on accounting-based valuation formulae*€¦ · tages of the aeg approach as compared to riv. these...

On Accounting-Based Valuation Formulae*

JAMES A. OHLSON

W. P. Carey Professor at the W. P. Carey School of Business, Arizona State University, Tempe, AZ, 85287-

3606, USA

Abstract. This paper considers accounting-based valuation formulae. Its initial focus is on two problems

related to residual income valuation (RIV). First, insofar valuation depends on theresent value of expected

dividends per share, applying RIV requires clean surplus accounting on a per share basis. Awkwardly,

equity transactions that change the number of shares outstanding generally imply eps „ D bvps ) dps. A

clean surplus equality holds only if one ‘‘re-conceptualizes’’ either end-of-period bvps or eps as a forced

‘‘plug’’. Second, one cannot circumvent the per share issue by evaluating RIV on a total dollar value basis

unless one introduces relatively subtle MM-type restrictions. In light of RIV’s unsatisfactory aspects, the

paper proposes an alternative to RIV. This new approach maintains a strict eps-focus. It derives by

replacing bvpst in RIV with epst +1 capitalized (i.e. divided by r). One obtains a formula such that the

current market price equals next-period expected earnings capitalized plus the present value of expected

abnormal earnings growth, referred to as AEG. A number of propositions then demonstrate the advan-

tages of the AEG approach as compared to RIV. These results follow because epst+1 capitalized generally

approximates market price better than bvpst.

Keywords: Equity valuation, EPS, EPS growth, Dividend policy

JEL Classfication: M41, G12, G14

Introduction and Summary

During the past decade, Residual Income Valuation (RIV) has become prominent inthe accounting literature.1, 2 Themain reason forRIV’swidespread acceptance rests onits apparent ability to procure a constructive role for accounting data in equity valu-ation. Traditional equity valuation, with its emphasis on future cash flows, in contrastsuggests a general irrelevance of future earnings and other accounting data.Accounting-oriented textbooks like to counter this impression, of course. They dem-onstrate how RIV derives from PVED combined with clean surplus accounting:Current book value acts as a ‘‘valuation-anchor’’, and the PV of expected futureresidual earnings reconciles the difference between intrinsic andbook values.While thisaccounting-based valuation formula seems simple enough, the literature also recog-nizes that GAAP’s earnings construct violates clean surplus accounting. It is thengenerally argued that the dirty surplus items have approximately zero expected values,which essentially eliminates the problem. Overall, the literature does not view such

*An earlier version of this paper was titled ‘‘Residual Income Valuation The Problems’’.

Review of Accounting Studies, 10, 323–347, 2005

� 2005 Springer Science+Business Media, Inc., Manufactured in The Netherlands.

practical issues as detrimental toRIV, and it positionsRIVas a robust and sensiblewayof placing accounting data at the center of equity valuation.A commentary byBernard(1995) appears to have been especially influential in popularizing RIV as the approachto equity valuation.This paper takes a critical look at RIV and its conceptual underpinnings. Most of

the analysis deals with problems associated with capital transactions which changethe number of shares outstanding. Such transactions undermine RIV’s standardfoundation by opening up issues related to the relevance and validity of clean surpluson a per share basis. Moreover, expected changes in shares outstanding touches thePVED concept itself because it can be applied on either a per share basis, or on atotal (dollar) equity basis. Capital transactions thus introduce intertwinedaccounting and finance theory issues. But this analysis of capital transactions andRIV brings out a broader question: To what extent do theoretical constructs bear onthe appeal of RIV as compared to alternative, accounting-based, valuation formu-lae? The paper addresses the question by introducing a framework that allows for acomparison of alternative (but PVED-equivalent) valuation formulae. One of thesealternatives usefully captures the spirit of practical equity valuation. That is, theformula works such that the growth in eps explains the price to forward-earningsratio. And one cannot reasonably claim that the underpinning concepts of RIVconnect to the earnings growth principle observed in practice. Perhaps more thananything else, this aspect of RIV ‘‘not conforming to observed practice’’ leaves aquestion mark as to what ought to be the formula’s status for purposes of equityvaluation.Major points of the paper are:

• The concept that value equals the present value of expected dividends derivesfrom a per share perspective, and where dividends are regular cash dividends.RIV and PVED equivalence therefore directs attention to clean surplus on aper share basis. Total dollar value clean surplus does not guarantee the sameon a per share basis because potential capital transactions change the numberof shares outstanding. ‘‘Regular’’, GAAP-type, accounting with respect to epsand bvps thus leads to the possibility D bvps „ eps ) dps. Only by introducingunorthodox, contrived, accounting can one achieve clean surplus accounting ona per share basis. Such accounting forces either eps or end-of-period bvps toact as a pure ‘‘plug’’.

• A potential way of finessing the per share problem starts from a D in PVEDthat represents the total dollar amount of dividends net of all capital contribu-tions. This comprehensive PVED approach, however, yields the correct equityvalue for the current shareholders only if the (expected) equity transactions areneutral/irrelevant to prospective ‘‘future new’’ shareholders.

• A general analytical machinery demonstrates that RIV is but one specificaccounting-based valuation formula within a broad class. Using this machinery,the paper develops a competing valuation formula with the same core mathe-

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matical structure as RIV. In contrast to RIV, the alternative formula makes noreference to book values; it focuses solely on future (expected) eps. Next-periodexpected eps capitalized— eps1=r—replaces current book value as the startingpoint. And epstþ1=r replaces bvpst as it appears in the book value representa-tion of residual earnings, bvpst þ dpst � 1þ rð Þ � bvpst�1.3 The resulting for-mula can be interpreted as one of abnormal earnings growth (AEG) valuation.4

• Though AEG and RIV both equal PVED, the 2 formulae can be compared interms of their truncation errors. It is shown that, under mild conditions, theAEG approach builds in smaller truncation—errors than the RIV approach. Inother words, if one estimates value by assessing the ‘‘near future’’ only, then afinite-terms AEG works better than a finite-term RIV. This dominance resultcaptures the idea that the expected earnings capitalized generally approximatesthe expected market value closer than the expected book value.

• A final result pertains to the text-book model which relates the price-to-bookratio to the next period’s residual earnings (Penman (2004) chapter 14, forexample). It is shown that this model is a special case of the Ohlson andJeuttner (2004) model which relates the price-to-forward earnings to the changein residual earnings. As a consequence, one can again claim that an earningsand growth in earnings perspective is more general than a book value andgrowth in book value perspective.

1. Basics The PVED Formula

Researchers and practitioners interest in RIV depends on its equivalence toPVED. The latter formula is virtually always taken as fundamental in valuationtheory. Nevertheless, the ‘D’ in PVED can refer to two distinct definitions. Onepossibility defines ‘D’ in terms of dividends per share (dps), in which case the lefthand side of the PVED formula targets price per share. A second possibilitydefines ‘D’ as total dividends net of capital contributions, in which case PVEDtargets the total (dollar) value of the equity. These two approaches differpotentially; the matter does not reduce to scaling for the number of sharesoutstanding at the date of evaluation. The acronym PVED therefore raises thefollowing question: Should it be expressed on a per share basis, a total-equitydollar basis, or do both approaches generally work?The details of any answer depends on the assumptions, of course. Under weak

assumptions consistent with the so-called ‘‘random walk hypotheses for stock pri-ces’’, it turns out that the per share approach provides the starting point.To appreciate why the ‘D’ in PVED should equal dps, consider an investor who

buys a share (ex dividend) at date t, paying Pt. At the subsequent date the totalpayoff equals Pt+1+dpst+1. Capital contributions do not influence the payoff be-cause the investor can refuse any such transactions. Nor can pre-existing owners

ON ACCOUNTING-BASED VALUATION FORMULAE 325

always partake in an offering (retirement) of shares. Owning a share thereforeguarantees only the ‘‘regular’’ cash dividend.Suppose further that the investment is a fair game. This equilibrium requirement

means that an investor cannot improve on his/her forecast of returns by expandingthe information set. Specifically, an investor who buys one share of a stock at date tcan expect a return of Et ~Ptþ1 þ d ~pstþ1

� �=Pt

� �¼ R; regardless of the date t informa-

tion. It is implicitly assumed that risk and risk-free rates are held constant over timeso that the expected equity return, R, requires no subscript ‘t’. Recursive substitutionnow implies,

Pt ¼X1

s¼1R�sEt d ~pstþs

� �ð�Þ

and where, to be sure, both sides of the equation are on a date tper share basis.Conversely, (*) implies Et ~Ptþ1 þ d~pstþ1

� �=Pt

� �¼ R: PVED on a per share basis

accordingly corresponds to the traditional dividend-adjusted ‘‘random walkhypotheses’’. Neglecting issues related to changes in risk and interest rates, the-oretical finance relies on this modeling when investors’ expectations are homog-enous.Consider next the validity of PVED as a representation of a firm’s total equity, i.e.,

the ‘D’ now stands for total (dollar) dividends net of capital contributions. PVED ona total equity basis is of course as valid as (*) if the shares outstanding never changein the future. If, on the other hand, shares outstanding can change then one mustrecognize that the issue of dividends per share versus total dividends net of capitalcontributions arises because capital contributions—as opposed regular cash divi-dends—perturbs the ownership structure by introducing ‘‘new’’ owners (or ‘‘elimi-nate’’ owners). A firm that operates as a proprietorship simply ‘‘taxes’’ the (single)owner if the firm requires a capital infusion; the number of shares outstanding neednot change. With these observations in place it becomes clear that PVED on a totalbasis may differ from the market value because PVED can include (expected) wealthcreated for individuals who are not currently shareholders. Dealing with this issuerequires the articulation of MM-conditions so that (*) reconciles with PVED on atotal equity basis. A subsequent section formalizes such analysis. Before undertakingthis task, it helps to identify conditions on the accounting such that RIV meshes withPVED on a per share, or (*), basis.

2. Per Share Accounting and Clean Surplus

It goes almost without saying that the change in book value per share Dbvpsð Þwill generally differ from earnings per share (eps) minus dividends per share (dps).This claim does not rest on dirty surplus items, or on complexities associated witheps due to potentially dilutive securities (like convertible bonds). Ruling out thesepossibilities, but allowing for (regular) capital transactions, still leads to Dbvpst „ epst ) dpst unless (i) number of shares outstanding does not change, or,

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(ii), issue price per share equals bvps at the transaction date. Neither exception isof practical interest. With respect to (i), firms sometimes announce plans toengage in acquisitions and capital structure changes or, as another example,maintain share repurchase programs to support the stock price. And GAAPmakes no attempt to satisfy condition (ii)—which corresponds to mark-to-marketaccounting—even as a approximation.Absent clean surplus on a per share basis, the usual analysis that leads to equiv-

alence in RIV and PVED falls apart. Hence, if one defines

repst � epst � r � bvpst�1;where r ” R ) 1; one can then not rule out

PVED 6¼ bvpst þX1

s¼1R�sEt ~repstþs

� �;

where ‘ED’ refers to Et~dpstþ1

h i;Et

~dpstþ2

h i. . .. This inequality, to be sure, can hold

even if capital transactions occur only at the end of periods so that there are noambiguities associated with epst and bvpst.Per share clean surplus does not hold if a firm issues/buys shares insofar the

transaction changes bvps. An end-of-period issuance of shares increases bvpst if Pt

exceeds the bvps prior to the issuance. But the transaction leaves epst and dpstunaffected, and it follows that D bvpst > epst ) dpst (assuming no other capitaltransactions).Clean surplus on a per share basis can be achieved by ‘‘reconceptualizing’’ either

of the two accounting variables, of course. The two reference schemes are as follows:(i) Define

^epst � Dbvpst þ dpst:

Hence, ^epst will not generally equal ‘‘regular’’ epst, even if clean surplus holds ona total dollar basis. As a further complication, bvpst may be ambiguous if thereare dilutive securities outstanding (GAAP does not prescribe how one determinesbvps, in sharp contrast to eps). But this issue is subordinated and separate.Approach (i) stems from the construct that within a per share context eps derivesas a ‘‘plug’’ from the change in bvps adjusted for dividends per share. Becausechanges in shares outstanding have a direct influence on bvps, this eps constructincludes ‘‘realized gains/losses due to capital transactions’’. The difference,epst � epst determines the net of this ‘‘expense’’ item, and its sign depends on thesign of P ) bvps and whether shares are issued or purchased. Put mildly, thiskind of accounting is unorthodox.(ii) Recursively, define

bvpst � bvpst�1 þ epst � dpst with the initialization bvps0 � bvps0:

In contrast to (i), approach (ii) specifies book value per share instead of earnings pershare as the ‘‘plug’’. Empirical papers applying RIV on a per share basis rely on thescheme.5 Forecasted eps thereby take on a central role, though there also has to be


some assumption about forecasted dps. Accounting gains/losses allocated to currentshareholders because of (forecasted) capital transactions are now no longer part ofthe eps construct. This feature perhaps suggests that (ii) is preferable to (i). Still, toclaim that the approach leads to an appealing residual earnings construct wouldseem to be farfetched.Both approaches, (i) and (ii), seem awkward in their reliance on mechanical plugs.

A requirement of clean surplus earnings on a per share basis makes the RIV modellook contrived rather than intuitive. Because the per share basis causes the smudge,one may instead consider a RIV that targets a firm’s total (dollar) equity value. Theaccounting then becomes more palatable by removing per share issues. Nevertheless,the next section shows how potential future capital contributions still mar theanalysis.

3. PVED on a Total Equity Basis

Consider PVED when the ‘D’ stands for total dividends net of all capital contri-butions. Without referring to any concepts of equity valuation, it is of course truethat PVED equals RIV given the clean surplus relation. To justify a total dollar basisRIV therefore requires an analysis of conditions such that PVED equals the marketvalue of the total equity at date zero. RIV lacks proper grounding if PVED on a totalbasis differs from an evaluation (*) of the expected dps—sequence. In analyticalterms, the issue arises whether potential changes in the anticipated number of sharesoutstanding allows for

n0 �X1

t¼1R�tE0 d~pst½ � 6¼ PVED

where ‘D’ = all dividends net of capital contributions and n0 denotes the number ofshares outstanding at the date of evaluation (t = 0).As an example below shows, the two expressions equate if, and only if, the issu-

ance/buy-back of shares is value-neutral from the point of view of expected ‘‘newfuture’’ shareholders. Accordingly, the condition represents an ‘‘outsider’’ capitaltransaction irrelevancy concept in the spirit of MM.To illustrate the subtleties associated with a total-equity PVED approach, consider

the following:(1) Regular cash dividends are declared and paid on a per share basis every 12/31.

Let dpst ‡ 0 denote these dividends.

(2) The current date is 1/1 and t=0. Only one share is outstanding. The firm plans toissue k new shares at a price of p* one year later, where k and p* depend on thecircumstances at that date. These two variables are therefore random from a datet=0 perspective. Without substantive loss of generality, assume further thatshares outstanding will not change beyond date t=1. Total dividends net ofcapital contributions equal

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t ¼ 1 : dps1 � k � p�t � 2 : dpst þ k � dpst ¼ ð1þ kÞ � dpst

One can next evaluate PVED on a total equity basis. Define it as V0:

V0 � R�1 E0~dps1

h i� E0

~k � ~p�h i� �

þX1

t�2R�tE0 1þ ~k

� �~dpst

h i

¼X1

t¼1R�tE0 d~pst½ � þ

X1

t¼2R�tE0

~k � d~pst

h i� R�1E0

~k � ~p�h i

( )

:

Because the PVED-formula on a per share basis, equation (*), equals P0, it followsthat V0=P0 if and only if the expression inside { } equals zero. The latter restrictionmeans that the ‘‘new future’’ shareholders view the issuing of shares as neutral exante: They are indifferent to the anticipated transaction. MM applies in this con-fining sense.For the ‘‘current’’ shareholders to potentially benefit it suffices if the anticipated

transaction affects the sequence of expected dividends per share positively. Thus thecondition P0=V0 allows for economically relevant capital transactions. But theallocation of welfare across the two groups of shareholders has to be extreme in thatonly the current shareholders can benefit. This is of little practical interest unlessneither group benefits. No benefits whatsoever can be thought of as capital-irrele-vancy in a traditional MM sense. Treasury stock transactions fit neatly into thiscategory. Acquisitions of business do not, since the ‘‘new’’ shareholders typicallyrequire a premium to surrender their business assets. (In this case kÆp* correspondsto the market value of shares issued prior to the announcement of the transaction.)In sum, PVED on a total equity basis, V0, identifies value created to all shareholdergroups, including those that in the future leave or enter as owners on less than fairterms. Contrast this approach to PVED on a per share basis, (*), which identifies thevalue accruing to a current owner who does not ever change his ownership status,except on fair-value terms.A final point about capital transactions relates to the requirement that all capital

transactions (and dividends) subsumed by ‘D’ must be measured by their marketvalues. GAAP violates this market value condition for some capital transactions,such as convertible bonds when converted and (compensation) options when exer-cised. Further, expected values of the GAAP-measures of the capital contributionswill most likely be materially less than the related expected market values givenGAAP’s inherent downward biases.Observations in this and the prior section suggest that neither of the two RIV

models, on a per share basis or on a total equity basis, satisfactorily circumventproblems caused by capital transactions. The per share model seems particularlyawkward with its contrived schemes to attain clean surplus. PVED on a total equitybasis, on the other hand, depends on a relatively reasonable condition of fair capitaltransactions. This condition can in some ways be weakened, a point developed byChristensen and Feltham (2003). They discuss a full range of capital transactions and


how these can be accounted for outside GAAP to reflect wealth-redistributionsacross ownership interests, all within a clean surplus framework. Whether theserefined accounting-schemes would aid practical equity valuation can be debated. Theissue actually deflects by suggesting that accounting-based valuation ought to honein on some version of RIV.RIV’s primary problem concerns the need to juggle two accounting measures,

earnings and book values. A model that relies on 2 value-attributes runs atobvious cross-purposes with how investors in the real world look at the future.Practical investment analysis emphasizes (expected) earnings (or eps) as a corevalue attribute; expected book values or residual earnings have no such role (orat least rarely so). These aspects make it troublesome to assign RIV the elevatedstatus of being the accounting-based valuation formula.Subsequent sections develop a reasonable alternative to RIV. This new formula

maintains an unequivocal eps-focus. Further, the analytical machinery on whichRIV depends remains essentially the same.

4. A Machinery for Accounting-Based Valuations

RIV’s fragile underpinnings suggests that there is need for alternative, accounting-based, valuation formulae. A powerful approach to the broad problem of identifyingvaluation formulae consistent with PVED on a per share basis should thus embedRIV merely as a special case. To develop the analytics that equates PVED and RIV,it helps if one first articulates a class of valuation schemes consistent with PVED andthereafter introduces book values, earnings and clean surplus. This approach willusefully demonstrate the arbitrariness of RIV as an accounting-based valuationformula. An alternative to RIV, which focuses on future eps without any reference tobvps, will become apparent.Equating RIV to PVED depends only on a simple scheme that requires no ref-

erence to economic or accounting concepts. Consider the following mathematicalequality: Let ytf g1t¼0 be any sequence of numbers that satisfies R�t � yt ! 0 as t!1;then

0 ¼ y0 þ R�1 y1 � R � y0ð Þ þ R�2 y2 � R � y1ð Þ þ � � �

Adding the last equation and (*) implies that

P0 ¼ y0 þX1

t¼1R�t � zt ð��Þ

where

zt � yt þ �dpst � R � yt�1

and �dpst � E0~dpst

h i. Though the left hand side of (**) does not depend on the y-

sequence,—that is, (**) always equals (*)—the zt ) sequence does. In what follows,zt should always be viewed as dependent on some specification of (yt, yt ) 1).

OHLSON330

Armed with the expression (**), nothing stops us from putting y0=bvps0 andyt ¼ bvpst; t � 1; so that zt ¼ bvpst þ dpst � R � bvpst�1: This approach demon-strates the centrality of book values in RIV. Earnings, by contrast, enter theanalysis via the ‘‘back-door’’—epst=D bvpst+dpst—and this incremental step ishard to justify unless one assigns a ‘‘conceptual label’’ to zt, namely, residualearnings (per share). Given the dubious nature of clean surplus on a per sharebasis, it would probably be more accurate to say define epst � Dbvpst þ dpst anddefine repst � epst � r � bvpst�1; and without attaching any labels to epst andrepst. RIV on a per share basis applies, but the letter ‘I’ in the acronym RIVmisleads insofar the formula is more of a book value than an income approach toequity valuation.The above steps bring us back to the clean surplus implication (i) in Section III

with its unorthodox eps-concept. An alternative to (i) relies on the construct (ii),which treats bvps as the plug. In terms of the Section III notation, put yt ¼ E0 bvpst½ �and re-define per share residual income, repst � epst � r � bvpst�1. This unrestrictedchoice of a bvps -construct reflects the saying that ‘‘accounting measurement/valu-ation principles do not influence the validity of RIV’’. Both constructs of bvps clearly‘‘work’’. But the utter lack of any conceptual restrictions on what determines thebvps -sequence inevitably points toward a more poignant truth: The mathematicalmachinery by itself provides no compelling reasons why the y-sequence should corre-spond to a book value-sequence.One can perhaps argue that a bvps -specification of y makes sense because bvps0

supplies a ‘‘natural starting point’’ in equity valuation. Such a claim, however,immediately raises the question whether one can identify ‘‘starting point’’ measuresother than current book value. Casual observation, if not theory, points towardcapitalized next period expected earnings as a much better first-cut indicator ofequity value. In other words, investment practice with its focus on forward epssuggests the pick y0 ¼ E0 ~eps� next year½ �=r. The difference between P0 andE0 e~ps� next year½ �=r should then relate to the expected increments in subsequentexpected eps. It seems reasonable, therefore, that one should entertain the sequence

yt ¼ E0 ~epstþ1� �

=r � epstþ1=r; t ¼ 0; 1; 2; . . .

For t ¼ 1; 2; . . .., substituting into the definition of zt results in

zt ¼1

r� epstþ1 þ r � dpst � R � epst� �

¼ 1

r� Depstþ1 � r � epst � dpst

� ��

The analytics is definitional; it applies regardless of the accounting rules whichinfluence the epst-sequence. One interprets rÆzt as the expected eps-increment in ex-cess of what should be expected due to earnings retained in the prior period. Positivezt result in a positive premium, P0 > eps1/r. Alternatively, the price-to-forward-earnings ratio, P0/eps1, exceeds the benchmark 1/r in case of above-normal futureexpected earnings increments.Every specification of ytf g1t¼0 maintains PVED, but conceptual issues arise if one

wants to approximate PVED by cutting (**) short in terms of the number of yearsthat enter the evaluation. Finite horizons appeal in a world of bounded rationality.6


In this spirit, one can potentially judge the usefulness of accounting constructs interms of the errors

PVED� y0 þXT

t¼1R�tzt

!

¼X1

t¼Tþ1R�szt � ErrðT; yÞ

where y=(y0, y1, ...) specifies the accounting measurements and their expected val-ues. As a boundary case, for T=0 define Err 0; yð Þ � P0 � y0.While Err T ; yð Þ ! 0 as T !1, finite T generally leads to errors. And the smaller

the errors across various horizons (T), the better. In what follows the analysisconsiders

y0 � eps1=r; eps2=r; eps3=r; . . .ð Þand

y00 � bvps0; bvps1; bvps2; . . .� �

:

There are no reasons why Err 0; yð Þ ¼ 0 should hold as a practical matter. But theoutcome is of theoretical interest for both bv0 and eps1/r. To avoid strange knife-edgecases, one may consider more generally the possibility of Err T ; yð Þ ¼ 0 for all T ‡ 0.This ‘‘no-error’’ outcome implies that some hypothetical accounting measurementshave succeeded in satisfying zt ¼ 0 and PVED ) y0=P0 ) y0=0. Addressing what ittakes to achieve the stringent no truncation-error regardless of the horizon T pro-vides a useful baseline when one compares book values and earnings (capitalized) asvalue attributes.

5. The Specification of ytf g‘t¼0: Book Value vis-a-vis Earnings

This section compares the implications of specifying the y-sequence as either earn-ings or book values on the truncation-error Err T ; yð Þ. The sign of

Err T; bvps0; . . .ð Þð Þj j � Err T; eps1=r; . . .ð Þð Þj jfor various T ‡ 0 thus evaluates the relative performance.The initial setup disallows equity transactions, other than cash dividends, to keep

matters simple. In other words, the firm will effectively operate as a proprietorshipwith only 1 share outstanding. The assumption makes the ‘D’ in PVED unambig-uous. Given this simplification the analysis can be tied to the prior literature onvaluation, and thus it helps to use more ‘‘standard’’ notation. Let xt denote earningswhere thus xt=epst and dt denotes dividends where dt=dpst=D. Further, let xt

a ”xt ) rÆbt where bt=bvpst and xt

a=repst. In what follows the notation xt, dt, bt, xta

will be used only if the firm operates as a proprietorship (one share is outstanding atall dates, but dt may be negative).With these preliminaries in place the analysis next addresses under what condi-

tions

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z0t � �xtþ1 þ r � �dt � R � �xt� �

=r

or

z00t � �bt þ �dt � R � �bt�1� �

equals zero. Which of the two seems the more likely candidate to satisfy this zero-condition?It goes almost without saying that both z0t and z00t equal zero if the firm corresponds

to a ‘‘saving-account’’. Next-period earnings will be certain, but subsequent earningscan be uncertain because next-period dividends may be uncertain. In spite of thisuncertainty, one readily shows that for all future dates t

z0t � r�1 � �xtþ1 þ r � �dt � R � �xt� �

¼ 0

and

z00t � �bt þ �dt � R � �bt�1 ¼ 0:

Alternatively, �Pt ¼ �xtþ1=r ¼ R=rð Þ�xt � �dt ¼ �bt ¼ R�1 � �btþ1 þ �dtþ1� �

and, for both ofthe two y-specifications, Err(T;Æ )=0 all T ‡ 0.The savings account serves as a useful starting point because it shows that the right

accounting constructs can yield a clean result. Thus one may next examine whathappens in case of mark-to-market accounting, which is somewhat less stringentthan a savings account.A powerful analytical observation handles a more general version of mark-to-

market accounting. Note that

Dz00tþ1 ¼ �btþ1 þ �dtþ1 � R � �bt� �

� �bt þ �dt � R � �bt�1� �

¼ �xtþ1 � R � D�bt � �dt ¼ �xtþ1 þ r � �dt � R � �xt ¼ r � z0tTo interpret the relation in terms of its underlying accounting constructs, recall thatz00t ¼ �xa

t and thus r � z0t � D�xatþ1: Thus, one moves from a focus on book values, z00t or

�xat , to one on earnings, z0t or D�xa

tþ1, by taking the first difference in z00t or �xat . It follows

immediately that z00t ¼ 0 for all t implies z0t ¼ 0: But the opposite is obviously false.Applying these observations to Errð�; �Þ yields the result below.

Proposition I Assume PVED and clean surplus accounting. Consider the truncation-error definition Err T ; yð Þ where y corresponds to either y0 � �x1=r;�x2=r; . . .ð Þ ory00 � bv0; bv1; . . .

� �. If one further assumes that Err T ; y00ð Þ ¼ 0 for all T ‡ T*, then

Err T ; y0ð Þ ¼ 0 for all T ‡ T* as well. The converse, however, is false. j

Traditional accounting concepts supply an intuitive and direct interpretation ofthe above proposition. Perfect balance sheets, which equate a firm’s book value to itseconomic value, result in perfect earnings measures if one uses clean surplusaccounting. Reversing this claim does not work, however, because of the ‘‘cancelingerror’’ concept. That is, if one deducts a constant from all perfect balance sheets thenthe earnings measure does not change and thus it remains perfect. One cannot,therefore, infer perfect balance sheets from perfect measures of earnings (and


dividends). These ideas, which go to the heart of accounting, impel the proposition:There should be no surprise why earnings, as opposed to book values, take on suchdominant role in fundamental analysis of equity value. RIV, as noted, runs at cross-purposes with this core idea.Does the proposition depend on no capital transactions and the absence of po-

tential new/former shareholders? The answer is no. If a firm’s market value alwaysequals its book value, then the issuance of shares leads to no implicit or explicitgains/losses to the new/former shareholders; clean surplus on a per share basistherefore applies if it does so on a dollar basis. With respect to perfect (expected)earnings— Err T ; y0ð Þ ¼ 0 but Err T ; y00ð Þ 6¼ 0; T ‡ T* > 1—one can of course arguethat capital transactions make the scenario implausible. But this matter of plausi-bility goes beyond the scope; the formal analysis goes through.

6. Illustrations of a Book Value vis-a-vis Earnings Focus

Via two examples, this section illustrates dynamic underpinnings of an earnings-focus vis-a-vis a book-value-focus. The first example shows how a book-value, orRIV, focus with Err(T;y¢¢ )=0 can be thought of as mark-to-market in expectation.The second example shows how an earnings focus with Err(T;y¢ )=0 but possiblyErr(T;y¢¢ ) „ 0 can be thought of as permanent earnings in expectation. Bothexamples rely on the information dynamic found in Ohlson (1995), slightly extended.

Example 1 Mark-to-Market in ExpectationConsider the dynamic

~xatþ1 ¼ t1t þ ~e1tþ1~t1tþ1 ¼ t2t þ ~e2tþ1~t2tþ1 ¼ þ~e3tþ1

where Et½~ektþs� ¼ 0; all s ‡ 1. Using RIV yields the valuation function:

Pt ¼ bt þ R�1 � t1t þ R�2 � t2tHence, for s ‡ 2,

Et½ ~Ptþs�~btþs� ¼ 0;

and where s ‡ 2 is also necessary for the equality. It follows that Err T ; y00ð Þ ¼ 0, allT ‡ 2, and, due to the proposition, Err T ; y0ð Þ ¼ 0 as well.To see more concretely why Err T ; y0ð Þ ¼ 0 for T‡2, note that Et½tktþs� ¼ 0 for s‡ 2.

Next, since Et½~xtþsþ1� ¼ r � Et½~btþs� one has Et½~Ptþs� ¼ r�1 � Et½~xtþsþ1�. The equilibriumcondition Et½~Ptþs þ ~dtþs� ¼ R � Et½~Ptþs�1� (all s ‡ 2) combined with the last relationimplies r�1 � Et½~xtþsþ1� þ Et½~dtþs� ¼ R � r�1Et½~xtþs�, which indeed corresponds to z0t ¼ 0for all t ‡ 2 and Err T ; y0ð Þ ¼ 0, T ‡ 2.The labeling of this setting, mark-to-market in expectation, appeals because

�P tþs ¼ �btþs beyond the next year (s ‡ 2). The special case t1t ¼ t2t ¼ e2t ¼ e3t ¼ 0 all

OHLSON334

t, defines strict mark-to-market accounting: Pt=bt and Err 0; y00ð Þ ¼ 0: More gener-

ally, one can extend the information dynamic so that �Ptþs ¼ �btþs, s ‡ T*, for any

preselected T*.

Example 2 Permanent Earnings in ExpectationConsider the dynamic

~xatþ1 ¼ t1t þ ~e1tþ1t1tþ1 ¼ t1t þ t2t þ ~e2tþ1t2tþ1 ¼ ~e3tþ1

where Et ~ektþs½ � ¼ 0 all, s ‡ 1. Using RIV and clean surplus it follows that

Pt ¼ bt þ r�1 � t1t þ R � rð Þ�1�t2t ¼ r�1 � Et �xtþ1½ � þ R � rð Þ�1t2t:

As to the last equation, note that Et ~xatþ1

� �¼ tit. Next, because Et½~t2tþs� ¼ 0 for

all s ‡ 1, it follows that Et ~Ptþs� �

¼ r�1 � Et ~xtþsþ1½ �; s � 1: The equilibrium condi

tion Et ~Ptþs þ ~dtþs

� �¼ R � Et ~Ptþs�1

� �; s � 0 now implies r�1 � E ~xtþsþ2½ � þ Et

~dtþsþ1� �

¼R � r�1 � Et½~xtþsþ1�. One sees that z0t � E0f ~xtþ1½ � þ r � E0

~dt� �� R � E0 ~xt½ �g=r ¼ 0, for all

t ‡ 1, so that Err T ; y0ð Þ ¼ 0 if T ‡ 1. On the other hand, Err(T;y¢¢) „ 0 sinceEt ~Ptþs� �

¼ Et~btþs� �

þ r�1tit for all s ‡ 1, which violates mark-to-market in expecta-tion.As to the terminology, its motivation stems from the permanent earnings concept.

The earnings dynamic now satisfies7 Et½~xtþ1� ¼ R � xt � r � dt; which is more

demanding than Et½~xtþs� ¼ R � Et½~xtþs�t� � r � Et½~dtþs�1�, s ‡ 2 but not necessarily s=1.Permanent earnings therefore suffices for permanent earnings in expectations, butthe converse is false. Finally, note that the dynamic generalizes so that Err(T;y¢¢ )=0for T ‡ T*, and where one can pick any number T* (not just T*=1).The proposition combined with the examples suggests the following summary.

Mark-to-market in expectation corresponds to �Pt ¼ �bt and Err(t;y¢¢ )=0, t ‡ T*.This model contrasts with permanent earnings in expectations which corresponds to�Pt ¼ r�1�xtþ1 ¼ ðR=rÞ � �xt � �dt for all t ‡ T* and Err(t;y¢)=0. Moreover–and this pointis indeed central—whereas mark-to-market in expectation implies permanent earn-ings in expectation, the converse is false. On the basis of this observation, at least, afocus on earnings clearly dominates a focus on book values.

7. Book Values vis-a-vis Earnings when the Accounting is Conservative

So far the analysis has stated assumption that imply no truncation-error beyondsome (future) date. This setup build in the benchmarking conditions �PT ¼ �bT or�PT ¼ �xTþ1=r. These exclude the more realistic case when both capitalized earningsand book values have a consistent downward bias relative to market values. Biasesof this kind can be thought of as manifestations of conservative accounting. Felthamand Ohlson (1995), Ozair (2003), Zhang (2000), and others, formalize the concept ofconservative accounting and relate it to such biases. Equivalently, conservative


accounting implies that a firm’s long run expected earnings can continue to driftupwards even if the (net) dividend payout is fixed at 100%. In this sense growth inexpected earnings and conservative accounting are two sides of the same coin.Proposition I, to be sure, excludes conservative accounting because of the propertyEt D~xtþs½ � ¼ 0 as s fi ¥ if xt=dt.Relying on Zhang (2000), Ohlson (2002) outlines how capitalized earnings dom-

inate book values as a value attribute when one evaluates truncation-errors in set-tings with conservative accounting. Again the traditional partially ‘‘canceling errors’’concept makes its presence felt. In analytical terms, the difference �Pt � �xtþ1=r issmaller than �Pt � �bt as t fi ¥.Granting the centrality of the asymptotic inequalities �Pt > �xtþ1=r > �bt, one can

usefully show how a partial ‘‘canceling error’’ concept applies in a setting with (a)long-run expected growth, (b) clean surplus, (c) conservative accounting. Define the(expected) ‘‘errors’’ in book values as

�et � �Pt � �bt,

where t=0 is todays date and t=1, 2, .... For large t, �et is positive due to conservativeaccounting. Further, assume that �etþ1=�et; �Ptþ1=�Pt; �btþ1=�bt K1þ g, g > 0; i.e., the 3variables in question have the same asymptotic growth rate, g. Convergence inPVED requires g < r, which is another mild regularity condition. Next, combiningclean surplus with the equilibrium condition, �Ptþ1 þ �dtþ1 � �Pt ¼ r � �Pt, implies

�Pt ¼ �xtþ1=rþ �etþ1 � �etð Þ=r:

Due to growth, �etþ1 > �et so that �Pt > �xtþ1=r; this proves the first inequality. Thesecond inequality states that �xtþ1=r > �bt, which is equivalent to �etþ1 � �etð Þ=r�et. Inother words, do the balance sheet errors cancel each other sufficiently such that thedifference capitalized is less than the error itself? The answer is ‘yes’ simply because�etþ1 � �etð Þ=�et Kg < r.The preceding paragraph motivates the following proposition.

Proposition II Assume PVED and clean surplus accounting. Further assume that theexpected dividend sequence satisfies the regularity condition lim

t!1�dt=�dt�1 ¼ gþ 1

were 0 < g < r. Similarly, assume limt!1

�bt=�bt�1 ¼ 1þ g.Conservative accounting in the sense of

limt!1

�Pt=�bt� �

> 1

then implies there exists some T* such that for all T ‡ T*

Err T; y00ð Þ > Err T; y0ð Þ > 0:

Conversely, the accounting is conservative if the last statement is true.

Proof One readily shows that Err T ; yð Þ ¼ R�T �PT � �yTð ÞHence RT � Err T ; y00ð Þ½ .�Err T ; y0ð Þ� ¼ �xTþ1=r � �bT . Zhang (2000), in his Proposition I, has shown that thereexists some T* such that for all T > T � �xTþ1=�bT

� �> r. Sufficiency follows.

OHLSON336

Necessity is straightforward because Zhang (2000), also in his Proposition I, showsthat �xTþ1=�bT

� �> r implies conservative accounting. j

Because the assumptions are relatively weak, it would seem that RIV as an(intrinsic) valuation formula lacks appeal as compared to the AEG formula whichfocuses on expected earnings and their subsequent expected increments. A completeevaluation of the result’s robustness nevertheless requires that one considers capitaltransactions. In its current version the Proposition’s PVED assumption implicitlyrequires no expected changes in shares outstanding, or the expected capital trans-actions have to be fair. If neither condition is met, then, of course PVED must beevaluated on a per share basis. And for this setting clean surplus will not generallyapply given ‘‘regular’’ accounting for epst and bvpst.A relative minor modification of the proposition handles the per share problem

and the absence of a ‘‘regular’’ clean surplus. Referring back to Section III and the

definition of bvpst; bvpst ¼ bvpst�1 þ epst � dpst, consider the case when both

‘‘regular’’ bvpst and bvpst are conservative, i.e., lim �Pt=max �bvpst;�bvpst

n o� �> 1.

Additional mild conditions ensure that epstþ1=r > max bvpst;�bvpst

n oand accord-

ingly the proposition generalizes to admit changes in shares outstanding.

8. The Modeling of Earnings and their Growth vs. Book Values and their Growth

In comparing earnings and book values in equity valuation one may usefully con-sider implications of parametric models. Via assumptions on the sequence of futureexpected residual earnings, such modeling instructs by expressing values in simpleterms. This section argues that the traditional focus on the market-to-book ratioshould be complemented, if not replaced, by a model that focuses on the price-to-(expected) forward earning ratio.The next paragraph reviews the popular, well-known, model that explains market

value when anchored to (current) book value. With these derivations in place the restof the section can show why it makes sense to replace book value with next-periodexpected earnings capitalized. Certain aspects of the analyses will be striking.Whether one anchors value to book values or to earnings, the assumptions and themath will be much the same. Yet the analysis builds in unambiguous conclusions asto why one wants to focus on earnings.8

Following Penman (2004) and other textbooks, consider the most straightforwardparameterization of the sequence of future expected residual earnings:

�xat ¼ c � �xa

t�1 t ¼ 2; 3; . . .

and, as always, �xat � �xt � r � �bt�1. The parameter c satisfies c < R, and it determines

the growth in the sequence of anticipated residual earnings. Given clean surplus,RIV applies so that


P0 ¼ PVED0 ¼ b0þX1

t¼1R�t�xat ¼ b0þ R� cð Þ�1��xa1 ¼ b0 � RoE1� g

� �= r� gð Þ

� �;

where RoE1 � �x1=b0 and g ” c ) 1. Hence the valuation equation explains themarket-to-book ratio in terms of the forthcoming accounting rate-of-return, dis-count-rate, and growth parameter g. In this form the model provides well-knowninsights about the market-to-book ratio. As RoR increases, so does the ratio. Fur-ther, P0=b0ð Þ > 1 if and only if RoE1 > r; P0=b0 corresponds to RoE ¼ r. Theseobservations have theoretical appeal, but it is noteworthy that book value ratherthan capitalized earnings acts as the anchor to value.It turns out that the assumptions of the model (P0=PVED, clean surplus and the

recursive residual earnings equation) also lead to an entirely different valuationexpression. This alternative expression explains value in terms of forward earningsand the subsequent growth in forward earnings. It makes no reference to bookvalues, explicitly or implicitly, even though the assumptions remain the same.Understanding how the derivations yield the conclusion will be critical. There arethree steps.First, taking the first difference of the recursive residual earnings equation yields

D�xatþ1 ¼ c � D�xat ; t ¼ 2; 3; . . .

Second, due to the clean surplus relations, as shown earlier

D�xatþ1 ¼ r � z0t; t ¼ 2; 3; . . .

where, as before,

z0t ¼ r�1 � �xtþ1 þ r � �dt � R � �xt� �

:

Thus one can write

z0tþ1 ¼ c � z0t; t ¼ 2; 3; . . .

Third, combining the AEG-model,

P0 ¼ �x1=rþX1

t¼1R�t � z0t;

with the last recursive equation yields

P0 ¼ �x1=rþ R� cð Þ�1�z01 ¼ �x1=r þ R� cð Þ�1� 1r� D�xa2

¼ �x1r� g2 � g

r� g

� �

where g ” c ) 1, as before, and

g2 �D�x2 þ r � �d1

�x1:

OHLSON338

One conceptualizes g2 as the growth in expected earnings, year 2 vs. year 1, andwhere r � �d1 corrects year 2 earnings for the earnings foregone due to year 1 divi-dends. This model, developed in Ohlson and Juettner (2004), thus explains value as afunction of g2 (short-term growth in earnings), r (the discount factor) and g whichwill, under reasonable conditions on the dividend policy, correspond to long-termgrowth in expected earnings, g ¼ D�xt=�xt�1 as t fi ¥. Book value is nowhere to befound, and, to be sure, it cannot be inferred from �x2;�x1; �d1

� �(the accounting data

that suffices for P0).Looking at the two valuation expressions reveals their different accounting vari-

ables yet similar structures. As noted, the first should be thought of as a book-valuemodel. Because RoE1=(b1+d1 ) b0 )/b0, one naturally writes P0 ¼ f b0; �b1; �d1

� �.

With respect to the second valuation expression, P0 ¼ f �x1=r;�x2=r; �d1� �

� h �x1;�x2; �d1� �

.In words, the second valuation expression should be thought of as an earnings modelof equity value. From a ‘‘structural’’ point of view, the second expression is the sameas the first in that �x1=r;�x2=rð Þ simply replaces b0; �b1

� �. To maintain this symmetry,

one can read RoE as ‘‘growth in book value adjusted for dividends’’ and g2 as‘‘growth in earnings adjusted for dividends’’.It is readily shown that the book value model cannot be valid unless �xa

tþ1 ¼ c � �xat ;

and the earnings model cannot be valid unless D�xtþ1 ¼ c:D�xat . Moreover—and this

point is critical indeed—whereas �xatþ1 ¼ c � �xa

t implies D�xatþ1 ¼ c � D�xa

t , the conversedoes not hold. (For a proof, simply note that �xa

tþ1 ¼ c � xat þ k; k 6¼ 0; also implies

D�xatþ1 ¼ c � D�xa

t ).By way of summary, the following has been proved.

Proposition III Assume P0=PVED and clean surplus accounting. Consider thefollowing two models

A � �xatþ1 ¼ c � �xat ; t � 1

which implies, and is implied by,

P0 ¼ b0 þ�xa1

R� c:

B � D�xatþ1 ¼ c � D�xat t � 2

which implies, and is implied by,

P0 ¼�x1rþ 1

r� D�xa2R� c

:

Model A then implies model B, but the converse is false. j

Disregarding the mathematical aspects of the two models, the following statementcaptures the Proposition’s essence. If the valuation expression P0 ¼ f b0; �b1; �d1

� �

applies, then so does P0 ¼ h �x1;�x2; �d1� �

. The converse does not apply, however. Themodel focusing on earnings is therefore distinctly more robust than the one focusing onbook-value.


No apparent reasons suggest why one should hone in on the book value model asa practical matter. This model is arguably of interest only because it becomes part ofa demonstration why earnings rather than book values provide the focus in equityvaluation.Model B is without question an earnings model: Its accounting data

input—earnings, dividends or change in residual earnings—do not suffice to inferany book value. If one writes Dxa

tþ1 as a function of bt+1,bt,bt - 1, and suppressesdt+1,dt, i.e., Dxa

tþ1 ¼ Dxa btþ1; bt; bt�1ð Þ, then Dxa btþ1 þ k; bt þ k; bt�1 þ kð Þ does notdepend on k. Again, the ‘‘canceling errors’’ concept manifests itself. In sharp con-trast, xt+1

a, by itself and without the delta-sign, satisfies no such property. Nor willany ‘‘canceling error’’ concept apply to the book-value valuation expression. GivenP0 ¼ f b0; �b1; �d1

� �, the function f b0 þ k; �b1 þ k; �d1

� �depends on k. (An exception

pertains to the singularity when P0 ¼ �x1=r and �xat ¼ D�xa

t ¼ 0).Allowing for capital transactions in the analysis has the usual implications. It

causes problems in a context of clean surplus accounting on a per share basis.Specifically, putting Model A on a per share basis leads back to the issues raised inSection III. The recursive residual earnings equation requires, at least in principle, aresolution to the awkward question how clean surplus applies. Does bvps or epsserve as a plug in the clean surplus relation? This question cannot be avoided unlessone dismisses concepts of earnings. That is, one has to assume there exists someappropriate bvps construct such that

bvpstþ1 þ dpstþ1 � R � bvpst� �

¼ c � bvpst þ dpst � R � bvpst�1� �

for t=1,2,.... Model A so modified avoids capital transaction problems, but at thecost of being devoid of any eps- concept except as a forced definition,epst � Dbvpst þ dpst.Model B avoids the capital transactions and clean surplus issue because it requires

no bvps-construct. Consistent with the real world, the model centers around eps andits future increments, without any reference to how it may reconcile with any bvps-construct. With respect to the underlying sequence of expected outcomes, it actuallymisleads to refer to the equation D�xa

tþ1 ¼ c � D�xat insofar it suggests a role for clean

surplus and residual earnings. Given the generic irrelevance of book values, the moregeneral, Ohlson-Juettner, version of Model B eliminates clean surplus. Thus, it re-duces to the following, which will be referred to as Model C:

z0tþ1 ¼ c � z0twhere z0t � epstþ1 þ r � dpst � R � epst implies, and is implied by, the valuationexpression

P0 ¼eps1r� g2 � g

r� g

� �:

ModelC raises legitimate questions about the extent towhichGAAP’smeasure of epsaligns with the model’s dynamic requirements. These questions are not easily resolved,especially in light of the complexities associated with the accounting for capital trans-

OHLSON340

actions. It is also unclear how theoretical concepts, such as matching or the role ofaccruals, fit into Model C’s restrictions. Nevertheless, in spite of these ambiguities, thebroader point needs to be underscored.9 Put succinctly,

Model A)Model B)Model C

whereas the converses do not hold. Because all 3 models have the same single degree offreedom—growth (or c)—only Model C is of interest as a practical matter. And thismodel makes no reference to the clean surplus relation or restricts how book values(adjusted for dividends) are expected to evolve over time.10, 11

9. Concluding Remarks

This paper has examined RIV in a somewhat critical light, but these observationsshould not overshadow that the RIV model has contributed to accounting theory.Two points deserve to be mentioned.First, one can use RIV to identify residual earnings as a measure of a firm’s

value creation. By taking the present value of the expected residual earnings, thedifference between market to book values comes into focus. Many students ofaccounting undoubtedly have found the underlying analyses insightful. Amongother things, these have shown that the market-to-book ratio relates directly to afirm’s expected profitability, and that a firm’s profitability in turn relates toconservative accounting. Second, RIV captures settings with dividend policyirrelevancy effectively. Under reasonable conditions, residual earnings do notdepend on dividends. This independence property usefully shows a close con-nection between accounting constructs and dividend policy irrelevancy. Moreover,RIV streamlines the analysis of formal models because there is a simple way ofevaluating PVED that does not require an explicit sequence of expected divi-dends. Ohlson (2001) emphasizes this point. But it also follows that core insightsof such models in no way hinge on RIV. Nor do these analyses implicitly suggestthat RIV is a ‘‘natural’’ or ‘‘preferable’’ way of looking at equity valuation.Though analytical expediency can potentially lead to conceptual and practicalimplications, one need to keep in mind RIV may not be the only convenientscheme available to evaluate PVED.RIV’s most deficient aspect pertains to the elevated status it assigns to the current

and expected book values. It would be more accurate to relabel RIV as ABG, wherethe new acronym stands for abnormal book values growth. That is, the model takesthe present value of above/below benchmark increments in expected book values,adjusted for dividends, to explain the market minus book value premium. There isno need to refer to earnings or residual earnings. In fact, earnings enter the model viathe clean surplus relation in a somewhat underhanded way. And capital transactionsillustrate the contrived nature of relying on clean surplus to introduce earnings.Taken in its totality, it seems difficult to argue that RIV appeals conceptually or thatinvestors ought to rely on it as a premier, practical, valuation tool.


The analytical framework that leads to accounting-based valuation formulae showsthat RIV provides but one out of many possibilities. Within the available class, onecan avoid all of RIV’s disadvantages. One simply replaces RIV’s expected book valuewith the subsequent period’s expected earnings capitalized. This valuation approach,AEG, anchors valuation to expected next-period earnings capitalized, and then itexpresses this premium in terms of subsequent increments in expected earnings, ad-justed for dividends. The Appendix summarizes all the major properties of the model.Compared to RIV, AEG embeds three distinct advantages:First, the model demands no book value construct. Nor does the model rely on

clean surplus. The relatively weak assumptions mean that eps is as easy to work withas total dollar earnings. And the possibility of changes in shares outstanding has noadverse implications per se.Second, a focus on earnings can never be worse than a focus on book value,

but the converse is false. As the three propositions of this paper suggest, thepower of the partial ‘‘canceling error’’ concept seems undeniable. It leads to theimportant idea that, ex ante, earnings capitalized approximate market value moreclosely than book value.Finally, investment practice revolves around earnings and their subsequent

growth, not book values and their subsequent growth. AEG, not RIV, builds in thiscentral organizing principle of equity valuation. RIV has arguably been ‘‘oversold’’by text-books as a practical tool in investment analysis; a switch to AEG seemswarranted. All the theoretical arguments support this contention.

Appendix

The Abnormal Earnings Growth (AEG) Model Summary

Formula

P0 ¼e�ps1rþX1

t¼1R�tzt

where

zt � epstþ1 þ r � �dpst � R � e�pst� �

=r ¼ De�pstþ1 � r � e�pst � �dpst� ��

=r

The term r � epst � �dpst

� �modifies Depstþ1 for ‘‘the expected increment of period

t+1 earnings due to earnings retained during the prior period, t’’. Hence the specialcase zt ¼ Depstþ1=r applies only if the per share payout is 100%. The payout doesnot, however, affect zt under suitable assumptions; see IV below.

Properties

(a) zt=0 if

(i) the ‘‘firm’’ is a savings-account

OHLSON342

(ii) the firm uses mark-to-market accounting (in expectation)

(iii) the firm uses permanent earnings accounting (in expectation) Note that (i) is aspecial case of (ii), which in turn is a special case of (iii).

(b)P0 > �eps1=r if zt > 0

That is, positive expected abnormal earnings growth implies the price-to-forward-earnings ratio exceeds the benchmark, 1/r.

Baseline Models

(a) Suppose that for all t‡1

ztþ1 ¼ c � zt; 1 � c:

Then

P0 ¼eps1r� g2 � g

r� g

� �

where

g2 ¼Deps2 þ r � �dps1

eps1(short-term growth)

g � c� 1 ¼ limt!1

Depstþ1=epst(long-term growth)

Solving for r leads to the cost-of-capital formula

r ¼ Aþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A2 þ eps1P0� Deps2

eps1� g

�s

where

A � 1

2gþ dps1

P0

�

As a first special case, if g=0 then r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPEG�1

p

where

PEG � P0=e�ps1ð Þ=g2As a second special case, if Deps2

eps1¼ g then the constant growth model applies and

r ¼ gþ dps1=P0:(b) An assumption of Clean Surplus added to ztþ1 � czt. If

xat � residual earnings


then clean surplus and the definition of zt imply

zt ¼ D xatþ1=r

It also follows that

P0 ¼e�ps1rþ 1

r� D�xa2r� g

Note: This model, (b), is strictly less general than (a).(c) �xa

tþ1 ¼ c � �xt in liu of D�xatþ1 ¼ c � D�xa

t . As one—but only one—of the solutions toD�xa

tþ1 ¼ cD xat , consider

�xatþ1 ¼ c � �xat :

In that case

P0 ¼ b0 �R�oR1 � g

r� g

� �¼ b0 þ

�xa1r� g

where b0 = date 0 book-value (per share).Solving for r leads to r ¼ g � P0 � b0ð Þ=P0½ � þ �x1=r; that is r is linear in the book-to-

market premium and the forward-earnings yield. (This model does not seem to beknown by finance researchers who suggest (somewhat vaguely) that the market-to-book ratio should relate positively to risk.)Note: The model above, (c), is strictly less general than (b).

MM-considerations

In I, div-policy irrelevancy applies if (i) ¶eps1/¶dps0=) r and (ii) @zt=@dt�s ¼ 0:These conditions correspond to the idea that the payment of dividends on the marginhas the effects of lowering next-period eps with )r for each dollar of dividends.In III (a), dividend policy irrelevancy applies in the following sense: Consider the

dynamic

epstþ1 ¼ R � epst � r � �dpst þ r � z0t

z0tþ1 ¼ c � z0t

dpstþ1 ¼ d1 � epst þ d2 � dpst þ d3 � z0t:Then PVED does not depend on d1, d2, d3.The above analysis does not depend on whether one assume clean surplus or not.

Thus, it applies to model III (b) as well.With respect to III (c), consider

�bþ �d� �

tþ1¼ R � �bþ �d� �

t�R � �dt þ z00t

z00tþ1 ¼ c � z00t

OHLSON344

dpstþ1 ¼ d1 � �bt þ d2 � �dpst þ d3 � z00tThen PVED does not depend on d1, d2, d3.

Acknowledgments

The author wishes to thank seminar participants at the Chinese University inHong Kong and Stockholm School of Economics for useful interactions. Specialthanks are due In-Mu Haw, Junghun Lee, Lee-Seok Hwang, Steve Penman, StinaSkogsvik and Nir Yehuda.

Notes

1. This paper depends significantly on some pre-existing papers of mine, in particular, Ohlson (1999) and

(2002), and Ohlson and Juettner (2004). Penman (2004) discusses many of the issues associated with

residual income valuation. Also, an earlier version of the current paper exists, ‘‘Residual Income

Valuation: The Problems.’’

2. Nowadays, a large number of text-books discuss RIV. Examples include Bernstein and Wild (1997),

Palepu et al., (2000), Penman (2004), Revsine et al., (1999), Soffer and Soffer (2003) and White et al.

(2003). Theoretical books concerned with RIV are Beaver (1998), Christensen and Feltham (2003),

and Christensen and Demski (2003). Peasnell (1981, 1982) provides the basic ‘‘modern’’ development

of RIV.

3. The expression defines residual earnings if one eliminates earnings via the clean surplus equation.

4. Another way of thinking about AEG recognizes that RIV ‘‘applies for all accounting principles’’.

Hence, one can ‘‘measure’’ bvpst in terms of epstþ1=r.5. See for example Gephardt, Lev and Swaminathan (2001), Claus and Thomas (2001), Easton, Taylor,

Shroff and Sougiannis (2002), Easton (2004).

6. Ohlson and Zhang (1999) analyze issues related to bounded rationality and the terminal value

problem. Penman (1997) develops analytical results showing how various terminal values reconcile

across valuation schemes.

7. This permanent earnings concept is due to Ryan (1988).

8. Reference to this model can also be found in Brief and Lawson (1992), Easton (2004), Fairfield (1994),

Gjesal (1999) and Hutton (2000). All of these papers use too restrictive assumption. Either they

assume �xtþ1 ¼ c(�xat for t=0 as well as t ‡ 1 or they put restrictions on the dividend policy.

9. One can potentially deal with this issue using the research methodology found in Ohlson and Zhang

(1988).

10. A number of papers exploit the Ohlson and Juettner model to estimate firms’ cost-of-capital: Botosan

and Plumlee (2002), Easton (2004), and Gode and Mohanram (2003).

11. Ozair (2003) identifies information dynamics that sustain the Ohlson and Juettner model.

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