1 Evaluating Implied Cost of Capital Estimates By Charles M. C. Lee ** Eric C. So Charles C.Y. Wang First Draft: April 8 th , 2010 Current Draft: August 4 th , 2010 Abstract We develop an analytical framework for assessing the quality of firm-level implied cost of capital (ICC) estimates when price is noisy. Using this framework, we derive a closed-form expression for the variance of ICC measurement errors. Our analyses nominate: (a) prediction of future realized returns and (b) stability of measurement errors, as key indicators of ICC quality. Empirically, we compare seven alternative ICC estimates and show that several perform well along both dimensions. Moreover, we find that the lagged industry median ICC computed using any of the successful models will both predict future firm-level returns and produce stable measurement errors. ** All three authors are at Stanford University. Lee ([email protected]) is the Joseph McDonald Professor of Accounting at the Stanford Graduate School of Business (GSB); So ([email protected]) is a Doctoral Candidate in Accounting at the Stanford GSB; Wang ([email protected]) is a Doctoral Candidate in the Department of Economics. We thank seminar participants at Stanford, Northwestern, University of Chicago, UC-Irvine, LSU, and especially Bhaskaran Swaminathan of LSV Asset Management, for helpful comments and suggestions.
Transcript
1
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: August 4th, 2010
Abstract We develop an analytical framework for assessing the quality of firm-level implied cost of capital (ICC) estimates when price is noisy. Using this framework, we derive a closed-form expression for the variance of ICC measurement errors. Our analyses nominate: (a) prediction of future realized returns and (b) stability of measurement errors, as key indicators of ICC quality. Empirically, we compare seven alternative ICC estimates and show that several perform well along both dimensions. Moreover, we find that the lagged industry median ICC computed using any of the successful models will both predict future firm-level returns and produce stable measurement errors.
** All three authors are at Stanford University. Lee ([email protected]) is the Joseph McDonald Professor of Accounting at the Stanford Graduate School of Business (GSB); So ([email protected]) is a Doctoral Candidate in Accounting at the Stanford GSB; Wang ([email protected]) is a Doctoral Candidate in the Department of Economics. We thank seminar participants at Stanford, Northwestern, University of Chicago, UC-Irvine, LSU, and especially Bhaskaran Swaminathan of LSV Asset Management, for helpful comments and suggestions.
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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. In this study, we address a recurrent problem that seems to stand in the way of broader adoption of ICCs as proxies for firm-level expected returns – that of performance evaluation. Specifically, 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 importance of this problem is highlighted by the myriad of (seemingly arbitrary) assumptions and valuation approaches used to forecast firm-level cash flows. At a minimum, sensible ICC estimates call for sensible cash flow forecasts. But an endless combination of apparently equally defensible forecasting assumptions can be made, each of which can lead to a different set of ICC estimates. When do these differences matter? How might we adjudicate between them? More importantly, will our failure to do so detract from the credibility of the entire approach?
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)).
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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)).3 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 prices and the poor performance of other risk proxies, such as beta). 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.” (page 501) 4 In this study, we develop an analytical framework for assessing the quality of firm-level ICC estimates when price is noisy. We show that under fairly general assumptions, it is possible to derive a closed-form expression for the variance of ICC measurement errors, even when the errors themselves are not observable. In this set-up, we show that higher-quality ICC estimates will: (a) predict future cross-sectional returns and, (b) have stable measurement errors (i.e. have low measurement error variance).5 This analysis then forms the basis of our comparison between ICC estimates. Using the dual criteria of greater predictive power for future returns and more stable measurement errors, we empirically assess the usefulness of seven different ICC estimates. Four of these ICC estimates are based on an earnings capitalization model (PEG, MPEG, OJM, and AGR), one is based on a residual-income model (GLS), and two are based on a Gordon Growth Model (EPR, 3 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. 4 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 more consistent with Botosan et al. (2010), this debate is not the focus of our paper. 5 Our approach is similar to the two-dimensional performance metrics used by Lee, Myers and Swaminathan (1999) [LMS] to compare alternative value estimates for the Dow30 stocks. However, whereas LMS is focused on evaluating alternative value estimates in a univariate time-series, we analyze the properties of expected return measurement errors in a cross-sectional context.
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GGM). To avoid potential problems associated with analyst earning forecasts, and to ensure the largest possible sample, we use the cross-sectional technique introduced by Hou et al. (2009) to forecast future earnings. Our sample consists of 11,981 unique firms (80,902 firm-years) spanning the 1971-2007 time-period. We find that four of these estimates (GLS, GGM, EPR, and AGR) have a statistically reliable ability to predict realized returns over the next 12 to 60 months, while three others (OJM, PEG, MPEG) do not. Among the four proxies with some predictive power, GLS, EPR, and GGM generate appreciably lower average time-series measurement error variance. EPR and GGM, in particular, perform exceptionally well, and are on the two-dimensional “efficiency frontier” established by predictive power and minimal error variance. Because EPR and GGM are simply Gordon Growth models with different forecast horizons (one-year and five-years, respectively), our evidence suggests that a simple Gordon Growth valuation model provides an attractive means of estimating ICCs. In addition, we show that an industry-based ICC estimate based on any of the four successful estimation models (GLS, GGM, EPR, or AGR) will reliably capture a significant amount of cross-sectional variation in future firm-level realized returns (i.e., has good firm-level predictive power). Moreover, as expected, these industry-based ICC estimates exhibit lower time-series error variance than the firm-specific ICC estimates. Collectively, our results provide support for the use of an industry-based ICC (or expected return) estimate in investment or capital budgeting decisions. 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 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.
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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 Revisited 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. One way to parse unexpected return is to decompose it into two components, relating to cash flow news (i.e. shocks to expected cash flows) and discount rates news (i.e. shocks to discount rates). Campbell (1991) and Campbell and Shiller (1988a, 1988b) adopt this approach for their analysis of aggregate market returns, and Vuolteenaho (2002) extended it to a firm-level analysis: , (2)
where cfnt+1 is cash flow news in period t+1 and rnt+1 is discount rate news in period t+1. This is the framework adopted by Easton and Monahan (2005) to test the validity/quality of ICC estimates. Reasoning that the bias and noise in realized returns can be removed (or at least reduced) by controlling for cash flow and discount rate news, EM attempt to estimate empirical proxies for cfn and rn. Alternative ICC estimates are then evaluated in terms of their association with the “corrected” realized return measure. Notice that this decomposition is based on a strong assumption about the source of stock return movements. Specifically, it assumes that Pt = Vt for all t, where Pt is price and Vt
is the present value of its expected future cash payoffs to shareholders. In other words, this decomposition does not entertain the possibility that prices move for any reason other than fundamental news. It does not consider market mispricing of any kind, whether
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from 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 equivalence of price and value at all points in time is a strong operating assumption, even for adherents of competitively efficient markets. As Shiller (1987; page 458-459) famously observed, this assumption is unlikely to hold even if markets are competitively efficient (i.e. even if returns cannot be easily forecasted, prices can still be substantially different from their fundamental values).6 In fact, by assuming that stock returns always fully reflect fundamental news, we may well have assumed away the problems that gave rise to the need for ICCs in the first place – i.e., the noisy nature of prices and realized returns. More to the point, as we show in the next section, this assumption is neither necessary nor germane to the task of evaluating ICC (or expected return) estimates. II.2 An Alternative Approach We begin with the same setup as Campbell (1991), but introduce two modifications. First, we interpret the “cash flow news” term more loosely, so that it captures all other innovations or shocks that cannot be forecasted ex ante. Second, we introduce the idea of expected return estimates (or ICCs), which we define as estimates of the true expected return, measured with error. Using similar notation, we express a stock’s realized return as: , (3)
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. As before, rnt+1 is discount rate news (i.e., rnt+1 reflects innovations that revise the market’s expectation of a firm’s future returns). The last term, unt+1 , captures all other innovations or shocks to price that cannot be forecasted ex ante. Note that in the special case where all other shocks to price are due to cash flow news, equation (3) is identical to equation (2). However, in our framework, the unforecastable component of realized return is a broader concept, and need not be related to cash flow news.
6 As an example, consider the case where Pt = Vt + t, where t is either a random walk or long-horizon mean-reverting process.
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Next, we introduce the idea of ICC estimates ( tre ), which are defined as the true expected
return at the beginning of period t+1 (ert), measured with error (t). In concept, tre does
not need to be an ICC estimate as defined in the accounting literature – it can be any ex ante expected returns proxy, including a firm’s Beta, its book-to-market ratio, or its market-capitalization at the beginning of the period.
Finally, we assume that expected returns (ert) and the ICC measurement error (t) 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) (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. 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 ’s they will generate, over time and across firms. In this setup, statements we make about the desirability of one ICC estimate over another 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
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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 – depending on the covariance of and er. Nevertheless, as we show in the next subsection, 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.3 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 the 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 closely track its true expected returns in time-series. Therefore, we can more reliably extract the true expected return for a given firm. 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 (see Proposition 1 below).
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However, these measurements errors need not be stable for a given firm over time. To assess this second distributional property of the error terms, we need to be able to
empirically identify, and compare, the 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:
2 (4)
Where:
, is the time-series variance of the ICC estimates for a given firm.
, , is the time-series covariance between the ICC estimate and next period’s realized returns
, , is the time-series autocovariance in realized returns k period lags.
Notice that all the terms on the right-hand-side of Equation (4) are observable, and can be empirically estimated for each firm. The first term on the right-hand-side shows that the variance of the error terms for a given ICC estimate 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
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( ). 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. Given these results, we are now in a position to make two propositions: Proposition 1: Better ICC estimates (defined as ICC estimates that track true expected returns more closely) will have greater power to predict realized returns. The intuition is straightforward. When the error terms are small, firms’ ICC estimates will mimic closely true expected returns, thus predicting realized returns. When the error terms are not necessarily small, we nevertheless would like these errors to preserve the ranking of firms by their true expected return. In other words, we want the error terms to minimally interfere with the cross-sectional relation between expected returns and realized returns. The reasoning is perhaps more transparent when applied to portfolios. Because idiosyncratic noise can be diversified away, a portfolio’s average realized returns should reflect the average expected return of the individual stocks. When ICC errors are small, or when they preserve cross-sectional rank, the average ICC estimate of the portfolio will also predict its average realized returns. More formally:
∑ , = ∑ , - ∑ , + ∑ ,
= ∑ , + ∑ , - ∑ , + ∑ ,
Recognizing er is the true expected return (which incorporates all information relevant to returns prediction), it follows that the last two terms are zero in expectation. Therefore,
when the ’s are also small (or when they do not interfere with cross-sectional rankings), the average ICC estimate will forecast average realized returns. Notice this proposition derives directly from (A1) and (A2), as well as the assumption that other firm-specific news (unt) is uncorrelated with ICCs.
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Proposition 2: Better ICC estimates will exhibit lower error variances. The concept of lower error variance is also quite intuitive. Lower cross-sectional variance in the error terms should lead directly to greater predictive power for future realized returns.7 Lower time-series variance is also attractive, as discussed earlier, because it allows us to extract more reliable estimates of the true expected return for each firm in time-series. The notion of lower time-series error variance is closely related to, but is not the same as, the idea of superior tracking ability in ICC estimates. Financial practitioners are often interested in predicting future market multiples (or, in our parlance, future ICC estimates). Therefore, we might be tempted to simply compare ICC estimates on the basis of their ability to predict themselves – i.e. based on how well each predicts future manifestations of itself over time. For example, Gebhardt et al. (2001) reports the ability of their ICC estimate to predict ICC estimates several years into the future (i.e. in terms of its “tracking ability”). 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. In most research applications, our main interest is in how well each ICC estimate tracks the underlying expected returns both in the cross-section and in time-series This property is better reflected in the “error variance” measure – i.e. ICC estimates that track true expected returns better should exhibit lower variance in their error terms. In sum, we have provided a rationale for an 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.
7 Note that lower cross-sectional variance in the error terms is useful, but not necessary, for returns prediction. This is because even if cross-sectional variance in the error terms is large, we can still achieve some measure of returns predictability so long as ICC errors do not fully obscure the cross-sectional relation between expected returns and realized returns (i.e., so long as the errors, to some extent, preserve rank).
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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. Our final sample consists of 80,902 firm-years and 11,981 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. 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. In fact, during their sample period (1967 to 2006), the adjusted R2s of the models explaining one-, two-, and three-year-ahead earnings are 87%, 81%, and 77% respectively. Hou et al. (2009) find 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. (2009) 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: , , , , , ,
, , , (5)
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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 (5) in our sample are provided in Appendix II. Our average coefficients are qualitatively similar to those reported in Hou et al (2009).8 We calculate model-based earnings forecasts by applying historically estimated coefficients from equation (5) to the most recent set of publicly available 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.
8 We use fundamental data from Compustat Express while Hou et al. (2009) use data from the historical Compustat research database (discounted after 2006). Some of the differences, particularly in the early years, are likely due to differences in firm membership across the two databases.
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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,
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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 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.
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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:
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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:
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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
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procedures in Gode and Mohanram (2003), who use the average of forecasted near-term
growth EPSt3 EPSt2EPSt2
and five-year growth EPSt5 EPSt4
EPSt4
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
12
1
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 measures are included in the sample. The number of firms varies by year and ranges from a low of 1,241 in 1971 to a high of 3,262 in 1997. The average number of firms per year is 2,187, indicating that the seven 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 9.02% (for EPR), to 14.36% (for GGM). Comparing these estimates to the average Treasury yield suggests that the median equity risk premium is between 2% and 7%. 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% risk premium from the GGM 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.
17
Table II reports the median implied risk premia for the 48 industries classified by Fama and French (1997). Recall that implied risk premia are calculated as the implied cost of capital minus the Treasury yield on a 10-year bond as of June 30. To construct this table, we calculate the median ICC estimate for each industry-year, and average the annual cross-sectional medians over time. For each valuation model, industries are ranked from 1 to 48, with the highest ranking corresponding to the highest risk premium. Industries are presented in order from highest to lowest in terms of their Mean Rank, defined as average rank across the seven valuation models. StdDev Rank is the standard deviation of these rankings across the seven valuation models. Obs. is the number of firm-years in each industry. Table II shows that the 3 industries with the highest mean implied risk premia are FabPr (fabricated products and machinery), Toys (recreational products), and Clths (apparel). The three industries with the lowest risk premia are Chems (chemicals), Beer, and Drugs. We supplement the Mean Ranks with the time-series standard deviation of the ranks for a given industry, Std Rank. The standard deviation of the rank for FabPr is 2.2, indicating that this industry receives a consistently high risk premium relative to other industries across all seven models. At the bottom of the table, the drug industry has a mean rank of 2.9, with minimum variation across the models, indicating this industry has consistently low risk premia. Overall, the evidence suggests that certain industries have consistently higher (or lower) implied risk premia across all seven models, offering some hope that industry-based ICC estimates might be of some use. We explore this possibility in more detail later. Table III reports the average annual correlations between the seven ICC measures. Correlations are calculated by year and then averaged over the sample period. Pearson correlations are shown above the diagonal and Spearman correlations are shown below the diagonal. All reported correlations are significant at the 1% level. PEG and MPEG have the highest Spearman correlation at 96.3, which is not surprising given the similarity in their construction. In the same spirit, GGM and GLS exhibit a Spearman correlation of 87.0, while AGR and EPR are correlated at 82.4. Most of the other Spearman correlations are between 45 and 65, with none under 40. 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,
18
superior estimates of ICC should possess stronger predictive power for future returns. Table IV reports the correlation between annual ICC estimates derived from seven valuation models 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.9 The dependent variables are 12, 24, 36, 48, and 60 month buy-and-hold returns. Following Gow et al. (2009), 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, and the R-Squares are shown in italics. Significance levels are indicated by *, **, and *** for 10%, 5%, and 1% respectively. Panel A results show that among the seven 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 1% increase in firm-specific cross-sectional ICC is associated with a 0.19% (for AGR) to 0.45% (for EPR) increase in future realized returns over the next 12-months. Future annual returns are steady, or 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 0.515% (for EPR). 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
9 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.
19
with lower (relatively more negative) ICC estimates.10 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 an average annualized realized hedge return that ranges from 3.8% (for AGR over the first 12 months) to close to 10% (for EPR, averaged over the next 60 months). Finally, Panel C provides evidence on the statistical significance of the difference in predictive power across the seven models. To construct this panel, we compute the annual pair-wise difference in three-year cumulative returns for each pair of ICC 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 ICC estimate displayed in the top row has stronger (weaker) predictive power for realized returns than the ICC estimate displayed down the left-hand-side of the table. 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 panel indicates that EPR and GGM are the best predictors of future returns. For example, looking down the columns for EPR and GGM, we see that these two measures dominate all the others, but are themselves not statistically distinguishable from each other. GLS and AGR generally perform better than PEG, MPEG, and OJM, but the differences are not statistically significant. In sum, the Panel B results using a hedge portfolio approach are quite consistent with the Panel A findings using a pooled regression with two-way cluster corrected statistics. Contrary to Easton and Monahan (2005), but consistent with
10 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.
20
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.11 In particular, we find that EPR and GGM have the strongest predictive power for future returns, followed by GLS and AGR.12 Our continuous weighted strategies show that average hedge returns range from 4% to 10% per year across the four successful models, which seem both statistically and economically significant. IV.3 Comparison of Measurement Error Variance A second implication of the theory presented in Section II is that better ICC estimates should generate measurement errors that exhibit lower variance. We calculate the variance of the error terms under two different sets of assumptions. First, we assume that cash flow news (or more broadly,all unforecastable news) is uncorrelated with ex ante expected returns of a given firm in time-series (i.e., unt is uncorrelated with ert for each given firm). We refer to these as tests of the time-series measures of error variance. Second, we impose a set of stronger assumptions. Specifically, we assume that news not related to expected returns 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). In addition, we assume that 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). We refer to these as test of the cross-sectional measures of error 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 its expected returns, and less realistic to assume that cash flow news across firms is unrelated to firms’ expected returns in the cross-section. Moreover, requiring the AR(1) parameters to be cross-sectional constants seems to us to be rather 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.
11 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. 12 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 ours.
21
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 887 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 887 firms (n=887). Panel B reports t-statistics corresponding to the pair-wise comparison of firm-specific measurement errors across the sample of 887 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. 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). 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-
22
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 I 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 887 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. OJM is an outlier in terms of measurement error variance at 0.0124 (for comparison, the second largest is 0.0019, the corresponding value for PEG) and is omitted from Figure I. This graph shows that both GGM and EPR are on the efficiency frontier. The best performing model in terms of returns prediction is EPR, while the best performing model in terms of minimal error variance is GGM. In general, models that perform well along one dimension also tend to do better along the other. GLS and AGR, for example, do appreciably better than PEG, MPEG, and OJM along both dimensions, both neither GLS nor AGR are on the frontier established by EPR and GGM. 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
23
Table IV, the t-statistics are calculated using two-way cluster robust standard errors (Gow et al. (2009)). 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 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).
24
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. V. Summary 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 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 show that under fairly general assumptions, it is possible to derive an expression for the variance of these error terms, even when the errors themselves are not observable. Our analyses nominate: (a) prediction of future realized returns and (b) stability of measurement errors, as key indicators of ICC quality. In our empirical work, we compare seven alternative ICC estimates and show that several perform well along both dimensions. We recognize that these seven alternatives do not remotely exhaust the list of possible candidates for testing. In particular, because all of our estimates use a model-based earnings forecast method rather than analyst estimates, we acknowledge that it is quite possible we have not included ICC estimates that will perform better than those we test here. 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. In particular, our results 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 be a useful instrumental variable in the derivation of ICC estimates
25
that does not rely on a firm’s current stock price. We believe this is a promising venue for future research.
26
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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.
11 ttt uerer (T1)
ttt erre ~1 (T2)
11 ttt v (T3)
We extend Campbell’s (1991) decomposition of realized returns
tj
jtj
ttj
jtj
ttttt dEErEErEr
011
01111
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
11t
t
urn (T4)
so that we can re-write the decomposition to be
tt
tt unu
err
11 (T4’)
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
tttttt
t
erCovVarerCoverVar
reVarc
,,
~
00
0
(T5)
tttttt
tt
erCovVarerCoverVar
rereCovc
,,
~,~
11
11
(T6)
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:
21
2 1
t
tttt un
uuerr
(T7)
32
12
3 1
t
ttttt un
uuuerr
(T8)
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:
ttt
ttr
erCoverVar
rerCovc
,
~,0
11
(T9)
ttt
ttr
erCoverVar
rerCovc
,
~,1
22
(T10)
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
t
ttrr
erVar
rrCovc
, 121
(T11)
31
t
ttrr
erVar
rrCovc
,2
132
(T12)
Step 4: Identifying Firm-Specific Time-Series Variance in ICC Measurement Errors Under the above,
tttr erCovVarcc ,10 (T13)
and
r
rrrr
tt c
cccerCov
2
111, (T14)
so that
r
rrrr
t c
ccccVar
2
1110 2 (T15)
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:
titi
tititititititit rerCov
rerCovrrCovrerCovreVarVar
,2,
,1,2,1,,1,, ~,
~,,~,2~
(T16)
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 80,902 firm-years and 11,981 unique firms spanning 1970-2007.
Filter Criterion # of Firm-
Years Lost Firm-
Years # of Unique
Firms Lost Unique
Firms
1
Intersection of CRSP and Compustat observations with data on book values, earnings, statement forecasts, and industry identification for fiscal years greater than or equal to 1970
141,615 14,901
60,713 2,920
2
Non-missing ICC estimates and ICC estimates between 0 and 100% for all 7 models
80,902 11,981
Final Sample 80,902 11,981 Total Loss 60,713
2,920
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.
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.3. 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%. RF Yield equals the yield on the 10-year Treasury bond on June 30th of each year.
This table reports the median implied risk premia for the 48 industries classified by Fama and French (1997). Implied risk premia are calculated as the implied cost of capital minus the Treasury yield on a 10-year bond as of June 30. To construct this table, we calculate the median ICC estimate for each industry-year and then average the annual cross-sectional medians over time. For each valuation model, industries are ranked from 1 to 48, with higher ranks corresponding to higher average risk premia. Industries are then presented in order from highest to lowest in terms of their Mean Rank, defined as average rank of the mean risk premia across the seven valuation models. StdDev Rank is the standard deviation of these rankings across the seven valuation models. Obs. is the number of firm-years in each industry.
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. 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 II.
Note: All correlation coefficients are statistically significant at the 1% level
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TABLE IV ICC Estimates and Future Realized Returns
This table examines the ability of alternative ICC estimates to forecast future returns. Panel A reports the pooled cross-sectional results obtained from regressing firm-specific future realized returns on firm-specific implied risk premiums derived from seven valuation models – GLS, PEG, MPEG, OJM, EPR, AGR, and GGM (a full description of each model is included in Section II). 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), and R-Squared values are shown in italics below the t-statistics. 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 ICC. Specifically, an asset’s weight in month t is:
wi,t
1
N(ICCi,t ICC
______
mkt,t )
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.
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 ICC 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 in our sample. Table values are positive (negative) when the ICC estimate displayed in the top row has stronger (weaker) predictive power for realized returns than the ICC estimate displayed down the left-hand-side of the table. Standard errors are computed using Newey-West HAC estimators with 2 year lags. For all three panels, significance levels are indicated by *, **, and *** for 10%, 5%, and 1% respectively.
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. Measurement error variances are calculated as follows:
titi
tititititititit rerCov
rerCovrrCovrerCovreVarVar
,2,
,1,2,1,,1,, ~,
~,,~,2~
where Var(t ) is the measurement error variance, tire ,~ is the ICC in year t, and rti is the realized
return in year t i . Panel A reports summary statistics for the error variance from each model, using a sample of 887 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 887 firms. Panel B reports t-statistics corresponding to the pair-wise comparisons of firm-specific measurement error variances within the sample of 887 firms used in Panel A. *, **, and *** indicate significance at the 10%, 5%, and 1% level, respectively.
Panel A: Time-Series Variance of Measurement Error (n=887)
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:
titi
tititititititit rerCov
rerCovrrCovrerCovreVarVar
,2,
,1,2,1,,1,, ~,
~,,~,2~
where Var(i,t) is the measurement error variance computed using a cross-section of firms (indexed by
i) in year t, ~
,tier 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 significance at the 10%, 5%, and 1% level, respectively.
Panel A: Cross-Sectional Variance of Measurement Error
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
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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:
wi,t
1
N(IMICCi,,t IMICC
__________
mkt ,t )
where IMICCi,,t is firm i’s industry-median implied cost of capital in year t , IMICC__________
mkt,t is the
equal-weighted average IMICCi,,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:
titi
tititititititit IMICCrCov
IMICCrCovrrCovIMICCrCovIMICCVarVar
,2,
,1,2,1,,1,, ,
,,,2
where Var( t ) is the measurement error variance and rti 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
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[Table VII: Continued] Panel B: Continuous Industry ICC-Weighted Mean Returns by Valuation Model
Panel C: Industry ICC Time-Series Variance of the Measurement Errors (n=887)
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
44
Figure I Efficient Frontier of Firm-Specific Cost of Capital Estimates
The figure below plots, for each of six ICC models, the average annualized three-year-ahead profit of a continuous weighted ICC portfolio along the Y-axis. The strategy holds assets in proportion to their market-adjusted ICC. Specifically, an asset’s weight in month t is:
wi,t 1
N(ICCi,t ICC
______
mkt,t )
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. The x-axis reflects the median of firm-specific measurement error variances (multiplied by 100), calculated as follows:
Var( t ) Var(ert )~
2*Cov(rt1,ert
~
) Cov(rr1,rt2)Cov(rt1,ert
~
)
Cov(rt2,ert
~
)
where Var(t ) is the measurement error variance, ert
~
is the ICC in year t, and rti is the realized return
in year t i . Firm-specific measurement error variances are calculated based on equation (4) using a sample of 887 unique firms that meet our data requirements.
