected earnings is measured as the difference of two estimates of expected earnings, the variance of the measurement error component of the estimated change is approximately twice the variance of the measurement error components of the estimated levels. Also, because earnings is strongly auto-correlated, the true (non- measurement error) variation in changes in expected earnings is substantially smaller than the corresponding variation in levels of expected earnings. Thus, while the total variation in the change in earnings is substantially smaller than the total variation in the level of earnings (compare the standard deviations for earnings changes and levels reported in table 1), the proportion of variation due to measurement error is much larger for the measured change in expected earnings (the independent variable in table 6) than for the measured level of expected earnings (the independent variable in table 3). Since coefficient bias induced by measurement error is generally proportional to the ratio of the measurement error variance to total variance, differences in relative measurement error could account for the substantially smaller earnings coefficient estimates in table 6 versus table 3.


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shape predicted by the theory. Tests reported in tables 4 and 5 relating market value to book value conditional on the level of earnings also strongly support the predicted convexity with pooled regressions, though evidence for the convex form is less clear-cut with the yearly regressions. Throughout tables 2-5, the magnitudes of the estimated coefficients are consistent with the prediction that as the relative value of E/BV increases, the coefficient on BV should decrease from a value near one and the coefficient on earnings should increase toward an earnings capitalization factor. The results of tests reported in table 6 which evaluate the relation between changes in value and changes in expected earnings are also generally consistent with the predictions of the theory.

V. CONCLUSION This paper develops an option-style model of equity value that incorporates the capitalized

value of the firm' s expected earnings (under the assumption that the firm continues its current way of employing resources) but also explicitly recognizes the value of the firm's adaptation option (i.e., the value of the option to convert the firm's resources to alternative, more productive uses). The primary prediction of the model is that the value of equity is a convex function of both expected earnings and book value. The empirical evidence strongly supports the prediction of convexity-the coefficient on earnings increases with the ratio of earnings to book value and the coefficient on book value decreases with the ratio of earnings to book value. Thus, the results imply that equity value is a function of both expected earnings and book value, in contrast to models which incorporate one or the other (e.g., see Solomons 1995; Barth and Landsman 1995), and that the form of the function is convex, in contrast to models which assume the two elements of value are simply additive (e.g., Ohlson 1995).

This evidence has implications for a variety of users. For example, it suggests that earnings response coefficients increase with the level of earnings to book value. However, there is a need for further theoretical development in this area, particularly with regard to how to incorporate changes in book value into earnings response coefficient models. An implication for financial analysts is that valuation methods which focus on price-to-book multiples are more appropriate for firms with low return on equity while a focus on price-to-earnings multiples is more appropriate for firms with high return on equity. Finally, valuation research, that has generally focused on either earnings-based valuation (e.g., Barth et al. 1992) or balance sheet-based valuation (e.g., Barth 1991), should incorporate both earnings and balance sheet measures of value, using a specification which allows the coefficients to vary with the ratio of earnings to book value.


APPENDIX The convexity of the relation between market value (MV) and earnings (E) for a given level of

adaptation value (AV) is established by examining the first derivative of market value in equation (2): ,AV/c 0

MV(E I AV) = J AV f(E I AV)dE +J cE f(E I AV)dE.

The derivative is, with respect to the mean of earnings conditional on adaptation value, denoted here by A :

dMV =AV=

J f'(E I AV)dE + cf E f'(E I AV)dE. (Al)

The derivative of market value with respect to g depends on the level of A. As Gu->--oo, both terms in equation (Al) approach zero, and therefore the derivative dMV/dg approaches zero. Intuitively, as g becomes increasingly negative, most of the probability mass of f (E I AV) in equation (2) is below AV/c and MV approaches the constant AV. Therefore, as Ha-> -Ao, dMV/dg --> 0. Similarly, as g -> +oo, the first term in equation (A 1) approaches zero and the second term approaches c. Intuitively, as g becomes increasingly positive, most of the probability mass of f (E I AV) in equation (2) is above AV/c and MV approaches c A. Therefore, as Ha-> oo, dMV/dg -> c. Thus, the first derivative is bounded between 0 and c as g varies between -oo and +oo.

Further, dMV/dg is strictly increasing in A. To see this, note that increases in g represent a shift in probability mass from the first term in (2), where the slope of the value function is zero, to the second term in (2), where the slope of the value function is c. This implies that the derivative increases monotonically from 0 to c, implying that market value is an increasing, convex function of A.


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