Residual income, value-relevant information and equity valuation: a simultaneous equations approach

Abstract

The paper uses Ohlson (Contemp Account Res 11:661–687, 1995) and compares the relative predictability of the proposed simultaneous model for contemporaneous stock price with a traditional single equation model used by the previous studies. The paper also explores how residual income and value-relevant information affect firms’ equity price. The main results of the paper suggest that the predictive ability and estimation efficiency of the simultaneous models in explaining contemporaneous stock prices are better than those of the traditional single models. Moreover, investors will use the value-relevant information beyond accounting earnings, namely analysts’ earnings forecasts, bankruptcy cost and agency cost, in equity valuation to make decision. Note particularly, the higher the bankruptcy or agency cost is, the more important the role it plays in equity valuation and, on average, the higher the accuracy of price prediction is.

Appendix 1

1.1 Dynamic LIM under traditional residual income model

We first assume that the simultaneous model of traditional residual income model is

$$ A_0 X_{t+1} =W_0 +A_1 X_t +\Uptheta_{t+1} $$

(A1)

and let $X_{t+1} =W_0^\ast +A_1^\ast X_t +\Uptheta_{t+1}^\ast,$ where $W_0^\ast =A_0^{-1} W_0, A_1^\ast =A_0^{-1} A_1,\ \hbox{and}\ \Uptheta_{t+1}^\ast =A_0^{-1} \Uptheta_{t+1}.$ On the other hand, because

$$ \begin{aligned} E_t \left( {X_{t+1}} \right)&=W_0^\ast +A_1^\ast X_t \\ E_t \left( {X_{t+2}} \right)&=W_0^\ast +A_1^\ast E\left( {X_{t+1}} \right)=W_0^\ast +A_1^\ast W_0^\ast +\left( {A_1^\ast} \right)^{2}X_t \end{aligned} $$

we have

$$ \begin{aligned} \sum\limits_{i=1}^\infty {\frac{E_t \left( {X_{t+i}} \right)}{\left( {1+r} \right)^{i}}} &=\frac{E_t \left( {X_{t+1}} \right)}{1+r}+\frac{E_t \left( {X_{t+2}} \right)}{\left( {1+r} \right)^{2}}+\cdots \cdot \cdot\\ &=\frac{W_0^\ast +A_1^\ast X_t} {1+r}+\frac{W_0^\ast +A_1^\ast W_0^\ast +(A_1^\ast )^{2}X_t} {(1+r)^{2}}+\cdots \cdot \cdot\\ &=\left[ {\frac{I}{1+r}+\frac{I+A_1^\ast} {\left( {1+r} \right)^{2}}+\frac{I+A_1^\ast +\left( {A_1^\ast} \right)^{2}}{\left( {1+r} \right)^{3}}+\cdot \cdot \cdot \cdot \cdot} \right]W_0^\ast +\left[ {\frac{A_1^\ast} {\left( {1+r} \right)}+\frac{\left( {A_1^\ast } \right)^{2}}{\left( {1+r} \right)^{2}}+\cdots \cdot \cdot } \right]X_t \\ &=CW_0^\ast +BX_t, \end{aligned} $$

(A2)

where $C=\left[ {\frac{I}{1+r}+\frac{I+A_1^\ast} {\left( {1+r} \right)^{2}}+\frac{I+A_1^\ast +\left( {A_1^\ast} \right)^{2}}{\left( {1+r} \right)^{3}}+\cdots \cdot \cdot} \right],$ $B =\left[ {\frac{A_1^\ast} {\left( {1+r} \right)}+\frac{\left( {A_1^\ast} \right)^{2}}{\left( {1+r} \right)^{2}}+\cdots \cdot \cdot} \right].$ From (A2),

$$ \begin{array}{l} \frac{A_1^\ast} {\left( {1+r} \right)}B=\frac{\left( {A_1^\ast} \right)^{2}}{\left( {1+r} \right)^{2}}+\frac{\left( {A_1^\ast} \right)^{3}}{(1+r)^{3}}+\cdots \cdot \cdot\\ \left( {I-\frac{A_1^\ast} {1+r}} \right)B=\frac{A_1^\ast} {\left( {1+r} \right)}\\ (I(1+r)-A_1^\ast )B=A_1^\ast , \end{array} $$

we obtain

$$ \begin{aligned} B&=(I(1+r)-A_1^\ast )^{-1}A_1^\ast\\ &=(I(1+r)-A_0^{-1} A_1 )^{-1}A_0^{-1} A_1 \\ &=\{A_0^{-1} [A_0 (1+r)-A_1 ]\}^{-1}A_0^{-1} A_1 \\ &=(A_0 (1+r)-A_1 )^{-1}A_1. \end{aligned} $$

On the other hand, the constant term C is

$$ \begin{array}{l} C=\frac{I}{\left( {1+r} \right)}+\frac{I+A_1^\ast} {\left( {1+r} \right)^{2}}+\frac{I+A_1^\ast +\left( {A_1^\ast} \right)^{2}}{\left( {1+r} \right)^{3}}+\cdots \cdot \cdot\\ \frac{A_1^\ast} {1+r}C=\frac{A_1^\ast} {(1+r)^{2}}+\frac{A_1^\ast +(A_1^\ast )^{2}}{(1+r)^{3}}+\frac{A_1^\ast +\left(A_1^\ast \right)^{2}+\left(A_1^\ast \right)^{3}}{(1+r)^{4}}+\cdots \cdots\\ \begin{aligned} \left( {I-\frac{A_1^\ast} {1+r}} \right)C&=\frac{I}{1+r}+\frac{I}{\left( {1+r} \right)^{2}}+\frac{I}{\left( {1+r} \right)^{3}}+\cdots \cdot \cdot\\ &=\frac{I}{1+r}\left[1+\frac{1}{1+r}+\frac{1}{(1+r)^{2}}+\cdots \cdot \cdot \right]=\frac{I}{1+r}\cdot \frac{1}{1-\frac{1}{1+r}}=\frac{1}{r}I \end{aligned}\end{array} $$

and

$$ \begin{array}{l} \frac{1}{1+r}[I(1+r)-A_1^\ast ]C=\frac{1}{r}I\\ {[}I(1+r)-A_0^{-1} A_1 ]C=\frac{1+r}{r}I\\ A_0^{-1} [A_0 (1+r)-A_1 ]C=\frac{1+r}{r}I\\ {[}A_0 (1+r)-A_1 ]C=\frac{1+r}{r}A_0 \end{array} $$

Therefore, $C=[A_0 (1+r)-A_1 ]^{-1}\left(\frac{1+r}{r}\right)A_0 =\frac{1+r}{r}\left[ {A_0 \left( {1+r} \right)-A_1} \right]^{-1}A_0.$ From (A1), $X_{t+1} =\left[ {{\begin{array}{l} {\omega_{10}} \\ {\omega_{20}} \\ \end{array}}} \right]+\left[ {{\begin{array}{ll} {\omega_{11}} & {\omega_{12}} \\ 0& {\omega_{22}} \\ \end{array}}} \right]X_t +\Uptheta_{t+1},$ and thus the constant term is

$$ \begin{aligned} CW_0 &=\frac{1+r}{r}\left[A_0 (1+r)-A_1 \right]^{-1}A_0 W_0\\ &=\frac{1+r}{r}\left[ {I\left( {1+r} \right)-\left( {{\begin{array}{ll} {\omega_{11}} & {\omega_{12}} \\ 0& {\omega_{22}} \\ \end{array}}} \right)} \right]^{-1}\left[ {{\begin{array}{l} {\omega_{10}} \\ {\omega_{20}} \\ \end{array}}} \right] \\ \\ &=\frac{1+r}{r\left( {1+r-\omega_{11}} \right)\left( {1+r-\omega_{22}} \right)}\left[ {{\begin{array}{ll} {1+r-\omega_{22}} & {\omega_{12}} \\ 0& {1+r-\omega_{11}} \\ \end{array}}} \right]\left[ {{\begin{array}{l} {\omega_{10}} \\ {\omega_{20}} \\ \end{array}}} \right] \\ \end{aligned} $$

Hence, the first element is $\frac{(1+r)\left[ {\omega_{10} \left({1+r-\omega_{22}} \right)+\omega_{12} \omega_{20}} \right]}{r\left({1+r-\omega_{11}} \right)\left( {1+r-\omega_{22}} \right)}.$ Moreover, from (A2),

$$ BX_t =\frac{1}{\left( {1+r-\omega_{11}} \right)\left( {1+r-\omega _{22}} \right)}\left[ {{\begin{array}{ll} {\omega_{11} \left( {1+r-\omega_{22}} \right)}& {\omega_{11} \omega_{12} +\omega_{12} \left( {1+r-\omega_{11}} \right)} \\ 0& {\omega_{22} \left( {1+r-\omega_{11}} \right)} \\ \end{array}}} \right]\left[ {{\begin{array}{l} {x_t} \\ {\upsilon_t} \\ \end{array}}} \right]. $$

Thus, the first row is $\frac{\omega_{11}} {1+r-\omega_{11}} x_t +\frac{\left( {1+r} \right)\omega_{12}} {\left( {1+r-\omega_{11}} \right)\left( {1+r-\omega_{22}} \right)}\upsilon_t.$

This general approach applies to all the linear information models considered in the paper.