Abstract
China Journal of Accounting Studies, 2014 Vol. 2, No. 3, 161–171, http://dx.doi.org/10.1080/21697213.2014.953381 Transitory noise in reported earnings: Implications for forecasting and valuation James Ohlson* Polytechnic University of Hong Kong, Hong Kong, China; CKGSB, Beijing, China The paper considers a setting where the present value of expected dividends determines price and the information to forecast the future includes reported earnings. In the model, reported earnings have been garbled by transitory noise, which cannot be inferred. ‘True’, but now unobservable, earnings are permanent as in Ohlson (1992). The stage setting result shows that capitalized expected reported earnings for the next period equals price regardless of the noise. More subtle is the inﬂuence of current reported earnings on the forecast of future expected earnings and, relatedly, the valuation in terms of the history of data. Because of the noise term, Bayesian updating implies that the investor uses the entire earnings history to learn about permanent earnings and to forecast future expected reported earnings. Speciﬁcally, the main result shows that the next peri- od’s expected earnings equal a weighted average of (i) current reported earnings and (ii) beginning-of-the-period expected earnings for the current period. This framework is often referred to as ‘adaptive expectations’ because there is gradual learning and updating. It depends critically on dividend policy irrelevance. The paper goes on to show that the weight on current earnings (term (i)) decreases as the noise increases. The model has testable implications for returns on earnings regressions and how one operationalizes value-relevance. Keywords: transitory earnings; valuation; earnings forecasting; noisy earnings 1. Introduction Graham and Dodd (1940), in their classic Security Analysis, argued that successful equity valuation should focus on assessing a ﬁrm’s ‘long run sustainable earnings’ or what is nowadays referred to as permanent earnings. The framework intrigues because it relies on an ideal earnings construct to evaluate whether a stock is over- or under- priced. An ideal earnings construct can of course never be observed directly. At best it can be estimated. To convert the idea into practice, guiding principles in Graham and Dodd suggest that reported earnings need to be adjusted for more or less transitory items to approximate permanent earnings. A further reﬁnement of the process considers past changes in reported earnings to reﬂect that such earnings generally evolve less smoothly than permanent earnings. Hence, one needs to smooth past earnings to ﬁnd the trend. An earnings history can thereby inform an investor concerned with approxi- mating permanent earnings. This paper deals with questions relating to Graham and Dodd’s approach to equity valuation. How does the presence of (unobservable) transitory noise in reported *Email: johlson@stern.nyu.edu. Paper accepted by Liansheng Wu. © 2014 Accounting Society of China 162 Ohlson earnings inﬂuence forecasting and valuation? How does this setting compare with the special case when there is no noise at all? Reported earnings may be worthwhile to forecast and they may be usefully informative, but something needs to be done to deal with the noise. These issues naturally arise in practice and in empirical research, and thus they should be of interest. Two critical assumptions underpin our model. First, there is an ideal but unob- servable construct of earnings, namely permanent earnings (Ohlson, 1995). Second, the observable reported earnings equal permanent earnings plus unobservable transi- tory noise. Combined with a present value of expected dividends (PVED) require- ment, the two assumptions are shown to imply that price equals next period’s expected reported earnings capitalized. We further show how an investor learns from reported earnings to forecast future earnings: the next period’s expected reported earn- ings are essentially determined by a weighted average of (i) current (reported) earn- ings and (ii) the expected earnings for the current(!) period that prevailed at the beginning of the current period. There is also a relatively minor third term that adjusts for an effect due to retained earnings. A full payout eliminates the last term: if e denotes expected earnings at date t for period t+1 and x denotes period t earn- t+1 t ings then e = w ⋅ x +(1- w) ⋅ e . This recursive forecasting scheme derives from t+1 t t Bayesian updating in light of new information. Kalman Filter techniques supply the analytical foundation to derive the result. We then show that w is small when the variance of the transitory component of x is large. Noisy earnings therefore mean that the current earnings are relatively less informative as compared to the history of earnings prior to the current period. Dividends are present in the model in addition to the (unobservable) permanent earnings and reported earnings. The structure of the basic model, however, ensures that the price, PVED, does not depend on the dividend policy. The model exploits this divi- dend policy irrelevance property to shift the analysis away from future dividends to future permanent earnings. Thus, the model distinguishes the distribution of wealth from its creation, although the latter variable cannot be observed. The theoretical analysis helps to frame certain empirical questions. Consider a regression where analysts’ (consensus) expected earnings for the forthcoming year is the dependent variable. How should it be explained? In other words, how should one conceptualize the right-hand side (RHS) of the regression? Given the dependent vari- able, one naturally introduces accounting variables observed at the end of the current period, such as reported earnings. But the theory here also suggests that it makes sense to add the expected value of the current period’s earnings at the beginning of the cur- rent period. Thus, the RHS of the regression is split into ‘new’ information and ‘old’– but still relevant – information. Moreover, in this regression, relatively more weight will be placed on ‘old’ information for ﬁrms with relatively ‘noisy’ new information. This kind of analysis is clearly doable, and it offers a new paradigm on how one can assess the value relevance of accounting data. The analysis also bears on the traditional value relevance studies where the market return is the dependent variable. We show that the more noise there is in earnings, the lower is the earnings response coefﬁcient (ERC). Thus, it follows that the modeling here builds in the classical errors-in-variables econometrics. We believe that no paper has shown how this conclusion derives from a setting in which the val- uation rests on the concept that value is determined by the present value of expected dividends. China Journal of Accounting Studies 163 2. Setup Our model rests on a multiple periods valuation framework with well-deﬁned informa- tion and objective probabilities. Like most valuation models, we deal with how one represents price as it relates to earnings, when the PVED determines the price. But, contrary to the prior literature, in our modeling the investor visualizes the future by looking at the history of reported earnings and dividends: prior periods’ information remains relevant and current realizations of earnings/dividends update an investor’s expectations. We use the following notation: d = dividends, date t, x = earnings, period t, p = price, date t, R =1 + r = the discount factor or one plus cost of capital, an exogenous constant, Ω ≡ {d , ..., d ; x , ..., x } = the information at date t. t 0 t 0 t The symbol Ω underscores that the relevant information refers to the history of realized dividends and earnings. This feature differs from models like Ohlson (1995), Feltham and Ohlson (1995) and the valuation literature discussed in Feltham and Christensen (2003). (These models do not depend on the entire history being relevant.). Unless we indicate otherwise, the information is understood to be Ω . To simplify ~ ~ the notation, we therefore write E½~x jX ¼ E ½~x and E½d jX ¼ E ½d and tþs t t tþs tþs t t tþs similarly for the expectation of any other random variable (such as ~ ~ E½~ p þ d jX ¼ E ½~ p þ d ). tþ1 tþ1 t t tþ1 tþ1 Our ﬁrst assumption is standard. Price equals the present value of expected divi- dends: p ¼ R E ½d (PVED) t t tþs s¼1 Our second assumption speciﬁes the dynamic for dividends. The dividend-forecast is a function of the dividend policy and the most recent information: d ¼ p x þ p d þ ~ u (DD) tþ1 1 t 2 t tþ1 where π (π ≠ 0) and π are the ﬁxed dividend policy parameters. The u are zero mean 1 1 2 t unpredictable disturbance terms. We place no other restrictions on the disturbance terms’ distributions (variances, for example, may depend on earnings or dividends). The DD assumption can be generalized to allow for non-linearity and other com- plexities on the RHS. But these generalizations would not be worth it insofar they merely introduce ‘house-keeping’ issues. The totality of assumptions – which includes a third one stated below – will ensure that the class of dividend policies (DD) work such that neither π nor π inﬂuence conclusions. That said, note that the forecasting of 1 2 dividends must depend on reported earnings; thus, we maintain the regularity condition π >0. Our third assumption, which speciﬁes the earnings’ dynamic, introduces the model’s main innovative concept. This dynamic revolves around the permanent earnings con- struct x*. The innovation is that investors cannot observe x*; that is Ω excludes x*. Reported earnings – which investors do observe – evolve stochastically according to two dynamic equations: 164 Ohlson ~ ~ x ¼ x þ d t t (ED) ~x ¼ R ~x rd þ ~ e t tþ1 tþ1 t 2 2 where d Nð0; r Þ, and e Nð0; r Þ are i.i.d. normally distributed disturbance terms t t d e ~ ~ and covðd ; e Þ¼ 0. We also assume covð~ u ; e Þ¼ 0. t s t s To appreciate the assumption ED, a number of observations help. (i) Using traditional terminology, one refers to x as a latent, or ‘hidden’, vari- able because it cannot (in general) be observed. In our model, x deﬁnes per- manent earnings. This terminology reﬂects that the retention of earnings alone accounts for the growth ðE ½~x =x ¼ 1 þ r ð1 d =x ÞÞ, which t t tþ1 t t equals one when there are no retained earnings. (ii) Combined with DD, ED implies a sequence of {E [d ]} conditional on the t t+τ τ information Ω . This setup means that PVED, ED, DD, and Ω yield some t t reduced form valuation function p = p(Ω ). t t (iii) The setting when there is no noise in earnings has been previously developed by Ohlson (1995). It serves as a benchmark for the more general setting (which turns out to be substantially more complicated). If δ = 0 for sure, then, trivially, x ¼ x and x becomes observable. Combined with PVED and DD, t t this observable permanent earnings setting implies p ¼ E ½~x =r t t tþ1 and p ¼ðR=rÞx d : t t A strictly positive variance term, var ðd Þ[ 0, or x 6¼ x , changes the model t t radically. Although the capitalized expected earnings relation perhaps remains valid (the issue is not obvious), the second equation cannot be true since the price, p , depends solely on information that excludes x (and x cannot be inferred from Ω ). t t Hence, the question arises as to how one identiﬁes the function p(Ω ), which replaces ðR=rÞx d . (iv) The i.i.d. random variable δ speciﬁes the transitory part of earnings. This component of earnings is, of course, unobservable; otherwise one could infer permanent earnings, x . An investor accordingly forecasts and infers value in a setting with incomplete information due to the transitory noise component in earnings. It makes sense to hypothesize that the information content of (reported) earnings should increase as the noise decreases. One should further expect the earnings response coefﬁcient (i.e., the usual ERC as it shows up in returns on earnings regressions) to increase as the noise decreases. These issues will indeed be addressed. (v) Value creation uncertainty shows up because of the second source of uncer- tainty, ɛ . We assume that the ɛ have a strictly positive variance. It avoids t t the boundary case when x can be inferred from the history of dividends and the initialization condition x =x . (An assumption of x ¼ x is not necessary 0 0 in our model. Still one can argue that at the ﬁrm’s inception x ¼ x = 0 and – d > 0 speciﬁes the start-up capital contribution). 0 China Journal of Accounting Studies 165 (vi) The normality assumption on the two noise terms in ED leads to Bayesian revisions that are linear in the observables. (vii) The requirement that the two noise terms in ED do not correlate could be relaxed. However, doing so would have led to a more elaborate analysis with the thrust of the conclusions being the same. (viii) It may be tempting to hypothesize that DD can be modiﬁed by letting x ,no less than x , appear on the RHS in DD. But this generalization changes the model non-trivially. (This is yet another point we will return to.) The three assumptions – PVED, DD, and ED – provide all of the model’s ingredi- ents. We now turn to the insights they yield. 3. Results We ﬁrst show that reported earnings works as an ex ante valuation attribute: the capital- ization of expected earnings equals price. The result relies on a subtle property of x . Given a forward-looking perspective, in expectation the x behave as if they are perma- nent. Further, because E ½d ¼ 0, the capitalization extends to permanent earnings as t tþ1 well as reported earnings. Proposition 1. Consider PVED, DD, and ED combined with the mild regularity R r condition max root of the matrix \R: Then p p 1 2 p ¼ E ½~x =r ¼ E ½~x =r: t t tþ1 t tþ1 Proof For the proof, see the appendix. The proof exploits the expectational equivalence of x and x in that ~ ~ E ½D~x r ð~x d Þ ¼ E ½D~x r ð~x d Þ ¼ 0 t tþs1 t tþs1 tþs tþs1 tþs tþs1 as long as s 2. In the expression the ‘=0’ part reﬂects the permanence of the x , and thus the x is no less permanent in expectation. (For τ = 1, the ﬁrst equality breaks down because x is observable while x is not, given the conditioning information Ω . t t Although this aspect is irrelevant in Proposition 1, it plays a role later.) The parameters π and π do not inﬂuence Proposition 1 (setting aside the regularity 1 2 condition). A certain robustness in the conclusion is therefore present. That said, we have not yet proved that p(Ω ) is independent of π or π . t 1 2 The next result articulates how the information Ω forecasts next period’s expected earnings. With this conclusion in place, one also obtains p(Ω ) via Proposition 1. Proposition 2. The earnings and dividends dynamics ED and DD with σ , σ >0 δ ɛ imply E ½~x ¼ Rðw x þð1 w ÞE ½~x Þ rd , where 1>w >0. The sequence fw g t tþ1 t t t t1 t t t t t¼1 evolves according to a difference equation w = h(w ; R, σ , σ ) independent of the div- t t-1 δ ɛ idend policy parameters π and π . 1 2 Proof For the proof see the appendix. An application of Kalman Filter techniques proves the proposition. Such tech- niques derive Bayesian revisions about future expected outcomes in light of the recent 166 Ohlson information. Although elaborate, these revisions can be worked out in a (X -conditioned) linear framework because the ɛ and δ terms satisfy normality. t t To interpret the earnings forecasting dynamic, recall that for observable x the per- manent earnings dynamic satisﬁes E ½~x jx ; d ¼ Rx rd . If the information X – t t t t tþ1 t t which excludes x – replaces the ideal information x and d , then the last proposition t t shows that one replaces x with a weighted average of (i) the current earnings, x , and (ii) those earnings that the investor expected at the beginning of the period, E ½~x .In t1 t this way the investor also updates her best assessment of permanent earnings since E ½~x ¼ E ½~x . t t tþ1 tþ1 A particularly simple expression is obtained if the payout satisﬁes d ¼ wx þð1 wÞE ½~x . On average, the scheme is a 100% payout t t t1 t (E ½d ¼ E ½~x ). Now t1 t t1 t E ½~x ¼ w x þð1 w Þ E ½~x t t t t t1 tþ1 t For any set of weights fw g , a direct backward recursive substitution shows that s¼1 E ½~x depends on the entire history of past earnings. t tþ1 We next consider the properties of the weights w . These require a time subscript because Bayesian updating of E ½~x includes an updating of the weight w . Implicit in t tþ1 t the analysis is some unspeciﬁed (exogenous) prior distribution for x . However, the prior distribution ceases to be relevant as t becomes large. This observation conforms to dynamic Bayesian models in which, as time passes, one learns more and more about the true structure that generates the data. In analytical terms, the sequence generated by h(w ) = w approaches a stationary point lim hðw Þ¼ w. It turns out that the properties of h t+1 t!1 (w ) are such that one solves for the stationary point w = h(w) via a quadratic equation. Thus, one derives w = f(θ, R) where h so that only the variance ratio affects w. (The qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2 quadratic solution equals w ¼ 0:5R ðR þ 1 þ hÞ ðR þ 1 þ hÞ 4R ). This equation reveals how the weight w depends on the exogenous parameters, σ and σ . ɛ δ Corollary 1. The solution lim hðw Þ¼ w exists for any 0<w <1. Moreover, w t 0 t!1 2 2 decreases as r increases and it increases as r increases. d e The result is intuitive. It says that the relative importance of updating for the most recently observed earnings depends upon the underlying noise in the earnings; the less the noise, the greater the weight assigned to current earnings (keeping the economic uncertainty ﬁxed). In a similar vein, w increases as the economic uncertainty increases because it makes the variances in the noise relatively unimportant. As to the dividend policy, it cannot affect the weight because it is unrelated to the resolution of the economic uncertainty (ɛ) or the noise (δ). Indeed, DPI obtains because of not only the assumption on earnings, ED, but also the assumption on dividends, DD. The latter does not include x* on the RHS of the dynamic. (The next section shows why putting x on the RHS changes the problem). Two additional properties of w are worthwhile discussing. Both are subtle and per- haps not so apparent. First, the weight on earnings satisﬁes the lower bound 1 − 1/R > 0 (given R >1)as the variance ratio approaches inﬁnity. In effect, regardless of the noise, earnings are always informative. This result is by no means obvious because, at the other extreme, w approaches one as approaches zero. But R > 1 implies that, on average, value is e China Journal of Accounting Studies 167 created as time passes, and the only way to learn about it is through earnings, no mat- ter how noisy. Second, the weight w on earnings increases as the discount factor R increases (which can be conjectured on the basis of the previous observation). This makes sense if one invokes the following heuristic argument. An increase in economic uncertainty, r , increases the weight w (as noted earlier). But an increase in economic uncertainty should also increase the risk, which is part of what should go into R. Hence, one can expect the signs of ∂w/∂R and ∂w/∂σ to be the same, which is the case. Although this - line of reasoning goes beyond the formal model, it has some economic plausibility by connecting the discount factor to the economic risk in earnings. Proposition 2 combined with PVED, i.e., the two propositions taken together, leads to the valuation function p(Ω ): p ¼ðR=rÞ½ w x þð1 wÞ E ½~x d : t t t1 t t The expression generalizes the permanent earnings case, p ¼ðR=rÞx d by replacing t t x with the estimate E ½~x ¼ w x þð1 wÞ E ½~x . In other words, for the general t t t1 t t t setting with Ω , E ½~x tþ1 p ¼ ¼ E ½~x d : t t t r r Although the variable x is not explicit in the last expression, it is part of the informa- tion used to determine E ½~x and E ½~x . A cohesive set of ideas thereby shows how t t tþ1 t an investor learns about permanent earnings from Ω , which in turn determines the esti- mate of the present value of subsequent expected dividends. The details of the ~ ~ sequence E ½d , E ½d ,.... need not concern an investor because of the model’s DPI t tþ1 t tþ2 property, precisely because permanent earnings is an ideal earnings construct. Given the expression for value we next derive the regression that explains the market return, ret =(p + d )/p , in terms of unexpected earnings (normalized by t t t t start-of-period price). While the result is interesting in its own right, it also bears on the large number of empirical studies of the returns on earnings regression. Corollary 2. The assumptions ED, DD and PVED imply ~ret ¼ a þ b½ ~x E ½~x =p t t t1 t t1 where α = R and b ¼ðR=rÞ w: The fact that the intercept equals R is unsurprising since PVED implies that the expected return equals R (and unexpected earnings has zero mean, of course). The intuition behind the slope-coefﬁcient, in contrast, depends directly on the tran- sitory noise in earnings. The dynamic ED can be thought of as an error-in-variables model, where the relative variance h determines the extent of the error. Hence, the last corollary shows that, since w decreases as θ increases, the slope coefﬁcient β decreases as the relative error in earnings increases. Standard econometrics teaches us that the slope-coefﬁcient in a univariate regres- sion decreases if the RHS variable is measured with error. Corollary 2 conforms to this idea if one keeps in mind that x E ½~x acts as an imperfect measure of the ‘ideal’ t t1 t measure x E ½~x . Only for the latter case will the slope coefﬁcient be equal to R/r, t1 t t which (for reasonable values of R) is far greater than that obtained in actual regressions (as has been long known). Thus, the discussion reveals that the error-of-measurement issue on the RHS may not only depend on an erroneous measurement of the ‘market’s’ 168 Ohlson expectation of the forthcoming earnings, as is commonly argued, but instead on the earnings construct itself. The response coefﬁcient R/r depends on earnings being perfectly aligned with permanent earnings. If there is an error in this regard – due to transitory effects in the accounting earnings measure – then there is another source of error that reduces the slope coefﬁcient. This point is not new, but we do not believe it 6,7 has been formalized in this way previously. Our ﬁnal comment in this section bears on the distributional assumption on ɛ and δ in the earnings dynamics ED. These random variables have a normal distribution to ensure analytical tractability. Bayesian analysis now implies that the forecast E [x ]is t t+1 linear in the information X . Thus, normality should be viewed as a convenience assumption. That said, one can also motivate the linear forecast by stipulating that the forecast must be the minimum variance linear forecast (which is the property that text- books discussing Kalman ﬁlter techniques most often stress). 4. The dividend distribution policy and value creation Our model sheds light on value creation as opposed to its distribution in a Bayesian framework. As noted previously, the DD assumption on the forecasting of dividends aligns with the DPI property, i.e., the policy parameters π and π do not inﬂuence the 1 2 weights in the value function. Given this fact, they have no inﬂuence either on the fore- cast E ½~x or the value function p(Ω ). There is an aesthetic neatness because noise in t tþ1 t earnings does not perturb the DPI inherent in the permanent earnings model. Noise in earnings causes no problems because, whatever the parameters of a dividend policy are, dividends do not bring to bear on the creation of value: nothing can be learned about the ɛ from knowing the policy. After all, DD ultimately represents past earnings and pure noise. Thus, one sees that DPI ceases if the investor can learn about the ɛ from dividends. As an extreme example, suppose that d ¼ const x þ noise; clearly, any evaluation of the weight w depends no less on the const and the variance of the noise. More generally, DPI does not apply if x had appeared on the RHS in the dividend dynamic or if the noise in the DD correlates with the disturbance terms in the ED. Because all of X inﬂuences the forecasting, it is of course true that the entire history of dividends inﬂuences the value function. The relevance of dividends in X reﬂects that the effect of retained earnings on the forecasting of (permanent) earnings cannot be neglected. Stated somewhat differently, if one forecasts x plus earnings foregone due tþ1 to past dividends, r d R , then neither the policy parameters nor the past divi- ts s¼1 dends will be relevant as long as one cannot learn about ɛ from the dividend dynamic. The above discussion of the absence/presence of DPI is not new. In many ways it can be traced to 1960s’ ideas that managers use dividends to convey information about the underlying ‘true’ or future reported earnings (e.g., see Pettit, 1972,or ﬁnance text- books such as Brealey & Myers, 2002). And now it helps to understand the dividend policy to explain returns or value. The modeling here thus formalizes what the litera- ture has long hypothesized as a possibility. 5. Concluding remarks From a conceptual perspective, the results in this paper try to provide insights related to the construct of permanent earnings. The construct poses problems, it can be argued, because permanent earnings are unobservable no matter how one modiﬁes GAAP, i.e., the construct exists only in the realm of ‘theory’ with its negative connotation. China Journal of Accounting Studies 169 The developments here modify this argument: reported earnings can take on a central role as information to approximate permanent earnings. In this framework, transitory items act as noise in ﬁnancial statement analysis, and the name of the game becomes one of trying to ﬁnd some earnings number that can serve as a starting point to forecast next period’s earnings. Moreover, the reasoning is consistent with analysts’ perception that the forecasting of earnings is a focal point of equity valuation. Yet such a focus on reported and ideal earnings is fully consistent with the concept that PVED determines value at all points in time. Hence, the model distinguishes value creation from its distri- bution, and it even recognizes the importance of dividends not providing information on the noise in reported earnings. We believe that viewing analysts’ forthcoming expected earnings as a dependent variable provides a paradigm of how to study value relevance empirically. The model- ing here further suggests that it makes sense to put the dependent variable lagged by one year on the right-hand side in the regression, and then ask what current period accounting data also explains (analysts’) expected earnings. A focus on explaining expected forthcoming earnings leads to the role of special items, indicators of the qual- ity of earnings, explaining cash ﬂows versus accruals, and a whole slew of issues familiar from recent accounting research. And the current analysis brings out that the results from such regressions bear on the returns to earnings relation. Speciﬁcally, if some measures of reported earnings play a modest incremental role in explaining the next period’s expected earnings, then these earnings should not contribute much to the returns to earnings regression either. In sum, one can ask questions about what account- ing information – with a speciﬁc focus on various measure of earnings – analysts use to update their forecasts and how this information must bear on value and returns in a consistent fashion. Acknowledgement The author wishes to thank Ken Yee for supplying some key insights in this paper. That said, the author alone is responsible for any errors. Notes 1. The permanent earnings concept has a long history in accounting thought, like Beaver (1997) and Black (1980), to cite well-known discussions. Ryan (1986) developed the basic relations in formal terms. Ohlson (1992) extends this analysis. Textbooks also consider per- manent earnings, e.g., Penman (2006) (who uses Graham and Dodd’s terminology, sustain- able earnings). For recent empirical work recognizing the centrality of permanent earnings in equity valuation, see Easton, Shroff, and Taylor (2006), Dichev and Tang (2007). 2. The model here differs from Ohlson (1999). This model allows for (idiosyncratic) transitory noise that informs if observed, but this noise is never part of reported earnings. 3. The literature often refers to this weighted average scheme as ‘adaptive expectations’ or as ‘exponential smoothing’. In spite of the vast accounting literature on earnings forecasting, it seems to have never been applied. However, 35 years ago Elton and Gruber (1972) pub- lished an empirical paper in Management Science comparing this approach to competing ‘mechanical’ forecasting models. They suggest adaptive expectations worked the best. It may further be noted that Elton and Gruber refer to ‘normal’ earnings rather than to ‘perma- nent’ earnings. See also Elton, Gruber, Brown, and Goetzmann (2007, pp 477–482) for a general discussion. 4. As an alternative way of expressing the permanence of earnings in expectation, note that ~ ~ E ½x ¼ R E ½x r E ½d for all s 1. However, in case τ =0 t tþsþ1 t t tþs tþs E ½~x 6¼ R x r d unless x = x . t tþ1 t t t t 170 Ohlson 5. Note that because x 62 X ,E ½~x 6¼ x : t t t t t 6. Easton et al. (2000) evaluate how the splitting of earnings into permanent and transitory parts inﬂuences the returns on earnings regression. 7. It may be noted that the modeling here has not allowed for growth and conservative accounting (including future positive NPV projects). This claim follows from Zhang (2000), which shows that P [ E ½~x =r is necessary and sufﬁcient for conservative accounting t t tþ1 given growth. However, we can work with a more general assumption to avoid this prob- lem; yet the thrust of all conclusions will remain. One can use the following assumption, which generalizes permanent earnings: ~x ¼ Rx rd þ v þ e , where t 1t 1;tþ1 tþ1 t ~v ¼ c v þ e speciﬁes the growth concept and v is observable. 1;tþ1 1;t 2;tþ1 1,t 8. To be sure, a dividend policy such as d ¼ K ~x þ ~ u also eliminates DPI. Now one learns t t t1 about lagged x , which in turn means that K and d bear on E ½~x as well as E ½~x : t-1 t t t t t1 9. Easterwood and Nutt (1999) evaluate the contributing factors of current accounting data in the revision of a FY2 forecast which a year later turns into a FY1 forecast. This approach differs from ours since we consider FY1 at the end and start of the current period. 10. Shroff (1995) shows how one identiﬁes ﬁrms that are likely to contribute to a relatively high R when one regresses returns on earnings in the cross-section. 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Appendix Proof of Proposition 1 Let y ¼ x =r and z = y + d - Ry . Then p ¼ E ½~y þ R E ½~z . The result follows t t t+1 t t t t tþ1 t tþs t s because ED implies E ½~z ¼ 0 and the regularity condition guarantees that no variable grows at t tþs ~ ~ a rate faster than R . Also, E ½x ¼ E ½x . t t tþ1 tþ1 Proof of Proposition 2 The proof relies on Sargent (1987, pp. 230, 231). The only difference pertains to how to deal with dividends. ts Let y ¼ r R d ; thus y deﬁnes the implicit earnings due to past dividends. Let t s t s¼1 0 0 0 0 x ¼ x þ y and x ¼ x þ y . The model (x ; x ) satisﬁes the one in Sargent (1987), i.e., his t t t t t t t t 0 0 equations (16) and (17), with c =1, ρ = R, h ¼ x , and z ¼ x . Due to DD, there is no informa- t t t t tion in d , d as one determines the best linear projection E ½h as a function of Ω . Now it fol- t t-1 t tþ1 t lows from Sargent, expressions (21) and (22) where K = w /R, that t t 0 0 0 0 0 E ½~x ¼ R½w x þð1 w ÞE ½~x . Noting that E ½~x ¼ E ½~x and plugging in the deﬁni- t t t t1 t t tþ1 t t tþ1 tþ1 0 0 tions for x and x the result follows, including 1>w >0 from Sargent’s equation (21). t t
Journal
China Journal of Accounting Studies
– Taylor & Francis
Published: Jul 3, 2014
Keywords: transitory earnings; valuation; earnings forecasting; noisy earnings