Bridging the gap between stock price and bottom-line accounting numbers

We develop a method for extracting “other information” from the articulation between bottom-line accounting numbers and stock prices. We posit that “other information” captures future earnings growth originating from conservative accounting recognition principles as demonstrated by Penman and Zhang (2020) and Penman and Zhu (2022), as well as nonzero net present value investment opportunities. Our findings confirm that “other information” is strongly associated with various proxies for expected future earnings growth and firm risk attributes. Furthermore, we show how a structural expected return model incorporating our “other information” estimate can predict out-of-sample future stock returns and generate sizeable long-short return spreads.


Introduction
A substantial number of accounting-based valuation studies recognize that a gap exists between the stock price and the valuation implied by bottom-line accounting numbers, the so-called "other information" (henceforth, ϑ). 1 Given evidence that the value relevance of bottom-line accounting numbers has declined and loss-making firms have become more common in recent decades, a better understanding of ϑ becomes increasingly important to financial statement users and standard setters. 2  The identification and measurement of ϑ, however, pose substantial challenges due to its unobservable nature and indeterminacy of dimensionality.In this paper, we uncover ϑ from the articulation between the stock price and bottom-line accounting numbers and validate its valuation implications.
While the stock price capitalizes the projection of a firm's future earnings and earnings growth, the bottom-line accounting numbers, namely earnings and (cumdividend) book value, only record the "certainty" portion of value-added from past transactions and defers the recognition of more risky components of economic benefits until the underlying risk is largely resolved (Penman and Zhang 2020;Penman and Zhu 2022).An "other information" term is consequently required to bridge the gap between the stock price and the valuation implied by bottom-line accounting numbers.In this paper, we develop a method to estimate ϑ.We start from a parsimonious linear valuation model that accommodates accounting conservatism and estimate ϑ as the difference between the stock price and a linear combination of bottom-line accounting numbers.By design, the ϑ estimate summarizes all value-relevant information beyond current earnings and (cum-dividend) book values, adjusted for risk.
The concept of ϑ captures the capitalized future earnings and earnings growth that are yet to be recognized in current earnings and book values.Its omission in the financial statements can be attributed to two related, but distinct, effects.First, according to conservative accounting principles, the riskier components of economic benefits associated with past transactions are deferred to the future until that risk is resolved; hence, the generation of future earnings growth is tied to risk.That is, holding the firm's lifelong earnings stream constant, such a deferral mechanism creates ϑ by matching the earnings recognition to the underlying uncertainty resolution process. 3Second, the firm may face nonzero-NPV investment opportunities that represent valuable operating options.These options, if exercised at a future date, would alter the firm's lifelong earnings stream.Holding the accounting policy constant, any 1 Ohlson (1995) is the first to introduce the term to summarize factors not captured by current abnormal earnings that contribute to the prediction of future abnormal earnings and thus equity value. 2 For example, Barth et al. (2019) find that bottom-line accounting numbers explain smaller fractions of stock price variations over time.Joos and Plesko (2005) show that as short-term losses become more frequent, an increasingly important component of valuation is attributable to the terminal value. 3In a mark-to-market accounting, all nonzero-NPV investments are captured in the book value, and ϑ becomes redundant.In that sense, ϑ is a natural consequence of accrual accounting.valuation changes due to changes in the firm's future investment opportunities would be reflected in ϑ. 4Our empirical examination focuses on assessing the validity of ϑ estimates.First, we show that firms that appear to exhibit more conservative accounting and face more valuable future investment opportunities tend to have a larger proportion of their equity values that is attributable to ϑ.These results suggest that both accrual accounting and real investment opportunities jointly explain equity valuation beyond bottom-line accounting numbers.In addition, we show that ϑ strongly predicts future earnings growth even after controlling for expected growth proxies, as identified by Penman andZhu (2014, 2022) (henceforth, PZ) and other studies.We also provide evidence that ϑ significantly predicts one-year-ahead equity risk attributes such as stock beta, downside beta, idiosyncratic volatility, and stock returns.Collectively, these tests validate the role of ϑ in conveying signals for evaluating expected earnings growth and the embedded riskiness in growth.
As an empirical application, we examine the usefulness of our ϑ estimate in constructing out-of-sample expected return measures.We derive a characteristic expected return expression that combines the book-to-market ratio, forward earnings yield and the ϑ-to-price ratio.It can be viewed as a concise version of PZ's expected return model. 5Unlike PZ, we do not use historical return data to estimate the returncharacteristic relation nor carry forward the weights attached to the characteristics in implementing our expected return model.Instead, we seek a forward-looking construction by utilizing the transformation of valuation parameter estimates as obtained through the Ashton and Wang (2013) regressions at the industry level.We show that our expected return estimates predict one-year-ahead realized stock returns almost one-to-one in the cross-section, consistent with expected returns being optimal statistical forecasts of future realized returns.These results hold even after controlling for future cash flow news and discount rate news, as well as other leading return predictors.In addition, the discount rate news proxy based on our expected return estimates exhibits highly significantly negative coefficients with contemporaneous returns, consistent with notion that a rise in the discount rate depresses the stock price (Easton and Monahan 2005;Lee et al. 2020). 6Further portfolio analysis reveals that our expected return estimates generate an ex ante near-monotonic decile ranking of one-period-ahead realized stock returns.Specifically, an investor following a longshort portfolio strategy based on the extreme deciles earns an 10.40% hedge return per annum on average, and there is no evidence of reversal in the hedge portfolio returns up to five years ahead.
We extend the valuation literature in two important aspects.First, we characterize and quantify the missing link between stock price and bottom-line accounting numbers.Our "other information" variable naturally connects to expected future earnings and earnings growth resulting from conservative accounting principles and future nonzero-NPV investment opportunities, and conveys risk.It complements bottomline accounting numbers in explaining risky earnings growth and expected stock returns.It is estimated from a reverse engineering procedure using the stock price, current book value, (forward) earnings, and dividends, as well as industry-wide information.We recognize implicitly the joint effect of future cash flow growth and the riskiness associated with the growth on valuation.This is in contrast to the conventional valuation approach that estimates growth in expected cash flows and riskiness as separate value attributes.
Second, by giving empirical content to "other information", our study directs fundamental analysis to evaluate long-term expected earnings growth.By relating a hypothesized source of growth to ϑ, one can validate how much the market is willing to pay for that growth.Given the evidence that growth stocks appear to outperform value stocks for over a decade, 7 understanding what the market is paying for beyond bottom-line accounting numbers becomes increasingly crucial for understanding the changing investment landscape.From the standard setter's perspective, it comes down to whether detailed financial statement analyses can add to the understanding of ϑ.Barth et al. (2019) show that while the value relevance of bottom-line accounting numbers has decreased over time, the decrease is compensated by detailed accounting line items.Our study further suggests that the incremental value relevance of an accounting line item likely depends on how the underlying accounting procedure handles risky expected growth or how it reveals future investment opportunities.
In addition, we propose a new approach for estimating firm-level expected stock returns.There are a few noticeable differences between our expected return estimates and those found in the existing literature.Unlike in applications of the implied cost of capital (ICC) models (e.g., Claus and Thomas 2001;Gebhardt et al. 2001;Easton 2004), implementing our expected return model does not require separate assumptions about risk and long-term growth.Instead, we rely on the ϑ term to capture the joint valuation implications of risky growth, to ascertain whether our ϑ estimate is internally consistent in the sense that it jointly reflects the co-determination of expected earnings growth and associated risk (Penman 2016).Moreover, our forward-looking estimation procedure does not rely on historical returns for its model calibration, contrary to existing characteristic models for expected returns (e.g., Lyle et al. 2013;Penman andZhu2014, 2022). 8We use only time t data and allow both the characteristics and valuation weights to be updated when new information becomes available.9While our firm-specific expected return estimates use the industry cost of capital as a component, we provide evidence showing that deviations of firmspecific characteristics from their respective industry averages play a more important role in predicting future returns.Hence, our expected return model can be thought of as a hybrid model that combines appealing features of both ICC and characteristic models.
The rest of the paper is organized as follows.Section 2 explains the motivation and relation to prior literature.Section 3 details the theoretical foundations and estimation procedure.Section 4 describes the sample and obtains empirical estimates of the valuation multiples.Section 5 examines the usefulness of the ϑ estimate for explaining expected growth and firm risks.Section 6 constructs the expected return estimates and validates their out-of-sample performance.Section 7 presents additional analyses and robustness checks.Section 8 concludes the paper.

Motivation and relevant literature
"Other information" (ϑ) has been recognized as an important valuation construct since Ohlson (1995).The fundamental analysis literature seeks to identify observable elements of ϑ to inform investment strategies.Value relevance studies also invoke the concept of "other information" to articulate the relevance of certain accounting measurements.Corporate finance studies often draw links between certain managerial practices and ϑ to study their valuation consequences.Conceptually, ϑ summarizes all value-relevant information that is beyond earnings and (cum-dividend) book values, adjusted for risk.Feltham and Ohlson (1995) assert that there are two major avenues in which earnings and book values fail to capture all NPVs of future earnings and earnings growth.First, conservative accounting principles require that the riskier components of economic benefits originating from past transactions are deferred to the future.The deferral can be exemplified by the immediate expensing of costs associated with internally generated intangibles, aggressive asset depletion schedules, asymmetric asset impairments, deferred revenue recognition, and so forth.In this manner, earnings are shifted across reporting periods, given a lifelong earnings stream.Therefore, accounting conservatism induces predictable future earnings growth, which is priced in by the market through ϑ.
Second, "virtually all accounting procedures do not recognize today the expected net present value of future investment projects" (Feltham and Ohlson 1995, pp.718-719).Future positive-NPV investment opportunities represent operating options that managers exercise at their discretion.Holding the accounting policy constant, the valuation may change when investors anticipate exercising such options, thus reflecting an additional source of expected future earnings growth.In other words, ϑ represents the NPVs added by not only the deferred expected earnings growth originating from past transactions but also the expected long-term earnings growth associated with future investment projects, adjusted for risk.In some instances, the valuation effects of the two sources of growth may not be easily disentangled.In fact, they are often jointly in action.For example, R&D activities may bring more investment opportunities within reach of the firm's competencies, but the underlying risk in the R&D outcome means that accounting must defer the recognition of the expected payoff as earnings and booked assets.
Risk and uncertainty are intrinsically linked to earnings growth attributable to accounting conservatism and future investment opportunities.The accounting deferral mechanism is essentially driven by the underlying risks of expected future earnings so that earnings are recognized only when the associated uncertainty is resolved (Penman and Reggiani 2013).Therefore, conservative accounting to a certain extent creates ϑ by matching the earnings recognition to the underlying uncertainty resolution process (Penman and Zhang 2020).While investment opportunities may have positive NPVs ex ante, there is often some significant risk that a firm may forgo profitable investment projects due to capacity constraints or even undertake projects that turn out to yield negative NPVs.In other words, the ex post value attributed to these investment opportunities eventually depends on the firm's capacity to exercise the option and other factors that may change the expected profitability of the investment projects, all of which are at risk today.Hence, ϑ also reflects risk from the firm's investment opportunity set.Furthermore, consistent with the modern economics of accounting, where book values of assets can be expected to generate a risk-free rate of return under conservative accounting principles, the associated future earnings/cash flows are then viewed as certainty equivalents (Christensen and Feltham 2009;Cochrane 2009).Consequently, the risk element measured by the covariance between risky earnings/cash flows and the stochastic discount factor must be included in ϑ.Therefore, ϑ reflects not only expected earnings growth but also the associated risk that determines the discount rate in valuation.
The conceptual and practical importance of ϑ in valuation has been largely overlooked in the extant literature.For example, studies often directly regress stock prices on contemporaneous bottom-line accounting numbers and treat the fitted value of the regression as the intrinsic equity value, where the error from estimation is attributed to mispricing.Using this approach, Rhodes-Kropf et al. (2005) decompose the market-to-book ratio into the market-to-intrinsic value and intrinsic value-to-book components and find that the former (latter) is a positive (negative) predictor of the firm's future mergers and acquisitions activities.We argue that ϑ constitutes a correlated omitted variable in this analysis.Omitting ϑ can lead to inconsistent estimates for the valuation weights of the accounting numbers and thus erroneous intrinsic value estimates.Indeed, it is expected that firms facing more investment opportunities and thus higher ϑ embedded in their stock prices are more likely to pursue growth through mergers and acquisitions (Levine 2017).A similar critique applies to several studies of how market mispricing affects corporate policies such as earnings management, corporate governance, corporate investments, and the value-versus-growth anomaly. 10owever, estimating ϑ has proved to be a challenging task.Two common approaches have been used to quantify it in the literature.The first approach attempts to extract ϑ from the difference between the expected abnormal earnings based on the consensus analyst forecasts of earnings and the abnormal earnings predicted from an AR(1) model in the Ohlson (1995) framework (e.g., Dechow et al. 1999).The second approach attempts to identify observable variables that may contain information for incremental future cash flows as proxies for ϑ (e.g., Myers 1999). 11 Unfortunately, these attempts to measure and incorporate ϑ in valuation models have failed to consistently deliver material improvement in applications for several reasons.First, as argued above, ϑ by its very nature embeds the market's expectations on both short-term and long-term earnings growth.It possesses a multidimensional property, and its nature is likely to be context-specific.12Calibrating an observable accounting proxy that sufficiently aggregates such information may be impractical.Second, as ϑ is defined relative to bottom-line accounting numbers, its identification requires a framework that allows for accounting conservatism.Third, ϑ must reflect the joint effects of expected earnings growth and the risk adjustment (i.e., the discount rate).It suggests that the expected earnings growth and the associated risk need to be simultaneously co-determined (Penman 2016). 13n a slightly different context, PZ motivate a valuation and expected return model that effectively incorporates some elements of ϑ.Specifically, they argue that the effects of conservative accounting principles can be partially captured by various line items from the financial statements.Therefore, accounting ratios based on these line items should bear signals about expected earnings growth and the associated risks as a result of conservative accounting practices.For instance, PZ argue that several balance-sheet growth measures, including operating accruals and change in net operating assets, imply partial realizations of growth potentials and resolution of risks originating from past transactions and thus are inversely related to future growth and risk.Based on their characteristic model, they find that, in addition to current book value and earnings, these accounting ratios can reliably predict twoyear-ahead earnings growth.They further show that the same set of ratios appear to be consistently priced in ex ante expected returns, beyond current earnings and book values.Their results provide support to the notion that ϑ should capture expected earnings growth and the riskiness in that growth.
In this paper, we develop a new approach to measuring ϑ based on a reverseengineering procedure.Our approach extends and complements PZ's work in several aspects.First, given that ϑ is latent, a bottom-up identification strategy (as in PZ) is unlikely to capture all the factors that contribute to it.There may be an inexhaustible list of accounting ratios that could be constructed to capture some of the conservatism effects.Furthermore, ϑ also captures nonzero NPVs that are added by future investment opportunities, which are beyond the recognition scope of almost all accounting procedures and thus largely unexplored in the PZ analysis.Therefore, we pursue a top-down approach and rely on the stock price as an efficient information aggregation mechanism to reduce the high-dimensionality of ϑ into an observable scalar value.After all, assuming market efficiency, any information is value-relevant if and only if it is priced.
Second, while PZ's analysis qualitatively links accounting conservatism with future growth and riskiness in that growth, we seek to quantify the net valuation effect attributable to ϑ.By construction, the ϑ estimate that is derived from our approach reflects expected growth only to the extent that it adds nonzero NPV.Conceptually, it is possible that a source of earnings growth is value-neutral if the risk premium associated with such earnings growth in the discount rate completely offsets the growth.14This makes our measurement potentially useful for evaluating managerial practices.For example, an option to enter a new product market may add future earnings and earning growth, but it may not necessarily add value if the underlying risk is too high.

Estimation procedure
Our estimation procedure is outlined as follows.First, we express stock price as a linear combination of the (cum-dividend) book value, earnings, and ϑ.We then follow Ashton and Wang (2013) and Wang (2018) to recast this general linear expression into a cross-sectional earnings forecasting model.Next, we estimate this model by industry-year to recover the industry average valuation parameters.To the extent that firms in the same industry operate in a similar business environment and apply similar accounting policies, these industry average parameter estimates serve as reasonable conditional averages for the firm-specific valuation multiples.Finally, we insert the obtained parameter estimates and firm-specific accounting numbers back into the linear valuation model.This yields an explicit estimate of ϑ as the difference between the observed stock price and the resultant linear combination of bottom-line accounting numbers.Ohlson (1995) and Feltham and Ohlson (1995) show that the market price of equity can be expressed as a linear function of current accounting fundamentals and a variable summarizing other value-relevant information under the following three assumptions.First, required return is defined by the no-arbitrage condition: E t [P t+1 + d t+1 ] = RP t , where P t is the stock price, d t is net dividend (net of capital contribution) at time t, R equals the gross rate of required return, and E t [.] is the expectation operator conditional on information available at time t.Second, the clean surplus accounting relation holds: b t = b t−1 + x t − d t , where b t is book value of equity and x t is accounting earnings.Third, the future abnormal earnings can be predicted by the current bottom-line accounting numbers and an unidentified ϑ in a linear fashion.Pope and Wang (2005) further extends this framework by incorporating accounting conservatism manifested in abnormal book value growth as an additional predictor of abnormal earnings.Under these assumptions, equity value can be summarized by the following simple linear pricing rule: where ϑ t denotes "other information" that is priced by the market but not yet captured by bottom-line accounting numbers. 15This specific form is consistent with the Miller and Modigliani (1961) dividend irrelevance proposition, as book value and dividend compensate each other one-for-one.
Given the linear combination of accounting fundamentals, b t + α 1 (b t + d t ) + α 2 x t , the term ϑ t represents the value-relevant other information reflected in the market but not booked in accounting.Equation 1 echoes the notion that, when buying a stock, investors not only buy the value implied by current earnings and book value, but also the expected earnings growth reflected in ϑ.For simplicity, we assume that ϑ t follows an autoregressive AR(1) process: 16 where φ > 0 is restricted by the transversality condition such that it is allowed to be higher than one plus the cost of equity capital for only some finite periods.ε ϑ,t+1 is a zero-mean error term.We view (φ − 1) as the firm's expected growth rate in ϑ, which changes every period based on the new information that becomes available.
To estimate ϑ, we need to estimate a set of valuation parameters, α 1 , α 2 , and φ, in Eqs. 1 and 2. We obtain these estimates using the method developed in Ashton and Wang (2013).Specifically, we first replace ϑ t = P t − (1 + α 1 )b t − α 2 x t − α 1 d t by invoking the linear pricing rule of Eq. 1.Under the no-arbitrage dividend condition 15 Throughout the paper, we suppress the firm subscripts when there is no chance of confusion.The linear pricing rule is more general than it may appear.Pope and Wang (2000) extend the linear pricing rule in a setting incorporating stochastic discount factors.Ang and Liu (2001) show that this linear form is preserved for a class of affine processes with any specification of the set of value-relevant firm fundamentals and any latent dynamics of the expected risk premiums under certain conditions. 16We could explicitly include other variables in the dynamic, such as a conservatism adjustment term as per Ashton and Wang (2013).Note, also, that ϑ t itself can be expressed as a nonlinear function of accounting and non-accounting variables.and clean surplus accounting, Eqs. 1 and 2 then imply that the expected earnings can be written as:17 Equation 3 establishes an intrinsic link between prices and expected earnings and reflects the notion that prices lead earnings.That is, prices forecast future earnings beyond the information reflected in current earnings and book values.Note that Eq. 3 holds without an error term, where the valuation parameters (α 1 , α 2 , φ, R) are the "true" firm-specific parameters conditional on time t information.While the firmspecific parameters in the model are latent, their sample averages can be estimated from a portfolio of reasonably homogeneous firms, given a measure for Hence, for each cross-section of firms at time t, Eq. 3 has the feature of a random coefficient model.The sample average estimates can then be used to approximate the firm-specific parameters.Following a longstanding capital market practice, we estimate the valuation parameters by industry-years.Specifically, we deflate both sides of Eq. 3 by P t and run the following regression model: where the dependent variable F E t,t+1 is the analysts' consensus one-period-ahead earnings forecast formed at t and serves as our proxy for market earnings expectations, E t [x t+1 ].This implies that the imprecision in recovering the true firm-specific parameters from Eq. 4 may arise from two sources: the measurement error in F E t,t+1 and the random deviations of the firm-specific valuation parameters from their industry averages.We denote the estimated industry-level parameters as α 1 , α 2 , φ, and R.
From Eq. 1, it follows that the firm-level ϑ can then be estimated as: ( 5 ) The estimated ϑ term, θit , captures future growth as reflected in current prices and the current state of the industry, but not as manifested in the firm's bottom-line accounting numbers.Notably, our estimation procedure uses only time t information, which ensures that the accounting numbers, stock prices, earnings forecasts, and valuation weights are based on the same information set.This is due to fact that our model is based on a partial equilibrium condition that holds at every industry-year.
Although the estimates for the valuation parameters and ϑ are subject to estimation error, to what extent the measurement error problem detracts from the usefulness of the model is an empirical question.If the industry-level valuation parameters (α 1 , α 2 ) are consistent estimates of the firm-specific valuation parameters (α i1 , α i2 ), then approaches zero, and, thus, on average, we have consistent estimates of ϑ.The measurement error described in Eq. 6 has the following properties: the error is independent of the current price; the variance of the error increases with either the variance of α 1 or α 2 ; and the consistent estimation of individual α 1 and α 2 is a sufficient but not necessary condition for the consistent estimation of ϑ.

Data description
Our empirical analysis draws data from three sources: (1) analysts' forecasts and stock price information from I/B/E/S; (2) accounting data from Compustat, and (3) stock return data from CRSP.We construct a base sample of all firms listed in NYSE, AMEX, and NASDAQ and identified at the intersection of the three databases from 1985 to 2015.We exclude all non-equity issues, such as ADRs, as indicated by the CRSP share code (SHRCD).Following prior studies, to mitigate the effect of extreme values, we delete observations with a book-to-price ratio (b t /P t ) lower than 0.01 or higher than 100, earnings yield (x t /P t ) lower than −1 or higher than 1, or stock We conduct our tests using per-share data adjusted for stock splits and stock dividends.The market expectation of one-year-ahead earnings, E t [x t+1 ], is measured using the median analyst forecast, F E t,t+1 , observed four months after the fiscal year-end.We accumulate one-year-ahead buy-and-hold stock returns, R t+1 , from May of year t + 1 to April of year t + 2. We follow the recommendation by Shumway (1997) and apply a −33% adjustment for delisting returns.
Table 2 provides sample descriptive statistics of the input variables used in the estimation of Eq. 4. The mean and median of book-to-market ratio (b t /P t ) are 0.61 and 0.51, comparable to those reported in prior studies.The forward earnings yield, F E t,t+1 /P t , is higher than the trailing earnings yield, x t /P t , at all reported percentiles of their distributions, consistent with expected earnings being less affected by negative accruals.Accruals (Acc t ) have a negative mean and median, consistent with prior findings that accruals tend to have more income-decreasing items.Net operating asset growth ( NOA t ), percentage sales growth rate (Saleg t ), investments in property, plant & equipment (I nvest t ), and net external financing (Exf t ) all have positive means and medians, indicating that firms typically experience positive growth.The average one-period-ahead buy-and-hold return, R t+1 , is 12% with a large standard deviation of 48%.Overall, the distributions of input variables and key firm characteristics are within the expected ranges reported in prior studies.

Model estimates
We estimate (4) by industry and year using the equal-weighting one-step generalized method of moments (GMM).Although the regression model is non-linear in the valuation parameters, Slutsky's theorem ensures that consistent estimation of the linear coefficients leads to consistent estimation of the re-parameterized model parameters (Hayashi 2000).In our baseline analysis, the estimation of Eq. 4 is conducted over 12 industries for each year, using the industry classification provided by Kenneth French's data library. 19We test the robustness of our results to alternative industry classifications and other grouping criteria in Section 7. Table 3 Panel A presents the average GMM estimates of the valuation parameters.Column (1) reports the time series averages of industry required returns R. The time series average expected return is 8.5%, similar to what is reported in prior studies. 20Column (2) shows that the average growth in ϑ, (φ − 1), is positive and statistically significant, suggesting that investors typically anticipate that the firm's long-term growth opportunities will expand.Table 3 columns (3) and (4) show that α 1 is statistically insignificant and close to zero in most years, and that α 2 is posi-19 https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html. 20See, for example, Claus and Thomas (2001), Gebhardt et al. (2001), Easton (2004), Nekrasov and Ogneva (2011), and Ashton and Wang (2013) tive in all years.Furthermore, these observations suggest that the incremental role of earnings increases over time when valuation anchors on book value.Table 3 Panel B compares the average valuation multiples by industry.For instance, average growth in ϑ, (φ − 1), is only 1.68% in the manufacturing sector but reaches 13.19% in the healthcare sector.They suggest that industry membership is an important source of variation in valuation multiples, thus lending support to our industry-specific estimation of valuation parameters.
We construct the other information estimate ϑ t according to Eq. 5. To capture the relative importance of ϑ in the stock price, we scale the estimate by stock price, ϑ t /P t .Figure 1 plots the annual averages of this ratio over time separately for the five industry sectors, also from French's data library.On average, ϑ accounts for approximately 50% of the stock price, indicating that ϑ is indeed an important valuation component.This ratio varies considerably over time and across industry sectors.For example, ϑ t /P t reached almost 100% in the healthcare industry in 1997, but dipped into the negative region in the finance industry (included in the "Others" sector) during the 2008 financial crisis.

"Other information" and risky earnings growth
In this section, we present evidence that delineates the drivers of ϑ and the implications for a firm's future earnings growth and associated risk attributes.

Accounting conservatism effect
While the stock price capitalizes the projection of a firm's lifelong earnings stream, accounting earnings and book value growth are credited only by the "certainty" portion of value-added from past transactions, and the recognition of more risky components of value creation is deferred until the underlying risk is largely resolved.Such deferrals under conservative accounting principles give rise to ϑ.We empirically examine whether ϑ manifests the accounting conservatism effect by regressing the estimate ϑ t /P t on several firm-specific proxies for accounting conservatism.
We apply three commonly used conservatism measurements in our analysis. 21We first use NOA t to (inversely) measure conservatism.PZ argue that the accounting recognition of balance sheet growth implies the resolution of uncertainty, reducing the deferral of earnings and book value growth.Therefore, we expect NOA t to be negatively associated with the ϑ estimate. 22We then use the difference between realized stock return and the ratio of current earnings over lagged stock price as the second measure of conservatism.This measure is essentially the difference between economic earnings and accounting earnings scaled by beginning-of-the-period stock price, and it captures the conservatism in earnings recognition (Zhang 2000).Finally, we consider the Cscore t developed by Penman and Zhang (2002).It estimates the "reserves" created by conservative accounting through aggressive expensing of R&D and advertising expenditures.Table 4 Panel A shows that ϑ t /P t is indeed negatively associated with the first measure and positively related to the other two, supporting the notion that accounting conservatism can induce ϑ.

Investment opportunity effect
Our ϑ construct not only reflects the conservatism effect but also captures the value attributable to the firm's future investment opportunities.Intuitively, the equity value of the firm depends not only on the "recursion value" of its assets-in-place but also, at least in part, on exercising future investment opportunities by the firm.While the former is partly proxied by the linear combination of (cum-dividend) book value and earnings (adjusted for accounting conservatism as discussed above), the latter is factored into stock prices without affecting the accounting numbers; thus, it must manifest in the ϑ term. 23To validate this interpretation, we examine whether our ϑ estimate is associated with reasonable proxies for the firm's investment opportunities.
We consider several proxies for investment opportunities used in prior studies.24Specifically, we alternately estimate the association of ϑ with the firm's patent stock, R&D intensity, and asset redeployability (Cao et al. 2008;Trigeorgis and Reuer 2017;Grullon et al. 2012;Kim and Kung 2017).Table 4 Panel B reports the regression estimates.In these regressions, we control for the PZ growth proxies to partial out the effects of accounting conservatism on ϑ.The estimates show that the firm's equity value attributable to ϑ is higher when the firm accumulates more innovation outputs, reinvests more in inventive activities, or has the ability to redeploy assets to alternative production technologies.

"Other information" and expected earnings growth
A key implication of ϑ is that it can forecast expected earnings growth.From the accounting conservatism perspective, accounting deferrals shift current earnings and book value growth to future periods, inducing expected earnings growth without affecting the firm's total lifelong earnings.From the perspective of investment LT G Q i denotes the i th quartile of Ltg t for i = 1, 2, 3, 4 opportunities, the firm can expand the scope and scale of its operations, charging incremental future earnings growth beyond the earnings stream implied by the status quo production technology.Both mechanisms suggest that ϑ should be positively related to expected future earnings growth.
Figure 2 shows the empirical distribution of θt /P t by the quintile of analysts' consensus long-term growth estimates (LT G t ), a proxy for analysts' expectation of the firm's earnings growth over the next three to five years.Strikingly, as LT G t increases, all quintiles of the distribution uniformly move rightwards, demonstrating a strong positive association between expected earnings growth and ϑ.
To examine the incremental predictive power of θt /P t relative to other growth proxies, we estimate the following cross-sectional regression model: where EGrowth t,t+h is a measure of the firm's forward earnings growth from year t to year t + h, Y t denotes a vector of control variables, and is a vector of corresponding coefficients.Following PZ, Eq. 7 controls for other determinants of earnings growth identified by PZ, including earnings yield (x t /P t ), book-to-market ratio (b t /P t ), change in net operating assets ( NOA t ), accruals (Acc t ), investments in PP&E and inventories (I nv t ), external financing (Extf t ), firm size (Mcap t ), and leverage (Lev t ).Admittedly, we can only observe future growth up to a finite horizon h in our empirical tests, so we cannot quantify all implications of θt /P t for growth in the far distant future.
We estimate Eq. 7 by year and report the Fama and MacBeth (1973) estimates in Table 5.If ϑ t /P t has incremental information content in explaining the expected earnings growth, then we expect β > 0. In Panel A, we use ϑ t /P t to forecast future In Column (4), we consider the 4-year compound annual growth rate in sales, as an additional growth proxy, because the growth embedded in ϑ may realize into sales before earnings in the finite horizon.The estimates in Panel A show that, for all four growth proxies, the β coefficients are positive and statistically significant at the 1% level.The coefficients are also economically significant.For example, the estimates in Table 5 column ( 1) imply that a one-standard-deviation increase in ϑ t /P t is associated with an incremental increase in EGR t,t+2 equivalent to 30.1% of its standard deviation.
In Panel B, we replicate the regressions with four alternative growth metrics implied by analysts' forecasts to test whether the market's current growth expectation is associated with ϑ.Through columns (1) to (4), we consider analysts' long-term growth estimates (LT G t ) and implied earnings growth (I EGR t,t+h , with h = 2, 3, 4).I EGR t,t+h is defined the same way as EGR t,t+h , except that realized future earnings (x t+h ) are replaced with analysts' h-year-ahead earnings forecasts observed at year t.The results based on these measures remain similar to those in Panel A. Overall, the results in Table 5 indicate that other information, as implied by our model, captures sizeable cross-sectional variation in expected future earnings growth.

"Other information" and firm risk
Both the accounting conservatism effect and investment opportunity effect suggest that ϑ should also indicate firm risk.On the one hand, conservatism is an accounting response to the risk in the economic benefits associated with past transactions.This implies that the larger the portion of value that is attributable to ϑ, the greater the risk that is associated with future earnings and earnings growth.On the other hand, the NPVs of future investment opportunities are uncertain and at the risk of not being realized.Therefore, ϑ indicates not only expected growth but also risk to the firm.
We extend our analysis to examine the ability of ϑ to capture future firm risk attributes.We estimate the following regression model: where Risk t+ is a proxy for the firm's forward risk and where the set of control variables Y t are the same as those specified in Eq. 7. Again, we conduct the Fama-MacBeth estimation of Eq. 10 and report the results in Table 6.
In Panel A, we examine how ϑ is associated with future stock market risk attributes.The first risk attribute is the firms' capital asset pricing model (CAPM) , which focuses on the firm's exposure to negative aggregate market news.The third measure is the idiosyncratic volatility, I V ol t+1 , which has been shown to bear asset pricing consequences after controlling for systematic risk.Lastly, we also consider one-year-ahead return, R t+1 , as a dependent variable, because its predictable component is a comprehensive measure of priced risk in asset pricing theory.For all measures of equity risk, we find highly significant positive coefficients of ϑ t /P t , suggesting that ϑ is indeed associated with elevated levels of riskiness.We note that a one-standard-deviation rise in ϑ t /P t implies an incremental 11.2% increase in a typical firm's downside beta (see Table 6 column 2).
In Panel B, we examine how ϑ relates to forward volatility in various fundamental performance measures.The results show that ϑ t /P t positively forecasts the volatility of quarterly return on equity, cash flow from operations, sales growth, and profit margin over the three-year period.This suggests that the stock market risk implications of ϑ are congruent with implications of ϑ for fundamental performance risks.Collectively, the results in Tables 5 and 6 support the role of our ϑ estimate in capturing expected earnings growth as well as the riskiness in that growth.

Application: estimating expected returns
To validate the usefulness of the estimates of ϑ, we construct a characteristic expected return model that incorporates ϑ and examine its out-of-sample predictive ability for stock returns.The no-arbitrage condition, clean surplus relation, and Eqs. 1 and 2 together imply the following total required return expression (with added firm subscript i for precision): where , ϑ it P it .Equation 11 describes the expected one-period-ahead return as a linear function of book-to-market ratio, forward earnings yield, and the price-deflated ϑ.Consistent with Eq. 1, we expect that the last term in Eq. 11, E t [ϑ t+1 ] = φϑ t , summarizes the implications of long-term earnings growth for risk and expected returns beyond book value and earnings.Penman and Zhang (2020) show theoretically that earnings growth and earnings yield convey information about expected stock returns under conservative accounting.The return expression also closely resembles the returncharacteristic relation employed in PZ, except that we do not require explicit proxies for long-term growth and we base our earnings yield on forward earnings rather than realized earnings.26We use three industry-level estimates, (α 1 , α 2 , φ), in place of the firm-specific parameters in Eq. 11 to construct expected firm-specific return estimates.In other words, we construct the firm-level expected return from Eq. 11 by combining crosssectional industry average parameter estimates and the firm-level book-to-price ratio, forward earnings yield, and the estimated ϑ-to-price, as follows: where P it , ϑ it P it .The above estimation procedure bears the notable feature that we use only time-t information to construct time-t expected future returns.Hence, this is a forwardlooking approach and can be used to describe ex ante market perceptions. 27If the industry-based parameters (α 1 , α 2 , φ) are all consistent estimates of the firm-specific valuation parameters (α 1 , α 2 , φ), then the expected difference between the true firm expected return of Eq. 11 and our expected return estimate of Eq. 12, R it − R it = βZ it − β Z it , would be zero.Hence, consistent estimation of all the valuation parameters yields consistent estimation of the firm's true expected returns.
Figure 3 visualizes the time-series evolution of the average expected return estimates and the implied risk premiums.The risk premium appears to rise after major market shocks such as the 2001-2002 dot-com bubble burst, the 2007-2008 financial crisis, and the 2010-2011 European sovereign debt crisis.This is consistent with the notion that risk premiums are higher in adverse market and economic environments.Importantly, the parameter φ − 1, which captures the expected expansion of ϑ, appears to contribute some sensitivity of the expected return estimates to temporal macroeconomic conditions.Specifically, all estimates of φ − 1 are positive and around 2.5% except for the years 2008-2009, when φ − 1 drops sharply to -3% due to the extremely depressed prospects in business activities.These observations seem to echo the Ashton and Wang (2013) interpretation of φ − 1 as a proxy for expected macroeconomic growth prospects.

Return forecast regressions
We examine the usefulness of incorporating the ϑ estimate in the context of predicting stock returns out-of-sample.Following the prior literature, our validation methodology is based on a tautological decomposition of realized stock returns (Vuolteenaho 2002;Easton and Monahan 2005).Specifically, the log-transformed realized return (r t+1 = log(1 + Ret t+1 )) is approximately the sum of the logtransformed expected return (er t ), cash flow news (cf n t+1 ), and discount rate news (drn t+1 ): While the expected return is known at time t, the two news components are contemporaneous and not predictable ex ante.We specify the following regression model: where er t = log(1 + R it ).If the empirical proxies on the right-hand side are perfect, Eq. 14 should coincide with 13, implying δ 1 = δ 2 = δ 3 = 1.Since empirical measures of these three return components involve measurement errors, the construct validity of the expected return estimates can be jointly assessed by how closely their regression coefficients in Eq. 14 are from their theoretical values of one.The cash flow news proxy is defined as follows: where roe t+1 = log(1 + x t+1 /b t ), f roe t,t+1 = log(1 + F E t,t+1 /b t ), and κ c is the expected persistence of roe t+1 estimated from a two-year cross-sectional hold-out sample for each industry. 28The number 0.96 is approximately equal to one minus the historical average of dividend yield.
28 All estimates of κ c are significantly below 1, thus rejecting the unit root null.
The discount rate news is calculated as where r and κ r are the constant and persistence parameters of an assumed first-order auto-regressive process for log-transformed expected returns, er t+1 = r + κ r × er t + ε μ t+1 .Because our measure of expected return admits time variation, our proxy for drn t+1 slightly differs from that used in Easton and Monahan (2005), which applies to models assuming constancy in expected returns.We estimate Eq. 14 alternately using panel regressions controlling for year fixed effects and Fama-MacBeth regressions.Table 7 reports the results.Column (1) shows that, after controlling for year fixed effects, er t positively predicts one-year-ahead stock returns on a univariate basis.The predictive coefficient is 0.87, which is statistically significant at the 1% level.In column (2), after controlling for contemporaneous news proxies, the coefficient of er t is raised to 1.03 and remains highly significantly different from zero but statistically indistinguishable from the theoretical value of one.Column (3) further shows that adding additional return predictors does not impair the loading on our expected return estimates.Columns (4) to ( 6) present qualitatively similar results and support similar inferences using Fama-MacBeth regressions.Importantly, 27 (23) out of the 30 annual cross-sectional regressions produce positive (and statistically significant) coefficient estimates for er t across the specifications in Columns ( 4) to ( 6), demonstrating the stability of its return predictability.Turning to our news proxies, we find that cf n t+1 (drn t+1 ) receives large and highly significant positive (negative) coefficients across all specifications.
Notably, the negative coefficients on drn t+1 imply that discount rate changes based on our expected return estimates induce large negative revisions in contemporaneous stock prices, further supporting the validity of er t as an expected return proxy.In Table 7 Columns ( 5) and ( 6) cf n t+1 (drn t+1 ) receives positive (negative) coefficients in all of the 30 annual cross-sectional regressions.In comparison, Easton and Monahan (2005) find that most existing implied cost of capital estimates do not load sufficiently large return predictive coefficients, and their derived discount rate news proxies exhibit weak explanatory power for contemporaneous returns.

Portfolio analysis
An important question in evaluating the usefulness of an expected return measure is whether investors can expect to earn economically meaningful gains if they form portfolios based on this measure.Therefore, we conduct portfolio analysis to test whether a long-short strategy based on the expected return estimates, er t , is associated with sizable out-of-sample gains.This approach is also useful in mitigating the measurement error problems in firm-level expected return estimates (which confound inferences in Table 7), as firm-level measurement errors are smoothed out at the portfolio level.Specifically, we form 10 portfolios of stocks based on decile rankings of firms' expected return estimates and observe the subsequent annual buy-and-hold returns for each of the years t + 1 to t + 5. Table 8 reports time series averages of these portfolio returns.At the one-year horizon, our expected return estimate, er t , generates a near-monotonic ranking of realized returns.An investor following a zerocost investment strategy going long on the tenth decile portfolio and short on the first decile portfolio is expected to generate a hedge return of 10.40% (t-statistic 2.64), which is both statistically significant and economically large.Furthermore, extending the buy-and-hold horizon to the fifth year, we find no sign of reversal in the gains earned in the first year.This suggests that the one-year-ahead return spread is unlikely to be caused by mispricing, corroborating the risk-based interpretation.Moreover, the apparent demise of return predictability beyond the first year after portfolio formation is mostly driven by the rebounce of the low extreme decile portfolios.If the investor buys the tenth decile and shorts on the second decile portfolios, she can still expect approximately 5% annual hedge returns from the second year onward, and these return spreads are marginally statistically significant.Therefore, our expected return estimate maintains a modest level of return predictability over the long term for most of the stocks.Collectively, the results in Tables 7 and 8 demonstrate the usefulness of the new expected return measure developed by incorporating our ϑ estimate.

Return predictability of industry costs of capital and firm-specific adjustments
One of the valuation parameters we uncover from estimating (4) by industry-year is the industry cost of capital R. We can further show that our firm-specific expected return estimates are equal to R adjusted by deviations in the firm fundamentals from their respective industry averages.Specifically, the expected return estimate for firm i in industry j can be re-expressed as:

Robustness tests
In this section, we discuss a number of robustness tests that we perform to examine the sensitivity of our results to alternative empirical design choices.

Ignoring "other information"
One may argue that the return predictive ability of our expected return estimates may be driven entirely by the information conveyed by forward earnings yield and book-to-market ratios.To check for this possibility, we estimate expected returns by restricting ϑ to zero.Untabulated results show that the cross-sectional return predictive regression coefficients are considerably lower, ranging from 0.21 to 0.39, and the average one-year-ahead extreme decile portfolio hedge return is about half as large as that reported in Table 8.These findings indicate that ϑ is an important contributor to the construction of meaningful expected return estimates.

The importance of industry grouping
One innovation in our estimation procedure is the use of industry average valuation multiples to approximate the firm-level counterparts.This method assumes that firms in the same industry have common exposure to similar input and product market conditions, and thus their accounting numbers and value drivers have more comparable implications for valuation.To examine the importance of this approach, we estimate (4) again in the cross-section without any industry grouping and repeat all subsequent analyses.In this case, the ability of ϑ to explain growth and firm risk and the return predictive ability of the resulting expected return estimates become considerably weaker.For example, we note a return predictive coefficient of 0.34 (t-stat 2.12) for the univariate specification and of 0.46 (t-stat 2.66) after controlling for cash flow news and discount rate news in the Fama-MacBeth return forecast regressions.This analysis elucidates the importance of industry grouping in improving model performance.

Alternative industry classifications
We also check whether our results hold for alternative industry classifications.In the main tests, we apply a 12-industry classification scheme.Intuitively, using the valuation parameters estimated from a portfolio of firms with greater homogeneity should generate more accurate estimates of firm-specific parameters.Clearly, there is a trade-off between more granular industry classifications and statistical reliability in estimations.We assess the sensitivity of our results to alternative industry classifications by repeating the analyses presented in Tables 5, 6 and 7 using the Fama-French 5-industry classification.Table 10 columns (1) to (3) report the regression estimates using the Fama and MacBeth (1973) regressions.Notably, we find that our ϑ estimate is strongly associated with expected earnings growth and future risk attributes.Moreover, the resulting expected return measures significantly predict the cross-section of future returns out-of-sample.
We also consider other criteria to achieve homogeneous groupings of firms.First, we consider tercile partitions based on firm size or book-to-market within each of the Fama-French five industries.Furthermore, in a smaller sample using panel regressions controlling for year fixed effects, we replicate the analyses using the Hoberg and Phillips (2016) text-based industry networks when estimating (4).All attempts return similar results to those in our main analyses.

Excluding loss firms
We examine the sensitivity of our results to a sample that excludes loss-making firms.One potential concern is that loss-making firms may have significant impact on the estimation results, as the connection between their accounting numbers and stock prices may differ from profitable firms.To address this concern, we repeat our analyses by excluding firms with negative earnings before estimating the industry-level valuation parameters.Table 10 columns ( 4)-( 6) report the Fama and MacBeth (1973) regression results.Again, the results suggest that the exclusion of loss firms does not alter our main inferences as reported in Tables 5, 6, and 7.

Conclusion
Established valuation theories have long celebrated the existence of an "other information" (ϑ) variable that bridges the gap between stock prices and the value embedded in bottom-line accounting numbers.We show how this other information captures the net present value-added by anticipated yet uncertain future investment opportunities, as well as risky earnings growth originating from conservative recognition principles.We also posit that although expected future earnings growth may add to stock valuation, the risk associated with that growth countervails its value implications.Therefore, ϑ manifests the net effect of future earnings growth and the riskiness in that growth.We characterize and quantify ϑ from the articulation between bottom-line accounting numbers and stock price.ϑ is estimated from a reverse engineering procedure using stock price, current book value, and forward earnings and dividends, as well as industry-wide information.A battery of empirical validation tests show that the ϑ estimate is highly associated with the firm's accounting conservatism and future investment opportunities, and that it yields significant predictive power for future earnings growth and risk attributes.
Finally, we demonstrate the economic significance of ϑ in an application of constructing out-of-sample expected return estimates.We derive a structural expected return expression that explicitly incorporates the ϑ term and examine its ability to predict stock returns out-of-sample.We find that our expected return estimates can significantly predict future realized stock returns (even after controlling for future cash flow news and discount rate news) and generate sizeable long-short return spreads.Furthermore, while both industry average required return and deviations in firm characteristics from the respective industry averages complement each other in predicting firm-specific returns, the return predictive ability is largely attributable to the latter component.Standard deviation of return on equity of the firm over the 12 quarters between year t and year t + 3.

σ (CF O) t,t+3
Standard deviation of operating cash flows deflated by total assets of the firm over the 12 quarters between year t and year t + 3.

σ (Saleg) t,t+3
Standard deviation of percentage quarter-over-quarter sales growth of the firm over the 12 quarters between year t and year t + 3.

σ (P M) t,t+3
Standard deviation of net profit margin of the firm over the 12 quarters between year t and year t + 3.

Ret t+k
Future

Fig. 2
Fig. 2 Empirical distribution of θt /P t by long-term growth quartiles.Note: This figure plots the kernel density distributions of θt /P t by quartile of analysts' consensus long-term growth estimates (Ltg t ).LT G Q i denotes the i th quartile of Ltg t for i = 1, 2, 3, 4

Fig. 3
Fig. 3 Time series plot of expected return estimates.This figure plots the expected return estimates (ER), expected risk premiums (RP ), and the growth rates in "other information", (φ −1), over the sample period 1985 to 2015.The solid line and the dashed line plot the average expected return estimates and the riskfree rates for each year.The dotted line plots the yearly averages of the estimated long-term growth rate of ϑ , φ − 1.The shaded bars between the solid and dashed lines indicate the annual averaged expected risk premiums

Table 1
Sample selection The estimation sample consists of 88,702 firm-year observations over 11,157 firms.Table 1 summarizes our sample selection procedure.The subsamples employed in subsequent validation tests may vary depending on availability of additional variables required.All variables used in the study are defined in Appendix.
Note: This table presents the sample selection procedure.We start with a dataset of 163,194 firm-year observations in the intersection of Compustat and CRSP databases and end up with the final sample of 88,702 firm-year observations that is used to estimate "other information" and expected returns price lower than $0.5.

Table 2
Summary statistics Note: This table presents summary statistics of the variables in our main empirical analyses.Mean is arithmetic mean, SD is standard deviation, Median is the sample median, and P k is the k th percentile.The estimation sample includes 88,702 firm-year observations.All variables are defined in Appendix

Table 3
Valuation parameter estimates Note: This table reports average parameter estimates from annual cross-sectional GMM estimations of Eq. 4. The estimates in columns (1) and (2) are reported in percentage scale, and the estimates in remaining columns are reported in raw scales.t-statistics are reported in parentheses.***, **, and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.Standard errors are estimated using the Newey-West procedure with eight lags.#sig.reports the number of annual estimates that are statistically significant at the 10% level

Table 4
Bridging the gap between stock price and bottom-line accounting... Determinants of θt /P t Fig. 1 Time series trend of θt /P t by industry.Note:This figure plots the time series of θt /P t over the sample period 1985 to 2015 across five industries.The industry classification is from Kenneth French's websiteNote: This table examines the determinants of θt /P t .Panel A reports the estimates from regressions of θt /P t on four proxies for accounting conservatism, and Panel B reports the estimates from regressions of θt /P t on four proxies for investment opportunities.t-statistics are reported in parentheses.***, **, and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.All variables are defined in Appendix

Table 5
In Table5columns (1) to (3), the dependent variables EGR t,t+h (h = 2, 3, 4) are h-year-ahead earnings growth defined as follows:25 This table presents annual average estimates from regressions of expected growth proxies on θt /P t .Dependent variables in Panel A are based on future realized growth.Dependent variables in Panel B are based on growth implied by analysts' forecasts.Coefficients on control variables are omitted to conserve space.t-statistics are reported in parentheses.***, **, and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.All variables are defined in Appendix realized growth.

Table 6
θt /P t and firm risk Beta t+1 , a conventional measure of the stock's systematic risk exposure.The second measure is the downside CAPM beta, Beta Down t+1 This table presents annual average estimates from regressions of future risk proxies on θt /P t .Dependent variables in Panel A are based on future stock market risk.Dependent variables in Panel B are based on future volatility in firm performance.Coefficients on control variables are omitted to conserve space.tstatistics are reported in parentheses.***, **, and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.All variables are defined in Appendix beta,

Table 7
Fama and MacBeth (1973)tional return forecast regressionsThis table presents the estimates from regressions of one-year-ahead realized stock returns on expected return estimates.Columns (1)-(3) report the results based regressions with year fixed effects and standard errors clustered by firm and year.Columns (4)-(7) report results based onFama and MacBeth (1973)regressions.t-statistics are reported in parentheses.***, **, and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.All variables are defined in Appendix

Table 8
Porfolio analysis based on expected return estimates This table presents results from portfolio analysis of the expected return estimates constructed based on Eq. 12.Stocks are sorted into 10 equal-sized portfolios in April of each year t, and buy-and-hold stock returns are accumulated over the five subsequent one-year intervals up to the year t + 5. t-statistics are reported in parentheses.***, **, and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.All variables are defined in Appendix

Table 9
Return predictability of industry cost of capital and firm characteristic adjustments This table examines the relative usefulness of the industry-and firm-specific components of the expected return estimates for predicting stock returns.Panel A presents the estimates from regressions of one-year-ahead realized stock returns on the two components.Panel B reports the average one-year-ahead stock returns of sequentially double-sorted portfolios by the industry-and firm-specific components of expected returns.t-statistics are reported in parentheses.***, **, and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.All variables are defined in Appendix

Table 10
This table partially replicates the results in Tables 5-7 when using the Fama-French five-industry classification and excluding loss firms.t-statistics are reported in parentheses.***, **, and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.All variables are defined in Appendix Consensus analysts' forecast of k-year earnings: I/B/E/S median forecast (MEDEST) formed on the third Thursday of April following year t financial year-end.Mcap t Log of one plus market capitalization on the analyst forecast date: market capitalization is I/B/E/S share price (PRICE) times the number of shares outstanding (SHOUT).Lev t Financial leverage: total financial debt divided by b t , where total financial debt Compustat debt in current liabilities (DLC) + debt in long-term liabilities (DLT).Mom t Momentum: CRSP Daily Stock File buy-and-hold stock return over the 12-month period prior to the earnings forecast date.Acc t Accruals: Compustat change in receivables (RECT) + change in inventory (INVT) + change in other current assets (ACO) − depreciation and amortization charges (DP) − change in other current liabilities (LCO).NOA t Percentage change in net operating assets: Compustat change in receivables (RECT) + change in inventory (INVT) + change in other current assets (ACO) + change in property, plant and equipment (PPENB)+ change in intangible assets (INTAN) + change in other long-term assets (AO), − change in payables (AP) − change in other current and long-term liabilities (LCO + LO).Saleg t Growth in sales: Compustat change in sales (SALE) divided by sales of prior year.I nvest t Investment in property, plant & equipment and inventory: Compustat change in gross costs of property, plant and equipment (PPEGT) + change in inventory (INVT).Exf t Net external financing: Compustat cash proceeds from long term debt issues (DLTIS) plus new equity issues (SSTK) + net change in current debt (DLCCH) − cash payments for retiring long term debt (DLTR) − share repurchases (PRSKC) − dividends (CDVC).LT G t Analysts' long-term growth forecast: I/B/E/S item measured on the third Thursday of April following year t financial year-end.EGR t,t+h h-year-ahead earnings growth rate, with k = 2, 3, 4, calculated based on Eq. 8. CAGR t,t+4 Four-year-ahead compound annual growth rate in sales, calculated based on Eq. 9. Beta t+1 Stock return beta: CAPM estimate using CRSP daily stock returns from May at t + 1 to April at t + 2. Stock return downside beta: CAPM estimate using CRSP daily stock returns from May at t + 1 to April at t + 2, using only negative daily market index returns.Idiosyncratic volatility: standard deviation of daily return residuals from CAPM using CRSP daily stock returns from May at t + 1 to April at t + 2. σ (ROE) t,t+3 buy-and-hold stock return: accumulated from May at t+k to April at t + k + 1. r t+1 Natural logarithm of one plus Ret t+1 .cfnt+1 Cash flow news, as defined in Eq. 15. drn t+1Discount rate news, as defined in Eq. 16.Natural logarithm of one plus the firm-specific component of expected return estimates