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Evaluating cross-sectional forecasting models for implied cost of capital

Abstract

The computation of implied cost of capital (ICC) is constrained by the lack of analyst forecasts for half of all firms. Hou et al. (J Account Econ 53:504–526, 2012, HVZ) present a cross-sectional model to generate forecasts in order to compute ICC. However, the forecasts from the HVZ model perform worse than those from a naïve random walk model and the ICCs show anomalous correlations with risk factors. We present two parsimonious alternatives to the HVZ model: the EP model based on persistence in earnings and the RI model based on the residual income model from Feltham and Ohlson (Contemp Account Res 11:689–732, 1996). Both models outperform the HVZ model in terms of forecast bias, accuracy, earnings response coefficients, and correlations of the ICCs with future returns and risk factors. We recommend that future research use the RI model or the EP model to generate earnings forecasts.

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Notes

  1. We observe that parsimonious models in general outperform complex models. For example, we test the forecast model of Abarbanell and Bushee (1997), which is based on the forecast approach by analysts, a model that forecasts future sales growth and profit margins, and a model that combines RI and HVZ. None of these more complex models outperforms the EP or the RI models.

  2. We estimate the HVZ model at the dollar level as it is specified in their paper. We also perform robustness test by estimating their model at the per-share level. The inference still holds.

  3. The model including Bt−1 produces similar results.

  4. This represents a departure from HVZ as they estimate earnings before extraordinary items, without excluding special items. We exclude special items because they are less predictable by nature. We perform a robustness test using earnings before extraordinary items. The forecast errors are bigger for all models. However, the inferences do not change as the rank ordering of the models in terms of forecast accuracy and ERCs is unaltered.

  5. We also perform robustness test by estimating the cumulative ERC the same way as the “Annual ERC” in HVZ. Specifically, we estimate ERCs by regressing the buy-and-hold returns over the next 1, 2, and 3 years on the unexpected earnings over the same horizon. The tenor of the results does not change.

  6. We further partition our sample into four time periods (1969–1978, 1979–1988, 1989–1998, and 1999–2008). We do not observe any systematic changes in forecast accuracy of the HVZ, EP and RI models. However, the forecast accuracy of the RW model deteriorates over time. This is not surprising as the naïve model is not well suited for more complex operations.

  7. This represents a departure from HVZ, who form calendar time portfolios starting on July 1. The advantage of our approach is that the financial statement information is equally timely for all observations. The disadvantage is the fact that the compounding period may not be identical for all firms in our sample. As a robustness test, we carry out all tests in a subset of firms with December fiscal year-ends (over 60 % of the sample) and find virtually identical results.

  8. Easton and Monahan (2005) recommend running regressions with the ICC measure and proxies for cash flow news and discount rate news. However, these proxies require forecast revisions, which are not feasible to estimate for cross-sectional models. Hence, we only run univariate regressions.

  9. A concern might be that the lower coefficients on the HVZ model in the return regressions might arise mechanically due to the greater magnitude and greater spread of the ICC estimates generated from the HVZ model. To account for this, we perform the following sensitivity test. We standardize all the ICC measures each year by subtracting the minimum and then dividing by the range (maximum–minimum) for ICC using that method in that year. In other words, we set each ICC = (ICC − min)/(max − min). We then re-estimate the regressions using the standardized ICC measures. We continue to find the weakest relation between ICC from the HVZ model and future returns. For instance, for 1-year-ahead returns, the average coefficients on ICCHVZ, ICCEP, and ICCRI from the Fama–MacBeth regressions are 0.230, 0.331, and 0.411, respectively. For 2-year-ahead returns, the average coefficients on ICCHVZ, ICCEP, and ICCRI are 0.190, 0.360, and 0.342, respectively. For 3-year-ahead returns, the average coefficients on ICCHVZ, ICCEP, and ICCRI are 0.120, 0.309, and 0.298, respectively.

  10. Easton and Monahan (2010) argue that the latter approach is logically inconsistent as ICC metrics are estimated precisely because of the flaws in conventional measures of risk that often rely on ex post returns. We present these results to ensure a comparison between our results and those presented in HVZ.

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Acknowledgments

We would like to thank Jim Ohlson (editor), Mei Feng (discussant), two anonymous referees, Patricia Dechow, Scott Richardson, Jacob Thomas, Franco Wong, and seminar participants at Boston University, Erasmus University, University of British Columbia, University of Miami, University of Toronto, and 2013 Review of Accounting Studies Conference for helpful comments.

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Correspondence to Kevin K. Li.

Appendices

Appendix 1: Variable definitions for models used to generate forecasts

HVZ model

$$ E_{i,t + \tau} = \alpha_{0} + \alpha_{1} A_{i,t} + \alpha_{2} D_{i,t} + \alpha_{3} DD_{i,t} + \alpha_{4} E_{i,t} + + \alpha_{5} NegE_{i,t} + \alpha_{6} AC_{i,t} + \varepsilon_{i,t + \tau} $$
Variable Definition Xpressfeed variable
Et+τ Earnings in year t + τ ib-spi
At Total assets in year t at
Dt Common dividend dvc
DDt A dummy variable that equals 1 for dividend payers and 0 otherwise  
NegEt A dummy variable that equals 1 for firms with negative earnings and 0 otherwise  
ACt Change in noncash current assets less change in current liabilities excluding change in short-term debt and change in taxes payable minus depreciation and amortization Δ(act-che)-Δ(lct-dlc-txp)-dp

EP model

$$ E_{i,t + \tau} = \beta_{0} + \beta_{1} NegE_{i,t} + \beta_{2} E_{i,t} + \beta_{3} NegE*E_{i,t} + \varepsilon_{i,t + \tau} $$
Variable Definition Xpressfeed variable
Et+τ Earnings in year t + τ divided by number of shares outstanding in year t (ib-spi)t+τ/cshot
NegEt A dummy variable that equals 1 for firms with negative earnings and 0 otherwise  
NegE*Et Interaction term of NegE and E  

RI model

$$ E_{i,t + \tau} = \chi_{0} + \chi_{1} NegE_{i,t} + \chi_{2} E_{i,t} + \chi_{3} NegE*E_{i,t} + \chi_{4} B_{i,t} + \chi_{5} TACC_{i,t} + \varepsilon_{i,t + \tau} $$
Variable Definition Xpressfeed variable
Et+τ Earnings in year t + τ divided by number of shares outstanding in year t (ib-spi)t+τ/cshot
NegEt A dummy variable that equals 1 for firms with negative earnings and 0 otherwise  
NegE*Et Interaction term of NegE and E  
Bt Book value of equity divided by number of shares outstanding ceqt/cshot
TACCt Richardson et al. (2005) total accruals, i.e., the sum of the change in WC, the change in NCO, and the change in FIN, divided by number of shares outstanding WC = (act-che)-(lct-dlc);
NCO = (at-act-ivao)-(lt-lct-dltt);
FIN = (ivst + ivao)-(dltt + dlc + pstk);
All variables deflated by csho

Appendix 2: Computing implied cost of capital

The implied cost of equity used in this paper is computed as the average of the four commonly used metrics, ICCGM, ICCPEG, ICCGLS, and ICCCT. We briefly describe how these four metrics are computed below.

ICC based on the Ohlson and Juettner-Nauroth Model: ICCGM and ICCPEG

Ohlson and Juettner-Nauroth (2005) show that the implied cost of capital can be expressed as:

$$ r_{e} = A + \sqrt {A^{2} + \frac{{eps_{1} }}{{P_{0} }}\left( {g_{2} - (\gamma - 1)} \right)} \,{\text{where}}\,A = \frac{1}{2}\left( {(\gamma - 1) + \frac{{dps_{1} }}{{P_{0} }}} \right) \,{\text{and}}\,g_{2} = \frac{{(eps_{2} - eps_{1} )}}{{eps_{1} }} $$

Gode and Mohanram (2003) make the following assumptions. They set (γ − 1) to Rf − 3 % where Rf is the risk-free rate. In addition, they use the average of short-term growth and analysts’ long-term growth rate (LTG) instead of g2 to reduce the impact of outliers.

If short-term growth (\( \frac{{eps_{2} }}{{eps_{1} }} - 1 \)) is greater than long-term growth rate (\( \root{4}\of{{\frac{{eps_{5} }}{{eps_{1} }}}} - 1 \)), we set g2 to equal the geometric mean of short term and long-term growth rate. If short-term growth is less than long-term growth, we set g2 to equal the long-term growth rate. Dividends are estimated by calculating current payout for all firms, defined as dividends (DVC) divided by income before extraordinary items (IB) for firms with positive current earnings or dividends divided by 6 % of total assets (AT) for firms with negative IB.

In addition, we compute an ICC from a simplified version of the Ohlson and Juettner-Nauroth model that ignores dividends and sets ICC to the square root of the inverse of the PEG ratio. We compute ICCPEG as:

\( {\text{ICC}}_{\text{PEG}} = \sqrt {\frac{{g_{2} }}{{\left( {{\raise0.7ex\hbox{${PRICE}$} \!\mathord{\left/ {\vphantom {{PRICE} {eps_{1} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${eps_{1} }$}}} \right)}}} \) where g2 is defined as it is for the RGM model

ICC based on the residual income model: ICCGLS and ICCCT

Gebhardt et al. (2001) use the residual income valuation model to estimate implied cost of equity. They use EPS estimates for future 2 years and the expected dividends payout (from historical data) to derive book value and return on equity (ROE) forecasts. Beyond the forecast horizon, they assume that ROE fades to the industry median by year 12. Industry median ROE is estimated as the median of all ROEs from firms in the same industry defined using the Fama and French (1997) classification over the past 5 years with positive earnings and nonnegative book values, where ROE is defined as the ratio of net income before extraordinary items (IB) to lagged total common shareholders’ equity (CEQ). Abnormal earnings are assumed to remain constant at year 12 levels for perpetuity. The cost of equity is computed numerically by equating current stock price to the sum of the current book value and the present value of future residual earnings, i.e., solving for r in the equation:

$$ P_{0} = B_{o} + \sum\limits_{{\tau = 1}}^{{12}} {\frac{{(eps_{\tau } - r* B_{{\tau - 1}} )}}{{(1 + r)^{\tau } }}} + \frac{{(eps_{{12}} - r* B_{{11}} )}}{{r(1 + r)^{{12}} }} $$

where eps is the forecasted eps (obtained either from explicit forecast or inferred from expected ROE and lagged book value); P0 is current price per share; B0 is current book value per share; and B1 through B11 are expected future book values per share obtained through the clean surplus relation, setting payout to equal current payout. Current payout is defined as dividends (DVC) divided by income before extraordinary items (IB) for firms with positive current earnings or dividends divided by 6 % of total assets (AT) for firms with negative IB. We depart from Gebhardt, Lee, and Swaminathan by using the model forecasts explicitly for years 1 through 5 and then applying ROE convergence.

Claus and Thomas (2001) also use the residual income model to estimate the implied cost of equity. They assume that earnings grow at the analyst’s consensus long-term growth rate until year 5 and at the rate of inflation thereafter. The implied cost of equity is estimated numerically by solving the following equation:

$$ P_{0} = B_{o} + \sum\limits_{\tau = 1}^{5} {\frac{{(eps_{\tau } - r*B_{\tau - 1} )}}{{(1 + r)^{\tau } }}} + \frac{{(eps_{5} - r*B_{4} )*(1 + g)}}{{(r - g)(1 + r)^{5} }} $$

where eps0 through eps5 are the forecasted future earnings per share; B0 is current book value per share; and B1 through B4 are expected future book values per share. Consistent with Claus and Thomas (2001), g is set to Rf − 3 %.

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Li, K.K., Mohanram, P. Evaluating cross-sectional forecasting models for implied cost of capital. Rev Account Stud 19, 1152–1185 (2014). https://doi.org/10.1007/s11142-014-9282-y

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Keywords

  • Earnings forecasts
  • Cross-sectional models
  • Implied cost of capital

JEL Classification

  • G12
  • G31
  • G32
  • M40
  • M41