###### Building a Model to Value a Firm

The Gordon growth model assumes that dividends grow indefinitely at a constant rate.

$$D_t= D_{t-1} (1+g)$$

Where:

\(g =\) Expected constant growth rate in dividends

\(D_{t}=\) Expected dividend at time *t.*

At any time \(t\), \(D_{t}=\)_{ }equals dividends at \(t = 0\), compounded at \(g\)* *for \(t\)* *periods.

$$D_t= D_o (1+g)^t$$

Under the Gordon growth model, the intrinsic value of a stock is calculated as:

$$\begin{align*} V_0& = \frac{D_0 (1+g)^1}{(1+r)^1}+\frac{D_0 (1+g)^2}{(1+r)^2} +⋯+\frac{D_0 (1+g)^n}{(1+r)^n} \\ \\ &= \frac{D_0 (1+g)}{(r-g)} \\ \\ &= \frac{D_1}{(r-g)} \end{align*}$$

The required rate of return on equity, \(r\), must be greater than the expected growth rate, \(g\). If the return rate required is less or equal to the expected growth rate, the Gordon growth model will not be valid.

The Gordon growth model can be used to estimate a stock’s required rate of return, \(r\)*.*

$$\begin{align*}r&= \frac{D_0 (1+g)}{P_0} +g \\ \\ & = \frac{D_1}{P_0} +g\end{align*}$$

The model values are very sensitive to the required rate of return, \(r\), and the expected dividends growth rate, \(g\).

The Gordon growth model is most appropriate for companies with earnings projected to grow at a rate similar to or lower than the economy’s growth rate. If a company’s earnings growth rate is above the growth rate of the economy, it is improbable that such a high rate of growth is sustainable into perpetuity.

For companies with an earnings growth rate above the economy’s growth rate, analysts should use a multistage DDM, where the final-stage growth rate reflects a rate more comparable to the economy’s growth rate.

In addition, analysts have often used the Gordon growth model to value broad equity market indexes.

The Gordon growth model can also be used to value fixed-rate perpetual preferred shares. For example, if the dividend on preferred stock is \(D\) and growth is zero, the value of the share using the Gordon growth model is:

$$V_{0}=\frac{D}{T}$$

The discount rate, \(r\)*, *is referred to as the capitalization rate because it capitalizes the dividend amount, \(D\).

The Gordon growth model implies a set relationship for the earnings, dividends’ growth rates, and stock value. With dividends growing at a constant growth rate \(g\)*,* holding the required rate of return stable, the stock value grows at \(g\).

Consider the following information:

$$\small{\begin{array}{l|l}\text{Risk-free rate} & 6\% \\ \hline\text{Equity risk premium} & 8\% \\ \hline\text{Beta} & 1.4 \\ \hline\text{Current dividend} & \$3 \\ \hline\text{Dividend growth rate} & 6\% \\ \hline\text{Current stock price} & \$30\\ \end{array}}$$

The intrinsic value of the stock is *closest to:*

$$V_{0}=\frac{D_{0}(1+g)}{(r-g)}$$

Using CAPM to find the required rate of return: \(r = 6\% + 1.4 (8\%) = 17.20\%\)

$$V_0= \frac{\$3(1+0.06)}{(0.1720-0.06)}=$28.39$$

## Question

Suppose a company’s annual dividend of $12 has just been paid. If the long-term growth rate is 4% and the required return on equity is 9%, using the Gordon growth model, the company’s value per share is

closestto:

- $130.
- $90.
- $160.

Solution

The correct answer is A.Using the Gordon growth model,

$$\begin{align*} V_{0} & =\frac{D_{0}(1+g)}{(r-g)} \\ \\ & =\frac{$5(1+4\%)}{(9\%-5\%)}= \frac{$5.2}{0.04}=$130 \end{align*}$$

Reading 23: Discounted Dividend Valuation

*LOS 23 (c) C**alculate the value of a common stock using the Gordon growth model and explain the model’s underlying assumptions.*