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Evaluating Implied Cost of Capital Estimates By Charles M. C. Lee** Eric C. So Charles C.Y. Wang First Draft: April 8th, 2010 Current Draft: November 1st, 2010 Abstract We propose a two-dimensional framework for assessing the quality of implied cost of capital (ICC) estimates as proxies for firm-level expected returns. Under fairly general assumptions, better expected return estimates should have higher cross-sectional predictive power for future stock returns and lower time-series measurement error variance. Using this framework, we evaluate seven alternative ICCs and find that several perform well along both dimensions, and all do significantly better than Beta-based estimates. Moreover, we find that the lagged industry median ICCs also predict firm-level returns and exhibit low error variance. Overall, ICCs appear to be attractive firm-level expected return proxies. ** All three authors are at Stanford University. Lee (professor.lee@gmail.com) is the Joseph McDonald Professor of Accounting at the Stanford Graduate School of Business (GSB); So (eso@stanford.edu) is a Doctoral Candidate in Accounting at the Stanford GSB; Wang (charles.cy.wang@stanford.edu) is a Doctoral Candidate in the Department of Economics. We thank Bhaskaran Swaminathan of LSV Asset Management, as well as seminar participants at Stanford, Northwestern, University of Chicago, UC-Irvine, LSU, National University of Singapore, University of Pennsylvania (Wharton), University of Toronto, Michigan University (Ross), Harvard, Yale, and Columbia, for helpful comments and suggestions. 1 I. Introduction The implied cost of capital (ICC) for a given asset can be defined as the discount rate (or internal rate of return) that equates the asset’s market value to the present value of its expected future cash flows. In recent years, a substantial literature on ICCs has developed, first in accounting, and now increasingly, in finance. The collective evidence from these studies indicates that the ICC approach offers significant promise in dealing with a number of long-standing empirical asset pricing conundrums.1 The emergence of this literature is, in large part, attributable to the failure of the standard asset pricing models to provide precise estimates of the firm-level cost of equity capital.2 An important appeal of the ICC as a proxy for expected returns is that it does not rely on noisy realized asset returns. At the same time, the use of ICC as a proxy for expected returns is not without its own problems and limitations. Currently at least three key problems stand in the way of broader adoption of ICCs as proxies for firm-level expected returns. First is the problem of firm-level cash flow forecasting. At a minimum, sensible ICC estimates call for sensible cash flow forecasts, but the literature has not converged on a particular approach to forecasting future earnings or cash flows. Second is the problem of performance evaluation. When prices (and therefore realized returns) are noisy, how can we assess the quality/validity of alternative firm-level ICC estimates? What are appropriate performance benchmarks? In other words, how do we know when we have a good ICC estimate? The third problem is the need for an instrumental variable. ICC estimates are measures of “actual yield” based on current prices, and therefore cannot be used as discount rates in an equity valuation exercise designed to challenge the current price. The question, then, is whether we can find a suitable empirical instrument that captures some dimension of the risk associated with firm-level ICCs, without relying on the current stock price. Using an analogy from bond pricing, can we 1 See Easton (2007) for a summary of the accounting literature prior to 2007. In finance, the ICC methodology has been used to test the Intertemporal CAPM (Pastor et al. (2008)), international asset pricing models (Lee et al. (2009)), and default risk (Chava and Purnanadam (2009)). In each case, the ICC approach has provided new evidence on the risk:return relation that is more intuitive and more consistent with theoretical predictions than those obtained using ex post realized returns. 2 The problems with using ex post realized returns to proxy for expected returns are well documented (e.g., see Fama and French (1997), Elton (1999), and Pastor and Stambaugh (1999)). 2 develop a “warranted yield” measure (or a “fitted” ICC estimate) based on firm characteristics that is systematically associated with firm-level expected returns?3 In this study, we focus primarily the issue of performance evaluation. Our main research question is: when prices are noisy, how do we know when an ICC estimate is a good proxy for true (but unobservable) expected return? A satisfactory answer to this question should precede (and inform) our approach to the other two problems. In fact, it is difficult to see how substantial progress can be made on either of the other two issues until we have some handle on this one. Prior studies that tackle the performance evaluation issue have resorted to one of two approaches: (a) by comparing the ICC estimates’ correlation with realized ex post returns, or (b) by comparing the ICC estimates’ correlation with perceived risk proxies, such as Beta, leverage, B/M, volatility, or size. In the first approach, ICCs that are more correlated with future realized returns (raw or corrected) are deemed to be of higher validity (e.g. Guay et al. (2005) and Easton and Monahan (2005)).4 In the second approach, ICCs that exhibit more positive (i.e., a more “stable and meaningful”) correlation with the other risk proxies are deemed to be superior (Botosan and Plumlee (2005), Botosan et al. (2010)). Although both approaches have some merit, neither weans us fully from the problems that gave rise to the need for ICCs in the first place (i.e. noisy realized returns and the poor performance of other risk proxies, such as Betas). Perhaps more importantly, prior results offer limited assurance that the resulting ICC estimates are truly useful for their intended purpose. For example, after examining seven ICC estimates, Easton and Monahan (2005) [EM] concluded that “for the entire cross-section of firms, these proxies are unreliable.” (p. 501) 5 In this study, we propose an analytical framework for assessing the quality of ICC estimates as proxies for firm-level expected returns. In our framework, a firm’s true expected return is the normative benchmark. We then explicitly model, and 3 This is in the spirit of the Gebhardt et al. (2001) analyses in which ICCs are “fitted” to a number of firm characteristics, and the forecasted ICC (or “fitted” value) is used to predict stock returns and forecast future ICC estimates. 4 Guay et al. (2005) examines the correlation between alternative ICC estimates and raw ex post returns. Easton and Monahan (2005)[EM] introduces a method for purging these returns of the estimated effect of future news. We discuss the EM approach in much more detail later. 5 Botosan et al. (2010) provide evidence that the EM finding may be due to the choice of control variables used to proxy for ex post realized returns, and is not an indictment of the ICC methodology. Although our evidence is consistent with Botosan et al. (2010), this debate is not the focus of our paper. 3 empirically estimate, key properties of the measurement errors generated by alternative ICC estimates relative to this benchmark.6 We show that under fairly general assumptions, it is possible to derive a closed- form expression for the firm-specific time-series variance of ICC measurement errors, even when the errors themselves are not observable. In this set-up, we propose that higher-quality expected return proxies should: (a) predict future cross- sectional returns and, (b) have low time-series measurement error variance.7 This two-dimensional framework then forms the basis of our comparison between ICC estimates. Our empirical analyses compare seven alternative ICC estimates and three traditional measures of expected returns (based on time-series Betas and the Fama-French 1-, 3-, or 4-factor risk models). Four of these ICC estimates are based on an abnormal earnings growth model (PEG, MPEG, OJM, and AGR); one is based on a residual-income model (GLS); and two are based on Gordon Growth Models in which the terminal value estimate is capitalized earnings after either one-year (EPR) or five-years (GGM). To ensure the largest possible sample, most of our analysis is conducted using the cross-sectional earnings forecasting technique introduced by Hou et al. (2009). In this approach, we use the same mechanical model to forecast future earnings for all seven valuation models (see Appendix II for details). For robustness, we also computed the same seven ICCs using future earnings estimates based on the mean analyst forecasts (one-year-ahead, two-year-ahead, and long-term growth rate) from the I/B/E/S database. Based on a two-dimensional evaluation framework, we show that all seven ICCs perform better than traditional Beta-based expected return estimates. Using the Hou et al. forecasting method, four of these ICCs (GLS, EPR, GGM, and AGR) have reliable predictive power for realized returns, and all seven ICC estimates have much lower mean time-series error variance than the Beta-based measures. 6 Note that the ICC approach assumes expected future cash flows are given exogenously, and computes a constant implied rate-of-return based on the current stock price. An alternative approach is to assume that cash flows and discount rates are both time-varying, and jointly estimate the two processes (see Hughes et al. (2009), Lambert (2009), and Callen (2009) for related discussions). Empirically, Pastor and Stambaugh (2009) and van Binsbergen and Koijen (2009) conduct such estimations at the aggregate market level. However, irrespective of the estimation technique, the question of performance evaluation remains. 7 Our approach is similar to the two-dimensional performance metrics used by Lee, Myers and Swaminathan (1999) to compare alternative value estimates for the Dow30 stocks. However, whereas their focus is on evaluating alternative value estimates in a univariate time-series, we analyze the properties of expected return measurement errors in a cross-sectional context. 4 Using mean analyst earnings forecasts, we find that six of the ICCs (GLS, EPR, AGR, OJM, PEG, and MPEG) have reliable predictive power for realized returns, and all seven have lower error variance than the Beta-based measures. Further analyses show that the relative performance of the ICCs is closely related to the compatibility of earnings forecasting methods to the valuation models. Specifically, an ICC estimate tends to perform poorly when the associated valuation model compounds the errors inherent in the earnings forecasts. For example, all four ICCs based on the abnormal growth model (AGR, OJM, PEG, and MPEG) perform relatively poorly using the Hou et al. forecasting method. Each of these models relies on heavily on the accuracy of the earnings growth forecast (in particular, the percentage increase from one-year-ahead to two-year- ahead earnings). Because the Hou et al. methodology is based on a cross- sectional pooled regression of earning levels, the earnings growth estimate from this forecasting technique tends to be noisy. Our evidence indicates the Hou et al. forecasting method is least suitable for use with the abnormal earnings growth models. Consistent with this conjecture, we find that when the earnings forecasts are based on mean analyst estimates from I/B/E/S, the four abnormal earnings growth models perform about as well as the other three. In fact, when mean I/B/E/S earnings forecasts are used, six out of the seven ICCs have reliable predictive power for returns. The only model that does not perform well with analyst forecasts is GGM, which depends heavily on the accuracy of the analysts’ long- term earnings growth forecast.8 Finally, we find that the lagged industry median ICC estimate based on any of the four successful estimation models (GLS, GGM, EPR, or AGR) will also reliably capture a significant amount of cross-sectional variation in future firm-level realized returns. Moreover, as expected, these industry-based ICC estimates exhibit lower mean time-series error variance than the firm-specific ICC estimates. These findings suggest that industry membership is a potentially useful instrumental variable for firm-level expected returns. Collectively, our results offer a much more sanguine assessment of ICC estimates than some of the prior literature. Like Botosan et al. (2010), our findings raise questions about the Easton and Monahan (2005) assertion that ICC estimates are not meaningfully correlated future realized returns. However, unlike Botosan et 8 In our computation of GGM, the noisiest analyst input (the long-term earnings growth rate) is compounded for three years to derive the key five-year-ahead earnings estimate used in this model. 5 al. (2010), our primary evaluation criteria do not require superior ICC estimates to necessarily exhibit stronger empirical correlations with estimated Beta or other presumed risk proxies. Our results are also relevant to a broader literature on expected return estimation beyond ICCs. Indeed, our framework allows researchers to independently assess the quality of firm-specific Betas and other empirically-inspired proxies for expected returns nominated in the empirical finance literature (such as B/M, Size, or Volatility). At the same time, our findings provide a bridge between academic findings and financial practice, which often implicitly employs industry-based ICC corrections for equity valuation purposes. The rest of the paper is organized as follows. In Section II, we develop the theoretical underpinnings for our performance metrics. In Section III we discuss data and sample issues, explain our research design, and review the construction of our seven ICC estimates. Section IV contains our empirical results, and Section V concludes. II. Theoretical Underpinnings II.1 Return Decomposition A firm’s realized returns in period t+1 may be thought of as consisting of an expected component and an unexpected component. Formally: , (1) where r t+1 is the realized return in period t+1, ert is expected return at the beginning of t+1 conditional on available information, and t+1 is unexpected return. We begin with a simple decomposition of unexpected return similar to Campbell (1991), Campbell and Shiller (1988a, 1988b) and Vuolteenaho (2002). Using similar notation, we express a stock’s realized return as: , (2) where rt+1 is a firm’s realized return in period t+1, and ert is its expected return at the beginning of the period. We then parse unexpected returns into two components: rnt+1 is discount rate news (i.e., rnt+1 reflects innovations that revise the market’s expectation of a 6 firm’s future returns), and unt+1 represents all other innovations or shocks to price that cannot be forecasted ex ante.9 Notice that in this framework, the true (but unobservable) expected return measure, ert, captures all ex ante predictability in stock returns (in fact, it is the only statistical object that predicts returns). Having defined our normative benchmark in this manner, we abstract away from the market efficiency debate. If one subscribes to market efficiency, then ert should only be a function of risk factors and expected risk premia associated with these factors. Conversely, in a behavioral framework, ert can also be a function of other non-risk-related behavioral factors. ˆ Next, we introduce the idea of ICC estimates ( ert ), which are defined as the true expected return at the beginning of period t+1 (ert), measured with error (t). In concept, ert does ˆ not need to be an ICC estimate as defined in the accounting literature – it can be any ex ante expected return measure, including a firm’s Beta, its book-to-market ratio, or its market-capitalization at the beginning of the period. The key is that, whatever “true” expected returns maybe, we do not observe it. What we can observe are empirical proxies with measurement error. Our goal here is to try to evaluate how good a job these proxies do in capturing/tracking er.10 Finally, we assume that expected returns (ert) and the ICC measurement error (t) may both exhibit some persistence over time. Specifically, we assume for a given firm, both follow an AR(1) process, with parameters and , respectively. Notationally: (A1) 9 Note that prior studies have generally interpreted the last term in (2) as cash flow news. We have in mind a broader concept, such that unt+1 embraces all other shocks to price, and need not be related to cash flow news. For example, it can arise due to model uncertainty (e.g. Pastor and Stambaugh (1999)), investor sentiment ((Shiller (1984), De Long et al. (1990), Cutler et al. (1989)), or any other source of exogenous noise ((Roll (1984), Black (1986)). The interpretation of this variable is not crucial to the analysis; more important is the fact that this term is cannot be forecasted ex ante. 10 Conceptually, if true expected returns are time-varying in a predictable fashion, the assumption of a constant discount rate ensures ICCs are not identical the true expected return of a stock (see Hughes et al. (2009) and Lambert (2009)). However, even given this conceptual distinction, the usefulness of ICCs as proxies for firm-level expected returns is an open empirical question. Our goal is to develop a framework for assessing the attractiveness of ICCs as expected return proxies, assuming a firm’s true (but unobserved) expected return is the primary variable of interest. 7 (A2) (A3) The first equation (A1) – examined by Campbell (1991) as a special case – simply acknowledges the fact that expected returns are time-varying and persistent. The second equation (A2) is not so much an assumption as a definition: we define ICC estimates as the true expected return measured with error. The third equation (A3) reflects the fact that the measurement errors themselves may also be time-varying and persistent. Given the nature of the cash flow forecasting mechanisms imbedded in the ICC estimates, this seems quite sensible. It would hold, for example, if an ICC estimation technique consistently understates the expected cash flows of certain companies relative to market expectations. In this setting, we would find measurement errors that persist from year-to-year, and exhibiting some mean-reversion over time as the nature of this error becomes transparent. Note that the differences between alternative ICC estimates will be reflected in the properties (time-series and cross-sectional) of their terms. Comparisons between different ICC estimates are, therefore, comparisons of the distributional properties of the ’s they will generate, over time and across firms. In this setup, statements we make about the desirability of one ICC estimate over another as an expected returns proxy are, in essence, expressions of preference over the alternative properties of measurement errors ( terms) that each is expected to generate. In other words, when we choose one ICC estimate over another, we are specifying the loss function (in terms of measurement error), that we would find least distasteful or problematic. The choice of ICC estimates becomes, therefore, a choice between the attractiveness of alternative “loss functions”, expressed over space. Under this setup, what would good ICC estimates look like? We doubt there is a single right answer for all research applications. For instance, in certain applications, we might be most interested in securing an unbiased forecast of future realized returns over fairly short horizons (say over the next 12 months). In other applications, we might be interested in forecasting what the market will likely use as a discount rate for the future earnings of a firm three to five years from now. The answer to these two questions might be different – e.g., one ICC estimate might be preferred for the first application while another might be preferred for the second. 8 Nevertheless, we believe it is possible to make some general statements about the attributes of “good” empirical proxies for firm-specific ICCs (i.e. those that track true expected returns well). While we cannot nominate a single criterion by which all ICC estimates should be judged (for that is impossible without specifying the researcher’s preference function over the properties of measurement errors), in our setup, all “well behaved” ICCs will exhibit certain empirical attributes. The extent to which they do so, thus, becomes a basis for comparison. II.2 Comparing ICC Estimates Recall our main objective is to produce ICC estimates that track true expected returns well, both across firms and over time. Ideally, we would like measurement errors to be small at all times (i.e. i,t ~ 0 for all i and all t). Unfortunately, this is not likely to hold, so we must choose between alternative error distributions, and specify those properties of that are most important to us as researchers. Given that the measurement errors (’s) are not necessarily small, two other properties become important. First, we prefer measurement errors that preserve ranks of the true expected returns in the cross-section. If this property holds, even though the ICC estimates are noisy, they are still informative about (i.e. “tracks”) firms’ true expected returns in the cross-section. Second we want measurement errors for a given firm to be relatively stable over time. This property is useful, because if i is stable over time, the ICCs for a given firm will more closely track its true expected returns in time-series. Therefore, we can more reliably extract the true expected return for a given firm. The notion of lower time-series error variance is closely related to the idea of superior “tracking ability” in ICC estimates. Financial practitioners are often interested in predicting future market multiples. In other words, they are interested not only in predicting future realized returns, but also in predicting what market multiple a particular firm will trade at in the future (or, in our parlance, future ICC estimates). For example, Gebhardt et al. (2001) examines the ability of their ICC estimate to predict ICC estimates several years into the future (i.e. in terms of its “tracking ability”). Although the concept of ICC tracking is quite similar to our low error variance criterion, the two are not exactly the same. In some applications, we might wish to compare ICC estimates on the basis of their ability to predict themselves – i.e. their “tracking ability” or how well each predicts future manifestations of itself over time. However, in most research settings, our main interest is in how well each ICC estimate tracks true (but unobservable) expected returns. This property is better reflected in the “error variance” 9 measure – i.e. ICC estimates that consistently track true expected returns better should on average exhibit lower time-series variance in their error terms.11 In sum, we have proposed two empirical properties by which to assess expected return proxies: higher predictive power for cross-sectional returns, and lower measurement error variance in time-series. Note that these two properties do not necessarily imply each other. Measurement errors that preserve the cross-sectional ranks of the true expected returns will, on average, exhibit a superior ability to predict future realized returns. However, these measurements errors need not be stable for a given firm over time. In fact, as we show in the next section, the ability to predict returns could also lead to higher time-series error variance12 II.3 Measurement Error Variance To assess the second distributional property of the error terms, we need to be able to empirically identify, and compare, the firm-specific time-series variance of the error terms (Var(i,t)) generated by different ICC estimates. A key objective (and we believe, contribution) of this paper is that we analytically disentangle the time-series properties of ICC estimates from those of the true expected returns, when both are time-varying, yet both are also persistent over time. We show in this sub-section (and in the Technical Appendix), that it is possible to derive a closed-form expression for the variance of the error terms (Var(i,t)), even when the errors themselves are not observable. This analysis then provides a foundation for our comparison between ICC estimates. In the Technical Appendix, we provide a detailed derivation of Var(t), and show how this measure can be identified by combining the time-series autocovariances of the ICC estimates, the time-series autocovariances of realized returns, and the time-series return- ICC covariances. Specifically, in the time-series case, where we assume cash flow news is uncorrelated with expected returms of a given firm in time-series, we show that: 11 To see the difference, note that when we regress one period's ICC on the previous, we capture the covariance between ICCt+1 and ICCt. However, the covariance between ICCt+1 and ICCt contains not only the covariance between ert+1 and ert, but also the covariance or tracking in the measurement error, as well as a covariance term between and er. 12 Consider a variable that captures mainly market mispricing (perhaps a price or earnings momentum indicator). This variable may have high predictive power for realized returns over some shorter time horizon, but will tend to exhibit low “tracking ability” (i.e. poor ability to predict future expected return proxies). For example, in the momentum literature, “Winners” do not generally predict future “Winners” over longer horizons. In our framework, this occurs because any predictive power deriving from market mispricing corrections will reduce the long-term stability of expected return proxies. 10 2 (3) Where: , is the time-series variance of the ICC estimates for a given firm. , , is the time-series covariance between the ICC estimate and realized returns k period in the future. , , is the time-series autocovariance in realized returns with k period lags. Notice that all the terms on the right-hand-side of Equation (3) are observable, and can be empirically estimated for each firm. The first term on the right-hand-side shows that the (time-series) variance of a given ICC estimate’s measurement error is increasing in the variance of the ICC estimate (c0). This is intuitive: as the variance of the errors increase, so will the observed variance of the ICC estimate. The second term on the right-hand-side shows that the variance of the error terms for a given ICC estimate is decreasing in the covariance of the ICC estimate and future returns ( ). This is also intuitive: to the extent that ICC estimates predict future realized returns, the variance of the errors are smaller. Finally, the third term shows that the error variance is a function of the time-series auto- covariance in returns. Specifically, it is increasing in the 1-period-lagged return covariance and decreasing in the 2-period-lagged return covariance. To the extent that returns are independent from period-to-period, this third term will play a small role in the estimation process. Note what we derive here is the time-series variance of the error terms, under the assumption that future news is uncorrelated with the ex ante expected returns of a given firm in time-series (i.e., unt+1 is uncorrelated with ert for each given firm). Alternatively, we could impose a set of stronger assumptions and derive a cross-sectional measure of error variance.13 This cross-sectional error variance estimate is similar to a measure 13 Specifically, we would need to assume that: (a) future news is uncorrelated with the level of the ex ante expected returns in the cross-section (i.e. uni,t is uncorrelated with eri,t for all firms i at a given time t), and (b) all firms share the same AR(1) persistence parameters for expected returns and measurement error (i.e., the and parameters, featured in equations (A1) and (A3) respectively, are cross-sectional constants). Under these assumptions, it is possible to derive a cross-sectional measure of error variance that can be empirically estimated (see Wang (2010)). 11 considered by Easton and Monahan (2005). We do not pursue this approach because: (a) low cross-sectional error variance does not ensure rank preservation of expected returns (which is, in our view, a more important attribute of ICCs), (b) the assumptions needed to derive this measure are restrictive and unrealistic, and (c) a time-series error variance measure better complements cross-sectional predictive power in a two-dimensional evaluation framework (because it allows us to assess both the cross-sectional and time- series performance of an ICC estimate). In sum, we have provided a rationale for a two-dimensional evaluation framework that compares ICC estimates under a set of minimalistic assumptions. In the following section we apply this evaluation framework to assess the merits of seven alternative ICC measures nominated by prior literature. III. Research Methodology III.1: Data and Sample Selection We obtain market-related data on all U.S.-listed firms (excluding ADRs) from CRSP, and annual accounting data from Compustat. To be included, each firm-year observation must have information on stock price, shares outstanding, book values, earnings, dividends, and industry identification (SIC codes). We also require sufficient data to calculate forecasts of future earnings based on the methodology outlined in Hou et al (2009), and to estimate ICCs for all seven models. In addition, we require firms to have non-missing expected return estimates computed using three Fama-French Beta-based models (FF1, FF3, and FF4). FF1 represents a CAPM-based ICC estimate for each firm computed at the end of June each year. To compute FF1, we first estimate each firm's beta to the market factor using the prior 60 months' data (from t-1 to t-60). FF1 is then obtained by multiplying the estimated beta to the most recent 12 months' compounded annualized market risk premium (provided by Fama and French), and adding the risk-free rate. Similarly, FF3 represents a 3-factor based ICC estimate computed using the Mkt-Rf, HML, and SMB factors and 60-month rolling beta estimates. Finally the FF4 model adds the momentum (UMD) factor in ICC estimation procedure above. Our final sample consists of 74,343 firm-years and 10,523 unique firms, spanning 1970- 2007 (see Appendix I). Note that this sample is considerably larger than those used in most prior ICC studies that require I/B/E/S analyst forecasts. 12 III.2: Earnings Forecasts In a recent study, Hou et al. (2009) use a pooled cross-sectional model to forecast the earnings of individual firms. They show that the cross-sectional earnings model captures a substantial amount of the variation in earnings performance across firms. Hou et al. finds that the model produces earnings forecasts that closely match the consensus analyst forecasts in terms of forecast accuracy, but exhibit much lower levels of forecast bias and much higher levels of earnings response coefficients. The ICC estimates they derive from these forecasts exhibit greater reliability (in terms of correlation with subsequent returns, after controlling for proxies for cash flow news and discount rate news) than those derived from analyst-based models. At the same time, the use of model-based forecasts allows for a substantially larger sample because it does not require firms to have existing analyst coverage. Moreover, the model-based approach allows us to forecast earnings for several years into the future while analyst forecasts are typically limited to one- or two-years. For all these reasons, we employ model-based forecasts of earnings throughout our analysis. Following the Hou et al. methodology, we estimate forecasts of earnings for use in all seven valuation models. Specifically, as of June 30th each year t between 1970 and 2007, we estimate the following pooled cross-sectional regression using the previous ten years (six years minimum) of data: , , , , , , , , , (4) where Ej,t+ ( = 1, 2, 3, 4, or 5) denotes the earnings before extraordinary items of firm j in year t+, and all explanatory variables are measured at the end of year t: EVj,t is the enterprise value of the firm (defined as total assets plus the market value of equity minus the book value of equity), TAj,t is the total assets, DIVj,t is the dividend payment, DDj,t is a dummy variable that equals 0 for dividend payers and 1 for non-payers, NEGEj,t is a dummy variable that equals 1 for firms with negative earnings (0 otherwise), and ACCj,t is total accruals scaled by total assets. Total accruals are calculated as the change in current assets [Compustat item ACT] plus the change in debt in current liabilities [Compustat item DCL] minus the change in cash and short term investments [Compustat item CHE] and minus the change in current liabilities [Compustat item CLI]. To mitigate the effect of extreme observations, we winsorize each variable annually at the 0.5 and 99.5 percentiles. The average annual coefficients from fitting estimating equation (4) in our sample are provided in Appendix II. Our average coefficients are qualitatively similar to those 13 reported in Hou et al (2009).14 We calculate model-based earnings forecasts by applying historically estimated coefficients from equation (4) to the most recent set of publicly available, non-winsorized, firm characteristics. We derive ICC estimates at the end of June each year by determining the discount rate needed to reconcile the market price at the end of June with the present discounted value of future forecasted earnings. Values of ICC above 100 percent and below zero are set to missing. For each earnings forecast, we calculate expected ROE as the forecasted earnings divided by forecasted beginning of period book values. Future book value forecasts are obtained by applying the clean-surplus relation to current book values, using forecasted earnings and the current dividend payout ratio. Finally, to ensure comparability of the results across alternative measures of ICC, we require firms to have non-missing ICC estimates across all seven models outlined in Section III.3. III.3 Alternative ICC Estimates In this section we discuss the construction of seven alternative ICC estimates. Because they are all based on the dividend discount model (or equivalently, the discounted cash flow model), given consistent assumptions, all should yield identical results. In practice, however, each produces a different set of ICC estimates due to differences in how projected earnings are handled over a finite forecasting horizon. The seven estimates can be broadly categorized into three classes: Gordon growth models, residual income models, and abnormal earnings growth models. Gordon Growth Models (EPR, GGM) Gordon growth models are based on the work of Gordon and Gordon (1997), whereby firm value (Pt) is defined as the present value of expected dividends. In finite-horizon estimations, the terminal period dividend is assumed to be the capitalized earnings in the last period (period T). Formally, 1 1 We consider two versions of this model, corresponding to T=1 and T=5. Specifically, EPR (where firm value is simply one-year-ahead earnings divided by the cost of equity) is a Gordon growth model with T=1, and GGM is a Gordon growth model with T=5. In 14 We use fundamental data from Compustat Express while Hou et al. (2009) use data from the historical Compustat research database (discontinued after 2006). Some of the differences, particularly in the early years, are likely due to differences in firm membership across the two databases. 14 each case, we use the Hou et al. (2009) regressions to forecast future earnings, and each firm’s historical dividend payout ratio to derive forecasted dividends. Residual Income Model (GLS) The standard residual income model can be derived by substituting the clean surplus relation into the standard dividend discount model: ∑ Where NIt+k is Net Income in period t+k, Bt is book value and Pt is the equity value of the firm at time t. Recent accounting-based valuation research has spawned many variations of this model, differing only in the implementation assumptions used to forecast long- term earnings (earnings beyond the first 2 or 3 years). Some prior implementations of the residual income models (e.g., Frankel and Lee (1998)) are essentially Gordon growth models. In this study, we use a version developed by Gebhardt, Lee, and Swaminathan (2001) [GLS]. In this formulation, earnings are forecasted explicitly for the first three years using the Hou et al. (2009) methodology. For years 4 through 12, each firm’s forecasted ROE is linearly faded to the industry median ROE (computed over the past ten years - minimum five years- excluding loss firms). The terminal value beyond year 12 is computed as the present value of capitalized period 12 residual income. Among the models we test, GLS alone uses industry-based profitability estimates. Abnormal Earnings Growth Models (PEG, MPEG, AGM, OJM) The third class of models is based on the theme of capitalized one-year-ahead earnings. Each member of this class of models capitalizes next-period forecasted earnings, but offers alternative techniques for estimating the present value of the abnormal earnings growth beyond year t+1. A standard finite-horizon abnormal earnings growth model takes the form: 1 1 1 1 15 PEG and MPEG: Easton (2004) shows that in the special case where T=2, and = 0, the standard abnormal growth model reduces down to what is known as the “Modified PEG ratio” (MPEG). For this model, we can extract the ICC is the value of re that solves: / Under the additional assumption that DPSt+1 =0, we can compute an ICC estimate that analysts commonly refer to as the “PEG ratio” (PEG): / A notable feature of PEG and MPEG is their reliance strictly on just short-term (one- and two-year-ahead) earnings forecasts. AGR: Easton (2004) also proposes a special case of the abnormal growth in earnings model with T=2, and a specific computation for long-term growth in abnormal earnings. Working out the algebra, the ICC estimate is the re that solves the following equation: 1 1 1 OJM: A final variation of the abnormal growth in earnings model is the formulation proposed by Ohlson Juettner-Nauroth (2005). In implementing this model, we follow the procedures in Gode and Mohanram (2003), who use the average of forecasted near-term EPS EPS EPS t 3 EPSt 2 t 5 t 4 growth EPSt 2 and five-year growth EPSt 4 as an estimate of short-term growth (g2). In addition, they assume , the rate of infinite growth in abnormal earnings beyond the forecast horizon, is the current period’s risk-free yield minus 3%. Solving for re we obtain the following closed form solution, referred to as OJM: 1 1 1 2 16 IV. Empirical Results IV.1 Descriptive Statistics Table I reports the medians of the seven implied ICC estimates for each year from 1971 through 2007. We compute a firm-specific ICC estimate for each stock in our sample based on the stock price and publicly available information as of June 30th each year. We also report the ex ante yield on the 10-year Treasury bond on June 30th. Only firms for which information is available to compute all seven ICC and three Beta-based measures are included in the sample. The number of firms varies by year and ranges from a low of 1,035 in 1974 to a high of 2,808 in 1997. The average number of firms per year is 2,009, indicating that the ICC estimates are available for a broad cross-section of stocks in a given year. The time-series mean of the annual median ICCs range from 9.10% (for EPR), to 15.36% (for OJM). Comparing these estimates to the average Treasury yield suggests that the median equity risk premium is between 1.5% and 7.5%. The lower end of this range is consistent with Claus and Thomas (1998) and Gebhardt et al. (2001) who find an implied market risk premium between 2% and 4%. At the high end of the range, the 7.4% risk premium from the OJM model is similar to the market risk premium reported by Ibbotson (1999), based from ex-post returns over the 1926-1998 period. In short, although our objective is not to estimate the market risk premium, these ICC estimates appear reasonable in aggregate. It is instructive to compare the results for these seven ICC estimates with the results for the Beta-based cost-of-capital estimate (FF1). Recall that we compute FF1 using firm-specific Betas estimated over the previous 60 months, and a continuously updated market-risk premium provided by Fama-French. The annual mean and median numbers for FF1 (13.08% and 14.02%) are similar to the ICC estimates. However, the time-series standard deviation of the annual FF1 estimates (16.72%) is 2 to 8 times larger than the standard deviation of the ICC estimates. Annual medians for FF1 range from -19.4% to 68.5%, and in 9 out of 37 years, the median FF1 cost-of-capital estimate is negative. The volatility of FF1 reflects the instability of the market equity risk premium estimated on the basis of historical realized returns. Table II reports the average annual correlations between the seven ICC measures and the three Beta-based measures. Correlations are calculated by year and then averaged over the sample period. Pearson correlations are shown above the 17 diagonal and Spearman correlations are shown below the diagonal. Not surprisingly, the seven ICC estimates are highly correlated among themselves, as are the three Beta-based estimates. All reported correlations among the seven ICC estimates are significant at the 5% level, as are the correlations among the three Beta-based measures. PEG and MPEG have the highest Spearman correlation at 96.1, which is not surprising given the similarity in their construction. In the same spirit, GGM and GLS exhibit a Spearman correlation of 87.3, while AGR and EPR are correlated at 82.9. Most of the other Spearman correlations are between 45 and 65, with none under 40. FF1, FF3, and FF4 are correlated at levels between 45.2% to 85.8%, but we find no significant correlation across the two groups – i.e., none of the seven ICC estimates are correlated with the three Beta-based measures. IV.2 Predictive Power for Returns As demonstrated in Section II, when measurement errors are small or when measurement errors preserve the ranks of ex ante expected returns, ICC estimates should display a positive correlation with ex post realized returns. Moreover, superior estimates of ICC should possess stronger predictive power for future returns. Table III reports the correlation between annual ICC estimates derived from each of the 10 expected return estimates and firm-specific buy-and-hold returns over the next 12, 24, 36, 48, and 60 month. We compute future realized returns several ways. First, Panel A reports the results of pooled cross-sectional regressions of firm-specific future realized returns on firm-specific implied risk premium measures.15 The dependent variables are 12, 24, 36, 48, and 60 month buy-and-hold returns. Following Gow et al. (2010), we compute t-statistics for this panel using two-way cluster robust standard errors (clustered by firm and year). The t-statistics are shown in parentheses. Significance levels are indicated by *, **, and *** for 10%, 5%, and 1% respectively. Panel A results show that among the seven ICC models, only four (GLS, EPR, AGR, and GGM) have significant predictive power for future returns across all five holding horizons. The estimate coefficients for these four models indicate that a one-percentage-point increase in firm-specific cross-sectional ICC is associated with a 0.203% (for AGR) to 0.425% (for EPR) increase in future realized returns over the next 12-months. Future annual returns are steady, or 15 To facilitate the pooled regression in the presence of time-varying risk-free rates, we compute a firm’s equity risk premium as its ICC minus the yield on the 10-year Treasury bond. 18 slightly increasing, in the holding period, suggesting that the ICC estimates capture a persistent component of expected returns which does not fade quickly over time. Averaged over the next 60-months, the same 1% increase in ICC is related, on average, to an annual increase in realized returns of 0.22% (for AGR) to 0.514% (for EPR). In contrast, the performance of the three Beta-based measures in forecasting future returns is quite poor. The 3-factor and 4-factor measures (FF3 and FF4) have no reliable power to predict cross-sectional returns. The 1-factor measure (FF1) actually predicts returns with the wrong sign. In other words, over the span of our 37 sample years, higher Beta firms on average earned reliably lower subsequent returns. Panel B reports the results of an alternative test. In this test, we use a continuous weighting scheme that holds each firm in proportion to their market-adjusted ICC (similar to Lewellen (2002)). Table values represent the buy-and-hold returns for a hedge portfolio in which each firm’s weight in month t is computed as: 1 , , , ______ where ICCi,t equals the firm’s implied cost of capital in year t , ICC mkt,t is the equal-weighted average ICC for all firms in year t , and N is the total number of stocks in the year t sample. In effect, this portfolio takes a long position in firms with higher (relatively more positive) ICC estimates and a short position in firms with lower (relatively more negative) ICC estimates.16 The weights are constructed so that the strategy is dollar-neutral at inception. Standard errors for the average 24, 36, 48, and 60 month portfolio returns are computed using Newey-West HAC estimators with 1 year, 2, year, 3 year, and 4 year lags, respectively. Panel B results show that, once again, the same four models exhibit consistently reliable associations with future returns. EPR and GGM display the highest level of predictive power for future returns, closely followed by GLS and AGR. In this panel, a long-short portfolio based on a continuous ICC-weighting scheme returns 16 Prior studies that test market pricing anomalies typically focused on extreme deciles, so that future realized returns reflect an equal-weighted “hedge” portfolio of Decile 10 returns minus Decile 1 returns. We believe our continuous-weighting scheme better measures the predictive power of an ICC estimate across the entire spectrum of firms (not just the extreme deciles). Predictive results using the extreme decile approach (not tabulated here) are similar or stronger across the models tested, but the key conclusions are qualitatively identical. 19 an average annualized realized hedge return that ranges from 3.9% (for AGR over the first 12 months) to close to 10% (for EPR, averaged over the next 60 months). In contrast, none of the three Beta-based measures exhibit any reliable ability to predict returns. Table IV provides evidence on the statistical significance of the difference in predictive power across these ten models. To construct this table, we compute the annual pair-wise difference in three-year cumulative returns for each pair of expected return estimates, where the returns are based on a continuous ICC- weighted hedge portfolio. Table values represent the time-series t-statistics of the difference between the strategies over the 36 years. Table values are positive (negative) when the expected return estimate displayed down the left-hand-side of the table has stronger (weaker) predictive power for realized returns than the expected return estimate displayed in the top row. Standard errors are computed using Newey-West HAC estimators with 2 year lags. Once again, significance levels are indicated by *, **, and *** for 10%, 5%, and 1% respectively. The evidence in this table indicates that EPR and GGM are the best predictors of future returns. For example, looking across the row for EPR and GGM, we see that these two measures generally dominate the others, but are themselves not statistically distinguishable from each other. GLS and AGR tend to perform better than PEG, MPEG, and OJM, but the differences are not statistically significant. The results in Panel B show that the seven ICC estimates have better predictive power than the Beta-based measures (negative table values throughout this panel). Once again, GLS, EPR, AGR, and GGM are the best predictors. In sum, the Table III Panel B results using a hedge portfolio approach are quite consistent with the Table III Panel A findings using a pooled regression with two- way cluster corrected statistics. Contrary to Easton and Monahan (2005), but consistent with Botosan et al. (2010), as well as Gebhardt et al. (2001), and Hou et al. (2009), we find substantial evidence that ICC estimates from several models have some ability to predict future realized returns.17 In particular, we find that, using the Hou et al. (2009) forecasting method, EPR and GGM have the strongest predictive power for future returns, followed by GLS and AGR.18 Our continuous weighted strategies show that average hedge returns range from 4% to 10% per 17 Although we do not explicitly control for cash flow news, the evidence from those papers that do so (including Easton and Monahan (2005), Botosan et al. (2010), and Hou et al. (2009)) all show that the inclusion of such a proxy has little effect on the ICC estimates’ ability to predict future returns. 18 Botosan et al. (2010) also report reliable predictive power for several ICC estimates. Most of their ICC measures, however, rely on analyst earnings forecast estimates and/or (in at least one case) Value-Line target price. This research design choice leads to substantial differences in sample firms as well as actual ICC estimates, so their results are not directly comparable to these. 20 year across the four successful models, which seem both statistically and economically significant. IV.3 Comparison of Measurement Error Variance As discussed in Section II, better ICC estimates should also generate measurement errors that exhibit lower variance. Conceptually, the time-series assumption is probably more realistic – that is, it seems more reasonable to assume that the cash flow (and other) news of a given firm is unrelated to the level of its expected returns, and less realistic to assume that cash flow news across firms is unrelated individual firms’ expected returns. Moreover, requiring the AR(1) parameters to be cross-sectional constants seems quite restrictive. However, one can readily envision applications in which lower error variance in the cross-section is a desirable attribute. Also, time-series estimations require each firm in the same to have multiple annual observations, thus introducing a potential survivorship bias. Given these trade-offs, we present results using both methods. Table V examines the time-series measure of error variance for each of the seven ICC estimates. Panel A reports descriptive statistics for the variance of the error terms. To construct this panel, we require each firm to have a minimum of 20 (not necessarily contiguous) years of data during our 1971-2007 sample period. A total of 836 unique firms met this data requirement. For each firm and each ICC estimation method, we then compute the variance of the firm-specific measurement errors based on equation (4). Table values in this panel represent summary statistics for the error variance (multiplied by 100) from each ICC estimate computed across these 836 firms (n=836). Panels B and C report t- statistics corresponding to the pair-wise comparison of firm-specific measurement errors across the sample of 836 firms used in Panel A. The results in Table V show that three ICC estimates (EPR, GLS, and GGM) generally have lower error variances than the other estimates. The ranking across these three measures vary somewhat depending on whether the mean or the median measure is used, but generally all three measures generate error terms that exhibit lower time-series variance. Panel B shows that EPR, in particular, generates measurement error variances that are reliably more stable over time than all the other models, except GLS and GGM. Panel C shows that any of the seven ICC estimates will have significantly lower error variance than the three Beta-based estimates. 21 Table VI reports results when the variances of the error terms are estimated in the cross-section. To construct this panel, we estimate a cross-sectional variance measure for each model/year, thus resulting in 36 variance measures (one per year) for each ICC estimate. Table values in this panel represent descriptive statistics for the error variance from each ICC estimate computed across these 36 annual estimations (n=36). For parsimony, we only compare the seven ICC estimates. Table VI shows that cross-sectional estimations of error variance are much noisier than the time-series results from the previous table. The Panel A results show that rankings across these seven models will differ, depending on whether the mean or median measure is used. For example, based on a ranking of means, the models with the lowest error variances are AGR, EPR, GLS, and MPEG, in that order. However, based on a ranking of medians, the best models are MPEG, EPR, AGR, and GLS. Panel B shows that none of the seven models are statistically distinguishable from each other in pair-wise comparisons. In short, no model is reliably better than any other on the basis of the cross-sectional error variance. To summarize, we find that using a time-series estimation technique, the error variances from EPR, GGM, and GLS are all reliably smaller than those generated by PEG, MPEG, and OJM. AGR ranks in the middle of the pack. The cross- sectional estimation technique results in much noisier error variance estimates that are statistically indistinguishable from each other. Interestingly, the best predictors of future realized returns (EPR and GGM) also rank best on the basis of the variance of the measurement error terms. Figure 1 provides a graphical representation of our results. The Y-axis represents the average annualized buy-and-hold returns for a continuous weighted ICC portfolio over the next 36-months. The X-axis depicts the median time-series variance of the measurement error terms estimated over 836 sample firms. Therefore, the upper left corner of the graph delineates an “efficiency frontier” where return prediction is maximized while time-series error variance is minimized. This graph shows that EPR is on the best performing model in terms of returns prediction and minimal error variance. In general, models that perform well along one dimension also tend to do better along the other. GGM, GLS and AGR, for example, do appreciably better than PEG, MPEG, and OJM along both dimensions. Strikingly, the seven ICC estimates all perform much better than the three Beta-based measures (FF1, FF3, and FF4). 22 IV.4 Industry-based ICCs Thus far, we have found that several ICC estimates have predictive power for firm-level returns, and also exhibit relatively low variance in their measurement errors. In this section we explore the usefulness of an industry-based ICC estimate when subjected to similar evaluation criteria. Fama and French (1997) attempted to derive industry-level cost of capital estimates using ex post realized returns. They concluded that the noise in the estimation of both factor risk premia and factor loadings rendered the task intractable. We now revisit this task, armed with ICC estimates and our new evaluation scheme. Panels A of Table VII reports the pooled cross-sectional results obtained from regressing firm-specific future realized returns on the firm’s industry-median implied risk premium derived from each of the seven valuation models. The dependent variables are 12, 24, 36, 48, and 60 month buy-and-hold returns. As in Table IV, the t-statistics are calculated using two-way cluster robust standard errors (Gow et al. (2010)). The evidence in Panel A shows that the industry median ICC estimates from each of the four successful models (GLS, EPR, AGR, and GGM) all have some predictive power for firm-specific returns. This predictive power is weak in the first 12-months, but become increasingly evident over longer holding periods. In terms of statistical reliability (t-statistics), the industry results are weaker than the firm-specific results reported earlier in Table IV. However, compared to the firm- specific results (Table IV), the coefficients on the industry ICCs are uniformly higher, indicating that the industry ICC estimates are less noisy. For example, focusing on the GGM model, a one-percent increase in the median industry ICC corresponds to a 0.396% increase in future returns over the next 12- months (compared to 0.220% for the same model in Table IV). Across these four models, a one-percent increase in the median industry ICC corresponds to a 3.00% to 5.30% increase in average realized returns over the next 60-months. Apparently some of the noise in the firm-level estimates is removed with industry portfolios. Panel B reports the returns from a buy-and-hold strategy where the firm-specific weights are based on the median ICC in its industry. Once again, the results are weaker than for the analogous firm-specific tests (see Table IV Panel B). Return 23 forecasts are not statistically significant until at least three years have passed. Nevertheless, the evidence in this panel confirms the fact that the same four models have some ability to predict future returns. The pattern of predictable returns persists over the next five years, as we observe steadily increasing coefficients over time across all four valuation models. The magnitude of these returns do not compare to some previously reported pricing anomalies (the price momentum effect is, for example, is approximately one percent per month across the top and bottom deciles). However, the consistency of the returns over the next five years is suggestive of a risk-based rather than a mispricing-based explanation. Finally, Panel C reports the time-series variance of the measurement errors associated with each model, when the firm-specific ICCs are replaced by their industry median. Not surprisingly, the error variances for the industry ICC estimates are generally lower than the firm-specific ICC estimates. For example, the mean error variance for firm-specific GLS is 1.778 (Table V Panel A), and is 1.488 for the industry-based ICC estimate (Table VII Panel C). In sum, the performances of the industry-based ICCs are quite encouraging. We find that the median industry ICC based on GLS, EPR, AGR, and GGM all have some ability to predict future firm-specific returns. Moreover, the industry-based ICC estimates have generally more stable error variance terms over time. These findings suggest an industry-based ICC estimate can be useful in investment and capital budgeting decisions. IV.5 Analyst-based ICC Estimates Thus far, the ICC estimates we have tested are all based on a mechanical earnings forecasting model. As explained earlier, our choice is motivated by recent evidence that the mechanical model produces reasonably reliable forecasts while allowing for a much larger sample. However, most of the prior literature has used an analyst-based approach to earnings forecasts when computing ICCs. We therefore now examine the performance of analyst-based ICC estimates using the same evaluation scheme. Figure 2 plots the efficiency frontier for seven ICC estimates computed using mean analyst earnings forecasts from I/B/E/S, as well as for the three Beta-based measures. For each ICC estimate, we used the mean one-year-ahead (FY1), two- year-ahead (FY2) earnings forecast. To derive future earnings for years 3 to 5 (as needed), we multiplied the two-year-ahead earnings forecast by (1+ LTG), where LTG is the mean long-term growth rate estimate. For example, to compute GLS, we derived the three-year-ahead earnings (FY3) by multiplying FY2 and 24 (1+LTG). To compute GGM, which requires a five-year-ahead forecast, we multiplied FY2 by (1+LTG)3. Figure 2 shows that using analyst-based forecasts, EPR and GLS are on the efficient frontier. The best predictor of future returns is GLS, but EPR has lower error variance. Interestingly, four other ICCs (AGR, MPEG, PEG, and OJM) all have some predictive power for future realized returns. Among the ICC estimates, only GGM does not have a significantly reliable ability to predict returns. As before, all seven ICC estimates have much lower error variance than the three Beta-based measures. Comparing Figures 1 and 2, we observe an interesting interaction between the earnings forecasting method and the valuation model used to compute ICCs. Specifically, we find that ICC estimates perform better when the valuation model in question complements the forecasting method. Conversely, the ICC estimates perform poorly when the valuation model compounds the noise in the input variables. For example, in Figure 1, all four of the worst performing ICCs are based on an abnormal earnings growth model (PEG, MPEG, OJM, AGR). Recall that these models depend critically on an estimation of the earnings growth rate, which is typically extracted from the change in earnings between FY2 and FY1. At the same time, recall that Hou et al. (2009) rely on a forecast of earnings levels in the cross-section. As a result, earnings growth rate forecasts from Hou et al. (2009) – i.e., FY2/FY1 – is likely to be particularly noisy. The evidence in Figure 1 suggests that the Hou et al. (2009) forecasting method is particularly unsuitable for the abnormal growth models. The results in Figure 2 seem to confirm this conjecture, while pointing out a similar problem with analyst-based earnings forecasts. This figure shows that when analyst forecasts rather than the mechanical model are used, the four abnormal earnings growth models perform much better. At the same time, we find that GGM performs quite poorly. The poor performance of GGM seems to be due to its strong reliance on LTG, which has been documented by prior studies to be the least reliable of the analyst inputs (LaPorta (1996), Lin and McNichols (1998)). In short, our evidence suggests that a careful matching of valuation models and forecasting models will lead to improved ICC estimates. V. Summary 25 The cost of equity capital is central in many managerial and investment decisions that affect the allocation of scarce resources in society. In this study, we have attempted to address a key problem in the development of market implied cost of capital estimates – how we might assess ICC performance as expected returns proxies when prices are noisy. In the theory section of this paper, we frame comparisons across alternative ICC estimates in terms of a comparison of the properties of their error terms. We propose: (a) cross-sectional prediction of future realized returns and (b) time-series stability of measurement errors, as key indicators of ICC quality. We show that under fairly general assumptions, it is possible to derive an expression for the time-series variance of these error terms, even when the errors themselves are not observable. In our empirical work, we compare seven alternative ICC estimates and show that several perform well along both dimensions. For robustness, we estimate each model using both a mechanical earnings forecasting model and the mean I/B/E/S analyst earnings forecast. We recognize that these seven (or even fourteen) alternatives do not remotely exhaust the list of possible candidates for testing. We have not, for example, attempted to adjust for any predictable bias in analyst forecasts when computing the ICCs (e.g., Gode and Mohanram (2008)). The ICC literature is still in its infancy and presumably all the nominees tested in this paper can be improved upon by future researchers. Our main goal is not so much to establish the dominance of certain ICC estimates, as to demonstrate the value of an assessment framework that relies only on a set of minimalistic assumptions. We do not assume here that Beta, or future realized returns, are normative benchmarks by which ICC estimates should be measured. Rather, we show that certain ICC estimates widely used in the literature actually perform quite satisfactorily as proxies for firm-specific expected returns. Indeed, all of the ICC estimates we tested perform much better than the Beta-based measures widely touted in finance textbooks. Our results also provide some evidence in support of an instrumental variable approach to ICC estimation. In particular, we show that the median industry ICC from several popular models (EPR, GGM, GLS, and AGR) also predicts firm-level returns, particularly over 3 to 5 year horizons. This result is important because it suggests that industry membership can potentially be a useful instrumental in the derivation of ICC estimates that does not rely on a firm’s current stock price. We believe this is a promising venue for future research. 26 References Black, Fischer, 1986, Presidential address: Noise, Journal of Finance 41, 529-543. Botosan, C., and M. A. Plumlee, 2005, “Assessing Alternative Proxies for the Expected Risk Premium,” Accounting Review 80, 21–53. Botosan, C., M. A. Plumlee, and H. J. Wen, 2010, “The relation between expected returns, realized returns, and firm risk characteristics,” working paper, University of Utah, March. Callen, J., 2009, “Shocks to shocks: a theoretical foundation for the information content of earnings,” Contemporary Accounting Research 26, 135-66/ Campbell, John Y., 1991, “A Variance Decomposition for Stock Returns,” Economic Journal 101, 157-179. Chava, S., and A. Purnanadam. 2009. Is default risk negatively related to stock returns? Review of Financial Studies, forthcoming. Claus, J. and J. Thomas, 2001, “Equity Risk Premium as Low as Three Percent? Evidence from Analysts’ Earnings Forecasts for Domestic and International Stocks,” Journal of Finance 56, 1629–66. Cutler, D., J. Poterba, and L. Summers, 1989, What moves stock prices? Journal of Portfolio Management, Spring, 4-12. Das, S., Levine, C. and K. Sivaramakrishnan, 1998, “Earnings Predictability and Bias in Analysts' Earnings Forecasts,” Accounting Review 73, 277-294. De Long, J. B., A. Shleifer, L. H. Summers, and R. J. Waldmann, 1990, Noise trader risk in financial markets, Journal of Political Economy 98, 703-738. Easton, P., 2007, “Estimating the Cost of Capital Implied by Market Prices and Accounting Data,” Foundations and Trends in Accounting Volume 2, Issue 4: 241- 364. ——— , 2004, “PE Ratios, PEG Ratios, and Estimating the Implied Expected Rate of Return on Equity Capital”, Accounting Review 79, 73–96. ——— and S. Monahan, 2005, “An Evaluation of Accounting-based Measures of Expected Returns”, Accounting Review 80, 501–38. Elton, J., 1999, “Expected Return, Realized Return, and Asset Pricing Tests”, Journal of Finance 54, 1199–1220. 27 Fama, E., and K. French, 1997, “Industry Costs of Equity,” Journal of Financial Economics 43, 153–93. Frankel, R. and C. M. Lee, 1998, “Accounting valuation, market expectation, and cross sectional stock returns”, Journal of Accounting and Economics 25, 283-319. Gebhardt, W., C. Lee and B. Swaminathan, 2001, “Towards an Implied Cost of Capital”, Journal of Accounting Research 39, 135–76. Gode, D. and P. Mohanram, 2003, “Inferring the Cost of Capital Using the Ohlson- Juettner Model”, Review of Accounting Studies 8, 399–431. Gode, D. and P. Mohanram, 2008, “Improving the relationship between implied cost of capital and realized returns by removing predictable analyst forecast errors”, Working Paper, Stern School – New York University. Gordon, J., and M. Gordon, 1997, “The Finite Horizon Expected Return Model,” Financial Analysts Journal (May/June), 52-61. Gow, I. D., G. Ormazabal, T. J. Taylor, 2010, Correcting for cross-Sectional and time- series dependence in accounting research, The Accounting Review, 85, 483-512. Guay, W., S. Kothari and S. Shu, 2005, “Properties of Implied Cost of Capital Using Analysts’ Forecasts,” Working Paper (University of Pennsylvania, Wharton School). Hou, K., M.A. van Dijk, and Y. Zhang, 2009, “The Implied Cost of Capital: A New Approach,” working paper, Ohio State University Hughes, J., J. Liu, and J. Liu, 2009, “On the relation between expected returns and implied cost of capital,” Review of Accounting Studies 14, 246-259. Lambert, R. A., 2009, “Discussion of ‘On the relation between expected returns and implied cost of capital,” Review of Accounting Studies 14, 260-268. LaPorta, R., 1996, “Expectations and the cross-section of stock returns,” Journal of Finance 51, 1715-42. Lee, C., J. Myers and B. Swaminathan, 1999, “What is the Intrinsic Value of the Dow?”, Journal of Finance 54, 1693–741. Lee, Charles M. C., David Ng, and Bhaskaran Swaminathan, 2009, “Testing international asset pricing models using implied costs of capital,” Journal of Financial and Quantitative Analysis 44, 307-335. 28 Lewellen, J., 2002, “Momentum and Autocorrelation in Stock Returns,” Review of Financial Studies 15, 533-563. Lin, H. W., and M. F. McNichols, 1998, “Underwriting relationships, analysts’ earnings forecasts and investment recommendations,” Journal of Accounting and Economics 25, 101-127. Ohlson, J., and B. Juettner-Nauroth, 2005, “Expected EPS and EPS Growth as Determinants of Value,” Review of Accounting Studies 10, 349-365. Pastor, L., and R. F. Stambaugh, 1999, “Costs of equity capital and model mispricing,” Journal of Finance 54, 67–121. Pastor, L., and R. F. Stambaugh, 2009, “Predictive systems: living with imperfect predictors,” Working Paper, University of Chicago and the Wharton School, University of Pennsylvania. Pastor, L., M. Sinha, and B. Swaminathan, 2008, “Estimating the Intertemporal Risk- Return Tradeoff Using the Implied Cost of Capital”, The Journal of Finance 63(6), 2859-2897 Petersen, M., 2009, “Estimating Standard Errors in Finance Panel Data Sets: Comparing Approaches,” Review of Financial Studies 22, 435-480. Roll, R., 1984, “Orange juice and weather,” American Economic Review 74, 861-880. Shiller, R. J., 1984, “Stock prices and social dynamics,” The Brookings Papers on Economic Activity 2, 457-510. Van Binsbergen, J. H., and R. S. J. Koijen, 2009, “Predictive Regressions: A Present- Value Approach,” Working Paper, Stanford University. Vuolteenaho, T., 2002, “What Drives Firm-level Stock Returns?” Journal of Finance 57, 233–264. Wang, C., 2010, “Relating Expected Returns and ICCs,” Working Paper, Stanford University. 29 Technical Appendix In this appendix, we derive a closed-form expression for the firm-specific variance of measurement error for any given ICC estimate Var(ωt) in terms of variables that can be empirically estimated. We do so by combining the time-series autocovariance of ICCs (Step 1), the time-series autocovariance in realized returns (Step 2), and the time-series return-ICC covariance (Step 3). We then combine these three steps to derive Var(ωt) in Step 4. We also lay out the assumptions needed to estimate the cross-sectional variance in ICC measurement errors (Step 5). Throughout this analysis, we have assumed that other firm-specific news (unt) is uncorrelated, in time-series, with the level of the ICC estimates for a given firm. Alternatively, we could assume that firm-specific news for firm i (uni) is uncorrelated, in cross-section, with the level of the ICC estimate at any given point in time t. The derivation is similar (see Wang (2010) for details). We assume that expected returns (ert) and the ICC measurement error (ωt) for a givenfirm follow an AR(1) process, with parameters φ and ψ, respectively. ert 1 ert ut 1 (T1) ~ e rt 1 ert t (T2) t 1 t vt 1 (T3) We extend Campbell’s (1991) decomposition of realized returns rt 1 Et rt 1 Et 1 Et j rt 1 j Et 1 Et j d t 1 j t j 0 j 0 Where we denote for simplicity the first, second, and third terms to be ert (ex-ante expected returns), rnt+1 (expected returns news), and cnt+1 (cash flow news), respectively. In addition, we extend the original Campbell decomposition by including a fourth term εt representing all other shocks (e.g. mispricing reversion) that is ex-ante unpredictable. We combine the last two terms and refer to them jointly as unforecastble news: unt = cnt + εt. Campbell (1991) shows that when expected returns follows an AR(1) process (T1) the innovation in expectations of future returns takes the form u t rnt 1 (T4) 1 so that we can re-write the decomposition to be u t rt 1 ert unt (T4’) 1 30 Throughout our analysis, we assume that realized cash flow other unforecastable news for a given firm is uncorrelated (in time-series) with its ex-ante expected returns. Under this setup, we can identify a firm's Var(ωt) by combining the time-series autocovariance of ICCs, time-series autocovariance in returns, and the time-series return-ICC covariance. Step 1. Time-Series Autocovariance of ICCs The first two time-series autocovariances can be written as c 0 Var e rt ~ (T5) 0 Var ert Cov ert , t 0 Var t Cov ert , t c1 Cov ~ rt 1 , ~rt e e (T6) 1 Var ert Cov ert , t 1 Var t Cov ert , t Step 2. Time-Series Return-ICC Covariance By combining our return decomposition (T4') with the AR(1) structure (T1) in expected returns, we can relate expected returns with future realized returns as follows: u t 1 rt 2 ert u t unt 2 (T7) 1 2 er u u u t 2 un rt 3 (T8) t 1 t 3 1 t t Using (T7) and (T8) and under the assumption of uncorrelated cash flow and other unforecastable news, we can write the time-series return-ICC covariance as: c1r Cov rt 1 , e rt ~ (T9) 0 Var ert Cov ert , t c 2 Cov rt 2 , ~rt r e (T10) 1 Var ert Cov ert , t Step 3. Time-Series Autocovariance in Returns The return decomposition of (T4’), together with (T7) and (T8), imply the following first and second order autocovariance in realized returns c1rr Cov rt 2 , rt 1 (T11) Var ert 31 c 2 Cov rt 3 , rt 1 rr (T12) 2 Var ert Step 4: Identifying Firm-Specific Time-Series Variance in ICC Measurement Errors Under the above, c0 c1r Vart Covert , t (T13) and c1r Covert , t c1r c1rr r (T14) c2 so that cr Var t c 0 2c1r c1rr 1r (T15) c2 Step 5: Identifying Cross-Sectional Variance in ICC Measurement Errors We can extend this framework to estimate a cross-sectional variance in measurement errors by making the following assumptions: 1. AR(1) parameters ψ and φ are common across firms in the cross-section, and 2. realized cash flow and other unforecastable news unt+1 is uncorrelated with the expected returns in the cross-section. Under these two assumptions, we can take all the above variances and covariances cross- sectionally, and all results follow. That is, the cross-sectional variance in ICC measurement error can be obtained by: Cov ri ,t 1 , e ri ,t ~ Var t Var e ri ,t 2Cov ri ,t 1 , e ri ,t Cov ri ,t 1 , ri ,t 2 ~ ~ ~ (T16) Cov ri ,t 2 , e ri ,t where variance and covariances are taken over the cross-section of firms (indexed by i) in a given year t. 32 Appendix I Sample Selection The table below details the sample selection procedure. The final sample used in our analysis consists of 74,343 firm-years and 10,523 unique firms spanning 1970-2007. # of Firm- Lost Firm- # of Unique Lost Unique Filter Criterion Years Years Firms Firms Intersection of CRSP and Compustat observations with data on book values, earnings, statement 1 141,615 14,901 forecasts, and industry identification for fiscal years greater than or equal to 1970 60,713 2,920 Non-missing ICC estimates 2 and ICC estimates between 80,902 11,981 0 and 100% for all 7 models 6,559 1,458 Non-missing Fama-French 3 74,343 10,523 Expected Return Measures Final Sample 74,343 10,523 33 Appendix II Regression Coefficients from Earnings Forecasts Regressions This table reports the average regression coefficients and their time-series t-statistics from annual pooled regressions of one-year-ahead through five-year-ahead earnings on a set of variables that are hypothesized to capture differences in expected earnings across firms. Specifically, for each year t between 1970 and 2007, we estimate the following pooled cross-sectional regression using the previous ten years (six years minimum) of data: , , , , , , , , , where Ej,t+ ( = 1, 2, 3, 4, or 5) denotes the earnings before extraordinary items of firm j in year t+, and all explanatory variables are measured at the end of year t: EVj,t is the enterprise value of the firm (defined as total assets plus the market value of equity minus the book value of equity), TAj,t is the total assets, DIVj,t is the dividend payment, DDj,t is a dummy variable that equals 0 for dividend payers and 1 for non-payers, NEGEj,t is a dummy variable that equals 1 for firms with negative earnings (0 otherwise), and ACCj,t is total accruals scaled by total assets. Total accruals are calculated as the change in current assets plus the change in debt in current liabilities minus the change in cash and short term investments and minus the change in current liabilities. R-Sq is the time-series average R-squared from the annual regressions. Years Ahead Intercept V TA DIV DD IB NEGE AC R-Sq 1 1.813 0.010 -0.008 0.318 -2.034 0.763 0.933 -0.018 0.858 (5.34) (44.69) -(33.42) (37.09) -(3.51) (163.97) (2.46) -(9.83) 2 2.996 0.012 -0.009 0.489 -2.792 0.686 2.358 -0.020 0.800 (6.35) (39.94) -(26.88) (39.51) -(3.61) (99.01) (2.89) -(7.80) 3 15.312 0.002 -0.001 0.600 -10.129 0.298 -0.316 -0.008 0.456 (24.55) (11.80) -(5.68) (44.33) -(9.93) (46.93) -(0.53) -(2.67) 4 21.290 -0.002 0.004 0.572 -13.066 0.190 -2.524 -0.004 0.329 (30.06) -(2.73) (6.57) (42.15) -(11.45) (30.36) -(2.03) -(1.20) 5 25.942 -0.001 0.003 0.509 -15.082 0.132 -4.746 0.007 0.257 (33.85) -(6.14) (8.64) (38.19) -(12.42) (21.36) -(3.26) (1.32) 34 TABLE I Implied Cost of Capital (ICC) Measures by Year This table reports the median implied cost of capital (ICC) estimates derived from seven valuation models (GLS, PEG, MPEG, OJM, EPR, AGR, and GGM). A full description of each model is included in Section III. We compute a firm- specific ICC estimate for each stock in our sample based on the stock price and publicly available information on June 30th of each year. ICC estimates are set to missing if they are either below zero or above 100%. FF1 is an expected return proxy computed using firm-specific Betas and the historical market risk premium provided by Fama-French. RF Yield equals the yield on the 10-year Treasury bond on June 30th of each year. Year Obs GLS PEG MPEG OJM EPR AGR GGM FF1 RF Yield 1971 1,117 15.92% 11.81% 13.96% 15.01% 7.57% 7.44% 19.02% 52.54% 6.52% 1972 1,180 16.30% 10.25% 12.49% 14.41% 7.24% 7.20% 18.92% 13.50% 6.11% 1973 1,133 17.63% 8.93% 11.69% 13.92% 11.65% 11.60% 21.15% -10.28% 6.90% 1974 1,035 19.56% 7.99% 11.37% 19.89% 16.34% 16.19% 23.69% -19.37% 7.54% 1975 1,629 21.32% 12.06% 14.57% 21.04% 13.86% 13.60% 26.54% 20.12% 7.86% 1976 1,722 20.46% 11.98% 14.19% 20.44% 12.54% 12.47% 25.66% 14.48% 7.86% 1977 1,956 18.94% 11.52% 13.59% 18.41% 13.14% 13.25% 23.74% 5.04% 7.28% 1978 2,048 18.29% 12.10% 14.39% 20.44% 12.91% 12.96% 23.34% 5.94% 8.46% 1979 2,052 18.03% 12.23% 14.47% 21.18% 13.65% 13.55% 22.22% 17.02% 8.91% 1980 2,071 18.90% 12.93% 15.58% 24.05% 14.68% 14.54% 22.36% 20.40% 9.78% 1981 2,334 16.52% 9.24% 11.46% 5.45% 12.18% 12.25% 18.61% 24.96% 13.47% 1982 2,431 17.72% 13.35% 15.89% 18.11% 15.81% 15.91% 18.74% -13.49% 14.30% 1983 2,701 14.50% 11.38% 13.25% 16.14% 8.34% 8.90% 14.99% 68.53% 10.85% 1984 2,488 15.74% 14.28% 16.26% 19.38% 10.25% 10.68% 16.67% -6.81% 13.56% 1985 2,424 14.84% 12.73% 14.41% 17.46% 9.58% 9.79% 15.69% 27.94% 10.16% 1986 2,382 13.02% 12.27% 13.55% 15.51% 7.35% 7.51% 12.50% 32.47% 7.80% 1987 1,929 12.57% 10.85% 12.86% 15.70% 7.52% 7.90% 10.47% 18.14% 8.40% 1988 1,945 12.50% 10.10% 11.91% 14.39% 8.00% 7.92% 11.79% -5.91% 8.92% 1989 2,101 12.35% 12.93% 15.43% 17.78% 10.07% 10.41% 10.83% 18.18% 8.28% 1990 2,068 13.19% 13.10% 15.98% 17.55% 10.26% 11.17% 12.44% 12.04% 8.48% 1991 1,940 12.57% 10.88% 13.86% 14.74% 9.00% 9.86% 12.21% 6.92% 8.28% 1992 1,849 11.57% 10.75% 13.10% 14.33% 7.51% 8.24% 10.56% 14.02% 7.26% 1993 1,776 10.25% 9.30% 11.30% 12.91% 6.72% 7.20% 9.06% 15.25% 5.96% 1994 2,166 11.49% 9.67% 11.63% 14.84% 7.90% 8.14% 10.99% 1.72% 7.10% 1995 2,293 11.04% 9.53% 11.06% 14.16% 7.81% 8.22% 10.52% 21.93% 6.17% 1996 2,778 10.40% 9.30% 10.53% 13.37% 6.60% 7.15% 9.81% 20.36% 6.91% 1997 2,808 9.64% 8.86% 9.99% 12.55% 5.84% 6.08% 8.75% 23.32% 6.49% 1998 2,784 9.17% 8.40% 9.46% 10.41% 5.38% 5.84% 7.17% 21.93% 5.50% 1999 2,526 9.60% 9.01% 10.12% 12.09% 6.13% 6.79% 7.51% 15.75% 5.90% 2000 2,344 10.48% 8.39% 9.76% 11.46% 7.39% 8.21% 8.45% 9.02% 6.10% 2001 2,092 9.36% 7.45% 8.64% 10.52% 6.29% 6.84% 7.19% -4.26% 5.28% 2002 1,880 8.96% 8.49% 9.47% 10.99% 5.30% 5.54% 6.91% -5.00% 4.93% 2003 1,392 9.30% 5.52% 7.39% 10.39% 6.55% 7.09% 6.80% 2.52% 3.33% 2004 1,385 9.26% 8.18% 9.58% 13.12% 6.69% 7.15% 8.16% 8.92% 4.73% 2005 1,471 9.50% 9.29% 10.70% 14.62% 6.84% 7.06% 8.24% 6.81% 4.00% 2006 2,036 9.32% 11.60% 12.64% 15.73% 6.24% 6.18% 7.54% 10.35% 5.11% 2007 2,077 9.29% 11.93% 13.05% 15.94% 5.52% 6.25% 7.15% 18.88% 5.10% Mean 2,009 13.50% 10.50% 12.42% 15.36% 9.10% 9.38% 13.96% 13.08% 7.56% Median 2,052 12.57% 10.75% 12.64% 14.84% 7.81% 8.21% 11.79% 14.02% 7.26% Std 479 3.85% 1.97% 2.26% 3.76% 3.17% 3.01% 6.29% 16.72% 2.52% Min 1,035 8.96% 5.52% 7.39% 5.45% 5.30% 5.54% 6.80% -19.37% 3.33% Max 2,808 21.32% 14.28% 16.26% 24.05% 16.34% 16.19% 26.54% 68.53% 14.30% 35 TABLE II Correlation between ICC Measures This table reports the average annual correlations between the seven implied cost of capital (ICC) measures derived from seven valuation models (GLS, PEG, MPEG, OJM, EPR, AGR, and GGM), and three Beta-based expected return proxies (FF1, FF3, and FF4). Pearson correlations are shown above the diagonal and Spearman correlations are shown below the diagonal. A full description of each model is included in Section III. GLS PEG MPEG OJM EPR AGR GGM FF1 FF3 FF4 GLS 1.000 0.551 0.620 0.339 0.527 0.342 0.899 -0.089 -0.031 -0.027 PEG 0.631 1.000 0.961 0.406 0.511 0.519 0.591 -0.110 -0.057 -0.044 MPEG 0.644 0.961 1.000 0.451 0.608 0.573 0.645 -0.138 -0.066 -0.058 OJM 0.458 0.495 0.539 1.000 0.417 0.396 0.331 -0.074 -0.049 -0.047 EPR 0.572 0.406 0.509 0.422 1.000 0.687 0.554 -0.147 -0.071 -0.075 AGR 0.493 0.439 0.526 0.447 0.829 1.000 0.348 -0.125 -0.068 -0.070 GGM 0.873 0.637 0.652 0.450 0.545 0.432 1.000 -0.103 -0.021 -0.014 FF1 -0.128 -0.147 -0.186 -0.121 -0.194 -0.172 -0.136 1.000 0.541 0.467 FF3 -0.043 -0.086 -0.096 -0.066 -0.099 -0.092 -0.037 0.521 1.000 0.858 FF4 -0.040 -0.078 -0.091 -0.065 -0.103 -0.096 -0.028 0.452 0.851 1.000 Note: Bolded correlation coefficients are statistically significant at the 5% level 36 TABLE III ICC Estimates and Future Realized Returns This table examines the ability of alternative expected return proxies to forecast future returns. Panel A reports the pooled cross-sectional results obtained from regressing firm-specific future realized returns on seven implied risk premium estimates (GLS, PEG, MPEG, OJM, EPR, AGR, and GGM) and three Beta- based expected return proxies (a full description of each model is included in Section III). The dependent variables are firm-specific 12, 24, 36, 48, and 60 month buy-and-hold returns. Regression intercepts are not shown, t-statistics are shown in parentheses below the coefficients and are calculated using two-way cluster robust standard errors (clustered by firm and year). Panel B reports the buy-and-hold returns for continuous firm-specific ICC-weighted strategies over the 12, 24, 36, 48, and 60 months following portfolio formation. The strategy holds assets in proportion to their market-adjusted ICC. Specifically, an asset’s weight in month t is: 1 ______ w i,t (ICCi,t ICC mkt,t ) N ______ where ICCi,t equals the firm’s implied cost of capital in year t , ICC mkt,t is the equal-weighted average ICC for all firms in year t , and N is the total number of stocks in the year t sample. T-statistics based on the 36-year time-series ICC-weighted mean 24, 36, 48, and 60 month returns are computed using Newey-West HAC estimators with 1 year, 2 year, 3, year, and 4 year lags, respectively, and are shown in parentheses. Two-tailed significance levels are indicated by *, **, and *** for 10%, 5%, and 1% respectively. Panel A: Regression of Future Firm-Specific Realized Returns on Firm- Specific ICCs 12 Month 24 Month 36 Month 48 Month 60 Month GLS 0.234** 0.542*** 0.961*** 1.313*** 1.715*** (2.42) (3.56) (4.32) (4.32) (4.57) PEG -0.096 -0.140 -0.098 -0.089 -0.104 (-1.41) (-1.29) (-0.73) (-0.50) (-0.49) MPEG -0.060 -0.055 0.018 0.070 0.090 (-0.95) (-0.56) (0.16) (0.48) (0.49) OJM 0.029 0.040 0.159** 0.218** 0.236* (0.76) (0.56) (2.07) (2.12) (1.73) EPR 0.425*** 0.959*** 1.533*** 2.120*** 2.570*** (3.31) (4.64) (5.70) (6.36) (5.97) AGR 0.203*** 0.442*** 0.752*** 0.998*** 1.171*** (3.29) (4.53) (6.04) (5.23) (4.82) GGM 0.229*** 0.491*** 0.809*** 1.105*** 1.397*** (3.47) (4.51) (4.95) (4.90) (5.08) FF1 -0.210*** -0.283*** -0.291** -0.488*** -0.474** (-2.72) (-2.96) (-1.72) (-2.29) (-1.75) FF3 -0.048 -0.080 -0.057 -0.094 -0.039 (-0.60) (-0.70) (-0.35) (-0.43) (-0.15) FF4 -0.048 -0.086 -0.075 -0.139 -0.104 (-0.64) (-0.83) (-0.54) (-0.75) (-0.47) 37 [Table III: Continued] Panel B: Continuous ICC-Weighted Mean Returns by Valuation Model 12 Month 24 Month 36 Month 48 Month 60 Month GLS 0.043*** 0.096*** 0.161*** 0.237*** 0.314*** (2.83) (2.64) (3.07) (2.87) (2.65) PEG 0.005 0.013 0.047 0.081 0.111 (0.25) (0.32) (0.71) (1.03) (1.15) MPEG 0.007 0.026 0.066 0.112* 0.152* (0.45) (0.81) (1.32) (1.90) (1.94) OJM 0.007 0.018 0.045 0.077** 0.096** (0.57) (0.71) (1.16) (2.04) (1.96) EPR 0.061*** 0.150*** 0.240*** 0.361*** 0.479*** (2.88) (3.61) (5.14) (5.53) (5.15) AGR 0.039*** 0.089*** 0.162*** 0.249*** 0.333*** (2.55) (3.08) (3.67) (2.86) (2.76) GGM 0.053*** 0.117*** 0.200*** 0.291*** 0.380*** (2.87) (2.71) (2.94) (2.80) (2.64) FF1 -0.015 -0.015 -0.009 -0.066 -0.144*** -(0.45) -(0.42) -(0.16) -(1.26) -(2.79) FF3 0.018 0.011 0.020 0.056 0.009 (0.57) (0.29) (0.44) (0.93) (0.15) FF4 0.019 0.011 0.009 0.026 -0.033 (0.67) (0.31) (0.19) (0.47) -(0.65) 38 TABLE IV A Comparison of Predictive Power for Future Returns across Models Panel C provides evidence on the statistical significance of the difference in predictive power across different models. To construct this panel, we compute the annual pair-wise difference in three-year cumulative returns for each pair of expected return proxies, where the returns are based on a continuous-weighted hedge portfolio. Table values represent the time-series t-statistics of the difference between the strategies over the 36 sample years. Table values are positive (negative) when the expected return proxy displayed down the left-side of the table has stronger (weaker) predictive power for realized returns than the expected return proxy displayed in the top row of the table. Panel A provides pair-wise comparisons for the seven ICC estimates. Panel B provides pair- wise comparisons for each of the seven ICC estimates to the three Beta-based estimates. Standard errors are computed using Newey-West HAC estimators with 2 year lags. Two-tailed significance levels are indicated by *, **, and *** for 10%, 5%, and 1% respectively. Panel A: Time-Series t-test Comparisons between seven ICC estimates GLS PEG MPEG OJM EPR AGR GGM GLS 0.94 1.155 1.16 -3.95*** -0.21 -3.58*** PEG -0.94 -0.096 -0.01 -2.41*** -1.21 -1.86** MPEG -1.15 0.10 0.05 -3.03*** -1.33 -2.14*** OJM -1.16 0.01 -0.049 -2.95*** -1.08 -1.95*** EPR 3.95*** 2.41*** 3.03*** 2.95*** 2.69*** 1.32 AGR 0.21 1.21 1.33 1.08 -2.69*** -1.31 GGM 3.58*** 1.86** 2.13*** 1.95** -1.32 1.31 Panel B: Time-Series t-test Comparisons of ICC estimates to Beta-based estimates GLS PEG MPEG OJM EPR AGR GGM FF1 -2.67*** -1.40 -1.58 -1.51 -3.70*** -2.59*** -3.60*** FF3 -3.30*** -1.21 -1.37 -1.31 -4.14*** -3.19*** -4.12*** FF4 -3.48*** -1.53 -1.73 -1.55 -4.27*** -3.50*** -4.11*** 39 TABLE V Time-Series Measurement Error Variances This table presents descriptive statistics for the time-series variances of measurement errors (multiplied by 100) of ICC’s derived from seven valuation models (GLS, PEG, MPEG, OJM, EPR, AGR, and GGM) and three Beta-based measures (FF1, FF3, and FF4). Measurement error variances are calculated as follows: Cov ri ,t 1 , e ri ,t ~ Var t Var e ri ,t 2Cov ri ,t 1 , e ri ,t Cov ri ,t 1 , ri ,t 2 ~ ~ ~ Cov ri ,t 2 , e ri ,t where Var( t ) is the measurement error variance, ~ri ,t is the ICC in year t, and rt i is the realized return e in year t i . Panel A reports summary statistics for the error variance from each model, using a sample of 836 unique firms with a minimum of 20 (not necessarily contiguous) years of data during our 1971-2007 sample period. Table values in this panel represent descriptive statistics for the error variance from each ICC estimate computed across these 836 firms. Panels B and C report t-statistics corresponding to the pair- wise comparisons of firm-specific measurement error variances within the sample of 836 firms used in Panel A. *, **, and *** indicate two-tailed significance at the 10%, 5%, and 1% level, respectively. Panel A: Time-Series Variance of Measurement Error (N=836) Mean t-Statistic StDev P25 Median P75 Skewness GLS 1.822 6.509 8.095 0.000 0.043 1.039 9.567 PEG 3.601 7.824 13.309 0.000 0.177 1.478 5.905 MPEG 2.829 7.405 11.045 0.000 0.151 1.369 7.122 OJM 3.660 11.075 9.555 0.000 1.251 3.911 7.287 EPR 1.674 8.116 5.964 0.000 0.050 1.054 7.973 AGR 2.696 7.190 10.842 0.000 0.174 1.225 6.994 GGM 2.146 6.066 10.231 0.000 0.000 1.003 8.365 FF1 5.584 14.691 10.990 1.008 2.880 5.975 5.887 FF3 6.551 15.855 11.947 1.576 3.561 7.101 5.495 FF4 7.192 17.960 11.578 1.867 4.158 7.958 5.046 Panel B: t-Statistics Corresponding to Differences in TS Variance of (N=836) GLS PEG MPEG OJM EPR AGR GGM GLS 0.000 -3.38*** -2.41*** -4.96*** 0.47 -1.98*** -0.832 PEG 3.38*** 0.000 1.46 -0.10 3.86*** 1.60 2.62*** MPEG 2.41*** -1.46 0.000 -1.80* 2.88*** 0.26 1.40 OJM 4.96*** 0.11 1.80 0.000 5.44*** 2.03*** 3.32*** EPR -0.47 -3.86*** -2.88*** -5.44*** 0.000 -2.85*** -1.26 AGR 1.98*** -1.60 -0.26 -2.03*** 2.85*** 0.000 1.23 GGM 0.83 -2.62*** -1.40 -3.32*** 1.26 -1.23 0.000 Panel C: t-Statistics Corresponding to Differences in TS Variance of (N=836) GLS PEG MPEG OJM EPR AGR GGM FF1 8.80*** 3.45*** 5.60*** 4.07*** 9.59*** 5.53*** 6.87*** FF3 11.16*** 4.94*** 7.20*** 5.75*** 11.14*** 7.29*** 8.55*** FF4 12.92*** 6.06*** 8.46*** 7.23*** 13.04*** 8.90*** 9.87*** 40 TABLE VI Cross-Sectional Measurement Error Variances This table presents descriptive statistics for the cross-sectional variances of measurement errors (multiplied by 100) of ICC’s derived from seven valuation models – GLS, PEG, MPEG, OJM, EPR, AGR, and GGM. For each model/year, measurement error variances are calculated as follows: ~r 2Cov r , e r Cov r , r Cov ri ,t 1 , e ri ,t ~ Var t Var e i ,t ~ ~ Cov ri ,t 2 , e ri ,t i ,t 1 i ,t i ,t 1 i ,t 2 where Var(i,t) is the measurement error variance computed using a cross-section of firms (indexed by ~ i) in year t, eri ,t is firm i’s ICC in year t, and ri,t+k is firm i’s return in year t+k. The covariance terms are also computed using a cross-section of firms in each year t. Panel A reports summary statistics for the error variance from each model, using a sample of 36 annual cross-sectional estimations, over the 1971-2007 sample period. Table values in this panel represent descriptive statistics for the error variance from each ICC estimate computed across these 36 years. Panel B reports t-statistics corresponding to the pair-wise comparisons of firm-specific measurement error variances within the sample of 36 years used in Panel A. *, **, and *** indicate two-tailed significance at the 10%, 5%, and 1% level, respectively. Panel A: Cross-Sectional Variance of Measurement Error Mean t-Statistic StDev P25 Median P75 Skewness GLS 4.8564 1.7251 16.8904 0.0000 0.8184 1.8039 5.4319 PEG 6.8464 2.1547 19.0645 0.1796 1.0041 2.5227 3.9916 MPEG 4.9234 1.6053 18.4016 0.0000 0.4382 1.8504 4.6574 OJM 5.5894 2.2777 14.7237 0.5517 2.2833 3.3243 4.8913 EPR 2.3790 2.8414 5.0237 0.0000 0.4907 1.8451 3.7065 AGR 1.5482 3.6459 2.5478 0.0000 0.6133 1.6224 2.3590 GGM 7.2415 2.0188 21.5218 0.0771 1.5467 2.5808 3.9409 Panel B: T-Statistics Corresponding to Differences in CS Error Variances (N=36) GLS PEG MPEG OJM EPR AGR GGM GLS 0.462 0.041 0.238 -1.130 -1.146 1.078 PEG -0.462 -0.443 -0.382 -1.359 -1.65* 0.080 MPEG -0.041 0.443 0.195 -1.002 -1.069 0.719 OJM -0.238 0.382 -0.195 -1.449 -1.69* 0.421 EPR 1.130 1.359 1.002 1.449 -1.021 1.501 AGR 1.146 1.65* 1.069 1.69* 1.021 1.563 GGM -1.078 -0.080 -0.719 -0.421 -1.501 -1.563 41 TABLE VII Performance of Industry-Median ICCs This table examines the performance industry-median ICC estimates as a proxy for firm-specific expected returns. Panel A reports the pooled cross-sectional results obtained from regressing firm- specific future returns on the firm’s industry-median implied risk premiums derived from each of 7 models. The dependent variables are firm-specific 12, 24, 36, 48, and 60 month buy-and-hold returns. Regression intercepts are not shown, test statistics are calculated using two-way cluster robust standard errors (clustered by firm and year), and R-Squared values are shown in italics. Panel B reports the buy-and-hold returns for continuous firm-specific ICC-weighted strategies over the 12, 24, 36, 48, and 60 following portfolio formation using the cross section of ICC’s. The strategy holds assets in proportion to their market-adjusted industry median ICC. Specifically, an asset’s weight in month t is: 1 __________ w i,t (IMICCi,,t IMICC mkt,t ) N __________ where IMICC i,,t is firm i’s industry-median implied cost of capital in year t , IMICC mkt,t is the equal-weighted average IMICC i,,t for all firms in year t, and N is the total number of stocks in the year t sample. T-statistics based on the 36-year time-series ICC-weighted mean 24, 36, 48, and 60 month returns are computed using Newey-West HAC estimators with 1 year, 2 year, 3, year, and 4 year lags, respectively, and are shown in parentheses. Panel C contains descriptive statistics for the time-series measurement error variances (multiplied by 100). Measurement error variances are calculated as follows: Cov ri ,t 1 , IMICC i ,t Var t Var IMICC i ,t 2Cov ri ,t 1 , IMICC i ,t Cov ri ,t 1 , ri ,t 2 Cov ri ,t 2 , IMICC i ,t where Var( t ) is the measurement error variance and rt i is the realized return in year t i . For all three panels, significance levels are indicated by *, **, and *** for 10%, 5%, and 1% respectively. Panel A: Regression of Future Firm-Specific Realized Returns on Firm-Specific ICCs 12 Month 24 Month 36 Month 48 Month 60 Month GLS 0.356 1.145** 2.496*** 3.342*** 4.349*** 0.001 0.005 0.011 0.012 0.014 PEG -0.353 -0.243 0.372 0.586 1.021 0.001 0.000 0.000 0.000 0.001 MPEG -0.226 0.013 0.768 1.086* 1.588** 0.000 0.000 0.001 0.001 0.002 OJM -0.052 0.084 0.904 1.124 1.536 0.000 0.000 0.002 0.002 0.002 EPR 0.566 1.605*** 3.074*** 4.159*** 5.300*** 0.002 0.008 0.014 0.015 0.017 AGR 0.565 1.610** 3.026*** 4.050*** 5.163*** 0.002 0.007 0.013 0.014 0.015 GGM 0.396* 0.966*** 1.836*** 2.412*** 2.995*** 0.003 0.007 0.013 0.014 0.015 42 [Table VII: Continued] Panel B: Continuous Industry ICC-Weighted Mean Returns by Valuation Model 12 Month 24 Month 36 Month 48 Month 60 Month GLS 0.008 0.046 0.101 0.150** 0.205*** (0.28) (0.94) (1.53) (1.98) (2.82) PEG -0.013 0.001 0.037 0.063 0.114 -(0.52) (0.02) (0.58) (0.96) (1.62) MPEG -0.007 0.014 0.050 0.076 0.125 -(0.25) (0.29) (0.75) (0.98) (1.43) OJM -0.014 -0.005 0.034 0.057 0.108 -(0.50) -(0.10) (0.51) (0.80) (1.29) EPR 0.012 0.065 0.110 0.178* 0.264** (0.33) (1.03) (1.31) (1.81) (2.29) AGR 0.013 0.064 0.107 0.173* 0.262*** (0.39) (1.10) (1.36) (1.91) (2.35) GGM 0.013 0.046 0.095 0.128** 0.154*** (0.46) (0.95) (1.57) (2.01) (2.60) Panel C: Industry ICC Time-Series Variance of the Measurement Errors Mean t-Statistic StDev P25 Median P75 Skewness GLS 1.488 7.828 5.660 0.000 0.036 1.027 11.640 PEG 2.469 7.672 9.586 0.000 0.149 1.440 8.147 MPEG 1.633 8.496 5.724 0.000 0.138 1.400 12.449 OJM 2.432 8.747 8.281 0.000 0.549 1.657 8.165 EPR 1.830 6.785 8.032 0.000 0.006 0.835 9.506 AGR 1.846 6.889 7.980 0.000 0.029 0.874 9.100 GGM 1.800 8.261 6.489 0.000 0.174 1.325 10.004 43 Figure 1 Efficient Frontier of Firm-Specific Cost of Capital Estimates (Mechanical Forecasts) The figure below plots for each of 10 expected return proxies, the average annualized three-year-ahead profit of a continuous weighted ICC portfolio along the Y-axis. All seven implied cost of capital estimates are based on a mechanical forecast of earnings. The strategy holds assets in proportion to their market-adjusted ICC. Specifically, an asset’s weight in month t is: 1 ______ w i,t (ICCi,t ICC mkt,t ) N ______ where ICCi,t equals the firm’s expected return proxy in year t , ICC mkt,t is the equal-weighted average ICC for all firms in year t , and N is the total number of stocks in the year t sample. The x-axis reflects the median of firm-specific measurement error variances (multiplied by 100), calculated as follows: ~ ~ ~ Cov(rt 1,ert ) Var( t ) Var(ert ) * Cov(rt 1,ert ) Cov(rr 1,rt 2 ) 2 ~ Cov(rt 2,ert ) ~ where Var( t ) is the measurement error variance, ert is the ICC in year t, and rt i is the realized return in year t i . Firm-specific measurement error variances are calculated based on equation (3) using a sample of 836 unique firms that meet our data requirements. 0.09 EPR 0.08 0.07 GGM Continuous-Weighed ICC Returns 0.06 GLS AGR 0.05 0.04 0.03 MPEG 0.02 OJM PEG FF3 0.01 FF4 FF1 0 -0.01 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 Time-Series Measurement Error Variance 44 Figure 2 Efficient Frontier of Firm-Specific Cost of Capital Estimates (Analyst Forecasts) The figure below plots for each of 10 expected return proxies, the average annualized three-year-ahead profit of a continuous weighted ICC portfolio along the Y-axis. All seven implied cost of capital estimates are based on mean I/B/E/S analyst forecasts of earnings. The strategy holds assets in proportion to their market- adjusted ICC. Specifically, an asset’s weight in month t is: 1 ______ w i,t (ICCi,t ICC mkt,t ) N ______ where ICCi,t equals the firm’s expected return proxy in year t , ICC mkt,t is the equal-weighted average ICC for all firms in year t , and N is the total number of stocks in the year t sample. The x-axis reflects the median of firm-specific measurement error variances (multiplied by 100), calculated as follows: ~ ~ ~ Cov(rt 1,ert ) Var( t ) Var(ert ) * Cov(rt 1,ert ) Cov(rr 1,rt 2 ) 2 ~ Cov(rt 2,ert ) ~ where Var( t ) is the measurement error variance, ert is the ICC in year t, and rt i is the realized return in year t i . Firm-specific measurement error variances are calculated based on equation (4) using a sample of411 unique firms that meet our data requirements. 0.08 GLS 0.07 0.06 EPR AGR Continuous-Weighed ICC Returns 0.05 MPEG 0.04 OJM PEG 0.03 0.02 0.01 GGM FF4 0 FF3 FF1 -0.01 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 -0.02 Time-Series Measurement Error Variance 45

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