The Relation Between Stock Returns and Earnings

THE ACCOUNTING REVIEWVol. 65. No. 1Januaiy 1990pp. 49-71

The Relation Between StockReturns and Accounting

Earnings GivenAlternative Information

Robert LipeUniversity of Michigan

ABSTRACT: This paper examines the relation between stock returns andaccounting earnings under the assumption that the market observes cur-rent-period information other than earnings. This assumption is motivatedby existing empirical evidence that stock returns lead accounting earnings.The analysis shows that the returns-earnings relation depends on the rela-tive ability of earnings versus alternative information to predict future earn-ings as well as the time-series persistence of earnings. Assuming that theresearcher does not observe the alternative information, the earningsresponse coefficient should be increasing both in the ability of pastearnings to predict future earnings and in earnings persistence. Thevariance of stock price changes during the announcement of earningsshould be decreasing in predictability and increasing in persistence. Empiri-cal tests of these four hypotheses are generally consistent with the theory.Also discussed is how the assumption of alternative information may beuseful in examining the information environment hypothesis, in assessingad hoc methods of reducing measurement error bias, and in formulatinghow economic earnings differ from accounting earnings.

T HIS study investigates the relation between stock retums and accountinge£imings under the assumption that the market observes current-periodinformation other than earnings that is useful in predicting future earnings.

I am grateful to Vic Bernard, Dan Collins. Gene Imhoff. Bill Kinney. Roger Kormendi, S. P. Kotharl.Jerry Lobo, Evelyn Patterson, Bill Ricks, Tom Stober. Sundararaman Thiagarajan. Robert Verrecchia,Jim Wahlen, and Dave Wright for helpful discussions and to the anonymous referees for theirInsightful comments. Also, the workshop participants at the University of Michigan, WashingtonUniversity, and Duke University provided many useful comments. Funding was provided by theUniversity of Michigan and the Peat Marwick Foundation.

Manuscript received July 1988.Revisions received February 1989 and June 1989.

Accepted July 1989.

49

50 The Accounting Review, January 1990

Beginning with the seminal work by Ball and Brown (1968). the returns-earningsrelation has been the focus of many studies. As the literature has progressed,researchers have used finance theories to make empirical predictions. Forexample. Kormendi and Lipe (1987) model stock returns as a function of therevisions in expectations of earnings, assume that earnings can be representedby a univariate time-series process, and then show that the time-series propertiesof earnings wiil be an important factor in the returns-earnings relation. Otherstudies also use explicit theoretical models (Beaver et al. 1980; Beaver et al.1987; Collins and Kothari 1989; Easton and Zmijewski 1989; Imhoff and Lobo1988; Lipe 1986). The goal of this area of research is to increase understanding ofhow earnings and other accounting information are related to stock prices.

This paper's focus on alternative information is motivated by recent empir-ical evidence and theoretical models. Beaver et al. (1987). Collins et al. (1987),and Collins and Kothari (1989). among others, find that imexpected earningsfrom year t-l-1 (as measured by the researcher) are correlated with returns fromyear t. The implication is that the market obtains alternative information in yeart which is a substitute for some of the "news" in earnings of year t-l-1. Further,Holthausen and Verrecchia (1988) use an information economics model todemonstrate that the relation between stock prices and a given source of informa-tion depends on the availability of other useful information.

In order to examine the role of information other than accounting earnings. Igissume that during year t the market receives a noisy signal of earnings for yeart+1. Combining this assumption with the model used in Kormendi and Lipe(1987). I show that the stock return reaction to earnings is a function of (1) thetime-series properties of earnings. (2) the interest rate used to discount expectedfuture earnings, and (3) the relative ability of earnings versus alternativeinformation to predict future earnings. The third factor is the result of allowingthe market (but not researchers) to observe alternative information.

Four testable hypotheses regarding the relations between stock prices andearnings are derived. The first two are that the coefficient which measures thestock return response to a one-dollar earnings shock (hereafter, the responsecoefficient) is an increasing function of both the "predictability of the earningsseries'* and the time-series persistence of earnings. The predictability of earningsis defined as the ability of past earnings to predict future earnings, and it isrefiected in the variance of the shocks in the univariate earnings process (as thevariance decreases, the predictability of earnings increases). As earnings predict-ability increases, the current earnings information becomes more useful in pre-dicting future earnings and. therefore, the response coefficient increases. Analternative interpretation is that as earnings predictability increases, the differ-ence between the univariate earnings shock and the market's £issessment ofunexpected earnings decreases and. therefore, the downward bias in the re-sponse coefficient decreases. The difference between predictability and persis-tence is that the predictability of earnings is a function of the average absolutemagnitude of the annual earnings shocks, whereeis the time-series persistence ofearnings refiects the autocorrelation in earnings.

The other two hypotheses are that the variance of stock price changes duringthe release of earnings is negatively related to earnings predictability and

Lipe—stock Retums and Accounting Earnings 51

positively related to earnings persistence. Holthausen and Verrecchia (1988) andBeaver (1968), among others, analyze the variance of price changes when infor-mation is announced. The negative effect of predictability occurs because thevariance of price changes measures the retum reaction to the typical earningsshock. As the predictability of earnings increases, the absolute magnitude oftheearnings shock will probably be smaller, leading to a lower variance of pricechanges despite the fact that as predictability increases, a given earnings shockis worth more (the response coefficient is larger). The effect of persistence is posi-tive because the greater persistence means a larger reaction to the typical eeim-ings shock.

The fovir hypotheses are empirically tested using the sample firms fromKormendi and Lipe (1987). The parameters of interest (the response coefficient,variance of price changes, and predictability and persistence of earnings) areestimated for each of 145 firms. Simple and partial rank correlations show thatthe response coefficient is positively related to both predictability and persistenceacross firms. The variance of price changes is negatively related to predictability,but the positive relation with persistence is very weak. Sensitivity einalysessuggest that the results are not due to ignoring cross-firm differences in risk orsize. However, errors in measuring the imivariate earnings shocks may causeoverstated significance levels.

The rest ofthe paper is organized as follows. The theoretical relation betweenretums and e£imings is developed and discussed in Section I, with the mathe-matical details presented in the Appendix. Section II presents the empirical tests.Section III discusses other insights gained from a theoretical model that assumesthat the market has altemative information. Section IV summarizes andconcludes the paper.

I. Theory

The relations between stock prices, accounting earnings, and altemativeinformation are represented by the following three equations:

. . ) , (1)

(2)

(3)

In equation (1), the retum on a share of common stock, R,, is a function of someexogenous expected return, R', and the unexpected retum due to the release ofaccounting earnings, X,, and the altemative information, I,. Lags of X, and /, areincluded in equation (1) because they may be useful in determining the imantici-pated information.

Three assumptions are imposed on equation (1): stock price equals thepresent value of expected future dividends; the discount rate, |3, is constant overtime; and the present value of the revisions in expectations of future dividendsequals the present value of the revisions in expectations of future earnings. To-gether, these assumptions imply that the unexpected stock retum in period t, R',


equals the present value of the revisions in expectations of current and futureearnings. The flrst and second assumptions are commonly used in finance andaccounting studies. Ohlson (1988) shows that the two represent special cases of amore general model based on state dependent dividends and a no arbitrage equi-librium. The third assumption can be thought of as an extreme version of thestatement, "accounting earnings provide information about the future dividendpaying ability of the flrm." While this is a stringent assumption, some linkbetween earnings and dividends is necessary in order to derive a relation be-tween returns and earnings. For example, Ohlson links earnings and dividendsby assuming that both are driven by the same underlying events or states ofnature. The model could be based on less restrictive valuation assumptions, butthese three are chosen in order to clearly demonstrate the effect of alternativeinformation.'

Equations (2) and (3) specify the information structure of the model. Equa-tion (2) describes the univariate characteristics of earnings. The b, are the autore-gressive coefficients, and e, is the serially uncorrelated earnings shock in periodt. Kormendi and Lipe (1987) demonstrate that larger bi coefficients cause thecurrent-period earnings shock to have a larger impact on future earnings (greaterpersistence). The "predictability of the earnings series" is captured by thevariance of the earnings shocks, oi. If aJ=O. then past earnings predict flitureearnings perfectly. The ability of past earnings to predict future earnings de-creases as ai increasies. Note that the predictability and the persistence of theearnings series are distinct concepts. Persistence describes the time-seriesrelation between the current-period earnings shock and future earnings. Predict-ability reflects the variation in the earnings shocks. One can imagine a random-walk and a white-noise series which have the same variance of shocks. Theformer has much more persistence, but the two are equally predictable (theforecast errors have the same variance). Alternatively, one can imagine two ran-dom-walk series with high and low variance of shocks, respectively. The formeris less predictable, but the two have equal persistence.*

Equation (3) shows that the alternative information equals next period'searnings plus noise. n,+,. The noise is assumed to be serially uncorrelated and isuncorrelated with e,*». for all k. In the spirit of Holthausen and Verrecchia (1988).investors have useful information other than current-period earnings.^ The alter-

' These are essentially the same assumptions used In Kormendi and Lipe (1987), and using themprovides some continuity. In addition, Ohlson (1988, 42) states that, ". . . useful empirical studiescan be conceived even when the concepts of what determine security value are unspecified or under-identified, or when the study maintains hypotheses that do not derive from more primitive assump-tions."

'The economic determinants of the persistence or the predictability of earnings are not investi-gated in this paper. Instead, the theory shows that these two parameters are potentially important,and emplrlcaJ evidence suggests that they are important. Given the results, an economic analysis ofthe peirameters appears useful. For example. Lev (1983) provides some intuition regarding the auto-correlation of earnings and the variability of earnings shocks.

^ Other studies have modeled some forms of alternative information. One approach Is todecompose earnings and show how the components can provide more information than earningsalone (Lipe 1986; Rayburn 1986: Wilson 1986; among others). In most of these models, assessmentsof expected future earnings do not involve trade-offs between the compwnents and, therefore,earnings predictability is not a factor. Another approach is to include both accounting earnings and

Lipe—stock Returns sind Accounting Earnings 53

native information could encompass production/investment decisions oranalysts' forecasts of Xî made at t. Alternatively, since X, and I. are the onlysources of new information, one could think of I, as representing all informationother than current and past accounting earnings. Since the empirical tests use ofannual data. I, and X, are assumed to be observed simultaneously in year t.*

The appendix transforms these assumptions into a model of returns as fol-lows:

S^Z^}^^tEM}£±, (4)

The term in brackets reflects the new information revealed in yeeir t. The unex-pected information regarding the current-period earnings shock equals e,—Mw,.In year t, the market observes X, and infers e,. But a portion of e, was anticipatedin year t-1. Speciflcally. by observing X^, and 7,-i. the market infers w,=e,+n,.The expectation of e, in year t-1 is Mw,, where:

M = — - — .

The other new information concerns next year's earnings. By observing X, and I,,the market infers luî • The expectation of e^, in year t is Mw,*i. Prior to year t, theexpected VEilue of ern is zero. Thus, the new information about e n equals Mwî.This is multiplied by the discount rate in order to obtain the present value of therevision in the expectation of X^n. PVR is the persistence of the earnings series. Ifthe univariate earnings series were white noise, then PVR=O and equation (4)would only contain the bracketed term divided by price. But if the time-series co-efficients differ from white noise, then the new information in X, and I, will affectthe expectations of earnings in periods beyond t+1. In other words, the newinformation persists into the future, and PVR captures this persistence. P,-i is thestock price at the end of year t-1.

Equation (4) presents the theoretical relation between returns and earnings.Before the hypotheses can be developed, some estimation issues must be re-solved. If researchers observe /,. then they can use the market's assessment ofunexpected current-period earnings. e,-Mw,, as the independent variable in esti-

"permanent" earnings In the model (Beaver et al. 1980; Beaver et al. 1987). The typical assumptionis that accounting earnings equals the market's perception of permanent earnings plus noise. In thiscase, permanent earnings appeair to be known or knowable without reference to accounting earningseind, therefore, the latter do not play a role in the valuation of securities. In contrast, under theHolthausen and Verrecchia (1988) assumptions and the assumptions discussed above, the marketrevises expectations of future dividends based on accounting eemiings and alternative information.

* As with the valuation assumptions, there are alternative ways to structure equations (2) and (3).For example, /, could be a vector of Information. Also, there could be Interactions between X, and /, byassuming that e, cUid n, are cross-correlated or that /,-i {X,-,) affects X, (/,). If one used a shorter cumu-lation period for stock returns, then the release of /, and X, would be sequential instead of simulta-neous, as discussed in the Appendix. The exact effect of changing the assumptions would depend onwhat new assumptions were made.

54 The Accounting Review. January 1990

mating the reponse coefficient as follows:

^^}^ (5)

But in order to be consistent with previous papers, I estimate the empirical rela-tion using only returns and earnings data as follows:*

R,=ki+ao—+UR,,P.-.

=.R' -K1+PVR)(1 - M)—P.-1

" ' - - ^ " ^ Y (6). -1

In equation (6), ao is the response coefficient, R' is the average expected returnover time, and UR, consists of n,, n,+,,eî. and any intertemporal variation in ex-pected returns. Note that UR, and e. are independent. Therefore, UR, representsuncorrelated noise.

Using equation (6) instead of equation (5) implicitly assumes that researchersobserve R, and e, but not /,. Alternatively, researchers observe noneamings infor-mation but choose to ignore it. Thus, the focus of this paper is on how the exis-tence of £ilternative information which is not used by researchers affects theobserved relation between returns and earnings. The tasks of identifying alterna-tive sources of information and incorporating them into the empirical tests areleft to future research. Results in Ou and Penman (1989) suggest that such effortscan be successful.

In equation (6), ao=(l-l-PVR)(l-Af). According to the Kormendi and Lipemodel, ao=(l-HPVR). The difference is due to including/, in this paper. Note that0 < M s l which implies that ao<(l-hPVR). Consider the two extreme cases ofM=0 and M= 1. First, suppose that /, is not useful in predicting X,+, becauseffj= 00 (or /, is all noise). As aJ-oo.M-O, in which case Kormendi and Lipe andequation (6) yield the same theoretical value for ao. This occurs because the dif-ference between the models is the inclusion of/,; but if/, is so noisy that it pro-vides no information, then the models are the same. Second, suppose that al=O,in which case /, predicts X,+i without error. In this case, M=l and ao=O: thereturn reaction to the earnings shock is zero because current-period earnings areuseless in predicting X,+,, given /,. Thus, Kormendi and Lipe's (1987) result is aspecial case of equation (6).

The result that ao is a function of (1 -M) can also be interpreted as an errors-in-variables bias. Since the only explicit information source in Kormendi andLipe is earnings, e, represents both the earnings shock and the market's assess-ment of unexpected earnings. In this paper, the latter equals e,-Mw,. Ue,-Mw,

»Some researchers estimate the response coeniclent using reverse regression In order to avoidthe bias In ao due to measurement errors. However. Section III shows that the reverse regressioncoefficients are also biased. Therefore, reverse regression Is not used In this paper.

Llpe—Stock Returns and Accounting Earnings 55

(deflated by price) is the independent variable as in equation (5), then ai-il-I-PVR) which is consistent with the prediction of Kormendi and Lipe. But if theearnings shock is the independent variable as in equation (6), then there is mea-surement error which equals e.-[e,-Mw,) = Mw,. The well-known errors-in-vari-ables result (Maddala 1977, 292-293) is that:'

a = a o ' ( l - M ) = ( l + P V R ) ( l - M ) .

Thus, the response coefficient estimated in equation (6) is a biased estimate of(1+PVR), and (1 -M) represents the bias.

This discussion of measurement error is not intended to trivialize the role ofthe predictability of earnings in the relation between returns and earnings. By in-cluding I, in the model, market participants have two competing sources of infor-mation to use in forming expectations of future earnings. The usefulness of eachsignal is determined by its relative ability to predict the future. Thus, R"ia equa-tion (4) is a function of M (and, therefore, predictability) whether researchers useI, to measure market expectations or not. The use or nonuse of I, simply deter-mines whether M appears in the independent variable in equation (5) or in the re-sponse coefficient in equation (6).

Hypotheses

The four hypotheses are now developed. Equation (6) shows that ao is afunction of persistence and the relative variability of n, and e,. Differentiating aowith respect to ai demonstrates the relation between the response coefficient andthe predictability of the earnings series:

aao (l+PVR)aidai

(7)

Since al and a] are positive, the derivative is negative as long as PVR > — 1. If esim-ings are a white noise process, then PVR=O. A negative PVR would mean thatearnings are less persistent than white noise which, based on prior empiricalresults, is unlikely. Thus, the derivative is negative. As the predictability of theearnings series increases (or as al decreases), the response coefficient increases.To understand the positive relation, recall that in period t - 1 , the market'sexpectation of e, equals:

a]Mw,= (e,+n,).

[ol+al)Decreasing a] while holding al constant implies that less of e, is anticipated. Thedecrease in al causes the market to rely less on the prior information and more onX,. In other words, as predictability increases, X, is more useful in assessingexpectations of future earnings. Alternatively, the measurement error perspec-

'Thls assumes that deflating the dollar returns and the dollar earnings metrics by P,-, does notaffect the expected value of the coefficients.


tive suggests that ao is positively related to predictability because increasingearnings predictability reduces the variance of the measurement error whichcauses a decrease in the downward bias in ao.

Since 0<M< 1, equation (6) shows that Oo is positively related to PVR exceptin the extreme case of M=l . An increase in the time-series persistence of earn-ings leads to an increase in the response coefficient.

An alternative method of analyzing the relation between stock prices andearnings is to examine the variance of price changes around earnings announce-ments. The change in price. AP,, equals the stock return, R,, multiplied by the be-ginning of period price. P^..' Equation (4) implies that the variance of pricechanges equals:*

(8)

Note that Var(AP,) is not conditional on the information released in year t. Itreflects the average squared change in price based on the mean and variance ofthe information system and not the price change in response to a given earningsshock. The relation between Var(AP,) and the predictability of the earningsseries is assessed by differentiating equation (8) with respect to ai. or:

Since the derivative is positive. Var(AP,) is a decreasing function of the predict-ability of the earnings series, even though the response coefficient is anincreasing function of predictability. Intuitively, if firm A's earnings shocks areless variable than firm B's. a one-dollar shock in each series would cause a muchlarger stock return for firm A. But on average, the absolute magnitude of theearnings shocks is smaller for firm A and. therefore, the average variation in theprice of firm A is smaller. Thus, analyzing stock returns conditional on the factthat earnings are announced during the period is very different from analyzing

'Actually, part of the return In period t may be In the form of a dividend and. therefore. R.P,.,may differ from AP.. However, the fonner measure may be more appropriate for addressing therelation between stock prices and earnings; the price changes associated with the ex-dividend dateare probably not a function of information released during the year. The empirical tests were alsoperformed using Var(P,,-P,,.,). and the results were similar.

' First, since the variance equals the actual minus expected price change squared. R: does notappear in equation (8). Second, from equation (4). the change in price due to X, and /, is:

RrP,.Ml+PVR][e.-Mw,+mw,.,].

The variance of the change in price is then:Var(R,"P,.,] =

=(1-I-PVR)'-

which yields equation (8).

Llpe—stock Returns and Accounting Earnings 57

returns conditional on the magnitude of the earnings shock. In particular, thesetwo constructs have opposite relations with the predictability of the earningsseries.'

A further examination of equation (8) reveals that Var(AP,) is positivelyrelated to PVR (again assuming PVR> -1). Thus, whether the relation betweenstock prices and earnings is assessed using the response coefficient or the veirl-ance of price changes, the impact of persistence is predicted to be positive.

To summarize, the theory predicts that the response coefficient will be posi-tively related to the persistence and the predictability of the earnings series. Theunconditional vari£ince of price changes is expected to be positively associatedwith persistence but negatively associated with predictability. These hypothesesare tested in the next section.

n. Empirical ResultsKormendi and Lipe (1987) estimate the relation between returns and earn-

ings and test their proposition that ao should be related to PVR. This section usestheir sample of firms to test whether Oo Is also related to the predictability ofearnings. First, the Kormendi and Lipe data and results are discussed. Second,the impact of the predictability of earnings is examined. Third, some sensitivityanalyses £U"e conducted.

Kormendi and Lipe use a two-equation system to estimate both the responsecoefficient, Oo, and the time-series properties of earnings, b,. Using the notationfrom Section I, they estimate the following:

^ (10)

(11)

The inclusion of the J subscripts indicates that a separate system is estimated foreach flrm,J. Also, the autoregressive model of earnings is limited to two lags. Inorder to reduce the cross-sectional correlation in the data, Kormendi and Lipe useflrm-speciflc returns and earnings. Rj. are the residuals from an annual marketmodel in which the percentage return for firm J*s common stock in year t is re-gressed on the percentage change in the CRSP value-weighted market index. Theannual returns are cumulated from April of year t until March of year t+1 ,

'The negative relation between the predictability of earnings and Var( AP,) may appear eontraryto the Holthausen and Verrecchia (1988) information economies model. They derive a positive rela-tion between the ability of an information souree to prediet the future liquidating dividend of the firmand Var( AP,). The difference in signs is due to a difference in assumptions. In their model, the firm'svalue is a function of the liquidating dividend, u. If the ability of earnings to predict u Increases, thenthe announcement of earnings provides more information. The present assumptions imply that therevisions in expectations of future accounting earnings are valued by the stock market. As mentionedabove, as the predictability of the earnings series increases, the release of X, cind /, contains less newInformation regarding future earnings and. therefore, Var(AP,) decreases. Indeed. Holthausen andVerreeehia show that decreasing the variance of u leads to a decrease in Var(AP,). Thus, the twomodels agree that as the variance of the attribute that is valued in the stock market decreases,Var{AP,) decreases,


Model:

Table 1Summary Statistics for Estimated Parameters'

(n= 145 Firms)

AXy, = it-i+ej.

(10)

(11)

Parameter

PVRj^1

bu

i>v\j

MEDVALJ

Mean

3.388.93

.7845.58-.08-.171.10

1239

StandardDeviation

3.123.411.84

86.69.29.24.34

3314

Minimum

-2.284.36

.002.33-.70-.79

.329.3

FirstQvuxrtile

1.416.92

.0613.20-.29-.31

.87115

Median

2.507.91

.1723.11-.07- .191.09397

ThirdQuartile

5.0710.12

.6843.32

.12- .061.28772

Maxtinum

17.9822.7117.83

774.4.92.66

2.0927695

' Ry,=the flrm specific (real) percentage return for flrm J In year t.AXy,=the flrm speclflc change in (real) earnings.P,,-,=the (real) stock price for flrmj at the beginning of year t.

ey,=the shock In the univeiriate earnings series.bu=the autoregressive coefficients.a(y=the response coefficient.

Other estimated parameters sire as follows:i*W?y=eamings persistence, derived from the estimated by, assuming T= 10%.

^^=variance ofthe estimated earnings shocks, 6j,.9iPj=vai\ance of Rj,xPj,.,.

Xy=estimated beta from an annual meirket model regression.MEDVALy=the median value of equity for each flrm.J, in millions.

because sample firms all have December 31 year-ends.'°P/,-i Is the stock price atthe beginning of the year t cumulation period. Similarly, AXj, are the residualsfrom a regression of dollar changes in earnings per sheire (before extraordinaryitems) for flrmJ on the changes in the Standard and Poor's index ofearnings. Rj,,Pjt-i, emd AXy, are adjusted by the consumer's price index in order to mitigate theheteroscedasticity caused by inflation (see Kormendi eind Lipe 1987 for a morecomplete description ofthe data). Data from 1947-1980 are used to estimate thecoefficients of the system (lO)-(ll) for each of 145 firms. A nonlinear weightedleast squares approach is employed.

As mentioned above, the existence of alternative information impliesao^d +PVR). Table 1 contains summary statistics for the estimated parameters.

'° The April-March cumulation has been used in prior studies so that the returns are contempora-neous with the three qucirterly and one annual earnings announcements for the yeeir. The correla-tions between earnings and returns are larger for cumulation periods which begin in year t - 1 andlast longer than 12 months (Collins et al. 1987). However, since the purpose of this paper is to ex-amine the contemporaneous relation between R, and e, under the assumption that the market hasalternative information, the April-March cumulation is appropriate.

Lipe—stock Returns and Accounting Earnings 59

Table 2Kendall r Rank Correlations

(n= 145 Firms)

%-Rj

PVR]MEDVALJ

. 211"

.018-.229"

.285"

.109'

-.256".286''*.083

-.211"- .083

PVRj

.005*-.157'

.584"

.025

-.001.015.209"

-.573"-.041

Pvki

.046

(t,=the estimated response coefilcient for flrmj.y = the time-series persistence of earnings for flrm J assuming T= 10%.j =the persistence of earnings based on flrm-specific interest rates.

^(/=variance of the estimated earnings shocks, 6j,.(^4Pj=variance ofRj,xPj,.i.

\y=estimated beta from an annual market model regression.MEDVALJ=the median value of equity for each firm, j , in millions.

' Signiflcant at the .001 level.' Signiflcant at the .01 level.' Significant at the .10 level.* Signifies the four direct tests of the model. These four significance levels are based on one-tailed

tests. All other significance levels are based on two-tailed tests.

The mean (median) of doy across the 145 firms Is 3.38 (2.50). The mean (median)of PVRJ is 8.93 (7.91). In addition, agyjs less than {1+PVRj) for 144 of the 145firms. Note that the calculation of PVRj requires an assumed interest rate fordiscounting expectations of future earnings; a rate often percent is used in Table1. The results are consistent with equation (6).'*

Tests of Earnings Predictability

The response coefficient, ao, is hypothesized to be an increasing function ofthe predictability of the earnings series. Predictability is reflected in the varianceof the ecimings shocks, ai (as ai increases, the predictability decreases). The em-pirical measure of ai for firmj is denoted ^,j, and it equ£ils Var(ey,), where &j, arethe estimated residuals from equation (11). The estimates CLOJ and ^.j can beviewed as random variables, the distribution of which depends on the "true" dis-tribution of aoj and aij, respectively, and the errors from estimating the param-eters for each firm. These estimates are analyzed across firms in order to infer therelation between the "true" aoj and aij. In testing the hypothesis, the null is thatCLoj and V,j are unrelated (or positively related), with the alternative hjrpothesisthat the two are negatively related.

" since PVR ls a decreasing function of the interest rate, assumlngân Interest rate substantiallyhigher than ten percent will result In Aq, being greater than (l + PVRj) for most firms. However.Kormendi and Lipe (1987,341) report that even If one assumes a rate of 30 percent, CiQi<(l+PVRj) for67 percent of the firms.

60 The Accounting Review. January 1990

Table 3Kendall Partial Rank Correlations

(n= 145 Firms)

-.335" .130'.297' .084'

6Q,=the estimated resjwnse coefficient for flrmJ.f*VRy=the time-series persistence of earnings for flrmj, assuming T= 10%.

O^=varlance ofthe estimated earnings shocks. &j,.OA,J=variance of Ry,xP,,.,.

' Significant at the .001 level (one-tailed test).' Significant at the .02 level (one-tailed test).' Significant at the .10 level (one-tailed test).

Because the functional form of the relation between OQ, and i^,j is nonlinear,the relation is examined using rank correlations." Table 2 presjents the Kendall Trank correlations for the data. The correlation between doj and V.j is - .371 with aZ-statistic of -6 .6 . The correlation is significantly negative at less than the .001level." The null is rejected in favor of the alternative.

Table 2 shows that the correlation between aoj and (1 +PVRj) is significantlypositive, which is the same result reported by Kormendi and Lipe.'* But note thati^.j and [1+PVRj) are significantly negatively correlated (rank correlation= -.256, Z= -4.57). Recall that PVR is a function of the autoregressive coeffi-cients ofthe earnings series. The negative correlation is apparently showing thatfirms with less persistent time-series have larger variances of earnings shocks.The theory in Section I makes no prediction regarding the sign or magnitude ofthe relation between i^.j and {1+PVRj), but the negative relation is consistentwith Lev's (1983) analysis of economic factors such as whether the firm producesdurables or nondurables^

Since ^.j and [1+PVRj) are significantly correlated, the simple correlationsbetween CLOJ and these two variables may suffer from an omitted variables bias. Inother words, the simple correlation between, say, aoj and (1 +PVRj) could be over-stated because movements in (1+PVRy) are proxy ing for movements in ^,j(Maddala 1977, 155-157). Therefore, partial rank correlations are presented inTable 3 in order to assess the incremental importance of persistence and predict-ability. The partial rank correlation between doj and ^.j, given (1+PVRj), is

" Scatter plots (not reported) show a strong curve in the relation as well as some outliers. Therank correlations are less sensitive to these problems than Pearson correlations or OLS regressions.

"Under the null, Z~N{O,1). While using rank correlations reduces the impact of outliers andnonllnearitles, the test does assume that each pair, d^, and P^. represent an independent draw. Cross-sectional correlations among the firms could result in the estimated parameters being cross-sectionally dependent, however, the market-wide movements were removed from earnings and re-tums to reduce this dependency.

" The table reports the^rrelations between various peirameters andf'WJy, which are the seime asthe correlations with H+PVRj).


- .335. with Z= -5.93 (significant at .001);" again the null is rejected which isconsistent with the hypothesis developed in Section I. The correlation betweenOoy and (1+PVRy), given i^.j, is .130. with Z=2.30. which is significant at .02using a one-tailed test. Thus, both predictability and persistence appear to be sig-nificant determinants of the response coefficients.

Section I also analyzed the variance of price changes. The theory shows thatVar( AP.) should be positively related to both aJ and (1+PVR). Let i^^pj representthe variance of Rj,xPj,.t, estimated over time for each firm. Table 2 shows asimple rank correlation between t py and 9^ of .286. Table^3 shows that thepartial rank correlation between i^^pj and ^^. given (1+PVRj), is .297. Bothcorrelations are significant at .001. Thus, the empirical evidence supports the hy-pothesis of a negative relation between the variance of price changes and the pre-dictability of the earnings series.

The simple rank correlation between (1 +PVRj) and P^PJ IS essentially zero.The partial correlation between these two variables, given \^.j, is .084 which issignificantly different from zero at .07. Persistence is not as strongly related tothe variance of price changes as it is to the response coefficient.

Sensittvity Analysis

The empirical tests are Joint tests of the null hypotheses and the as-sumptions. If the assumptions are poor representations of the interactions be-tween accounting earnings, stock prices, and alternative information, then equa-tion (6) will be a poor description of ao. The major concern is whether the resultsare spurious. Note, however, that deviations between the model's assumptionsand the real world would likely result in a failure to reject the null instead of aspurious rejection. An exception to this general rule is if the assumption involvesan omitted correlated variable, as discussed below. Also, because this study usesannual data, it can only test whether the association between returns and earn-ings are consistent with the hjrpotheses; the tests cannot determine whetherearnings cause returns.

Measurement errors have a subtle effect on the tests. As discussed in SectionI. &0J is a downwardly bisised estimate of (1 +PVRj) because one cannot observethe noneamings infonnation used by the market. But the inability to measuremarket expectations does not bias OQ, away from its theoretical value of(1 -I-PVR>)(1 -Mj). On the contrary, the hypothesized positive relation between a<vand 9.J explicitly incorporates the informational superiority of the market.Therefore, the traditional errors-in-variables bias mentioned in most responsecoefficient studies (Kormendi and Lipe 1987. 331) can not lead to spuriousresults in this paper.

The estimate of aoy will be affected by errors in specif3ring the univariate time-series model ofearnings. Suppose the estimated earnings shock. e>,. contains thetrue shock, ej,, plus measurement error, ej,. The efiiect of ej, is to bias the Ootowards zero, and theîas is an Increasing function of al. But the measurementerror also means that V^=oij+al. If there is substantial cross-firm variation in al.

" Maghsoodloo and Pallos (1981) describe hypothesis testing with partial rank correlation. Thenormal approximations used here are Interpolated which might slightly understate the Z-statistic.


then firms with larger al will have larger P.j but smaller ap,. Therefore, the nega-tive correlation between cioj and i^.j could be due to measurement errors in theearnings shocks instead of due to a negative relation between ao and al.

Unfortunately, the true earnings shocks and their variance are not observ-able and, thus, direct resolution of this issue is impossible. However, the restiltsregarding the variance of price changes can provide indirect evidence. Note thatPAPJ is not a function of al because the former is not conditional on the estimatedearnings shocks. If a large portion of the cross-sectional variation in V,j is due toal, then the correlation between i^^pj and Vej would be indistinguishable fromzero. The^mpirical results presented above show a sigificant correlation between9APJ and V.J, which suggests that \^.j captures significant cross-firm variation inal. This suggestsjbut does not guarantee) that the significant negative relationbetween cioj and V,j is not spurious.

One assumption that could lead to spurious correlation is the cross-section-ally constant interest rate assumption. Differences in interest rates would refiectdifferences in risk. Omitting finn-specific risk affects the tests in two ways. First,if r varies across firms, then PVRj as calculated above is incorrect. To assess theseverity of this error, the estimated market model beta of firm J, denoted %, isused to calculate a fimi^pecific interest rate, fy." Table 2jhows that the newpersistence measure, PVRJ, and the original measijre, PVRj, are significantlycorrelated. The simple rank correlations between PVRJ and both do> and V^pjactually increase, and the partial correlation between PVRj and aoj, given V. , ismore significant than before. Thus, incorporating firm-specific risk into persis-tence appears to provide stronger results.

The second effect of differential risk is that if my valuation assumptions arenot valid, ao may be related to risk in ways other than in equation (4). Indeed,Collins and Kothari (1989) find that the estimated response coefficient is nega-tively related to the firm's stock market beta. Table 2 shows that the rank correla-tion between dq/ and %j is - .229 (significant^t the .001 level). However, thepartial correlation between do and Xj, given PVRJ, is only - .084 (significant atthe .07 level, using a one-tailed test). Thus, once PVR is adjusted for differences inXy, the incremental explanatory power of Xy is greatly reduced. Another concern isthat V.y could be proxying for %. But this is not the case as the rank correlationbetween V.j and % is .083 which is not significantly different from zer(a at the . 10level (two-tailed test). In addition, the partial correlations of doy and PVRj, doy andV,j, and Vipy and V.y, given \j, are all significant in the predicted direction. Thus,the primary results of this paper are probably not driven by omitting cross-sec-tional differences in risk.

Another potential omitted factor is firm size. The contemporaneous relation

" The cedculation of fy Is as follows:

145

Thus, the_ average fj is still ten percent. But now, 90 percent of each firm's Interest rate is determinedbased on %, and ten percent is a constant. Using factors other than .01 and .09 yields similar results.


between returns and earnings has been shown to differ with size (Collins et al.1987; Freeman 1987). The firms are ranked according to their median marketvalue of equity over the 1947-1980 period, denoted MEDVALj. Table 2 showsthat the correlation between MEDVALj and both ctoy and V^ ^ are significantlypositive. However, size is not significantly correlated with PVRj or V.j- Partialrank correlations (not reported) confirm that the inferences drawn above regard-ing the effects of persistence and predictability do not appear to be driven bysize.'^ These results regarding size may not generalize to the population of firms,however, because this sample contains NYSE firms that survived for at least 34years.

m. ExtensionsPrevious sections developed and tested specific hypotheses regarding the im-

pact of alternative information. This section examines some ofthe other benefitsof assuming that the market has information other than earnings. First, a modelwhich includes alternative information can address the information environmenthypothesis and other issues in a more formal way. Second, since the differencesin the infonnation sets observed by the market and the researcher are explicit inthe model, one can assess whether ad hoc ways of dealing with measurementerrors will be useful. Third, while the assumed link between future earnings andfuture dividends is without error, the existence of alternative infonnation sug-gests that economic earnings differ from accounting earnings, and the differenceis negatively autocorrelated.

Adding alternative information to an existing model of returns implies thatthe relation between returns and earnings is a function ofthe predictability oftheearnings series. But equation (4) also shows that the relation depends on al, theability of the alternative infonnation to predict future earnings. For example, anincrease in ai means that J, is not as useful in predicting e^, and, therefore, ao in-creases. The negative relation between the predictive ability of alternative in-formation and the response coefficient is similar to the information environmenthypothesis (e.g., Atiase 1985, 1987; Collins et al. 1987; Collins and Kothari 1989;Freeman 1987; McNichols and Manegold 1983). The difference is that the latter isusually stated in terms of the quantity of alternative information, whereas theformer is based on the predictive ability or "quality" ofthe alternative informa-tion. Thus, including proxies for how well future earnings can be predicted byalternative infonnation may be useful in testing the infonnation environmenthypothesis. Also, one could test whether cross-sectional differences in size, thepredominant proxy for differences in infonnation environments, becomesinsignificant when cross-sectional differences in ai are controlled. Note that priorstudies of the infonnation environment hypothesis generally do not derive theirtests from a theoretical model of the relation between returns and earnings,whereas the importance of al is a direct implication from equation (4).

" The results of the four hypotheses tests are similar If mean, beglnnlng-of-perlod, end-of-period.or mlddle-of-period value Is used instead of the median. However, the correlation between dq, and sizeis not significantly positive for all definitions.


As to measurement error, ao is a function of (1 -M) because the researcherdoes not perfectly measure the information used by the market. One way of ex-pressing this is that ao is a biased estimate of the true parameter of interest,(l-l-PVR). But note that reverse regression will also yield bieised estimates of(1 -I-PVR). To see this, rewrite equation (4) as:

=kx+k.R'.+k.[l+PVR) ^'M^'^^M^'^^ ûe,, (12)P.-1

Ue, is the residual, and fci and ki are the reverse regression coefflcients. Assum-ing that Rf is a constant and that scaling doUeir retums and dollar earningsshocks by P,-i does not affect the expected veilue of fc2,'* the value of the reverseregression estimate of ao,ao, can be determined as follows:

,13)

Thus, ah is a biased estimate of (1+PVR), and the bias depends on M. Since

1-M

the bias is away from zero. The bias occurs because R, includes PMwî which isthe revision in expectations of erf i. Since i8Mu;,+t is uncorrelated with e,, a portionofthe independent variable in equation (12) is uncorrelated with the dependentvariable leading to the classic errors-in-variables bias in IC2. Therefore, even if theearnings shocks and stock returns are measured without error, the reverseregression produces biased estimates of (1 -I-PVR) because stock returns reflectinformation about next year's earnings shock." Again, this becomes apparentwhen the market is assumed to observe altemative information.

Brown et al. (1987) suggest an alternative control for measurement errorsusing retums prior to the cumulation period for R, as a second independent vari-able in equation (6). This procedure should reduce the errors-in-variables bias ifprior returns are correlated with the measurement error in the proxy for unex-pected earnings but are uncorrelated with current-period retums or the market'sunexpected earnings. Here the difference between e, and the market's assess-ment of unexpected earnings equals MUJ,. Equation (4) shows that R,., will becorrelated with Mio,. Further,R,-i is uncorrelated withe,—Mu;,andR,. Therefore,the model in this paper suggests that the Brown et al. method should be useful.^"

" These assumptions simplify the analysis. If they do not hold, the bias in kz will be more com-plex, but still exists.

" Beaver et al. (1987. 150-151) state that the reverse regression will not provide unbiased coeffi-cients if stock price changes reflect information other than current-period earnings. This analysisshows that such a bias is to be expected.

^ Using quarterly data and a maximum return cumulation period of 40 trading days. Brown et al.(1987) report modest improvements from including lagged returns.

Lipe—stock Retums and Accounting Earnings 65

Equation (4) also has implications for studies which analyze the differencesbetween economic earnings and accounting ecirnings (Beaver et al. 1980; Beaveret £il. 1987).^' In these models, stock price is a function of economic earnings(which is unobservable to the researcher) and accounting eeimings equals eco-nomic earnings for the period plus noise. This paper does not explicitly addressthe noise in accounting earnings because I assume a direct link between the revi-sions in expectations of dividends and earnings. However, the new informationprovided by accounting earnings alone, e,, is not equal to economic earnings.The difference between the economic earnings of year t and e, is:

{e,-Mw.+0Mwî)-e,=PMwî-Mw,. (14)

Equation (14) demonstrates that economic earnings lead accounting earnings.Further, the difference is negatively autocorrelated. The reason is that theincome from a given transaction will be recognized by both accounting earningsand economic earnings, but the recognition can take place in different years. Forexample, if economic earnings recognize a portion of a transaction's income inyear t and the remaining portion in year t+\ while accounting earnings recog-nize all ofthe income in period t+1, the difference between the two must be nega-tively autocorrelated.^^ Thus, by assuming that the market has alternative infor-mation, one difference between economic and accounting earnings becomesclear.

IV. Conclusions

This study examines the theoretical relation between stock returns and ac-counting earnings, assuming that the market has a second source of current-period information in addition to earnings. Theoretically, the stock return duringthe period is a function of (1) the time-series persistence ofthe earnings series, (2)the interest rate used in discounting expected future earnings, and (3) the relativeability of earnings versus alternative information to predict future earnings.Comparative statistics show that the response coefficient is an increasing func-tion of the ability of past earnings to predict future earnings and an increasingfunction of persistence. In addition, the variance of stock price changesconditional on earnings being announced is a decreasing function of the predict-ability of the earnings series and an increasing function of earnings persistence.The predictability/response-coefficient effect is positive because the valueattached to a one-dollar current-period earnings shock is an increasing functionof predictability. The predictability/variance-of-price-changes effect is negativebecause the average quantity of unexpected information released during theperiod is a decreasing function of predictability.

The sample from Kormendi and Lipe (1987) was used for direct tests of thesefour hypotheses, with three being supported by the data. The exception is that

" Some studies refer to economic earnings as "permanent" earnings. Whatever the name, theconstruct is that number which is valuation sufficient. Also, the difference between economic earn-ings and accounting earnings is usually termed "garbling."

"See Beaver (1970) for a more detailed discussion of how differences between accounting andeconomic earnings will be serially correlated.


the variance of price changes is only weakly related to earnings persistence. Sen-sitivity analysis suggests that the results are not driven by omitting risk and sizefrom the empirical tests. Also, by explicitly including alternative information informulating the hypotheses, the results cannot be attributed to errors in mea-suring market expectations. But other econometric problems cannot be resolvedand, therefore, the strength of the results may be overstated.

The model presented and tested in this paper extends the results from pre-vious work. First, existing empirical evidence that earnings shocks in year t+1are correlated with stock returns in year t suggests modelling the alternativeinformation as a noisy signal of future earnings. Second, previous models and theassociated empirical tests demonstrate the importance of earnings persistence.Third, models of how rational individuals assess expectations based on compet-ing sources of information suggest that predictive ability determines the extentto which a given signal is used. Combining these elements provides new insightsinto the relation between stock prices and accounting earnings.

Appendix

This appendix contains detailed derivations of the theoretical model. Stockprice is assumed to equal the present value of expected future dividends, or:

In equation (A. 1), P, is the stock price per share in period t, E, represents the ex-pectations conditional on all information available at time t, D,*, is the dividendpaid to equity holders in period t+s, and /3= 1/(1 -I-T), where T is the appropriaterate of interest for discounting expected future dividends. Dividends are used inequation (A. 1) because, as Ohlson (1988,29) points out, "ultimately only payoffscount, and dividends alone can be consumed." The discotmt rate is asstunedconstant across time for simplicity.

Total stock retvirns can be decomposed into expected and unexpected com-ponents as follows:

—J\,TPr-l Pfl

This analysis focuses on i?,", the portion of the return in period t that is unex-pected based on the information available in period t - l . Combining equations(A.1) and (A.2) yields:

s=0 Pt-i

In order to bring accounting earnings directly into equation (A.3), the present


value of the revisions in expected dividends are assumed to equal the presentvalue of the revisions in expected accounting earnings, in which case:

(A.4)

Obviously the assumption will not hold exactly and, thus, there is some errorintroduced in equation (A.4). These errors will affect the empirical tests, but with-out knowing exactly how the present value ofthe revisions in expectations of ac-counting earnings and dividends differ, it is impossible to accurately predict themagnitude or even the direction of the effect.

Equation (A.4) determines the magnitude of the return conditional on thenew information released during period t. Equations (2) and (3) in the textdescribe the information structure. The earnings series is assumed to follow afirst-differenced, finite-order autoregressive process. The autoregressive coeffi-cients, bi, axe assumed to be known by both the market and the researcher. Theshock (or innovation) in the earnings series, e,, is assumed to be unconditionallydistributed JV(O,o ). The normality assumption is convenient later in calculatingconditional expectations. The alternative information provides a noisy signal offuture earnings. The noise, n,+,, is assumed to be distributed N(0,o^). Also, e,+,and n,+» are assumed to be independent, for all I and k. I, and X, are observedsimultaneously because tests in the paper use annuEil data.

To understand the contemporaneous relation between returns and earnings,the revisions in the expections of X,*, in equation (A.4) must be restated. It ishelpful to rewrite X, in its infinite-order moving-average form, as follows:

X.=e{L)e., (A.5)

where fl(L) = 1 +diL+e2L^+. . ., and L is the lag operator. The d coefficients comefrom inverting equation (2) into a moving-average process. The relation betweendj and bi will be demonstrated later. Using equation (A.5), the revision in the ex-pectation ofX,^ is:

= g ejlE.{e.^.j) -E..rle.^j)]. (A.6)

Equation (A.6) can be simplified by determining what new information isreleased in period t. Note that all past realizations (X,-, and /,.(, 1=1 —oo) areknown prior to period t. Thus, the release of X, implies that the value of e, isrevealed. Further, combining equations (2) and (3) yields:

N

I,=X,n+nî=X,+J^ b.AX,«-,-i-e,+,+a,+,. (A.7)( = 1

Since the only unknowns on the right-hand side of equation (A.7) are e,+, and n,*,,the release of X, and /, together implies that the sum (e,+,-«-n,«) is also revealed.Note that investors do not know e,+, or n,+, individually in period t. For notationalconvenience, let u;,t,=e,+,-l-n,«. Expanding the right-hand side of equation (A.6)


shows that:

s-2

J=0

(A.8)

The summation overj=0 to s - 2 equals zero because e, and iu,+, provide no newinformation about the earnings shocks in period t-i-2 or thereafter. The summa-tion over /c= 1 to 00 also equals zero because earnings shocks in or before periodt - 1 are already known. Thus, only the two middle terms will be nonzero, whichyields:

+&î[E.(em)-E^,(e,«)]. (A.9)Disclosing e, and u>,+, causes the market to revise its expectations of e, and eî.Equation (A.9) shows that the revision in the expectation of e,[eî) leads to arevision in the expectation of X,^, and the magnitude ofthe latter revision is de-termined by d.(6.-i).

Substituting equation (A.9) into equation (A.4) and rearranging terms showsthat:

Equation (A. 10) can be simplified in two ways. First, the discounted sum of themoving average coefficients captures how current information impacts expecta-tions of future earnings. This sum is referred to as the time-series persistence ofearnings and is denoted PVR, in order to be consistent with the notation in Kor-mendi and Lipe (1987). They demonstrate that this sum is a function of the b,coefficients, as follows (pp. 329-330):

Second, under the assumption that e, and n, are distributed N{O,ai) andN{O,ai), respectively, E,-,(e,) has the following convenient representation:

E...(e,)=—ÛJ,. (A. 12)ai+ai

Lipe—Stock Returns and Accounting Earnings 69

To prove this, recall that the unconditional expectation of e, equals zero. But attime t—l, the market knows w,=e,-i-n,. Since e, suid n, are independent and nor-mally distributed, w, is also normally distributed. The expectation of e,conditional on w, is then:

In equation (A.13), n, In,,) equeds the unconditional expectation of e, [w,); a. (a.)equals the standard deviation of e, {w,); and p equals the correlation between e,and w,. Since

and p=-

equation (A. 12) holds. For convenience, let

Using equation (A. 12), the revisions in expectations of e, and e,*, are as follows:

These two simplifications transform equation (A. 10) as follows:

Equation (A. 15) represents how new information impacts stock returns.The relation between returns, earnings, and alternative information in equa-

tion (A. 15) presumes that X, and J, are observed simultaneously. While this isappropriate when annual returns are used, the model could be specified differ-ently. For example, if returns are cumulated over a few days surrounding theearnings announcement, then one would naturally assume that X, is releasedduring the cumulation period. However, the relation in equation (A. 15) wouldhold for the short cumulation period only if the alternative iniformation isreleased during the cumulation period and no information is released betweencumulation periods. A more reasonable assumption is that /, is revealed duringthe time between cumulation periods and X, is released during the cumulationperiod. This leads to a sequential release of information that is similar to thesequence considered by Holthausen and Verrecchia (1988).

If information is released sequentially, then the form of equation (A. 15)changes, and the change depends on the alternative information assumption (eq.[3]). For example, one could replace equation (3) with:

I,=X.+n.. (A. 16)

In this case, /, reveals e,+n, at the beginning of period t instead of revealinge,*i+n,*i. The relation between returns cumulated over the short period of timeand e, will still be a fimction of M, so the predictability effect is still present. But


the correlation between R, and e,+i Is zero. Thus, the lagged return effect thatmotivates the form of equation (3) is eliminated by assuming equation (A.16).

If one maintains equation (3) and Edso assumes sequential information, thenthe model becomes much more cumbersome. At the beginning of period t, I, re-veals information about e, and e,+,. At the end of the period, X, resolves all uncer-tainty regarding e, and provides additional information about e,+i. Thus, thereare multiple sources of information regarding e,+i which complicates theconditioned expectations. In general, the relation between returns and earningsis still determined by both predictability and persistence. Also, R, is still corre-lated with

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