###### Covariance Stationary Property

A time series is said to be covariance stationary if its properties, such... **Read More**

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

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:

- 37.14.
- 35.50.
- 39.23.

Solution

The correct answer is A.$$\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}}$$$$\text{V}_{0}=22.00+\frac{4.52}{(1.09)^{1}}+\frac{5.75}{(1.09)^{2}}+\frac{7.98}{(1.09)^{3}}=$37.14$$

Reading 26: Residual Income Valuation

*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.*