Analysts’ earnings forecast errors and cost of equity capital estimates

Abstract

This study investigates the relation between analysts’ forecast errors and cost of equity capital estimates implied from analysts’ earnings forecasts and price. My analysis predicts and removes forecast errors from analysts’ earnings forecasts on an out-of-sample basis and then uses these adjusted analysts’ forecasts to reverse-engineer cost of equity capital estimates. While the correction for predictable analysts’ forecast errors meaningfully lowers each of three firm-level implied COEC estimates employed in this study and commonly used in the literature, I do not find that this correction improves their association with realized returns.

Appendices

Appendix 1

See Table 9.

Table 9 Variable definitions

Appendix 2

2.1 Empirical estimation of return news and earnings news

Invoking the clean surplus assumption and using a Taylor series approximation, Vuolteenaho (2002) expands the log book-to-market ratio to extend the return decomposition framework of Campbell and Shiller (1998a, 1998b), Campbell (1991), and Campbell and Ammer (1993). Vuolteenaho (2002) defines realized returns as expected returns plus earnings news (N _ROE ) less returns news (N _RET ):

$$ RET_{t + 1} = E_{t} (RET_{t + 1} ) + N_{ROE} - N_{RET} $$

(16)

In other words, associating realized returns with only expected returns ignores revisions or shocks to expected returns (N _RET ) and to expected earnings (N _ROE ).

Vuolteenaho (2002) defines earnings news, which encompasses both earnings surprises and revisions to future expected earnings over the firm’s lifetime, as:

$$ N_{ROE} = \Updelta E_{t} \sum\limits_{j = 0}^{\infty } {\rho^{j} roe_{t + j} } $$

(17)

where ρ is a discount factor of around 0.96; roe represents the natural logarithm of one plus return on equity (i.e. net income before extraordinary items over last year’s common equity); and the revision or shock is written as ∆E _t (·) = E _t (·) – E _t−1 (·). Similarly, Vuolteenaho (2002) defines returns (or discount rate) news, which encompasses shocks to expected future discount rates, as:

$$ N_{RET} = \Updelta E_{t} \sum\limits_{j = 1}^{\infty } {\rho^{j} ret_{t + j} } $$

(18)

where ret represents the natural logarithm of one plus returns.

To estimate N _RET and N _ROE , I follow the methodology of Vuolteenaho (2002) and Callen and Segal (2010) in using a log-linear vector autoregression (VAR). If we define z_i,t to be a vector of firm-specific state variables, the state vector can be assumed to follow the following multivariate log-linear dynamic:

$$ z_{i,t} = \Upgamma z_{i,t - 1} + \eta_{i,t} . $$

(19)

The VAR matrix (Γ) of coefficients is assumed to be constant over time and over firms, and the error terms (η _i,t, ) are assumed to be independent of variables known at time t − 1.

If we let $z_{i,t} = \left( \begin{gathered} ret_{i,t} \hfill \\ bm_{i,t} \hfill \\ roe_{i,t} \hfill \\ \end{gathered} \right)$, $\Upgamma = \left( \begin{gathered} \alpha_{1} \alpha_{2} \alpha_{3} \hfill \\ \beta_{1} \beta_{2} \beta_{3} \hfill \\ \gamma_{1} \gamma_{2} \gamma_{3} \hfill \\ \end{gathered} \right)$, and $\eta_{i,t} = \left( \begin{gathered} \eta_{i,1t} \hfill \\ \eta_{i,2t} \hfill \\ \eta_{i,3t} \hfill \\ \end{gathered} \right)$ in Eq. 19, with ret equal to the natural logarithm of returns, bm equal to the natural logarithm of the book-to-market ratio, and roe equal to the natural logarithm of return on equity, this gives rise to the short VAR:

$$ \begin{gathered} ret_{t} = \alpha_{1} ret_{t - 1} + \, \alpha_{2} bm_{t - 1} + \, \alpha_{3} roe_{t - 1} + \, \eta_{1t} \hfill \\ bm_{t} = \beta_{1} ret_{t - 1} + \, \beta_{2} bm_{t - 1} + \, \beta_{3} roe_{t - 1} + \, \eta_{2t} \hfill \\ roe_{t} = \gamma_{1} ret_{t - 1} + \, \gamma_{2} bm_{t - 1} + \, \gamma_{3} roe_{t - 1} + \, \eta_{3t} \hfill \\ \end{gathered} $$

(20)

The residuals from the VAR estimation are combined with Eq. 18 to express N _RET as:

$$ N_{RET} = e_{1}^{\prime } \rho \Upgamma (1 - \Upgamma )^{ - 1} \eta_{i,t} $$

(21)

where $e_{1} = \left( \begin{gathered} 1 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)$ and ρ is a discount factor. Similarly, earnings news can be expressed residually:

$$ N_{ROE} = e_{1}^{\prime } (1 - \rho \Upgamma )^{ - 1} \eta_{i,t} . $$

(22)

To assess the relation between realized returns and expected returns, I thus employ the following test:

$$ RET_{t + 1} = \partial_{0} + \partial_{1} E_{t} (RET_{t + 1} ) + \partial_{2} N_{RET} + \partial_{3} N_{ROE} + \varepsilon_{t + 1} $$

(23)