# Residual Income Model

The residual income model analyzes the intrinsic value of equity as the sum of:

• The current book value of equity; and
• The present value of expected future residual income.

Residual income is computed as net income minus an equity charge:

$$\text{RI}_{\text{t}}=\text{E}_{\text{t}}-(\text{r}\times\text{B}_{\text{t}-1})$$

Using the residual income model, the intrinsic value of common stock can be expressed as follows:

$$\text{V}_{0}=\text{B}_{0}+\sum_{\text{t}-1}^{\infty}\frac{\text{RI}_{\text{t}}}{(1+\text{r})^{\text{t}}}=\text{B}_{0}+\sum_{\text{t}-1}^{\infty}\frac{\text{E}_{\text{t}}-\text{rB}_{\text{t}-1}}{(1+\text{r})^{\text{t}}}$$

Where:

$$\text{V}_{0}=$$ Value of the share today

$$\text{B}_{0}=$$ Current book value of equity per share

$$\text{B}_{\text{t}}=$$ Expected book value of equity at time t

$$\text{r}=$$ Required rate of return on equity

$$\text{E}_{\text{t}}=$$ Expected EPS for period t

$$\text{RI}_{\text{t}}=$$ Expected per share residual income

#### Example: Forecasting Residual Income

A company has the following information:

• Current book value per share equals $9.00. • For the next three years, EPS is expected to be$3.25, $4.00, and$5.50, respectively.
• Dividends for the next three years are expected to be $2.25,$3.00, and $16.50, respectively. • The year 3 dividend is expected to be a liquidating dividend, i.e., the company is expected to close down its operations at the end of year 3 and distribute its entire book value in a dividend. The required rate of return on equity of 8%, a) The per-share book value and residual income for the next three years are: $$\small{\begin{array}{l|c|c|c}\textbf{Year} & \textbf{1} & \textbf{2} & \textbf{3} \\ \hline \text{Beginning book value per share}\\ (\text{B}_{\text{t}-1}) & \9.00 & \10.00 & \11.00 \\ \hline \text{Net income per share (EPS)} & 3.25 & 4.00 & 5.50 \\ \hline \text{Less dividends per share (D)} & 2.25 & 3.00 & 16.50 \\ \hline \text{Ending book value per share}\\ (\text{B}_{t-1}+\text{EPS} – \text{D})& \10.00 & \11.00 & \0.00 \\ \hline \text{Net income per share (EPS)} & 3.25 & 4.00 & 5.50 \\ \hline \text{Less per share equity charge}\\ (\text{rB}_{\text{t}-1}) & 0.72 & 0.80 & 0.88 \\ \hline\textbf{Residual income} & \textbf{\2.53} & \textbf{\3.20} & \textbf{\4.62} \\ \end{array}}$$ Using the residual income model, intrinsic value can be calculated as: $$\text{V}_{0}=9.00+\frac{2.53}{(1.08)}+\frac{3.20}{(1.08)^{2}}+\frac{4.62}{(1.08)^{3}}=17.75$$ Using the dividend discount model, intrinsic value can be calculated as: $$\text{V}_{0}=\frac{2.25}{(1.08)^{1}}+\frac{3.00}{(1.08)^{2}}+\frac{16.50}{(1.08)^{3}}=17.75$$ The residual income model and other valuation models like the dividend discount model should give the same value. There is an alternative way of computing residual income besides, $$\text{RI}_{\text{t}}=\text{E}_{\text{t}}-(\text{r}\times\text{B}_{\text{t}-1})$$ This involves applying the difference between the actual return on equity (ROE) and the required return on equity ($$r$$) to the beginning-of-period book value. $$\text{E}_{\text{t}}=\text{ROE}\times\text{B}_{\text{t}-1}$$ Therefore, $$\text{RI}_{\text{t}}=(\text{ROE}-\text{r})\times \text{B}_{\text{t}-1}$$ Using this formula, the residual income model equation can be expressed as: $$\text{V}_{0}=\text{B}_{0}+\sum_{\text{t}-1}^{\infty}\frac{(\text{ROE}_{\text{t}}-\text{r})\text{B}_{\text{t}-1}}{(1+\text{r})^{\text{t}}}$$ ## Question Assuming a company has the following information: • Current book value per share equals$22.00.
• For the next three years, EPS is expected to be $6.50,$8.00, and $10.50, respectively. • Dividends for the next three years are expected to be$3.50, $5.00, and$12.50 respectively.
• The Year 3 dividend is expected to be a liquidating dividend, i.e., the company is anticipated to cease its operations at the end of Year 3 and distribute its entire book value in a dividend.
• The required rate of return on equity of 9%.

Its intrinsic value would be closest to:

1. 37.14.
2. 35.50.
3. 39.23.

### Solution

$$\small{\begin{array}{l|l|l|l}\textbf{Year} & \textbf{1} & \textbf{2} & \textbf{3} \\ \hline \text{Beginning book value per share}\\ (\text{B}_{\text{t}-1})& \22.00 & \25.00 & 28.00 \\ \hline \text{Net income per share (EPS)} & 6.50 & 8.00 & 10.50 \\ \hline \text{Less dividends per share (D)} & 3.50 & 5.00 & 38.50 \\ \hline \text{Ending book value per share}\\ (\text{B}_{t-1}+\text{EPS} – \text{D}) & 25.00 & 28.00 & 00.00 \\ \hline \text{Net income per share (EPS)} & 6.50 & 8.00 & 10.50 \\ \hline \text{Less per share equity charge}\\ (\text{rB}_{\text{t}-1}) & 1.98 & 2.25 & 2.52 \\ \hline \textbf{Residual income} & \textbf{\4.52} & \textbf{\5.75} & \textbf{\7.98} \\ \end{array}}$$

$$\text{V}_{0}=22.00+\frac{4.52}{(1.09)^{1}}+\frac{5.75}{(1.09)^{2}}+\frac{7.98}{(1.09)^{3}}=37.14$$

LOS 26 (c) Calculate the intrinsic value of a common stock using the residual income model and compare value recognition in residual income and other present value models.

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