a synthesis of security valuation theory and the role of...

A Synthesis of security valuation theoryand the role of dividends,cash flows, and earnings*

JAMES A. OHLSON Columbia University

Abstract. The paper reviews and synthesizes modem finance valuation theory and theways it relates to the valuation of firms and accounting data. These models permit un-certainty and multiple dates, and the concept of intertemporal consistency in equilibriabecomes critical. The key conclusions are (1) the basic theoretical insight derives from apowerful condition of no arbitrage; there is no role for complete markets in basic valua-tion theory; (2) only anticipated dividends can serve as a generically valid capitalization(present value) attribute of a security; (3) the notion of risk is general, and models suchas the CAPM occur only as special cases; (4) the notion that one can capitalize cashflows rather than dividends requires additional (relatively stringent) assumptions; (5) ex-isting theory of "pure"eamings under uncertainty lacks unity regarding their meaning andcharacteristics. It is argued that only one concept of "pure" eamings makes economicsense. In this case eamings are sufficient to determine a security's pay-off, price plusdividends, consistent with some prior research but inconsistent with others.

Resume. L'auteur procede a I'examen et a la synthese de la theorie modeme derevaluation financiere et de la fagon dont elle se rapporte a revaluation des entrepriseset des donnees comptables. Les modeles utilises prevoient les cas d'incertitude et dedates multiples, et la notion d'uniformite intertemporelle en situation d'equilibre revetune importance critique. Les principales conclusions de l'auteur sont les suivantes: 1) leprincipe theorique fondamental derive d'une forte situation de non-arbitrage; 2) seuls lesdividendes anticipes peuvent servir d'attribut de capitalisation (valeur actualisee) valided'un titre sur le plan generique; 3) la notion de risque est generale, et les modeles tels quele modele d'equilibre des marches financiers ne se vedfient que dans des cas particuliers;4) la notion de capitalisation des fiux mon^taires plutot que des dividendes necessitedes hypotheses supplementaires (relativement rigoureuses); 5) la theorie existante desbenefices « purs » en situation d'incertitude manque d'unite en ce qui a trait au sens etaux caracteristiques de ces benefices. L'auteur afflrme que la revue qu'il a effectvee detravaux anterieurs I'amene a conclure qu'un seul concept de benefices « purs » se justifiesur le plan economique pour determiner le produit d'un titre qui comprenne a la fois lepdx et les dividendes.

* The author has benefitted from comments of the participants of workshops at the Universityof Chicago and Washington University (St. Louis). Special thanks are due the reviewer of thispaper, Jerry Feltham, and Trevor Harris, Russ Lundholm, Shashi Murthi, and Steve Ryan.

Contemporary Accounting Research Vol. 6 No. 2 - 11 pp 648-676

A Synthesis of Security Valuation Theory 649

Introduction and SummaryAccounting research frequently views accounting data as relevant in securityvaluation. This infonnation perspective motivates the empirical studies thatinvestigate the relation between security prices and accounting variables. In spiteof the considerable success of this research, much of it is confined because thehypotheses investigated do not relate to any theoretical consti'ucts.

The absence of theory use in empirical research impinges on its futuredevelopments, and progress will probably require more sharply delineatedhypotheses than typically has been the case. A number of recent papers—^notablyBeaver, Lambert, and Morse (1980); Beaver, Lambert, and Ryan (1987); Collins,Kothari, and Raybum (1987); Collins and Kothari (1988); Daley (1984); Easton(1985); Easton and ZmijewsM (1987); Kormendi and Lipe (1987); Lipe (1985);and Ryan (1986)—^recognize the limits to "brute empiricism," and they considerformal security valuation models prior to the empirical hypotheses. However,this research taken in its totality leaves a disjoint and ad hoc impression regard-ing the underlying theoretical constructs. Some papers embed the analysis inthe Litzenberger and Rao (1971) valuation model, and the capital asset pricingmodel (CAPM), while others do not. Another source of confusion arises becausethe security valuation models make reference to at least three different capitaliza-tion (discounting) attributes: cash flows, eamings, and dividends. Moreover, theprecise capitalization formula used to discount a given attribute is by no meansstandardized. The cited papers neither reconcile the variety of valuation modelsnor explain the extent to which they derive from different assumptions aboutthe economy. Yet another problem stems from notions that complete marketssomehow make a difference.

To clarify the issues raised by the lack of theoretical unity, this paper reviewsand synthesizes the theory of security valuation for multiple-date settings withUQcertainty. The key concepts presented constitute an integral, well-accepted,part of modem financial economics. I believe that only by extending theseconcepts to explicitly model the behavior of infonnation variables can one expectto extract theoretically sound and rich implications concerning an informationperspective on accounting.

The focal point of the discussion concerns the most complex aspect of se-curity valuation, namely, intertemporal valuation consistency. A multiple-datesetting requires that sequences of information realizations map into sequencesof endogenous security prices, and yet the derived expected returns must reflectthe underlying market risk. This instrinsically dynamic analysis cannot place adhoc or tautological restrictions on the factors that determine security returns.Following the development of an appropriate model framework one can derive(1) valid capitalization (discounting) concepts as they relate to security price;(2) the relation between current security price and current infonnation (such asearnings); (3) the relation between cuixent security price and expectations of fu-ture information realizations; and (4) the relevance of, respectively, cash flows,dividends, and eamings in security valuation.

650 J.A. Ohlson

The major results can be summarized as follows.

1 The central theoretical insight derives from a powerful condition of noarbitrage (within perfect markets). This weak equilibrium requirement on thesecurity prices leads to the existence of implicit (consumption) prices thatultimately determine the value of an uncertain dividend stream. Completemarkets are redundant in this analysis, and, more generally, such an assump-tion provides no additional results.

2 As a corollary of point 1, only (anticipated) dividends can serve as a generi-cally valid capitalization (present value) attribute of a security.

3 The theory results in a formula that determines security value as a function ofexpected dividends adjusted for their risk and discounted by the term-structureof risk-free rates. The notion of risk is general yet meaningfully identified.Models such as the CAPM occur only as special cases. The same can be saidfor the fonnula that capitalizes expected future dividends using compoundedexpected future security retums as discount factors.

4 The notion that one can capitalize some measure of anticipated cash flows(rather than dividends) holds only under extremely restrictive assumptions,and dividend policy irrelevance (i.e., conditions for "MM") is necessary butnot sufficient. Moreover, the cash flow measures capitalized are in generalex-post unobservable.

5 Existing theory of "pure" eamings tinder uncertainty lacks consistency re-garding its meaning and characteristics. Three distinct approaches can beidentified, and two of these are arguably conceptually deficient. (1) Eamingsused as a capitalization attribute i.e., the (discounted) present value of fu-ture (risk-adjusted) expected eamings leads to the same value as capitalizeddividends. This valuation scheme applies in special cases, but the condi-tions required are stringent and seemingly devoid of economic content. Thesemodels do not make a clear economic distinction between eamings and div-idends. (2) A more promising idea views pure eamings as an infonnationvariable that suffices to determine a security's payoff, price plus dividends.This eamings concept leaves the dividend policy unrestricted, and it does notdirectly depend on notions of eamings capitalization. (The two pure eamingsconcepts (1) and (2) are mutually exclusive in some settings.) (3) Eamingsas an infonnation variable that suffices to determine a security's value, price,exclusive of dividends. This concept is deficient because it contradicts basicnotions of dividend policy irrelevance, and the implications become unaccept-able in the certainty case. In sum, only concept (2)—which is due to Ryan(1986)—-satisfies properties that make good economic sense.

The paper develops the necessary theoretical constmcts from scratch. Thenext section considers security valuation within a simple two-date economy.The analysis of this setting identifies the central concept of no arbitrage andthe existence of implicit (consumption) prices. The third section extends themodel to multiple dates, and it develops Rubinstein's (1976b) fundamental re-


suit of how to capitalize anticipated dividends.' The fourth section considerssimplified versions of the dividend capitalization formula and the fifth section re-views linear Markovian models, showing how security prices depend on realiza-tions of information variables. The examples presented illustrate the workings ofintertemporal valuation consistency requirements. The sixth section analyzes thevalidity of cash flow (present value) capitalization. The subsequent section relieson the linear Markovian models to discuss the role of "pure" earnings conceptsin valuation theory. The final section briefly remarks on the use of theory inempirical research.

Security vaiuation within a two-date economyTo develop the theoretical underpinnings for security valuation, a simple two-date economy provides a useful point of departure. An economy that ceases toexist after one period clearly eliminates the need to determine future prices, andeach security's (dollar) payoff equals its terminal dividend payment. Multiperiodsettings, on the other hand, require more complex analyses because a security'spayoff equals the dividend pies the security price at the next trading date. Thetheory must determine prices for a sequence of dates rather than just one date,and this dynamic feature raises intertemporal equilibrium issues. Thus, to keepmatters simple, this section considers valuation at one date whereas the nextsection extends the analysis to multiple dates.

For the two-date economy, assume that the uncertaint;/ consists of a finitenumber of states, s = l,...,S. Each security, jj — 1 , . . . , / pays a dividend ofdjs in state s, and dj = {dji,..., djs) defines the row vector of dividends acrossstates for security j . The J x S matrix

D = [dj,] =

summarizes the dividends over securities (rows) and states (columns).The analysis leaves D unrestricted, except that dj — ( 1 , . . . . 1). In other

words, the investment opportunity set includes a risk-free security (j = / ) .The current price of security/ is denoted by Pj, so that in a two-date economy

the retum (one plus rate-of-retum) on security j in state s is defined by r-js =djsjPj. The price Pj values a claim on the dividend pattern dj, and it excludesany dividend paid at the initial date. For j = / , /" / equals the cost of a unitcertain payoff; hence, Pj^ = RF is the risk-free retum. Let P = (P ] , . . . ,Pj)denote the column vector of prices.

The most basic issue in the theory of valuation addresses the following ques-tion.

1 The material in the second and third sections is we!! known in finance theory (e.g., see Gar-man, 1978, and Rubinstein, i976b).

652 J.A. Ohlson

Ql; What is the relation between the vector P and the matrix D? That is, givensome configuration of dividends across securities and states, can we saysomething about the price vector?

Because this question relies on few ingredients, more interesting results couldpossibly obtain if one also postulates the existence of probabilities (i.e., "be-liefs") over the states. Let n , > 0, J^^ 11 = l,s = 1 , . . . , 5, denote the exoge-nous state probablitities. Given these probabilities, one poses a slightly differentversion of question Ql by focusing on the collection of random variables {dj}j.

Q2: Given some probability distribution for {dj}j, what can be said about theprice vector F?

Question Q2 does not deal directly with the concepts of risk and retum,which are central in the modem theory of finance. This "parametric" perspectivesuggests that one should be able to determine a security's price in terms of threevariables; (1) the expected payoff (i.e., E[dj]), (2) the risk-free rate (RF), and (3)a variable that relates to a security's risk. Hence, one can raise a third question.

Q3: If Pj = f{Rf,E[dj],nskj), then what can be said about the function/ ( . , . , . ) , and about the nature of security risk, i.e., "risk/'?

The anaylysis next proceeds to answer questions Q1-Q3. The key economicconcept used powerfully restricts the investment opportunities; P and D jointlymust satisfy a no arbitrage (NA) condition. It goes almost without saying thatan equilibrium cannot exist unless the economy excludes (pure) arbitrage op-portunities. To develop the concept of NA, let the row vector A, = (A,i,..., A,/)denote a portfolio with Xj units of shares invested in security /. For Xj > 0the position is "long," and Xj < 0 represents a "short" position. Markets areassumed perfect, i.e., the model disregards transaction costs, including such realworld phenomena as taxes and margin requirements for short positions. The costof any portfolio X therefore equals X-P. The payoff of a portfolio X in state sequals Ylj Xjdjg-, thus, to sununarize the row payoff vector across states, writeYljXjdj or AZ).

The definition of no arbitrage simply means that one cannot get "somethingfor nothing." Formally, define NA as follows.

Definition (No arbitrage)There exists no portfolio X such that J2j h'^k - 0, all s, Ylj h^J ^ 0, and whereat least one of the 5 + 1 inequalities is strict.

It should be emphasized that NA makes no reference to probabilities; if NAapplies for a vector 11' > 0, then it applies no less for any other vector II" > 0.NA is not a "subjective" matter given that the model unambiguously suppliesthe set of possible states.

The NA condition permits a direct link between P and D.


Lemma 1 (Steimke's Theorem; see Mangasarian (1969, p. 32). There exists apositive vector R = {Ri,... ,Rs,... ,Rs) > 0 such that

if, and only if, NA holds.Lemma 1 answers question Ql: given NA, one values a dividend pattern dj

by multiplying 4s with Rs and then adding up over the S states. More generally,the R vector values any portfolio's payoff pattern because the value of 52/ '^j<^jequals YlsC^j h^p) ' ^s = Ylj '^M " ^) = J2j h^i- The economics andfinance literature generally refers to R as implicit (consumption) prices, state-contingent prices, or support prices. Thus, Rs is the implicit price of a claimto one unit of dividends (consumption) in states s. A unique if obtains if, andonly if, markets are complete (i.e., the collection of vectors {dj}j generates ao S-dimensional space). (The result follows from basic linear algebra.) The potentialnonuniqueoess property of i? introduces no problems. Moreover, the dichotomybetween complete versus incomplete markets does not influence the substantiveconclusions in the theory of valuation, a point which will become apparent later.

In contrast to any discussion of question Ql, to deal with questions Q2 and Q3the probability vector IT comes into play. Its role is surprisingly modest, and thestraightforward manipulations that follow may at first glance appear odd becausethese introduce 11 through the "backdoor." The procedure and related analyticalsimplicity will highlight the essential equivalence between the "random variableperspective" on dividends and the apparently more primitive "mapping fromstates to payoffs perspective."

Define

The random variable Q satisfies Q> 0 with probability one. Further, EIQ] = 1since E ' ^ J = E^s^/^ = -P/ = Rp^. (Recall that dj = (1,. . . ,1)). One caninterpret Qs as a state i' consumption price normalized by the uncertainty of thestate and the discount factor.

Using Lemma i it follows that

and, since cov[4-, Q] = EldjQ] - EldjMQ] = E[djQ] - E[dj],

Pj = Rj'{E[dj] + coY[dj, Q]}, all J = 1 , . . . , / .

The last expression answers question Q2 in the following sense: AssumingNA, there always exists a random variable Q,Q > 0 and E[Q] = 1, such that—cov[5j, Q] naturally indicates a security's risk. The general valuation formulaalso partially answers Q3, because the formula bears on the structure of the

654 J.A. Ohlson

function Pj = f(Rp,E[dj],mkj). Even so, compared with Ql and its answer,the above analysis merely replaces the valuation operator R with the randomvariable Q. In other words, given some Rf and 11, one infers Qs from Rs, andconversely. And similar to R, Q is unspecified aside from its existence and basicproperties ( e > O , £ [ G ] = 1).

Sharper insights regarding the determinants of value obviously require thatmore be said about Q (or R). The CAPM serves as a pertinent illustration. Thisequilibrium model specifies/(/?f,£"[5,], risky) as

where ^ > 0 is some constant independent of;, and 5^ = Y^-dj denotes theaggregate dividend.^ (Without loss of generality, assume unit supply for eachsecurity.) Hence, to identify the CAPM within the more general framework oneputs Qs = Ki+ K2d,ns, where I = Ki-¥ K2E[d.rn\ and K2 = -K < 0. TheCAPM is therefore based on assumptions such that the uncertainty normalizedconsumption price, Q , is a linear function in the aggregate dividend, d^s-

The CAPM yields a statement about Q because the model effectively assumesan economy in which a "representative individual" with quadratic utility deter-mines the prices.^ The idea of a representative individual permits generalization.Specifically, represent the preferences of this individual by

pt/(c) + £

where c denotes current consumption, Ylj h^j is future (uncertain) consump-tion given portfoho A,, U{-) denotes the utility function, and p is a patienceparameter. As is standard in finance theory, U(-) is monotonically increasingand differentiate i.e., dU{x)/dx = u(x) > 0.

Maximizing the preference function with respect to c, X and simplifying therelated optimality conditions one obtains'*

2 The CAPM is normally expressed as Elrj] —Rf= cov[ry, f^] x K, where ry = dj/Pj and K ={£[rm] — Sp}/Var[rm]. However, the above relation implies the one in the text if one definesK asK = K/P^, where P^ = market value of all sectffities.

3 See Rubinstein, 1974.4 Specifically, the optimality conditions follow from

rmaxpU(c) + E \ U [

^ V j

subject to the budget constraint

c-^X-P ^ w ,

where w is endowed wealth.


P- — F

Further, since Pj — Elu('}2j Xjdj)]/pu(c) = Rj^ one derives that

/E

Note next that aggregate supplies of shares equal the equilibrium holdings,A,* = ( 1 , . . . . 1). One thus obtains

The last expression clearly resembles Pj = Rp^E[djQ]. More precisely, one canput

and it follows immediately that Q satisfies its two basic properiies: Q >0, since«() > 0, aod E[Q] = \. The variable Q relates directly to the preferences ofthe representative individual and the payoff of the market portfolio.

CAPM, as usual, illustrates the concepts. Suppose that lj(x) — — j(0—x)^, sothat u(x) = 0 —X, where 9 denotes the representative individual's risk-aversionparameter. In this case «(a«)/£[M(4)] = (9 - dm)l(^ - E[d„,]) = K^-i- Kjd^where Ki = 6/(0 - £[5^]) and -i^z = 1/(6 - £[5^]).

Question Q3 has now been answered; The notion of security risk is logicallycaptured by —cov[5,-, u(drn)/E[u(drn)]].^ The analysis therefore identifies risk asa concept the precise determination of which depends on the preferences andaggregate dividends in the economy. Although this concept of risk is easy tointerpret, it does not suggest that one must identify Q as u(dm)/Elu(dm)]. Ifmarkets are incomplete, one can find other Q that lead to the same measure ofrisk. But this possibility is of no economic signficance and one might as well putQ equal to u(dm)/E[u(dm)], which is necessary in the complete markets case.Altemativeiy, because the space spanned by the vectors {dj}j is irrelevant in thederivations, an assumption of complete markets—including a complete set ofArrow-Debreu securities—entails no loss of generality.

Security valuation within a multiple-date economyThe simple two-date model generalizes to handle multiple dates with surprisingease. The flow of ideas and concepts—^the use of NA in particular—parallelsthose of the previous section. The major new difficulty relates to the development

5 More genera!ly, absent a representative individual, one can put Q = Ui{d')/E[ui{d')] where /refers to the ;* individual: —«,• is margina! utility and d' is the dividend from the equilibriumportfolio. Note that one can use any i, yet coy[dj, Q] does not depend on which / one picks.We further note that prefereeces of a representative individual always exist if one assumes thatthe economy achieves (ful!) Pareto efficiency. See Huang and Litzenberger (1988).

656 J.A. Ohlson

of a framework that resolves uncertainty with the passage of time. Such modelunderpinnings are necessary because the resulting theory must determine thestructure of security values at all dates and under all circumstances. As willbe seen, information about securities takes on a concrete role in a multipledates setting. This model enrichment occurs because the passage of time isconceptually and mathematically equivalent to changes in the environment asdescribed by the available information.*

To generalize the two-date theory, let Zt denote the available information atdate t. Thus, the economy's uncertainty is resolved by observing Z(. The infor-mation, or "environmental description," is generic, so that z, in its most generalform defines a set, i.e., an "event." A sequence of realizations z\,...^z, gener-ates a history of the economy from its inception at date 1 through t. Withoutloss of generality, z, can be defined as a subset of some z when t > x. How-ever, to maintain this subset relationship becomes notationally burdensome inparsimonious Markovian settings, and in such cases one usefully thinks of z, asexcluding those aspects of the environment that do not affect the valuation ofsecurities.

Because the state description includes all relevant aspects of the environment,there exists j exogenous functions dp = dj(zt) determining the dividends foreach security j at date t given event z . Hence, one can view the dividends aseither part of the event description, or equivalently as a function of the eventdescription.

Functions F,-, = Pj{zt) determining the security prices must also exist becausethe security prices depend on the available information (event description). Un-like the dividends' functions, however, these price functions are endogenousand depend on all facets affecting tlie equilibrium. We will refer to Pjiz,) as thevaluation function of security j .

The NA two-date analysis can now be applied without difficulty for a se-quence of dates. A security's payoff given the event description z, equalsPj{z,)-¥dj{z,). Ruling out arbitrage between two adjacent dates, t, ?+ 1, Lemma1 implies that

for some positive implicit prices i?(z,+i; z,), all z,+i, z,. The notation J ^ ^ shouldbe read as the sum over all conceivable z,4.i given z,. Note that R depends onZt as well as z<+i, thereby conforming with the requirement that Pjt must be afunction of Zf Also, J^z ^fe+i;2() = RF(t+ l;z,)"', i.e., the sum defines theinverse of the risk-free rate over f, f + 1 given Zf.

The valuation function appears on both sides in the last expression. By sub-stituting recursively, one derives the cumbersome expression

6 The modeling of multiple-dates uncertainty economies is due to Radner (1968).

A Synthesis of Securitj' Vjiluation Theory 657

where

Riz,+2; z.) = ^ /f (z,+2; Zf+i )i?(z,+i; z,).

Continuing with this recursive substitution for Pj(zt+2),Pj(z,+3),... implies moregenerally that

(1)

where

hT-i;Zr).

The derivations clearly generalize the two-date model: depending on the cur-rent event description, the implicit prices R(zx', Zt), x > t, value any anticipatedstream of dividends (consumption) over future dates and across conceivableevent descriptions. For example, from the implicit prices one infers the term-stmeture of interest rates; Rpix; z?)"' ^ "^^ R(zx;z,) equals the cost of a certain(unconditional) unit payoff (consumption) at date x given the current event de-scription Zf

From the development of the two-date model, it should be apparent how onedeals with probabilities in the multiple-date setting. Let H(zx | z,) > 0 denotethe conditional probability of date x event description Zj given the current eventdescription z,, x > t. (The laws of probability apply in the usual fashion, andIl(z-c I Zt-i) = Y2^ H(zx I Zi)JJ(zt I Zt-i); recall that z, is a subset of z i or, moregenerally, provides a sufficient description of the environment at date /.)Define

Given z, and n(zt j Z(), the above definition induces a conditional probabilitydistribution for the random variables Qt+i{z,), Qr+2(zr),..., GxCzr),..., x > ?, suchthat Qt(Z() > 0 with probability one for each z,, and

E[Qn-{zt) I Zj] = 1, all x>t, and z,.

658 J.A. Ohlson

To simplify the notation, we suppress z, in Qx(zt) when the probability measureis conditioned on z,; thus.

Straightforward manipulations of expression (1)—which extend the two-datemodel in an obvious fashion—^now yield the expression that determines the valueof a security in terms of the probability distribution of future dividends giventhe current event description z,:

oo

Pj(Zt) = J2 RF(T,Zt)~^{E[djr I Z,] + COY[djr,Q^ I Z,]}.•c = r+l

The fonnula shows that one discounts the future expected dividends using theterm-structure of risk-free rates only after first having deducted the risk-measurescov[4(+i, —G;+i I Zt], cov[djt+2, —Qt+2 | z,],... fwm the expected dividends as ofrelated dates. Hence, in the general theory, the dividends capitalization formuladeals with risk by adjustments in the numerators, and not in the denominatorsas is frequently done in the literature.

Next, suppose that the economy consists of a "representative" individualwhose preferences are given by

where c, denotes consumption at date t. In equilibrium the individual holds themarket portfolio: Yljdjizt) = Ct = dm — dm(zt). From the first order optimalityconditions and some routine manipulations it follows that one can always put

Q(z,-z,) = u\d^(z,))lEW(d,n,) I zj,

where u'(-) = dU'(c,)/dc,.In summary, the following proposition answers questions Ql, Q2, and Q3 in

a multiple-date setting.

Proposition I (Rubinstein, 1976b): Consider an economy that excludes arbi-trage (within perfect markets). Let z, denote the event description, or availableinformation, at date f.

1 Then there exist functions R(z.^; z,) > 0, x > t, such that the value of eachsecurity j given the information Zt can be expressed as

CXI

Pj(^t) =

2 Given the infonnation evolution conditional probabilities n(zx | Z/) there existpositive random variables Q(zt;Z() satisfying E[Qx \ z,] = 1, all x > f, suchthat


where %(x; Zt) denotes the risk-free rate over an x — f date(s) interval.3 Given the preferences

U'{ct)+

of a representative individual who determines the equilibrium prices one canput

efe;Zr) = u\d,n{z^))lE[u\d^^) I Zj,

where dU'ict)/dct = u'(-) > 0, and d^t = dm{zt) — J^j^ji^t) denotes aggre-gate (random) dividends.

Possible simplifications of the dividends capitalization formulaIn the literature one frequently encounters valuation formulae that use (com-pounded) expected retums as discount factors rather than the risk-free rates,and they leave out any risk adjustment in the numerator (e.g., Christie, 1987;Collins and Kothari, 1988). These approaches to valuation seem to be motivatedby the idea that expected retums must reflect risk, and because the discountingby expected retums incorporates the risk factor, a numerator adjustment such asthat in Proposition I, part 2, is unnecessary. Specifically, one may consider therelevance of the formula

z j ,

where rp = (Pjizt) + dj(zt))/Pj(z,-i) denotes the retum on security j over thedate interval from / — 1 to t. However, this scheme is valid only under restrictiveassumptions (see below), and it should not be accorded any theoretical stand-ing. One can constmct reasonable (albeit tedious) examples violating the aboverelationship.^ (These examples also show that interchanging n and E in theformula still leaves it invalid.)

To salvage the above approach to valuation, one generally needs to assumeconstant expected retums, i.e., Eirj,+i | Zt] = jj.;, independently of z, for all t.Conversely, if the value of a security equals expected dividends discounted bysome fixed parameter, the parameter equals the expected retum.

Proposition II (Samuelson, 1965): The following two statements are equivalent:

7 AE example showing that the valuation formula is false is available from the author uponrequest.

660 J.A. Ohlson

A. E[fjt+i I Zt] = \ij independently of z, for all ?;

and

The hypothesis of Z(-independent expected returns is difficult to obtain in theo-retical analyses unless one also restricts the term stracture of interest rates to be"flat" and nonstochastic, i.e., Rf(x;z,) = R]r' independently of z . In this case,lij — Rf = cov[r/(+i,—g(+i I Zt] Gj equals security risk. The result followsfrom the analysis in the previous section since

which futher simplifies to

i.e., |Li;- — Rp = cov[r,v+i, —Qt+i j z,] as claimed. Hence, one now can expressthe value of a security as

(jRf + Oy) E[djx I Zj],

so that the discount factors incorporate risk.In spite of the derivations' simplicity and economic appeal, we emphasize

that they depend on constant (information-independent) expected returns. Themore general approach captures risk by adjusting the expected dividends asopposed to the discount factors, and only this approach attains validity in all no-arbitrage economies. (See Bar-Yosef and Leland, 1982, for further discussionof risk-adjusted discounting.)

Another limitation associated with the simplified dividends capitalization for-mula of Proposition II should be pointed out. The model treats the expectedreturn (and risk) as an exogenous parameter, and thus an important questionremains unanswered: What factors determine expected returns? In contrast, thegeneral valuation theory implies an endogenous expected return. As some of theexamples in the next section illustrate, the question raised permits systematicanalysis, even if the expected return happens to be zrdependent.

The analysis of linear modelsThe general theory can be used to discuss several important aspects of an infor-mation perspective on accounting. To make these points as concrete as possible,this section reviews parsimonious models of valuation. All of these models re-sult in linear, "closed form," valuation functions, which one derives withoutdifficulty given appropriate linear modeling of the stochastic behavior of the in-formation variables. The models' simplicity does not sacrifice any rigor because


the valuation derives from Proposition I, part 2, or Proposition IL Thus, theexamples also facilitate an understanding of the previously developed, relativelyabstract concepts.

Each of the examples embodies three points that should be kept firmly inmind. First, because dividends constitute the relevant valuation attribute, theanalysis derives a valuation function by modeling the stochastic behavior of div-idends. Second, the valuation relevant information, Zt, relates to the informationthat predicts the present value of dividends, or, more precisely, the informationthat affects the quantities £[5f+i | Zt],E[dt+z j Z;], . . . , and cov[Jj+i, —Qr+i | z,],cov[5^ .2, ~Qn2 \ ^t\, • • •• Third, the valuation function is endogenous, as is the(conditional) probability distribution of security returns. Thus, in the examplesbelow, one infers the expected rate of return, which may, or may not, dependon the information. Of course, such an exercise becomes impossible if the ex-ample departs from Proposition II, because such cases assume an exogenous,information-independent, expected return. This simplification also eliminates anyrole for the risk-inducing measures cov[5(+i, —g,. i | z , ] , . . . .

The examples are cast within a partial equilibrium framework. The term-structure of risk-free rates is flat and nonstochastic, and the distri:bution of theQi, Qt^i,... variables as they relate to the information and dividends is exoge-nous. The partial equilibrium approach rules out and makes it unnecessary toconsider how Q, and the risk-free rate relate to (expected) aggregate dividends.Part 3 of Proposition 1 showed how the analysis extends to identify fully risk interms of the preferences of a representative individual and aggregate dividends.In analyzing the relation between security price and information, the examplestherefore usefully illustrate the absence of a need to postulate general equilib-rium models such as the CAPM.^ Even so, some readers may find it instructiveand helpful to think of Qt as (a linear function of) the market portfolio return;the examples that follow will retain their substantive content.

Example I. (Adaptation of Miller and Rock, 1985) The setting has three dates:t = 0A,2. Terminal dividends are paid at date two, and "intermediate dividends"are paid at date one.^ Market values obtain as of the first two dates, zero andone. The stochastic evolution of dividends, and implicitly z,, is as follows:

8 See Rubenstein (1976b) and Huang and Litzenberger (1988) for discussions of muitiperiodversions of the CAPM.

9 TMs example modifies the Miller and Rock [1985] model, which discounts "earnings" ratherthan dividends at date one. In my mind, their model cannot make sense given any reasonablyprior ideas conceming the meaning of earnings.

The point becomes obvious in case of certainty. The Miller and Rock model implies thatX(=2 = RFPI=I where Rp is the risk-free rate plus one (i.e., x,=2 > ^r=i)- The result seemsodd because under certainty next-period earnings and current price relate by Xt+i = (RF — i)Pt,all t. More generally, given that x,+i = (Rjr — 1)P, under certainty it is readily shown that thepresent value of future earnings calculation J^t^i Rj''xt-n does not relate meaningfully to Pi.Of course, an exception to the latter occurs when x, = di, but this would seem to be of onlymodest interest.

662 J.A. Ohlson

where E[ii | ZQ] = E[e2 | zi] = 0; 0i, 82 and 83 are fixed and known parameters(83 relates to the so-called "persistence factor"). Hence, ZQ is "no information,"and z\ — {d\). The variable ei represents date one unexpected dividends. Onecan also equate z\ to {di,ei}, but the scheme is redundant because knowing disuffices to determine ei. The model implies that date one unexpected dividendsaffect the prediction of date two dividends:

£•[52 161] = 02-(-6183+8361.

By assumption, the model disregards risk: coN[Q2.,dz \ z\\ = cov[Q2.,d2 \zo] = cov[2i,5i I zo] = 0. Applying Proposition I.2., the specification impliesthe following equilibrium values:

As an exercise, one easily verifies that E[f\ \ zo] = E[f2 \ zi] = RF for all zi.

Example 2. (Rubinstein, 1976b, and Ohlson, 1979) This Markovian model has aninfinite number of dates. The information set is given by z, = {dt}, i.e., currentdividends are sufficient to predict dividends at all future dates. Specifically,

where it[l,. T | z j = 0, all x > 0 and z,, and cov[e(+x, —Gr+i: | z j = 0 forall X and z,. In this model 8 and o are known parameters, and 8 determinesthe expected growth in dividends since iiK+x | dt] =• Q^d,. The greater 8 andcurrent dividends are, the greater the expectation of future dividends. Also, aswill become apparent, a determines the risk inherent in the anticipated dividendstream.

Similar to the previous example, there is no point in expressing z, asZt = {rf;,e(} because e, can be inferred from dt,dt-i, and dt^i is irrelevantin predicting dividends paid at dates t+l,t + 2.,...

For e, s 0 one obtains the familiar dividends growth model due to Williams(1938). The example therefore generalizes this model by allowing uncertaintyand risk.

Using Proposition 1.2, a tedious but direct computation then shows that

P, = P(dt) = M , (2)

where

- (8 - a)).


IT current price therefore equals a fixed multiple of current dividends, wherethe multiple discounts the growth adjusted for risk, i.e., B = Rp^{Q)R^(Q

An easier, but indirect, way to solve for the last expression (2) is as follows.From the third section we know that the NA condition and the definition ofQ(zt+i; Zt) implies that in equilibrium

Piz,) = Rp^{E[P(zt+i) + dt+i I z,] + cov[P(z,+i) + 4+1,a+i I Zt]}. (3)

Hence, if one conjectures a linear solution P{zt) — P(dt) — Bdt for some yet tobe determined constant B that depends on i?F,8, CT, then

The dt variables scale the RHS and LHS of the equation and they can beeliminated:

Solving for B one obtains (2).It is important to note that solving (3) is the same as solving for the expres-

sion in Proposition 1.2 directly. Expression (3) meets the intertemporal valuationconsistency requirements. Subsequent examples also exploit this solution tech-nique.

Having solved for P{dt) one can derive the expected retum:

(S + 1)8/5 = i?f 8/(8 - 0).

Similarly, security risk is determined by

co¥[r(+i, - a + i I Zt] Elrt+i \ z,] - Rf =^ RFO/{Q - a).

The model therefore leads to a z^-independent expected retum and risk. Thelatter quantities increase as o increases, which makes economic sense.

Example 3. (Ohlson, 1983) As an elaboration of the previous example, considerthe information dynamics

Xr+i = (8] +

where £"[6 + | z,] = 0, A: = 1,2, x > 0, and cov[efo+T;5 —G«+x | '^t] = c* for allZt. Thus, the primitive information variable x, alone predicts future dividends,Zt ~ {xt\. For this dynamics one may think of Xt as a measure of earnings thatsuffices to determine price. The latter follows because jc, suffices to predict thefuture dividends. One then interprets %i as the expected grov/th in earnings, and©2 as a pay-out factor relating next date expected dividends to current earnings.

664 J.A. Ohlson

Using the same scheme as in Example 2, one conjectures that the solutionPt = P{xt) is linear in x,, i.e., P{xt) = Bxt for some 6. Inserting the conjecturedsolution into (3) implies that

Bx, = R

Eliminating Xj and solving for B yields

5 = (62 - a2)/iRF - (6; -

The solution thus shows that the value is a fixed multiple of earnings, and thismultiple reflects the risk-adjusted growth in earnings scaled by the risk-adjustedpay-out factor.

The zrindependent expected return equals

E[r,^i I 2,] = (Be, + Q2)x,/Bx, = 81+ Bz/fi.

Again, note that the (conditional) distribution of returns derives from the in-formation dynamics and the equilibrium valuation function. One shows withoutdifficulty that E[rf+i \ z,] > Rp if 01,02 > 0 (and Bi,01,92 > 0), and theequality holds if and only if Oi — G2 = 0. The risk in market returns thereforereflects the "primary" risk in the stream of dividends and earnings.

Example 4. (Gaiman and Ohlson, 1980) This model generalises both of theprevious examples by retaining the linear information dynamics but expandingZt to an n-dimensional vector. Let Z( s (xj , , . . . , Xn^-it, dt) denote a column vector.Using matrix notation, the information dynamics is given by

where Elegi+x \ z,] =0 and covleyt+x, —Qt+x | ^t] — Oy- One interprets Xj, as oneof n — 1 potentially relevant information variables, out of which one (/' = 1, say)may represent an earnings measure.

The solution Pizt) is linear in z,:

P{zr) = Bz, =

where B = (Bj , . . . ,Bn) is a row vector. Using the same solution method as inprevious examples, one can derive the solution for B:

B ' = [Rpln - [Qij - Oyf ]-'{Qij - Oij^'en,

where /„ is an « x « identity matrix, «„ = (0,0, . . . ,0,1) is an «-dimensionalcolumn-vector, and the superscript t denotes matrix (or vector) transposition. Theexpected rate of return is generally Zf-dependent because it equals the ratio oftwo functions that are linear in z . However, this expression becomes exceedingly


elaborate for large n because one must invert a matrix of size n to solve for theB-vector.

The above result shows that linear information dynamics implies linear valu-ation functions. The model is as general as one reasonably can expect becausethe analysis of nonlinear dynamics is unlikely to be fruitful. There are no rea-sons suggesting that other kinds of models can be used to determine the valueas a "closed form" function of a vector of information variables.

In contrast to previous examples, the information dynamics include casesin which current dividends affect future values of the information variables,i.e., dE{xia+i I Zt]/ddt ^ 0. This feature of the model is of potential economicimportance: descriptors of the fimi, such as eamings, may natuirally depend onpast dividends.

Ohlson (1983) analyzes the case for n = 2, that is.

4+1 = (621 + h\t*i)xi + (O22

and Pt = P(xt, d,) is linear

Let 0,y = Qij — aiji the solution for (^{,^2) equals

Bi =--B2IRFK-\ (4a)

B2 = [(Rp - 611)622 + QuenW'^ (4b)

and where K = (Rf- Bn)iRf - §22) -This model can be used to discuss the nature of concepts of "pure" earnings.

The solutioB for {Bi^BzX (4a) and (4b), is manageable, yet the model providessome richness in the ways current eamings and dividends can relate to futureeamings and dividends. Specifically, standard accounting suggests that d£[Jc,H.i |Xi,di]/ddt < 0 if one wants to interpret X[ as eamings, i.e., ceteris paribus, anincrease in current dividends should penalize future expected eamings. (See thenext section.)

Example 5. (Beaver, Lambert, Morse, 1980, as extended by Ohlson, 1989) UsingProposition II, this model postulates an exogenous and zrindependent expected

P(Zi) = li

The information dynamics is determined by

t+i = (63 + £it+i)xt + (04 + €2t+i)at +

666 J.A. Ohlson

where Z; = (x,, at) and E[at+i \ Zt] = E[lkt+\ | z,] = 0, = 1,2,3. In this modeldt does not predict future dividends given {xt.,at);Pt is therefore independent ofdt. Beaver, Lambert, and Morse (1980) interpret xt as "permanent" earnings andat as unexpected permanent earnings. Date t unexpected permanent earningstherefore affect the prediction of date t +1 permanent earnings. In a solutionPt = BiXt + B2at,Bi reflects the multiple associated with permanent earnings,and B2 the multiple associated with unexpected permanent earnings.

Conjecturing a solution Pt — B{Xt+B2at one obtains

t -i- 82a,) + (83X, -f 84a,)}.

The equation must hold for all values of Xt and at. Hence,

To solve for Bi and B2 in this system of equations obviously poses no problems.In this example it follows that the multiples Bi and B2 generally derive from

83 and 84 as well as from 81 and 82. The earnings process, by itself, does notdetermine the equilibrium solution. (As Example 4, special cases of this modelwill illustrate concepts of "pure" earnings.)

Note that linear information dynamics combined with the assumption onreturns in Proposition II yield a solution Pt = Bz, where

Qijjn, and £„ are as defined in the previous example. The use of Proposition IIrather than I implies that |J, replaces Rp, and 8y replaces 8,j — ay. Consistentwith the discussion in the fourth section conceming Proposition II, the notionof risk in this example is ad hoc (and exogenous) since the analysis does notconsider the Oy parameters, i.e., the risk inherent in the dividend stream itself.

The role of anticipated cash flows in valuationThe theory demonstrates clearly why future dividends across events and datesserve as the relevant valuation attributes: ultimately only payoffs count, anddividends alone can be consumed. The no arbitrage concept can be put to workgiven this payoff aspect of dividends, and one cannot replace dividends with cashflows, or earnings, as a capitalization attribute and thereby prove the existence ofsome (possibly alternative) set of implicit prices. As an empirical matter, cashflows or earnings might well take on an important function in the predictionof the present value of future dividends and thus in the valuation of securities.But this information function of cash flows or earnings does not automaticallyelevate any of the information variables to the status of a relevant attribute in apresent-value calculation.

This simple point conceming prediction of information variables must beappreciated because intertemporal valuation consistency requires the prediction


of all information variables that may affect values. The prediction of informationvariables merely reflects the recursive nature of predicting dividends further outin the future. Example 3 illustrates the process: the prediction of x,+i,Xr+2, • • • atdate t occurs because they relate directly to dividends at dates f -h 2, ? H- 3 , . . . .Aitematively, intertemporal valuation consistency requires the prediction of Xj+iat date t because x +i (potentially) affects Pf+i, and to determine F, we mustpredict P,+\ (and 4+i), i.e., the analysis exploits expression (3).

To focus on dividends as the capitalization attribute is not only a matter ofmore or less esoteric theory. In the real world this role of dividends is seen at thetime when a stock goes ex dividend. These events make a stock less valuable,and, on the average, the price declines approximate the dividends. This outcomeis entirely consistent with the valuation theory developed. Of course, empiricalresearch in finance and accounting recognizes the relevance of dividends at leastimplicitly by making an adjustment for dividends in the definition of the retumon a security. Proposition II similarly illustrates the relevance of dividends asa vaiuation attribute. As a practical matter a stationary expected retum may bequite reasonable, provided that one adjusts for dividends in the definition ofreturns. But this simple model then implies that the capitalized expected futuredividends determine value.

In dealing with valuation issues, the accounting literature generally refers tocash flows—^not to dividends—as the appropriate capitalization attribute. Al-though rarely (if ever) discussed in any detail, two closely related ideas seem tomotivate this concept: (1) dividends must be paid out of cash flows, and (2) anyreduction in current dividends leads to increased future dividends because theextra cash retained can be invested in bonds. Of course, Modigliani and Miller(1961) (MM) considered (2) in particular; they showed that, under certainty,all dividend policies are equally optimal. Rubinstein (1973, 1976a) generalizesthis result for mean-variance (CAPM) and Arrow-Debreu (complete markets)settings.

The conditions that allow for an appropriate concept of cash flows as a cap-italization (present value) attribute are, in fact, stringent, and dividend policyirrelevance is only necessary and not sufficient. An additional condition re-quires future investments (and borrowings) to have, ex ante, zero net presentvalue, i.e., the investment—and borrowing—policy is irrelevant, and a valuemaximizing firm might as well cease making any new investments. It shouldfurther be explained that the (net) cash flows capitalized are those generatedby existing assets (and debt); cash flows deriving from assets (investments) ac-quired (planned) in the future must be excluded. Thus, given new investments,investors and accounting researchers never get to observe the realizations of the(random) variables that were capitalized in the first place.

To formalize the above ideas, consider the sources of funds equals uses offunds identity:

668 J.A. Ohlson

where Q = cash flows from operations, B, = net borrowings, and // = netinvestments. Given date t, the generally uncertain cash flows at date x,x > f,derive from two sources, namely from investments made at or prior to datet,c\f, (read "e" as "from existing assets"), and from investments planned atfuture dates, c{j (read "/" as "from assets acquired at/uture dates"). Hence,c^ = cl, + c{i for all t <x, and the distributions of Cx,c^t< ^"' ^(t depend onthe information at the current date t (z,).'"

For ease of exposition, consider next a pure equity firm, that is, Bt = 0 forallr:dr = clt + cl^-I-,,x>t.

Substituting the RHS into the valuation formula of Proposition I.2., one imme-diately concludes that

F(z,) = Y^ Rp(x;zt)-'{E[clt I Zt] + cov[c',^,,Q, \ z,]},x=r+l

if and only if

Y, RFiv, Zt)^'{E[c{, - /, I Zt] + cov[c(, - h, Q, I z,]} = 0.

In other words, cash flows from existing assets can be used as a capitalizationattribute if and only if the risk-adjusted net present value of future investmentsequals zero. The latter condition is obviously extremely stringent unless the firmgradually liquidates, and Ix — d^t — ^- '^"^' ^^ course, if future investments dooccur, then it never makes sense to capitalize total future operating cash flows.

The above analysis by no means uncovers a flaw in the MM dividend policyirrelevance hypothesis because the latter result follows from weaker conditions.Given a z,-dependent but otherwise fixed investment policy, possibly with posi-tive net risk-adjusted present value, a change in the borrowing policy will gen-erally change the pattem of future dividends. Such a policy change, however,would have no effect on the current price, P,, provided the net present value ofincremental borrowings equals zero. In other words, the irrelevance of dividendpolicy refers to incremental analysis, whereas the concept that the cash flowsfrom existing assets and liabilities can serve as a capitalization attribute requiresfuture investment/borrowing to be irrelevant in absolute terms.

One could perhaps argue that as a practical matter the assumptions necessaryfor discounting future cash flows (from existing assets) are at least approximatelysatisfied. >' Nevertheless, this possibility is only of limited consolation because.

10 The z, set remains primitive throughout this section. Specifically, it is not required that Z(include any of the cash flow or investment/borrowing variables.

11 The cash flow capitalization concept always works in the extreme case when one defines"existing assets" sufficiently broadly. That is, one may view "existing assets" as including allfuture opportunities that yield positive net present value. Thus "superior management abilities"can be viewed as an existing asset. This concept of cash flow capitalization turns into a virtualtautology, and under the circumstances it seems doubtful that the concept conveys any usefulinsights.


as explained, one still deals with ex post unobservable variables. While ex antecash flow capitalization may well be useful in capital budgeting, it does notfollow that the concept is also central in the theory of valuation. Much of theaccounting literature is, unfortunately, oblique regarding this point.

The role of earnings in valuation theoryWhat can be said about eamings in the theory of valuation? The broad an-swer is disappointing but unsurprising: very little. No published research hassucceeded in integrating a model of accounting eamings with modem financetheory, and in this regard the absence of substantive insights is striking. Todevelop a theory of historical cost accounting and security vaiuation ought tobe of extraordinary importance in accounting research. A natural departure forsuch research is the work by Paton and Littleton (1940), which so usefully—^butinformally—conceptualizes the structure of historical cost accounting and themeasurement of eamings. Although their central doctrine of "value surrenderedequals value acquired" obviously derives from notions of an equilibrium, thiskind of reasoning has not found its way into "modem" research.

The literature often concems itself, however, with "pure" eamings concepts,as is exemplified by the terminology permanent earnings, ungarbled earnings,economic income, and earnings as a sufficient information variable. The com-mon thread of this research deals with the issue of how one generalizes theclassical certainty analysis of earnings. The ideas and results are merely sug-gestive in the sense that these direct attention to the attributes that accountingeamings perhaps ought to satisfy under "ideal" but hypothetical circumstances.The formal analyses represent eamings as a primitive datum, and notably lack-ing in these models are substantive accounting ingredients such as "stocks andflows," "transactions," "property rights," and "contracts."

Keeping the above caveats in mind, I next review and discuss the models ofpare earning concepts in terms of the genera! valuation theory (Propositions Iand II). I use the examples in the fifth section to illustrate the various eamingsconcepts and how these relate to specific information dynamics assumptions.Each eamings concept places certain restrictions on eamings as it relates todividends. The perspective invoked on "pure" eamings is therefore as an in-formation variable with respect to future dividends. We avoid tautological orinconsistent concepts of "pure" eamings, which may occur if one relates Pt to(current or future) eamings a priori without consideration given to the informa-tion dynamics.

In the literature one can discern three distinct characterizations, or criteria,that a pure concept of eamings, "x," ought to satisfy:(i) Eamings as a capitalization attribute; an appropriate present value calculation

of future risk-adjusted expected eamings thus determines a security's value

(ii) Eamings as a sufficient information variable that determines a security'spayoff: Pt + dt = Bxt (i.e., P, = Ex, - 4 ) .

670 J.A. Ohlson

(iii) Eamings as a sufficient information variable that determines a security'svalue: Pt = Bxt.

Analyses of case (i) can be found in Beaver, Lambert, and Morse (1980), inOhlson's (1989) extension of that paper, and in Ohlson (1983). These paperspostulate linear information processes that include a primitive variable, Xt, andidentify conditions such that a capitalization of (risk-adjusted) expected valuesof Xt+i,Xt+2,. •; given z,, results in /",.

As a first illustration of case (i), Ohlson (1983) uses Example 4 with n = 2,i.e., Zt = (x,, d,). For this setting, Ohlson proves that

P, = [(821 -

if, and generally only if, 822 = O22 = 0. The condition therefore restricts thedividend policy because current dividends cannot influence next period expecteddividends. (Example 3 holds as a special case.) The motivation behind the anal-ysis derives from eamings capitalization under certainty. Thus, 821 — O21 as itappears in the formula "corrects" the dividends capitalization formula for thepayout coefficient of anticipated dividends (and their risk), and the summationmns from x = ? rather than x = t + I. These "correcting" procedures of theformula in Proposition 1.2 are necessary under certainty as well. Evidently, atleast for this model, "eamings capitalization" involves more than replacing dtwith Xt in the present value calculation.

As a second illustration of case (i), Ohlson (1989) considers Example 5 andshows that under appropriate parametric restrictions on the (xt,at,dt) processone obtains

for some constant p. The discount factor p is implicitly assumed to reflect therisk in the eamings stream, just as Proposition II. But p does not generally equalyi (the expected security retum). In fact, given the above model, it follows fromProposition II that pP, = E[P,+i +x,+i | z,], and thus p = ^ if, and only if,E[dx I Zt] = E[xx I Zt]. More generally, Ohlson (1989) shows that there exists aconstant X such that XE[xt+x | z,] = E[dt+x j Zf], x > 0, and where one interpretsA, as a payout factor. Similar to the previous example, the eamings capitahzationformula is not fully identical to the underlying dividends capitalization formula,and the discounting depends on a payout factor that restricts dividends as itrelates to eamings.'^

12 Of course, if the model requires |i (the expected market retum) to equal p (the capitalizationfactor), then one must answer the awkward question: Why should we expect dt to differ fromXt when E[dt — jf-t | z,] = 0? This implication of proposition II applies in other contextsas well. Many papers (e.g., Collins and Kothari, 1988) capitalize "cash flows" using (i as a


Earnings as a sufficient determinant of the security's payoff, i.e., case (ii), hasbeen considered by Ryan (1986). Although Ryan does not require certainty, thecertaintj' setting can be used to motivate the criterion. Pure earnings in this modelare equal to (or are defined as) x,+i = (Rf — l)Pt. Further, because in equilibriumP^Rp = P,^i + d,.n it follows that x, == Ki{P, + dt) where Kx = {Rp - l)/Rp.Hence, Xt and the security's payoff have a one-to-one relation. Alternatively, forthis earnings concept the valuation function satisfies

where Si is a constant {Bi ~ Rp/(Rp — 1)). Pt therefore depends on dt and oneother "primitive" information variable, Xf.

Example 4 with n — 2 usefully illustrates that the above valuation rela-tion holds only for specific restrictions on the parameters of the ixt,dt) pro-cess. Specifically, given the general solution Pt = BiXt + B2<i/ and the re-lated expressions for Bi,B2, (4a) and (4b), one shows easily that B2 = ~1if and only if 9n — an = Rp. The latter condition means that the nextperiod risk-adjusted earnings growth rate equals Rp when current dividendsare zero. It furJlier follows that Bj = —Rp/{Qi2 — <Ji2). The last restrictionimplies that the parameters associated with the second predictive equation,5,4.1 = (621 + e2i<+i)- i + (622 + e22r+i)4. can take on any values without affectingthe equilibium value Pt. Alternatively, given On — <S\\ = Rp (or B2 = —1) thevaluation parameter B\ depends only on the parameters that affect the predictionof next period earnings. The criterion (ii) for Example 4 with n — 2 thereforeleads to dividend policy irrelevance in valuation. Given B2 = — 1, it may alsobe noted that the restriction Bi = Rp/iRp — 1) implies ©12 — 012 = 1 — Rp.

Cases (i) and (ii) may hold simultaneously. The previous example works ifone combines it with ©22 = O22 = 0. On the other hand. Example 5 doesnot satisfy case (ii), and one may infer that the analyses in Beaver, Lambert,and Morse (1980) and Ryan (1986) are mutually exclusive.'^ (This implicationfollows immediately by the nature of dynamic valuation theory, dt cannot affectPt unless dt also relates to the prediction of dt+i^dt^a,... A direct examinationof the tvi'o predictive equations in Example 5 reveals that dt is irrelevant as aninformation variable because dt is absent in both equations' RHS).

Criterion (iii) has been proposed by Black (1980). This concept of pure earn-ings is equivalent to an intertemporally constant price/earnings ratio. Clearly,Such a relation holds if, and generally only if, Xt suffices to predict dt+i,dt+2,The model in Example 3 illustrates this case: x, alone predicts Xt+i and dt+i.

discouBt factor, and this valuation scheme makes no more sense. These papers seem unawareof the fact that E[dx — Ix ! Z(] = 0 where x^ now denotes "cash flows," and they never addresswhy one should not simply equate dividends and "cash flows." (Note that d^ — c^g = Cx only

13 Contrary to the requirements of criterion (ii), in the Beaver, Lambert, and Morse (1980) modelPI depends on two primitive variables, (x,,at), and P, is independent of rf,.

672 J.A. Ohlson

and, via recursive substitution, x, suffices to predict <i(+2,4+35 •. • as well. TheP/E-iatio equals (62 — <J2)/(RF — (61 - Oi)), where the parameters 62 and 02relate to the anticipated dividend payout, and 9i, Oi determine the stochasticbehavior (risk-adjusted growth) of the earnings process. The valuation settinghas some empirical appeal because the risk-adjusted growth in earnings, 9] — Oi,enters logically into the valuation multiplier of x,. The model to some extentdistinguishes between dividends and ("pure") earnings because the payout ratio,df/xt, is stochastic. Also, as noted, the expected return in excess of the risk-freerate has a simple expression: E[ri.î \ Xt] — Rp — Gi+B~â2 for all Xt, andwhere B = Pt/xt equals the multiplier. Hence, if B^Ô2 is small relative to(Ji (which seems empirically reasonable), then the risk in the earnings process,cov[(X(+i — Xt)/x,., —Qt+i I z,] = Ci induces and approximates the risk in thesecurity's return.

What do we make out of the above "pure" earnings concepts? Put briefly,both criteria (i) and (iii) embed severe theoretical problems, and I am inclinedto view these problems as "fatal." Only criterion (ii) would seem satifactory,although certain limitations are present in this case as well. First, with respectto criterion (i), the notion of earnings as a capitalization attribute yields few, ifany, sharp insights. The earnings capitalization formulae require adjustments de-pending on how dividends relate to earnings, and these adjustments do not flowfrom intuitive economic concepts. The analyses leave a "mechanical" impressionbecause the theory becomes unworkable for complicated dividend policies. Thepayout constructs are too contrived, and to accommodate a richer set of modelsrelating earnings to dividends seems, at best, difficult. It may further be notedthat though criterion (i) (capitalization of earnings) can be combined with (ii)or (iii), the implications of the two latter criteria do not facilitate an understand-ing of why criterion (i) should be an economically meaningful restriction onpure earnings. Of course, none of these negative observations should be surpris-ing because "x," describes the environment only in a predictive but otherwiseprimitive sense.

Second, concerning criterion (iii), we note that a constant P /E ratio violatesbasic MM precepts in an extreme sense: the price at date t does not dependon dividends paid at date /. Such a relation between current price and currentdividends obviously makes little theoretical sense. From a somewhat differentperspective, one should expect current dividends to affect future (expected)earnings, but this property is conspicuously absent in example 3. In a technicalsense, these problems can be avoided if one assumes that "pure" earnings overthe interval f — 1, ? depend on dividends at date t. But to require such a propertyof pure earnings would seem to be exceedingly contrived (to say the least). Thepreviously discussed appealing aspects of example 3 do not counter its moresubstantive theoretical problems.

Third, concerning (ii), this characterization of pure earnings satisifes basicMM concepts. A dollar of additional dividends reduces the market value by


a dollar because dPt/ddt = — 1 (and current dividends do not affect cunentearnings). As shown previously, this property of the valuation function mirrorsthe dividend policy irrelevance. Also note that current dividends affect futureearnings, a relation that would seem to make economic sense. Another appealingaspect of (ii), which is emphasized by Ryan (1986), focuses on the fact thatBj = RP/(RF — 1) can be interpreted as a multiplier for current eamings. Andif dt = Xt (i.e., the payout ratio is 100 percent), then one arrives at the properperpetuity solution Pt = Xt/(RF — 1).

The discussion indicates that eamings as a sufficient determinant of pay-off(criterioE (ii)) has none of the disadvantages of eamings as a sufficient determi-nant of price (criterion (iii)) because it aligns with basic and compelling MMprecepts dealing with dividend policy irrelevance. On that basis alone, (ii) makesmore theoretical sense than (iii). Similarly, (ii) does not rely on the contriveddividend policy assumptions necessary for eamings capitalization (criterion (1)).The latter concept seems unworkable if one allows for unrestricted dividend poli-cies. (In any event, (ii) does not always mle out (i).) We are therefore inclinedto conclude that Ryan's pure eamings characterization is the only theoreticallyviable concept out of the three proposed.

The major limitation with criterion (ii) relates to its "incompleteness." Onemay, for example, ask to what extent this concept relates to future cash flows(rather than dividends) as discussed in the sixth section. This issue of how cur-rent eamings/dividends reflect anticipated cash flows seems to be an unansweredquestion. Another problem relates to the dimensionless characterization of eam-ings. As a matter of accounting, one tends to think of eamings as a "flow"variable, but the analysis does not by itself allude to this property of eamings.The notion of flows then naturally leads to stock variables, thereby suggestingthat a theory of earnings cannot exist without a theory of owners' equity (i.e.,"net worth" or "book value"). The core of a more fully developed theory wouldthen perhaps explain the logical relevance behind the equation that makes finan-cial accounting cohesive: eamings minus the change in owners' equity equals(net) dividends (i.e., debits equal credits and the accounting model satisfies theclean surplus doctrine). But this approach toward accounting variables and valu-ation requires that owners' equity is put on a co-equal status with eamings, andone must question the adequacy of exclusive focus on "pure" eamings concepts.

Concluding remarks concerning the empirical literatureFrom the analysis one can infer the limitations of the theoretical constructsused in much of the empirical literature. In broad terms, all problems relateback to a oon-recognitioe of the central fact in valuation theory: the price of asecurity is determined by the present value of its dividends, and every valuationfunction satisfies intertemporal consistency requirements to exclude arbitrageopportunities.

First, most papers generally finesse the linkage between the infonnation vari-

674 J.A. Ohlson

ables studied and dividends. Only Easton (1985) and Easton and Zmijewski(1987) make an attempt to tackle this empirically complex issue. Others, suchas Beaver, Lambert, and Morse (1980) and Kormendi and Lipe (1987), do notexplore or identify an explicit role for dividends in valuation. Thus, in thesecases the valuation models are best thought of as incomplete or underidentified.Ohlson (1989) discusses this point in some detail as it relates to Beaver, Lambert,and Morse (1980).

Second, to stipulate a role for unobservable valuation attributes—^such as "un-garbled eamings" or "cash flows from existing assets"—-in some ways introducesmore problems than it solves. From the view point of theory, the scheme stillnecessitates a precise link to dividends; otherwise, such models become meretautologies. And, in any event, to develop empirically testable propositions usingthis framework one must consider the difficult task of linking the "unobservable"variables to "observable" ones. (Without such a model the theoretical constructssuperficially embellish the research.)

Third, it should be clear that the frequently applied Litzenberger and Rao(1971) model lacks both of the ingredients necessary to derive a theoreticallyvalid valuation function. This CAPM-related model neither provides a linkage todividends nor flows from a multiple-dates environment.'^ The latter means thatany related valuation function typically does not satisfy the no arbitrage inter-temporal consistency requirements around which valuation theory revolves. Tobe sure, this negative comment has to be put in perspective: Litzenberger andRao published their paper in 1971, and they obviously did not have "access" toRubinstein's seminal results, which were published five years later.

The above observations about theory use in empirical research should not beconstrued as a critique invalidating the studies. This paper merely synthesizesvaluation theory in terms of modem finance economics and explicates how thevarious parts of the theory fit together. But such theoretical analyses do notpredetermine the questions empirical researchers may wish to address. Theoryis of limited relevance for most questions, and thus useful empirical studies canbe conceived even when the concepts of what determine security value are un-specified or underidentified, or when the study maintains hypotheses that do notderive from more primitive assumptions. There is therefore nothing intrinsicallywrong with «c>/ specifying an explicit Uiik between the independent (accounting)variables and expectation of dividends or some other valuation attribute. Balland Brown (1968), for example, rely on no such link. Given the question theyraise, it would seem odd (to say the least) to suggest that their study suffers fromdeficiencies because no "linkage theory" embeds the empirical hypotheses.'^

14 The Liizenberger and Rao (1971) model also includes a term that purportedly captures"growth opportunities." This term has no rigorous foundation under uncertainty.

15 Christie (1987) apparently disagrees. He concludes that "therefore, regardless of whether theanalysis is conducted in levels or retums, a model of future cash flows is required" (p. 233,emphasis added).


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