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.

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, ICC_GM, ICC_PEG, ICC_GLS, and ICC_CT. We briefly describe how these four metrics are computed below.

ICC based on the Ohlson and Juettner-Nauroth Model: ICC_GM and ICC_PEG

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 R_f − 3 % where R_f 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 ICC_PEG 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 R_GM model

ICC based on the residual income model: ICC_GLS and ICC_CT

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 R_f − 3 %.