01# earning, adaptation and equity valui - burgstahler, dichev

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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 REVIEWVol. 72, No. 2April 1997pp. 187-215

Earnings Adaptation

n d q u i t y a l u e

David C. BurgstahlerUniversity of Washington

Ilia D. DichevUniversity of Michigan

ABSTRACT: This paper develops and tests an option-style aluation model, whosemain prediction s that equity value is a convex function of both earnings and bookvalue, where the function depends on the relative alues of earnings and book value.Earnings provides a measure of how the firm's resources are currently sed. Bookvalue provides a measure of the value of the firm's resources, independent of howthe resources are currently sed. When he ratio arnings/book alue is high, he firmis likely o continue its current way of using resources, and earnings is the moreimportant eterminant f equity value. When earnings/book value is low, the firm smore likely o exercise the option o adapt ts resources to a superior alternative use,and book value becomes the more important eterminant f equity value. Evidencefrom a variety of empirical pecifications s consistent with he convexity prediction.

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

Data Availability: Data used in this study are collected or purchased from publicsources identified n the study.

I. INTRODUCTIONvidence rom a variety f studies hows hat quity value s related o accounting arnings(e.g., Ball and Brown 1968; Barth et al. 1992; Collins and Kothari 1989). These studiesusually 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 Fundat 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, RossWatts, 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 TerryShevlin are gratefully acknowledged.Submitted February 1995.

Accepted October 1996.

187


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188 The Accounting Review, April 1997

characterization of expected future earnings and dividends. At the same time, a number of studiesshow that equity value is related o balance sheet measures of assets and iabilities (e.g., Landsman1986; Barth 1991; Shevlin 1991). These studies use the book values of the firm's assets andliabilities to draw inferences about the value of the firm, imbedding an assumption that measuresof 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 bookvalue and measures of earnings are redundant alternatives or valuation (see, for example, Beaverand Demski 1979; Barth and Landsman 1995). However, in more realistic settings with marketimperfections, accounting systems can provide information about book value and earnings ascomplementary, rather than redundant, components of equity value. Book values from thebalance sheet provide information about the net value of the firm' s resources. This informationis based on (primarily historical) market prices and s therefore argely independent of the successwith which the firm currently employs its resources.1 In contrast, earnings from the incomestatement provide a measure of value which reflects the current irm-specific results of employing

firm resources. Accordingly, earnings are a relatively more important determinant of value whenthe firm' s current activities are successful (earnings are high relative to book value) and are to becontinued, while book value is more important when the firm's resources are likely to be adaptedto some superior alternative use. However, in general, the value of the firm is a function of bothearnings and book value because the firm has the ongoing option to either continue its presentactivities or adapt its resources to alternative uses.

This paper develops a model of firm value which explicitly recognizes that the potential toadapt firm resources to an alternative use represents an option available to the firm and the ex antevalue of the option should be reflected in market value.2 The firm is viewed as a combination ofa 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 twocomplementary elements of value are recursion value and adaptation alue. Recursion valueis the value which results from discounting the stream of future earnings under the assumptionthat the firm continues to apply its current business technology to its resources. Adaptation valueis the value of the firm's resources independent of the firm's current business technology; thisvalue exists whenever resources can be adapted to alternative uses. The alternative uses mayinvolve external adaptations (where the resources are sold to another entity) or internal adapta-tions (where the firm retains the resources and adapts them to an alternative use).3 Because thefirm has the option to adapt firm resources to uses other than their current use, both adaptationvalue and recursion value are determinants of equity value.

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

2 The potential ex ante importance of the adaptation option for equity valuation s supported by a broad iterature n financeand 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 capitalinvestments) 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 ofbusiness technology is similar to what Barth and Landsman (1995) label managerial skill. Our concept of recursionvalue corresponds roughly to Sterling's (1968) stationary tate for a firm, in which the firm will continue in muchthe same manner as in the past. Sterling (1968, 489) contends that the stationary state is what many accountants havein 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 s informativeabout adaptation value, and adaptation value could be either resale value or replacement cost depending on what typeof adaptation (external or internal) is more likely.


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Burgstahler and Dichev-Earnings, Adaptation and Equity Value 189

We combine earnings and net assets in an option-style model of equity value, where thelikelihood of exercising the adaptation option depends on the relative values of earnings and bookvalue 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 convexfunction of earnings holding book value constant and, at the same time, equity value is a convexfunction 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 predictedby the model. Results on the relation between changes in equity value, expected earnings and netbook value also provide evidence of the predicted convexity.

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

II. MODELIn 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 currentbusiness technology to its resources) and adaptation value (the value of the firm's resourcesadapted to an alternative use). The relative weights on the two components of value reflect theprobability 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 recursionvalue 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 s described by the multivariate normaldensity parameterized by a vector of means and a variance-covariance matrix, f(E, AV) = f( {RE,9AV } GE'EAV'GEAVEI}).ll available information relevant to the evaluation of expected futureearnings and adaptation value is assumed to be captured by the parameters of the multivariatedistribution. The covariance between E and AV will generally be positive, i.e., firms with higherlevels of earnings typically employ more resources (AV), and firms with more resources typicallygenerate higher levels of earnings.

The recursion value component of equity value is the discounted present value of expected future

earnings,which is assumed o be the

productof an earnings capitalization actor c and expected future

earnings E. The capitalization actorc mplicitly reflects risk-adjustment, he schedule of future nterestrates, and other features which are sometimes explicitly included n valuation models.

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

MV(E, AV) = Efmax(cE, AV] = 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 isrestricted to a single point in time, it follows that the value derived is a conservative estimate of the value which wouldresult 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 wealthmaximization is sufficient to characterize their preferences.


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

Market Value as a Function of Recursion Value for Fixed Adaptation ValueIn 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 Eby f(E IAV), and expanding equation (1) for a fixed AV, the market value of equity is given by:5

MV(E IAV) = J AVf(E IAV)dE + J cEf(E IAV)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 tooperate the company using the current technology, while for E < AV/c, they prefer to adaptresources 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 thesum of the adaptation value plus the value of an option on the recursion value. Specifically, theintegral term in equation (3) is effectively a call option, whose value is increasing in theconditional mean of expected earnings.

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

Proposition 1: Market value is an increasing, convex function of expected earnings, for agiven 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, he revision in market value dueto 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 fixedadaptation value. The horizontal line in the figure represents a fixed adaptation value and theupward sloping line represents recursion value which is linear and increasing in the level ofexpected earnings. The two lines intersect at the indifference point E = AV/c. The market valueof equity curve approaches the fixed adaptation value on the left and approaches capitalizedexpected earnings on the right. For low expected earnings, most of market value is attributableto adaptation value and a revision in

expected earnings matters little for valuation. For highexpected earnings, most of the market value is attributable o recursion value and a revision ofexpected earnings has a large effect on market value. For intermediate levels of earnings, themarket value of equity depends on a more balanced combination of adaptation and recursionvalue. Thus, holding adaptation value constant, the market value revision in response to a changein 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 tothe present. However, the discount factor is constant for a constant interest rate and constant time to exercise, andtherefore, to simplify the exposition, we omit the explicit discount factor. In the appendix, the discount factor is againomitted because the constant discount factor has no effect on the proofs.


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FIGURE 1Market 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 ncreases over the range of expected earnings oward ts limiting value, the earningscapitalization actor.6

6 Though we do not explore the issue in this paper, there could also be an eventual decrease in the slope of the relationat the uppermost evels of expected earnings because high levels of expected earnings for a given resource base are likelyto attract competition and consequently may not be sustainable. Miller (1994, 33-34) describes this phenomenon asfollows: Above normal profits always carry with themthe seeds of theirown decay. They attractcompetitors, both fromwithin 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) forevidence 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 andleads to the prediction of an S-shaped relation between unexpected returns and unexpected earnings. Thus, the Freemanand Tse (1992) non-linearity is different from our predicted convex relation in motivation, type of non-linearity, andpredicted effects.


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Market Value as a Function of Adaptation Value for Fixed Recursion ValueThe 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 ofAV, for a fixed E, yields:

MV(AVIE) = cE + J(AV - cD)f(AV

IE)dAV. (4)

In this form, the market value of equity conditional on a fixed level of expected earnings can beinterpreted as the sum of capitalized earnings plus the value of an option on downsideprotection, i.e., the value of the firm's option to elect the adaptation value if it proves superiorto the value of the firm, based on the earnings stream generated with the current businesstechnology. Equation (4) provides the basis for a second proposition (proof is analogous to proofof 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 givenexpected earnings.

Intuitively, Proposition 2 holds because, for a fixed level of expected earnings, an increasein adaptation value provides additional downside protection. Further, the higher the level ofadaptation value relative to the fixed level of expected earnings, the larger the revision in marketvalue associated with a revision in adaptation value. If the adaptation value is large relative tocapitalized expected earnings, the current business technology of the firm is inferior to alterna-tives and adaptation value becomes the primary determinant of market value; recursion value isonly 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 firmvalue is a function of expected future dividends, which in turn are assumed to be functions ofexpected future earnings and current earnings. Much of this research focuses on empiricalspecifications 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 anoption-like framework. Berger et al. (1996) posit that the value of the firm reflects the value ofthe option to abandon the firm for its liquidation value. For a sample of firms selected becausethe abandonment option is especially relevant, they find that equity value increases in liquidationvalue after controlling for expected going-concern cash flows. However, they do not test forconvexity as predicted here. Hayn (1995) considers situations where earnings are considered tobe sufficiently low as to make liquidation of the firm preferable to continued operation. Shefocuses on the effect of inclusion of negative earnings (losses) on measures of the informationcontent of earnings and presents evidence that stock price movements are much more stronglylinked to positive earnings than negative earnings. Subramanyam and Wild (1996) investigatehow the informativeness of earnings is affected by the validity of the going concern assumptionand find that the informativeness of earnings is negatively related to the risk of failure (measuredusing the Altman (1968) Z-score).

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


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

Our model also differs from other accounting-based valuation models which specify equityvalue 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 specificationsderived from this model (e.g., Easton and Harris 1991; Sougiannis 1994) typically also assumea 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 andbook value where the roles of earnings and book value depend on their relative values. Whenearnings are low relative to book value, the firm is likely to adapt assets to alternative, moreproductive uses; for these firms, book values are especially relevant. When earnings are highrelative to book value, the firm is likely to continue the current, highly productive uses; for thesefirms, earnings are especially relevant in valuation.

IV. EMPIRICAL TESTSOverview and Choice of Proxies

This section defines empirical proxies for the theoretical variables and specifies the form oftests of the predicted relations. The main focus of the tests is to determine f the form of the relationbetween equity value, recursion value and adaptation value displays the convexity predicted bythe 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 n the relative values of recursion value and adaptation value.In the language of our paper, it is variation n business technology (i.e., variation in the way thatfirms use resources to produce earnings) that leads to variation in the relative values of recursion

value and adaptation value. Because the cross-sectional variation n business technology is likelyto be substantially larger than the variation n business technology of a particular irm over time,our primary ests are cross-sectional. Additionally, we examine a second specification in changes.

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

Recall that recursion value is defined as the value of capitalized future earnings, assumingthat the firm continues to apply its current business technology. We use current earnings as a proxyfor recursion value for two reasons, the first based on a priori reasoning and the second based onprevious empirical research. First, current earnings reflect, by definition, the results of applyingthe firm's current technology. Although prior research indicates that analyst forecasts or otherproxies can provide marginal mprovements over current earnings as a proxy for observed futureearnings in general (including future earnings which are the result of various adaptations), currentearnings is arguably a better proxy for the future earnings expected if the irm continues to applythe current business technology. Second, most univariate time-series research shows thatearnings, on average, follow a random walk. In fact, Bernard (1994) finds that it is difficult toimprove on current earnings as a predictor of future earnings. For these two reasons, currentearnings is the main proxy for recursion value used in the empirical tests. Later, in the resultssection, 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 theorder of approximately one percent or less. Thus, it appears hat few firms would have more than a trivial ex ante valueof the real liquidation option.


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

We use book value as our proxy for adaptation value because adaptation value is closelyrelated to book value by design. The concept of adaptation value is defined as the value of thefirm's net assets, independent of the firm's current business technology, while book value is acost-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 mightbe used (e.g., proxies for replacement cost and net realizable value) are not readily available andmust be estimated.8 Further, there is little reason to expect substantively different results usingalternative proxies for adaptation value because these proxies typically begin with book value andincorporate relatively minor adjustments. n fact, research n other settings documents that resultsbased 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 EAll 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 proxyfor adaptation value and E is a proxy for recursion value:

MVt= lBVt+y2Et+e (5)

where e is a normally distributed error erm with mean zero and unspecified variance (see footnote

12 for further discussion.) The empirical version of equation (1) approximates market value ofequity for a given firm at time t as a linear combination of book value and expected earnings attime t. As E becomes extremely low relative to BV, BV becomes the sole determinant of MV andy2 approaches zero while y, approaches unity.'0 On the other hand, as E becomes extremely highrelative to BV, E becomes the sole determinant of MV and y1approaches zero while y2 approachesthe earnings capitalization factor. However, the model in section II implies that market value ingeneral will be a function of both BV and E. Thus, equation (5) must be operationalized in a formwhich provides for the convexity predicted by the theory so that the coefficients on BV and E areallowed 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 bewilling 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 foradaptation if existing activities are to be contracted, whereas replacement cost measures capacity for adaptation ifexisting activities are to be expanded.

9 For example, results in Beaver and Landsman (1983) suggest that measures of current cost-adjusted book valuesperform 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 thereplacement 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 recisely, the coefficient on BV should approach he factor relating BV and adaptation value. If BV were a perfectproxy 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 ofadaptation value, the coefficient on BV would be expected to exceed one.


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Tests of Proposition 1 require that we hold book value constant while testing for theincremental effects of earnings. Tests of Proposition 2 require that we hold earnings constantwhile testing for the incremental effects of book value. Thus, in tests of Proposition 1, we holdbook value constant by dividing market value and earnings by book value. In this form, the resultscan be interpreted as a test of the incremental value relevance of earnings by examining marketvalue per dollar of net assets as a function of earnings per dollar of net assets. Similarly, in testsof 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 examiningmarket 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 attime t) and Et (as a proxy for expected future earnings at time t using the current businesstechnology). We, however, choose to use BVt 1 as the measure of adaptation value for year t. Bydefinition, BVt includes Et as a component. Therefore, empirical tests using BVt-1 will moreclearly separate the effects of E and BV.1I

To test for the convex form implied by Proposition 1 where market value is viewed as afunction of expected earnings for a fixed level of book value, we divide through by BVt-1 as themeasure 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 thecoefficient on BV in the original relation and the estimated slope is an estimate of the coefficienton E. The proposition implies that the values of the coefficients depend on the level of E/BV. Atthe lowest levels of E/BV, y, should be approximately one and y2 should be close to zero whileat higher levels of E/BV, y1 should move toward zero and y2 should move toward the earningscapitalization factor. Results of these tests are reported n tables 2 and 3.

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

MVt/Et = y1 BVt- JEt + Y2EtEt + E* (7)= y2 + Y1 BVt-I/E t+ ;**

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

I'In any event, the operational choice of BVt 1versus BVt has little influence on the results. The results reported n tables2-5 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 fromthe alternative assumptions that the standard deviations of the error erms in equation (5) are either proportional o book

value, which leads to equation (6), or proportional o earnings, which leads to equation (7).


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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 observationsin each of the three ranges. Within each range, the estimated coefficients are restricted to beconstant but the coefficients are allowed to vary across the three ranges. The number of rangesin 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 highestrange but would also reduce the number of observations available to estimate the coefficients foreach range. The choice of three ranges seems to be a reasonable compromise.

Results of Cross-Sectional Tests in LevelsThe 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 tofirms 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 adjustedfor Special Items net of tax (Compustat item 237 less Compustat item 17 adjusted by the statutorytax rate). 13 Market value is number of shares outstanding multiplied by the market value per shareat 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 ValuePlots 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 topthree 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 timefrom about 1,200 for 1976 to more than 4,100 for 1994. The early years had comparatively fewfirms with negative earnings but the proportion increased significantly beginning in about 1981and grew to approximately 30 percent for years after 1986. Average earnings as a percentage ofbook value declined fairly steadily over the sample period. The average ratio of market to bookvalue hovered around a value slightly above one during the first five years of the sample periodand then fluctuated more widely around a range of two to three for the remainder of the sampleyears.

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 bookvalue, for all of the individual observations in the study. The plot is consistent with the predictedshape 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 ifthe firm continues its current mode of operations. However, the results are substantially the same for unadjustedearnings.

'4The three percent trimming resulted in an overall deletion of six percent of the observations in the original sample anda 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 bottomone percent of the observations produced a sample ranging from zero to 105.61 for MV/BV and -3.58 to 2.01 forE/BV. These values and plots of the data convinced us that one percent trimming would leave extreme observations withundue influence in the regression analysis. (Using one percent trimming, the estimated coefficients for the modelsdiscussed below were generally consistent with, but more significant than, the reported results for three percenttrimming.) The three percent trimming rule used here eliminates fewer observations than the trimming rules used inrelated previous studies. For example, Collins and Kothari (1989) trimmed all observations with market value to bookvalue in excess of five.


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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 BVTotal Earnings E_/BV, MV7BV1JYear N N Mean Std. Dev. Median Mean Std. Dev. Median

76 1,208 62 .15 .09 .15 1.39 1.01 1.0577 1,249 61 .15 .09 .15 1.34 .90 1.0878 1,335 32 .17 .10 .17 1.53 1.17 1.1679 1,423 63 .18 .11 .18 1.77 1.59 1.2580 1,512 108 .17 .12 .16 2.41 2.78 1.4381 1,694 186 .14 .14 .15 2.17 2.45 1.3682 1,783 349 .09 .15 .11 2.12 2.09 1.3983 1,907 364 .11 .18 .12 2.99 3.16 1.9184 2,124 436 .09 .19 .12 2.24 2.04 1.6385 2,219 558 .07 .19 .10 2.47 2.29 1.7386 2,450 674 .05 .22 .09 2.97 3.30 1.9387 2,758 750 .06 .22 .09 2.60 2.72 1.7588 2,850 800 .06 .22 .10 2.23 2.01 1.6789 2,941 875 .05 .23 .09 2.44 2.48 1.6590 3,049 942 .03 .24 .08 2.08 2.30 1.3091 3,294 1,045 .02 .25 .07 3.01 4.08 1.6392 3,681 1,125 .03 .27 .07 3.19 3.96 1.8693 4,127 1,231 .04 .28 .08 3.73 4.41 2.3094 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 EarningsObservations

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 7.7377 1,155 7.05 4.04 6.01 8.25 3.61 7.4178 1,260 6.61 3.96 5.62 8.21 3.98 7.3079 1,314 6.18 3.87 5.20 8.65 4.96 7.3280 1,354 6.74 4.51 5.39 12.06 8.63 9.4481 1,455 7.27 5.05 5.70 11.38 7.72 9.0982 1,395 9.10 7.48 6.85 15.29 11.19 11.9883 1,501 7.83 6.09 6.01 18.35 13.01 14.3684 1,638 7.98 6.21 6.11 14.23 8.77 11.8185 1,627 9.27 7.90 6.80 18.91 13.75 14.9886 1,740 9.62 8.71 6.78 20.93 14.87 16.7587 1,964 9.28 8.81 6.54 17.62 12.25 14.1788 2,005 8.85 8.05 6.13 15.84 10.59 13.1389 2,019 8.99 7.83 6.56 17.03 11.35 13.9090 2,070 9.94 9.06 7.09 15.44 10.36 12.5791 2,203 11.24 11.30 7.36 21.95 16.05 17.7692 2,508 10.78 11.28 7.15 21.73 14.56 17.9793 2,826 9.17 8.46 6.62 23.18 15.72 18.5094 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)


13/30


TABLE 1 (Continued)

Panel C: Scaled Changes in Market Value (AMV), Scaled Changes in Earnings (AE), and Scaled Changesin 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 .1177 1,166 .12 .40 .06 .03 .06 .02 .12 .12 .1178 1,206 .27 .49 .13 .04 .07 .03 .15 .13 .1379 1,266 .37 .70 .16 .04 .08 .03 .16 .16 .1480 1,361 .69 1.20 .27 .02 .08 .02 .18 .24 .1381 1,441 .03 1.01 .01 .02 .09 .02 .17 .28 .1182 1,607 .34 1.02 .14 -.02 .10 -.01 .09 .20 .0783 1,629 .71 1.27 .38 .03 .11 .03 .19 .36 .1084 1,822 -.24 .90 -.11 .03 .11 .03 .10 .24 .0985 1,955 .42 1.03 .21 .00 .12 .00 .10 .27 .0886 2,057 .36 1.14 .14 .01 .15 .01 .12 .34 .0887 2,280 -.05 1.09 -.08 .04 .17 .02 .13

.34 .0988 2,450 .10 .87 .06 .03 .14 .02 .08 .26 .0889 2,501 .31 1.06 .07 .02 .15 .01 .08 .29 .0790 2,578 -.20 1.00 -.22 .00 .15 .01 .07 .30 .0591 2,704 .86 2.01 .23 .01 .17 .00 .12 .40 .0592 3,014 .32 1.32 .08 .02 .17 .02 .12 .44 .0593 3,371 .60 1.62 .21 .02 .17 .02 .15 .46 .0794 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 t the end of year (Compustat tem 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 tems available or common adjusted or special tems net of tax (item 237 less item 17adjusted by the statutory ax rate) for year t.

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

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

level of earnings increases, the slope of the market value function clearly increases.15 Thepredicted convexity is apparent even at the individual firm level for this comprehensive sampledrawn 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 ofscaled market value on scaled earnings specified in equation (6) above. The first regressionassumes a simple linear relation over the entire domain of E/BV while the second regressionassumes piece-wise linearity as an approximation o 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/Bookratios against the mean ROEs shows the convex shape predicted here, especially for the decile 2-9 means. The lack offit for the first decile in Bernard's (1994) data which includes negative earnings firms is consistent with our empirical

results (discussed further below).


14/30

Burgstahler and Dichev-Earmings, Adaptation and Equity Value 199

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

1976-1994

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15/30


first regression tests the overall value-relevance of earnings and provides a benchmark againstwhich to compare both the results of previous studies and the results of the second regression.

Table 2 contains estimates of the coefficients for the simple linear form relating MVt/BVt_ toEt/BVt 1 in equation (6). Results are presented for individual year regressions, followed by theaverage of the individual year regression results. Results are also presented for a pooled regressionwith all firm/years pooled together. The coefficient on earnings is highly significant for allregressions, 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 (andrange from a high of 8.60 to a low of 1.66), and thus are lower than would be expected for anearnings capitalization factor, especially in later years. The decline of the coefficient estimatesand the adjusted R2s over time suggests that either the constraint imposed by the simple linearform 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 regressionprovides a test of the predicted convex shape by allowing the slope and the intercept of the

TABLE 2Coefficient 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 .3577 .43 9.62 6.00 19.78 .3578 .26 4.11 7.33 19.59 .3779 .50 5.59 7.05 14.39 .2480 .99 6.53 8.60 10.13 .1581 1.15 11.50 7.13 10.65 .1682 1.75 31.95 4.11 9.31 .0983 2.29 27.37 6.53 10.74 .1384 2.01 36.70 2.57 7.13 .0685 2.29 41.81 2.85 6.91 .0586 2.83 40.02 2.78 5.45 .0387 2.45 44.73 2.48 6.40 .0488 2.14 47.85 1.66 5.87 .0389 2.33 52.10 2.27 7.57 .0490 2.03 45.39 1.80 6.36 .0391 2.96 41.86 2.11 3.92 .0292 3.15 44.55 1.76 3.59 .0193 3.65 51.62 2.24 4.89 .0294 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 ovariance matrix White 1980).Variable Definitions: see table 1.


16/30


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17/30


regression line to vary with the magnitude of Et/BVt . Operationally, we divide the domain of

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 middleand high group intercept (b2 and b3) and slope (b5 and b6) coefficients are estimated incrementalto 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 middleand high groups are for tests of incremental significance relative to the low group. However,coefficients reported n the tables are the total intercept and slope coefficients for the middle group(i.e., bl+b2 for the intercept and b4+b5 or the slope) and the total coefficients for the high group(i.e., bI+b3 for the intercept and b4+b6 or the slope). Thus, for example, t5 shown in the table isthe relevant t-statistic for a test of the difference between the slope coefficients for the middlegroup and the low group (i.e., a test of the hypothesis that b5=0) while t6 is the relevant t-statisticfor a test of the difference between the slope coefficients for the high group and the low group.16In addition to the t-tests of incremental significance for the middle and high groups relative to thelow 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 differencebetween 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/BVincreases 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 generallyincrease as E/BV increases-the t-statistics reveal that b5 and b6 are uniformly significantlypositive. Additionally, comparison of the estimates shows that b6 is greater than b5 in every yearbeyond the first two years and the asterisks ndicate that b6 s significantly greater than b5 n every

year beyond the first four years. The magnitudes of the estimated slope coefficients for the highearnings group are generally in the range from 10 to 20, consistent with the magnitude expectedfor an earnings capitalization factor. The piece-wise regression also fits the data much better thanthe simple linear form. The adjusted R2 ncreases from .02 to .27 for the pooled regressions. Theincreases in R2 for the individual year regressions are also large in the latter half of the sampleperiod, though the increases are less dramatic for the first half of the sample period. Also, incontrast to the simple linear form, the R2 s 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/BVfor the low earnings interval are significantly negative-the model predicts that the coefficientsshould 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 nvestigate this phenomenon, we ran a corresponding set ofregressions 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 andhigh ranges are not for a test of whether the coefficient is equal to zero. For example, in table 3, the overall interceptfor 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 incrementalcoefficient for the high range is negative (though the magnitude of the t-statistic in this case suggests that the incrementaldifference is not significant).

17Similar puzzling results associated with negative earnings observations have been reported n other research, e.g., Janand 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.


18/30


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

Sensitivity AnalysisWe 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 inour proxy for expected future earnings. The second group of tests considers the effect of theconstant capitalization factor assumption on our results.

Measurement Error Sensitivity AnalysisSince 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 theproxy 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 biasedtoward zero. Thus, the effect of measurement error on the results is a potentially importantconcern. At the same time, there are reasons to believe that measurement error cannot fullyexplain the results in table 3. First, in order to explain the large differences across intervals inestimated slope coefficients, large differences in measurement error would be required. Second,

the measurement error explanation would also predict smaller estimated coefficients in thehighest earnings interval, yet the estimated coefficients in the highest earnings interval aregenerally larger, rather than smaller.

To evaluate the effects of measurement error, we form portfolios based on lagged scaledearnings, i.e., Et1/BV t2. Transient events which induce measurement error in Et/BVt_1 as ameasure of future expected earnings are, by definition, those which do not persist into the futureso the measurement error in Et-1/BV t2 should be relatively uncorrelated with the measurementerror 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 shouldbe relatively unaffected by measurement error in the individual observations. For each year,observations are sorted on

EtJI/BVt-2to form 30

portfolios.The median values of

MVt/BVt-Iand

Et/BVtI are then computed for each portfolio.'9 A plot of the portfolio medians, which isanalogous 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 forthe 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 hat the resultsare not attributable to the effects of outliers. However, this same procedure applied to the portfolio means yieldedvirtually 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 datashows that the observations omitted at the left side of the plot, as a resultof the trimming, continue the slight trend upwardto the left which can be seen in the plot in figure 2, and which is also apparent from the negative estimated coefficientsfor the lowest E/BV intervals in table 3.


19/30


to the individual-observation-level regressions reported n tables 2 and 3 using portfolio mediansfor the dependent and independent variables. The estimated coefficients show the same patternsand approximate magnitudes as those reported n tables 2 and 3. The R2s are much higher for thethree-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-intervalportfolio-level regressions) than for the corresponding ndividual-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 s animportant determinant of the magnitude of the coefficient estimates reported n tables 2 and 3.22

Capitalization Factor Sensitivity AnalysisThe 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 reasonsincluding risk and size. Thus, it is important o obtain some assurance hat the observed convexityresults are not attributable o an interaction with factors related to cross-sectional differences inearnings capitalization factors. We partition he sample into more homogeneous leverage and size(measured by total assets) subsamples within which earnings capitalization factors are expectedto be relatively constant. If the results observed in the subsamples are similar to those observedin the overall sample, we obtain some assurance that the overall results are not attributable ovariation 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, andrerun the regressions from table 2 and table 3 within each subsample. For the two leveragesubsamples, results are qualitatively consistent between the two subsamples and consistent withtables 2 and 3. The predicted convexity is also observed in both size subsamples. However, thesingle-interval regressions for the large firms demonstrate appreciably larger slope coefficientsthan 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 convexityremains highly significant, accompanied by an R2 on the order of 40-50 percent. The small-firmthree-interval regressions show an even more pronounced improvement in fit over the single-interval regressions, demonstrated by a dramatic ncrease in R2 (in the range of 10-20 percent)and clear evidence of convexity in the slope coefficients.

Summary of Sensitivity AnalysisOverall, 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 homogeneoussubsamples 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 agged E/BV. The convexshape was still clearly discernible with these longer lags, although it predictably became less clear as the time lagincreases.

22We also used the average of the actual earnings realizations for the following three years as a proxy for ex ante earningsexpectations. This proxy for expected earnings has been used in other accounting research e.g., Bernard 1994), but thereare reasons to believe that the proxy is less appropriate here. Our model requires a measure of expected future earningsassuming that the firm continues to apply its current business technology to its resources, but actual future earningsrealizations may reflect either the results from continuation of current activities or the results of adaptation. Thus, theproxy may be particularly nappropriate or 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 werequalitatively the same as the primary results based on current earnings.


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Burgstahler and Dichev-Earnings, Adaptation and Eciuity Value 205

Market Value as a Function of Levels of Adaptation Value for Fixed Expected EarningsFollowing 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 isconstant across firms, we test the significance of the slope coefficients relating market valuescaled by earnings and book value scaled by earnings. This approach provides a differentperspective on the valuation relation in equation (5) though the results are clearly not statisticallyindependent of the evidence presented in the previous section. Thus, the results here are to beinterpreted as alternative specifications of tests of the theory, rather han as independent tests.

Market value and book value in our sample are always positive and, therefore, all observa-tions 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 3Median Market Value/Book Value versus Median Earnings/Book Value for

Portfolios Formed on Lagged Earnings/Book Value6 - 1977-1994

c 4-

1 ;

04_I ..

-0.2 1 0.0 01 0 .2 0.3

Median Earnings IBook Value


21/30


earnings). However, zero is a discontinuity point for the relation. Holding earnings constant andmoving from left to right, scaled book value increases in the first quadrant and decreases in thethird quadrant. Similarly, holding earnings constant and moving up, market value increases in thefirst quadrant and decreases in the third quadrant. Thus, it is clearly inappropriate o examine thepredicted relation over the combined domain of positive and negative earnings. As explainedbelow, the relation for the set of observations scaled by positive earnings has a straightforwardinterpretation. In contrast, the interpretation of the relation for negative earnings is complicatedby the anomalous relation between equity value and negative earnings documented in table 3 andother 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 andscaled book value indicates the value relevance of book value. Within the positive region, smallvalues of scaled book value represent a high level of earnings relative to the book value. These areexpected to be firms where the largest component of equity value is recursion value-the adaptationvalue should be relatively unimportant. Thus, the relation between scaled market value and scaledbook value should be relatively weak for small values of scaled book value. As the level of scaledadaptation value increases, the slope of the relation should gradually ncrease 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, andthen a piece-wise linear version where the observations are sorted on BV/E into three equal-sizedgroups 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 valueand scaled book value. Therefore, the upper ive percent of the observations for both variables aretrimmed.23 Descriptive statistics for the scaled and trimmed market value and book valuevariables are presented in panel B of table 1.

Table 4 contains the coefficient estimates for the simple linear form of equation (7). The slopecoefficients 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 ofearnings. As one might expect when earnings and book value are complementary determinantsof market value, changes in the incremental explanatory power of earnings in table 2 are mirroredby changes in the incremental explanatory power of book value in table 4. Specifically, the patternof 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 coefficientestimates from the pooled and individual year regressions are consistent with the convex formpredicted by the theory. The estimated coefficients on earnings (the intercepts n (7)) and on bookvalue (the slopes in (7)), are similar to those reported n table 3. For the pooled regression, theintercept s 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 forthe middle region of BV/E, and .85 for the high region. Thus, the estimated coefficients onearnings and book value are generally consistent with their theoretical values (and consistent withthe estimates from the specification reported in table 3). The results for the individual yearlyregressions show substantial variability. Nonetheless, the individual year results provide signifi-cant evidence of the predicted convexity in comparing the coefficients on book value for the low

23The ive percent trimming results in an MV/E variable ranging from 0 to 115 and a BV/E variable ranging from 0 to71. 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.


22/30


TABLE 4Coefficient 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 .0177 7.43 29.14 .12 3.79 .0278 7.70 31.44 .08 2.53 .0179 7.91 25.01 .12 2.68 .0180 11.19 21.95 .13 1.84 .0081 10.06 25.15 .18 4.02 .0182 10.97 22.39 .47 8.58 .1083 14.37 23.31 .51 7.21 .06

84 9.87 28.49 .55 12.30 .1585 12.00 22.68 .75 11.86 .1886 13.16 26.32 .81 14.79 .2287 11.50 29.69 .66 14.76 .2288 9.41 31.37 .73 23.08 .3089 11.06 30.67 .66 14.76 .2190 11.15 33.62 .43 13.60 .1491 16.48 35.14 .49 13.10 .1192 16.15 41.70 .52 16.44 .1693 17.27 39.62 .64 14.31 .1294 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 ovariance 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 lesspowerful comparisons predictably show less consistent results. Comparisons of the slope

coefficients of the low and middle groups provide significant evidence of convexity in eight of19 years, and comparisons of the slope coefficients of the middle and high groups show significantevidence of convexity in 12 of 19 years.

Note that the empirical results in tables 2-5 are not simply a reflection of results from previouspapers 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 theconvexity results because the lowest and highest three percent of earnings have been eliminatedfor the tests in tables 2 and 3. Second, a large transitory component in extreme values of earningswould suggest a weak relation with equity value for both low and high earnings. However, herewe observe a weak relation for low earnings but a strong relation for high earnings. Similarly, the

empirical resultsin

tables2-5

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 iswidespread evidence of convexity throughout the entire range of earnings. In table 3, there is aclear difference in the coefficients for the medium and high earnings ntervals, which include onlypositive earnings. Additionally, in the auxiliary analysis where the range of earnings wassegmented 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 4and 5, where all of the negative earnings observations have been eliminated. Finally, the visualevidence in both figures 2 and 3 shows clear evidence of convexity throughout the entire rangeof earnings.

Results for Changes Over TimeEquation (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 orm. An obvious alternative othe levels specifications used in the previous section is to difference equation (5) over timeto 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 adaptationvalue is more complicated than the levels relation. A change in net resources affects both theadaptation value and recursion value components of equity value. For a firm with a low level ofexpected earnings, since a change in resources will generally translate directly into a change inadaptation value, the change in equity value will be approximately equal to the change inresources. On the other hand, a change in resources will also change recursion value through achange in the level of expected earnings, where the change in recursion value is the capitalizedmarginal expected earnings provided by additional resources. For a firm with a high level ofexpected earnings, the change in recursion value will again be approximately equal to the changein resources, as long as the firm has previously invested in resources up to the point where themarginal return is equal to the marginal cost of capital. Consequently, across all levels ofexpected earnings, we expect the change in equity value to be approximately equal to the changein resources.

In contrast, a change in expected earnings affects recursion value by definition but does notnecessarily affect adaptation value. Thus, the effect of a change in expected earnings on equityvalue will vary with the level of expected earnings. For a firm with a low level of expected earningswhere equity value is primarily determined by adaptation value, a change in expected earningswill have a small effect on equity value. For a firm with a high level of expected earnings, the effecton equity value should be approximately the earnings capitalization factor times the change inexpected earnings. Thus, the relation between changes in equity value and changes in expectedearnings should reflect the convexity observed in the previous section related to the level ofexpected earnings.

We adopt a specification for the change in equity value which allows the effect of either thechange in book value or the change in expected earnings to vary with the level of expectedearnings (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


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To allow the relation between changes in market value, changes in book value and changesin 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 inearnings and the change in book value for three groups formed by sorting the observations onEt-1/BVt_1. Observations with a higher ratio of earnings to book value are predicted to be firmsfor which recursion value is a larger component of value, i.e., the magnitude of the earnings tobook ratio should be directly related to the magnitude of the coefficient relating changes in marketvalue to changes in earnings.

Table 6 contains the results of estimating equation (10). The coefficient on the change in bookvalue is significant. Estimates of b4 are significantly different from zero for all but the first yearand are generally close to one. The estimates for the middle and high groups, b4 + b5 and b4 + b69are also close to one but slightly smaller on average. In most cases, differences among the groupsare not significant so there is little evidence that the coefficient depends on the level of expectedearnings relative to book value. Thus, the estimated coefficients on the change in book value aresignificant and close to our expectations.

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

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

plotted in figures 2 and 3 and the statistical tests reported n 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 intable 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 themeasurement 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 variationin levels of expected earnings. Thus, while the total variation in the change in earnings is substantially smaller than thetotal variation in the level of earnings (compare the standard deviations for earnings changes and levels reported in table1), 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 table3). Since coefficient bias induced by measurement error is generally proportional to the ratio of the measurement errorvariance 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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27/30


shape predicted by the theory. Tests reported n tables 4 and 5 relating market value to book valueconditional on the level of earnings also strongly support the predicted convexity with pooledregressions, 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 theprediction 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 capitalizationfactor. The results of tests reported n table 6 which evaluate the relation between changes in valueand changes in expected earnings are also generally consistent with the predictions of the theory.

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

value of the firm' s expected earnings (under he assumption that the firm continues its current wayof 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 predictionof

convexity-the coefficient on earnings increases with the ratio of earnings to book value and thecoefficient on book value decreases with the ratio of earnings to book value. Thus, the resultsimply that equity value is a function of both expected earnings and book value, in contrast tomodels 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 elementsof value are simply additive (e.g., Ohlson 1995).

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


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

adaptation alue (AV) is established by examining he 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 s, with respect to the mean of earnings conditional on adaptation alue, denoted

here by A :

dMV=AV= f'(E IAV)dE + cf E f'(E IAV)dE. (Al)

The derivative of market value with respect o g depends on the level of A. As Gu->--oo, both termsin equation (Al) approach ero, and therefore he derivative dMV/dg approaches ero. Intuitively,as g becomes increasingly negative, most of the probability mass of f (E IAV) in equation 2) is belowAV/c and MV approaches the constant AV. Therefore, as Ha-> Ao, dMV/dg --> 0. Similarly, asg -> +oo, the first erm n equation A 1) approaches ero and he second term approaches . Intuitively,as g becomes increasingly positive, most of the probability mass of f (E IAV) in equation 2) is aboveAV/c and MV approaches c A. Therefore, as Ha-> oo, dMV/dg -> c. Thus, the first derivative is boundedbetween 0 and c as g varies between -oo and +oo.

Further, dMV/dg is strictly ncreasing n A. To see this, note that ncreases n g represent a shiftin probability mass from he first erm n (2), where he slope of the value function s zero, to the secondterm in (2), where the slope of the value function is c. This implies that the derivative increasesmonotonically from 0 to c, implying that market value is an increasing, convex function of A.


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REFERENCESAli, A., and P. Zarowin. 1992. The role of earnings levels in annual earnings-returns studies. Journal of

Accounting Research (Autumn): 286-296.Altman, E. 1968. Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. The

Journal of Finance (September): 589-609.Ball, R., and P. Brown. 1968. An empirical evaluation of accounting income numbers. Journal ofAccounting

Research (Autumn): 159-178.Barth, M. 1991. Relative measurement errors among alternative pension asset and liability measures. The

Accounting Review (July): 433-463., and W. Landsman. 1995. Fundamental issues related to using fair value accounting

01# earning, adaptation and equity valui - burgstahler, dichev

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