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Using earnings forecasts to simultaneously estimate firm-specific cost of equity and long-term growth

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Abstract

A growing body of literature in accounting and finance relies on implied cost of equity (COE) measures. Such measures are sensitive to assumptions about terminal earnings growth rates. In this paper we develop a new COE measure that is more accurate than existing measures because it incorporates endogenously estimated long-term growth in earnings. Our method extends Easton et al. (J Account Res, 40, 657–676, 2002) method of simultaneously estimating sample average COE and growth. Our method delivers COE (growth) estimates that are significantly positively associated with future realized stock returns (future realized earnings growth). Moreover, the predictive ability of our COE measure subsumes that of other commonly used COE measures and is incremental to commonly used risk characteristics. Our implied growth measure fills the void in the earnings forecasting literature by robustly predicting earnings growth beyond the five-year horizon.

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Notes

  1. This growth rate is often referred to as the terminal growth rate or the growth rate in perpetuity. Throughout the paper we use the terms long-term growth, terminal growth, and growth in perpetuity interchangeably.

  2. Valuation textbooks emphasize that firm valuation can be highly sensitive to the assumed terminal growth rate of earnings (Penman 2009; Whalen et al. 2010). For example, Damodaran (2002) states that “of all the inputs into a discounted cash flow valuation model, none can affect the value more than the stable growth rate.”

  3. Another commonly used COE measure developed by Gebhardt et al. (2001) assumes a convergence in profitability to an industry benchmark over 12 years with a zero terminal growth thereafter. But as Easton (2006) points out, this approach creates systematic biases to the extent that firms with certain characteristics have other expected growth patterns.

  4. Developing a more accurate and less biased implied COE measure is important given the increasing use of implied COE measures in accounting and finance literature. Implied COE measures have been used to shed light on the equity premium puzzle (Claus and Thomas 2001; Easton et al. 2002), the market’s perception of equity risk (Gebhardt et al. 2001), risk associated with accounting restatements (Hribar and Jenkins 2004), dividend taxes (Dhaliwal et al. 2005), accounting quality (Francis et al. 2004), legal institutions and regulatory regimes (Hail and Leuz 2006), and quality of internal controls (Ogneva et al. 2007), as well as to test intertemporal CAPM (Pastor et al. 2008), international asset pricing models (Lee et al. 2009), and the pricing of default risk (Chava and Purnanadam 2010).

  5. Specifically, we use the CAPM beta, size, book-to-market, and momentum as the observable risk characteristics, and we use analysts’ long-term growth forecast, the difference between the industry ROE and the firm’s forecasted ROE, and the ratio of R&D expenses to sales as the observable growth characteristics. We take the part of COE (growth) that is not explained by these observable risk (growth) characteristics to be due to unobservable risk (growth) factors. Examples of such risk factors may include the risk of increased competition and extreme weather, credit risk, and litigation risk as perceived by market participants but not fully captured by the set of observable risk characteristics that we consider. We acknowledge that the set of risk and growth characteristics that we use in the estimation may be incomplete, however, the flexibility of our method allows incorporating any number of additional factors pertinent to a specific study.

  6. Specifically, we use a cross-sectional prediction model that first regresses past realized returns (growth) on past risk (growth) characteristics and then applies the resulting coefficients to current return (growth) characteristics to arrive at a return (growth) forecast.

  7. We are not aware of any papers that construct and validate forecasts of terminal growth or even growth beyond 5-year horizon. However, several papers forecast earnings over horizons beyond 2 years. For example, Chan et al. (2003) and Gao and Wu (2010) forecast earnings growth over the next 5 years, while Hou et al. (2010) forecast three-year-ahead earnings. Estimates from these models may serve as an alternative to short-term analysts’ forecasts.

  8. The residual income model is equivalent to the discounted dividend model assuming the clean surplus relation, i.e. the book value of equity at the end of year t + 1 is equal to the book value of equity at the end of year t plus net income for year t + 1 minus dividends for year t + 1.

  9. Empirically, we use the CAPM beta, size, book-to-market ratio, and momentum as observable risk drivers, and we use the analyst long-term growth forecast, R&D expenditures, and the deviation of firm’s forecasted ROE from the industry target ROE as observable growth drivers.

  10. The component due to unobservable risk (growth) factors is defined as the part of COE (growth) that is not explained by the observable risk (growth) drivers. For example, unobservable risk factors may include the risk of increased competition, liquidity risk, credit risk, litigation risk, and political risk as perceived by market participants but not fully captured by the above observable risk drivers.

  11. Regression (5a) assumes that independent variables are exogenous, i.e. E[ε i|MB i, MB i x i R , (1 − MB i )x iG ] = 0. A sufficient but not necessary condition for the exogeneity is the assumption that ε i R and ε i G are independent of MB i, x i R , and x i G .

  12. Note that the WLS regression restricts neither the magnitudes nor the signs of the risk premia and growth weights, λR and λG, which are determined endogenously based on earnings forecasts and stock prices.

  13. These weights assume equal weighting of the COE and growth components due to unobservable factors in (4), that is w i1  = w i2  = 1. As a robustness check, we vary the ratio of the weights (w i1 /w i2 ) from 0.5 to 2. Our inferences are robust to these variations.

  14. Leverage is another characteristic associated with equity risk. We do not include leverage in the estimation because Fama and French (1992) show that the power of leverage to predict future stock returns is subsumed by the CAPM beta, size, and book-to-market ratio.

  15. Our search of growth drivers reveals that the literature on forecasting growth in earnings over long horizons is very sparse. To our knowledge, there are no empirical papers that would forecast growth in residual earnings. There are also no papers documenting growth in accounting earnings over horizons exceeding 10 years into the future. Chan et al. (2003) explore growth over the ten-year horizon. However, their cross-sectional prediction model forecasts earnings growth only 5 years into the future. In our sensitivity tests, we have also included other growth predictors suggested in Chan et al. (2003), including past sales growth, earnings-to-price ratio, and alternative conservatism proxies used in Penman and Zhang (2002). Our results are not sensitive to including them in the estimation, and we opt for a parsimonious set of variables to avoid additional sample restrictions.

  16. Note that numerical estimation of implied COE is typical for the models that assume different short-term and long-term growth rates in earnings (for example, Gebhardt et al. 2001; Claus and Thomas 2001). The method proposed here is not more computationally complex than the extant COE estimation methods.

  17. We substitute missing Ltg with E 2 /E 1 − 1. Values of Ltg greater than 50% are winsorized.

  18. We would like to thank Partha Mohanram for sharing his forecast error adjustment codes.

  19. Hughes et al. (2008) suggest that the trading strategy based on exploiting predictable analyst forecast errors does not produce statistically significant returns, which is consistent with the market not being subject to the same biases as analysts. However, it is possible that in some instances stock prices may incorporate earnings expectations biased in the same direction as analyst earnings forecasts. If this is the case, adjusting earnings forecasts for such predictable errors leads to implied COE estimates that do not represent the market’s expectations of future returns but instead are equal to the market’s expectation of future returns plus the predictable return due to subsequent correction of the mispricing. The adjusted COE measure then represents the total COE that the firm faces due to both risk and mispricing. In our empirical analyses, we do not distinguish between the two interpretations of implied COE.

  20. To avoid the influence of extreme observations, we winsorize all variables except future realized returns at the 1st and 99th percentiles.

  21. Regression (6) is estimated using WLS. As a robustness check, we have replicated estimation using an OLS regression. The results are similar—implied COE measures predict future realized returns with coefficients significantly different from zero—but the predictive ability is weaker (the coefficient on unadjusted COE measure is significantly different from one). This deterioration in COE predictive ability underscores the importance of utilizing theoretically correct weights for the regression residuals.

  22. The insignificant relation between the CAPM beta and stock returns is a key motivation for alternative asset-pricing models (Merton 1973; Jagannathan and Wang 1996; Lettau and Ludvigson 2001).

  23. When analyst forecasts are sluggish, they do not incorporate the recent positive (negative) earnings news and are therefore biased downward (upward) following recent positive (negative) stock returns. The bias in forecasts mechanically leads to downwardly (upwardly) biased implied COE estimates following positive (negative) stock returns.

  24. Some risk (growth) drivers are not loading significantly in either Unadjusted or Adjusted Forecast regressions. These drivers include CAPM beta, analysts’ long-term growth forecast, and size. When we perform estimation excluding these drivers, all our results are predictably very similar.

  25. This result is consistent with Gode and Mohanram (2009) and Larocque (2010) who show that COE based on the PEG model improves its return predictability when analysts’ forecasts are adjusted for predictable errors.

  26. We further explore the role of observable risk characteristics in the sub-section on statistical prediction of returns and growth rates.

  27. A more direct validation requires estimating realized growth in residual earnings. We choose not to use growth in residual earnings in our main tests for two reasons. First, if our implied growth and COE estimates are correlated, using our COE estimate to calculate realized residual earnings may cause the latter to be spuriously correlated with our implied growth estimate. Second, when we use risk-free rates to calculate realized residual earnings, over 50% of cumulative residual earnings before extraordinary items (EBEI) over the first 4 years are negative and thus cannot be used as a base to estimate growth. Percentage of negative observations is lower when operating income before depreciation (OI) is used to estimate residual earnings. Accordingly, we replicate analyses presented in this subsection using growth in residual OI and obtain a qualitatively similar set of results (untabulated).

  28. By using a risk-free rate we avoid spurious correlations with implied growth rates that could arise had we used previously estimated implied COE estimates. The results are robust to using a uniform 10% rate as in Penman (1996), or a 0% rate that assumes no dividend reinvestment.

  29. We do not use annualized growth rates in the analysis because we cannot annualize four-year growth rates that are less than negative 100%.

  30. In this subsection, we focus on COE measures adjusted for predictable forecast errors.

  31. To derive growth in earnings using growth in residual earnings, we use the formula from the appendix in ETSS. Since we assume a constant future dividend payout while ETSS assume constant future dividends, we adjust the formula to make it consistent with our assumption.

  32. The estimate of the average historical rate is based on the data for aggregate nominal earnings of the S&P 500 firms from 1871 to 2009 provided by Robert Shiller at http://www.econ.yale.edu/~shiller/data/ie_data.xls.

  33. Risk premia are often measured relative to the rate on one-month Treasury bills. Based on this measure of the risk free rate, the average implied risk premium from the ETSS model is 5.82% (3.89%) compared to 3.89% (1.17%) from our model before (after) correction for analyst forecast errors.

  34. Hughes et al. (2009) provide a ballpark estimate of the difference between expected returns and implied cost of capital of 2.3%. They note that the actual difference can be larger.

  35. The Easton and Monahan (2005) test has proven to be a high bar for estimating construct validity of COE measures. Most conventional implied COE measures are negatively correlated with realized stock returns after controlling for cash flow and discount rate news and have significant measurement errors.

  36. The sample attrition for growth in EBEI is higher than for OI due to more frequent negative growth base (growth in EBEI cannot be calculated when four-year cum-dividend earnings for [t + 1, t + 4] are negative).

  37. Note, that the percentages of delisted firms do not add up to the total percentage of “nonsurvivors” from Panel A of Fig. 3. The difference is due to the cases where earnings are available, but growth cannot be computed due to negative four-year cum-dividend earnings for [t + 1, t + 4].

  38. The two other commonly used approaches to estimating COE (multiplying historical estimates of factor risk premia on historical factor loadings and using ex post realized returns) have their own merits and demerits. The first approach is problematic given the ongoing debate about the appropriate asset pricing model and substantial measurement errors in the estimates of factor risk premia and risk loadings (Fama and French 1997). The second approach requires a very large sample spanning dozens of years (which is often not available to the researcher), since more risky stocks can underperform less risky stocks for multiple consecutive years (Elton 1999). Also, ex post returns approach does not allow estimating the (ex ante) COE in real time necessary for capital budgeting and other decisions.

  39. A quadratic function \( w_{1}^{i} (\varepsilon_{R}^{i} )^{2} + w_{2}^{i} (\varepsilon_{G}^{i} )^{2} + w_{3}^{i} \varepsilon_{R}^{i} \varepsilon_{G}^{i} \) is positive (semi-)definite if it is positive (nonnegative) for any nonzero argument, \( \varepsilon_{R}^{i} \varepsilon_{G}^{i} \ne 0 \), which holds if and only if \( w_{1}^{i} > 0 \, ( \ge 0) \) and \( 4w_{1}^{i} w_{2}^{i} - (w_{3}^{i} )^{2} > 0 \, ( \ge 0) \).

  40. The proof is available from the authors upon request.

  41. To avoid using very high terminal growth in years with high risk-free rate we winsorize g CT at the 3% level. When we do not winsorize g CT , r CT performs worse and none of the inferences regarding our COE measure change.

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Acknowledgments

We thank James Ohlson (editor), an anonymous reviewer, Pervin Shroff, K. R. Subramanyam, and especially Steven Monahan (discussant), for their insightful comments and suggestions. The paper also benefitted from the comments received from the participants of research seminars at UC Irvine, Santa Clara University, Southern Methodist University, and the 2010 Review of Accounting Studies conference. Maria Ogneva acknowledges financial support from the Michelle R. Clayman Faculty Scholar endowment. We are grateful to Suhas Sridharan for her excellent research assistance.

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Authors and Affiliations

  1. University of California, Irvine, Irvine, CA, 92697, USA

    Alexander Nekrasov

  2. Stanford University, Stanford, CA, 94305, USA

    Maria Ogneva

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  1. Alexander Nekrasov
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Correspondence to Alexander Nekrasov.

Appendices

Appendix 1

1.1 Simultaneous estimation of COE and long-term growth

In this appendix, we derive expressions for implied COE and growth. Combining Eq. 3b with assumption (4) from Sect. 2 yields the following system of equations:

$$ \left\{ {\begin{array}{*{20}c} {\mathop {Min}\limits_{{\varepsilon_{R}^{i} ,\varepsilon_{G}^{i} ,\varepsilon^{i} ,\gamma_{0} ,\gamma_{1} ,\lambda_{R} ,\lambda_{G} }} \sum\limits_{i} {w_{1}^{i} (\varepsilon_{R}^{i} )^{2} + w_{2}^{i} (\varepsilon_{G}^{i} )^{2} } } \hfill \\ {s.t. \, X_{cT}^{i} /B_{0}^{i} = \gamma_{0} + \gamma_{1} MB^{i} + \varepsilon^{i} } \hfill \\ {\varepsilon^{i} = (G^{i} - \bar{G})(1 - MB^{i} ) + (R^{i} - \bar{R})MB^{i} } \hfill \\ {\gamma_{0} = \bar{G} - 1} \hfill \\ {\gamma_{1} = \bar{R} - \bar{G}} \hfill \\ {R^{i} = \bar{R} + \lambda_{R} x_{R}^{i} + \varepsilon_{R}^{i} } \hfill \\ {G^{i} = \bar{G} + \lambda_{G} x_{G}^{i} + \varepsilon_{G}^{i} } \hfill \\ \end{array} } \right. $$
(8)

Next, we simplify the problem in (8) so that it can be solved using standard regression analysis. Substituting the expressions for \( \varepsilon^{i} \), R i, and G i into the second equation in (8) and defining \( \nu^{i} = \varepsilon_{G}^{i} + (\varepsilon_{R}^{i} - \varepsilon_{G}^{i} )MB^{i} \), we express the above system of equations as follows:

$$ \left\{ {\begin{array}{*{20}c} {\mathop {Min}\limits_{{\varepsilon_{R}^{i} ,\varepsilon_{G}^{i} ,\nu^{i} ,\gamma_{0} ,\gamma_{1} ,\lambda_{R} ,\lambda_{G} }} \sum\limits_{i} {w_{1}^{i} (\varepsilon_{R}^{i} )^{2} + w_{2}^{i} (\varepsilon_{G}^{i} )^{2} } } \hfill \\ {{\text{s}} . {\text{t}} . { }X_{cT}^{i} /B_{0}^{i} = \gamma_{0} + \gamma_{1} MB^{i} + \lambda_{R} MB^{i} x_{R}^{i} + \lambda_{G} (1 - MB^{i} )x_{G}^{i} + \nu^{i} } \hfill \\ {\nu^{i} = \varepsilon_{G}^{i} + (\varepsilon_{R}^{i} - \varepsilon_{G}^{i} )MB^{i} } \hfill \\ \end{array} } \right. $$
(9)

Substituting \( \varepsilon_{G}^{i} = (\varepsilon_{R}^{i} MB^{i} - \nu^{i} )/(MB^{i} - 1) \) from the last equation, we obtain

$$ \left\{ {\begin{array}{*{20}c} {\mathop {Min}\limits_{{\varepsilon_{R}^{i} ,\nu^{i} ,\gamma_{0} ,\gamma_{1} ,\lambda_{R} ,\lambda_{G} }} \sum\limits_{i} {w_{1}^{i} (\varepsilon_{R}^{i} )^{2} + w_{2}^{i} ((\varepsilon_{R}^{i} MB^{i} - \nu^{i} )/(MB^{i} - 1))^{2} } } \hfill \\ {{\text{s}} . {\text{t}} . { }X_{cT}^{i} /B_{0}^{i} = \gamma_{0} + \gamma_{1} MB^{i} + \lambda_{R} MB^{i} x_{R}^{i} + \lambda_{G} (1 - MB^{i} )x_{G}^{i} + \nu^{i} } \hfill \\ \end{array} } \right. $$
(10)

Finally, substituting the expression for \( \varepsilon_{R}^{i} \) that satisfies the first order conditions, \( \varepsilon_{R}^{i} = w_{2}^{i} MB^{i} \nu^{i} /(w_{1}^{i} (MB^{i} - 1)^{2} + w_{2}^{i} (MB^{i} )^{2} ) \), we obtain the following weighted least squares regression:

$$ \left\{ {\begin{array}{*{20}c} {\mathop {Min}\limits_{{\nu^{i} ,\gamma_{0} ,\gamma_{1} ,\lambda_{R} ,\lambda_{G} }} \sum\limits_{i} {\frac{{w_{1}^{i} w_{2}^{i} (\nu^{i} )^{2} }}{{w_{1}^{i} (1 - MB^{i} )^{2} + w_{2}^{i} (MB^{i} )^{2} }}} } \hfill \\ {{\text{s}} . {\text{t}} . { }X_{cT}^{i} /B_{0}^{i} = \gamma_{0} + \gamma_{1} MB^{i} + \lambda_{R} MB^{i} x_{R}^{i} + \lambda_{G} (1 - MB^{i} )x_{G}^{i} + \nu^{i} } \hfill \\ \end{array} } \right. $$
(11)

Combining Eq. 11 with the above expressions for \( \bar{R} \), \( \bar{G} \), \( \varepsilon_{R}^{i} \), \( \varepsilon_{G}^{i} \), \( R^{i} \), and \( G^{i} \), we have the following WLS regression and equations that uniquely determine firm’s COE and expected growth rate:

$$ \left\{ {\begin{array}{*{20}c} {\mathop {Min}\limits_{{\nu^{i} ,\gamma_{0} ,\gamma_{1} ,\lambda_{R} ,\lambda_{G} }} \sum\limits_{i} {\frac{{w_{1}^{i} w_{2}^{i} (\nu^{i} )^{2} }}{{w_{1}^{i} (1 - MB^{i} )^{2} + w_{2}^{i} (MB^{i} )^{2} }}} } \hfill \\ {s.t. \, X_{cT}^{i} /B_{0}^{i} = \gamma_{0} + \gamma_{1} MB^{i} + \lambda_{R} MB^{i} x_{R}^{i} + \lambda_{G} (1 - MB^{i} )x_{G}^{i} + \nu^{i} } \hfill \\ {\bar{G} = \gamma_{0} + 1} \hfill \\ {\bar{R} = \gamma_{1} + \gamma_{0} + 1} \hfill \\ {\varepsilon_{R}^{i} = \nu^{i} \frac{{w_{2}^{i} MB^{i} }}{{w_{1}^{i} (MB^{i} - 1)^{2} + w_{2}^{i} (MB^{i} )^{2} }}} \hfill \\ {\varepsilon_{G}^{i} = \nu^{i} \frac{{w_{1}^{i} (1 - MB^{i} )}}{{w_{1}^{i} (MB^{i} - 1)^{2} + w_{2}^{i} (MB^{i} )^{2} }}} \hfill \\ {R^{i} = \bar{R} + \lambda_{R} x_{R}^{i} + \varepsilon_{R}^{i} } \hfill \\ {G^{i} = \bar{G} + \lambda_{G} x_{G}^{i} + \varepsilon_{G}^{i} } \hfill \\ \end{array} } \right. $$
(12)

The first equation specifies the weights \( w^{i} = w_{1}^{i} w_{2}^{i} /(w_{1}^{i} (1 - MB_{{}}^{i} )^{2} + w_{2}^{i} (MB_{{}}^{i} )^{2} ) \) that should be used in the WLS regression \( X_{cT}^{i} /B_{0}^{i} = \gamma_{0} + \gamma_{1} MB^{i} + \lambda_{R} MB^{i} x_{R}^{i} + \lambda_{G} (1 - MB^{i} )x_{G}^{i} + \nu^{i} \). Having found the intercept, slopes, and residuals from the regression, the third and the fourth equations can be used to obtain the sample mean R and G, the fifth and the sixth equations can be used to calculate the components of \( R^{i} \) and \( G^{i} \)due to unobservable risk and growth factors, and finally the last two equations can be used to calculate the firm’s COE and growth rate.

1.1.1 Comparison between our model and ETSS

Recall that our minimization problem outlined in Sect. 2 is specified as:

$$ \left\{ {\begin{array}{*{20}c} \begin{gathered} \mathop {\rm Min}\limits_{{\bar{R},\bar{G},\lambda_{R} ,\lambda_{G} ,\varepsilon_{R}^{i} ,\varepsilon_{G}^{i} }} \sum\limits_{i} {w_{1}^{i} (\varepsilon_{R}^{i} )^{2} + w_{2}^{i} (\varepsilon_{G}^{i} )^{2} } \hfill \\ \end{gathered} \hfill \\ {R^{i} = \bar{R} + {\varvec{\uplambda}}_{{\mathbf{R}}} \,{}^{\prime }{\mathbf{x}}_{{\mathbf{R}}}^{{\mathbf{i}}} + \varepsilon_{R}^{i} } \hfill \\ {G^{i} = \bar{G} + {\varvec{\uplambda}}_{{\mathbf{G}}}^{i} \,{}^{\prime }{\mathbf{x}}_{{\mathbf{G}}}^{{\mathbf{i}}} + \varepsilon_{G}^{i} } \hfill \\ \end{array} } \right. $$
(4)

Estimating regression (3b) in ETSS implies a different minimization problem. Because OLS minimizes the sum of squared residuals, the deviations of \( R^{i} \) and \( G^{i} \) from the sample means are jointly minimized in the following way:

$$ \begin{gathered} \mathop {\rm Min}\limits_{{\bar{R},\bar{G},\varepsilon^{i} }} \sum\limits_{i} {(\varepsilon_{G}^{i} (1 - MB^{i} ) + \varepsilon_{R}^{i} MB^{i} )^{2} } \hfill \\ \left\{ {\begin{array}{*{20}c} {R^{i} = \bar{R} + \varepsilon_{R}^{i} } \hfill \\ {G^{i} = \bar{G} + \varepsilon_{G}^{i} } \hfill \\ \end{array} } \right. \hfill \\ \end{gathered} $$
(13)

The key difference between ETSS’ and our minimization problems is that ETSS’ minimization function (13) does not increase even as \( \varepsilon_{R}^{i} \) and \( \varepsilon_{G}^{i} \) go to infinity as long as their linear combination, \( \varepsilon_{G}^{i} (1 - MB^{i} ) + \varepsilon_{R}^{i} MB^{i} \), remains the same. In contrast, our loss function (4) always increases in the magnitude of \( \varepsilon_{R}^{i} \) and \( \varepsilon_{G}^{i} \). Mathematically, our minimization function is positive definite while that in ETSS is positive semi-definite.Footnote 39 The assumption of a positive definite function is a standard assumption in the definition of a loss function. We find that the minimization of any positive definite quadratic function of \( \varepsilon_{R}^{i} \) and \( \varepsilon_{G}^{i} \) is sufficient to uniquely identify firm-specific R and G.Footnote 40

Appendix 2

2.1 Benchmark COE measures

Implied COE from Claus and Thomas (2001), r CT , is an internal rate of return from the following valuation equation:

$$ P_{0} = B_{0} + \sum\limits_{\tau = 1}^{4} {\frac{{E_{\tau } - r_{CT} B_{\tau - 1} }}{{(1 + r_{CT} )^{\tau } }}} + \frac{{E_{5} - r_{CT} B_{4} }}{{(r_{CT} - g_{CT} )(1 + r_{CT} )^{4} }}\quad \quad (r_{CT} ) $$

where P 0 is the stock price as of June of year t + 1 from I/B/E/S; B 0 is the book value of equity at the end of year t from Compustat divided by the number of shares outstanding from I/B/E/S; E 1 and E 2 are one- and two-year-ahead consensus earnings per share forecasts from I/B/E/S reported in June of year t + 1; E 3, E 4, and E 5 are three-, four-, and five-year-ahead earnings per share forecasts computed using the long-term growth from I/B/E/S as E 3 = E 2(1 + Ltg), E 4 = E 3(1 + Ltg), and E 5 = E 4(1 + Ltg); B τ is the expected per-share book value of equity for year τ estimated using the clean surplus relation (B t+1 = B t  + E t+1 − d t+1); g CT is the terminal growth calculated as the 10-year Treasury bond yield minus 3%.Footnote 41

Implied COE from Gebhardt et al. (2001), r GLS , is an internal rate of return from the following valuation equation:

$$ P_{0} = B_{0} + \sum\limits_{\tau = 1}^{11} {\frac{{(ROE_{\tau } - r_{GLS} )B_{\tau - 1} }}{{(1 + r_{GLS} )^{\tau } }}} + \frac{{(IndROE - r_{GLS} )B_{11} }}{{r_{GLS} (1 + r_{GLS} )^{11} }}\quad \quad (r_{GLS} ) $$

where ROE τ is expected future return on equity calculated as earnings per share forecast (E τ ) divided by per share book value of equity at the end of the previous year (B τ−1); ROE 1 and ROE 2 are calculated using one- and two-year-ahead consensus earnings per share forecasts from I/B/E/S reported in June of year t + 1; ROE 3 is computed by applying the long-term growth rate from I/B/E/S to the two-year-ahead consensus earnings per share forecast; beyond year t + 3, ROE is assumed to linearly converge to industry median ROE (IndROE) by year t + 12.

Implied COE from Gode and Mohanram (2003), r PEG , is calculated as:

$$ r_{PEG} = \sqrt {\frac{{E_{1} }}{{P_{0} }}g_{2} } \quad g_{2} = \frac{{(E_{2} /E_{1} - 1) + Ltg}}{2}\quad \quad (r_{PEG} ) $$

where P 0 is the stock price as of June of year t + 1 from I/B/E/S; E 1 and E 2 are one- and two-year-ahead consensus earnings per share forecasts from I/B/E/S reported in June of year t + 1; Ltg is the long-term earnings growth forecast from I/B/E/S reported in June of year t + 1. This measure is a modified version of the Easton (2004) PEG measure, which assumes g 2 = E 2 /E 1.

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Nekrasov, A., Ogneva, M. Using earnings forecasts to simultaneously estimate firm-specific cost of equity and long-term growth. Rev Account Stud 16, 414–457 (2011). https://doi.org/10.1007/s11142-011-9159-2

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