Chapter 7, Problem 30QP

PE and Terminal Stock Price. In practice, a common way to value a share of stock when a company pays dividends is to value the dividends over the next five years or so, then find the “terminal” stock price using a benchmark PE ratio. Suppose a company just paid a dividend of $1.15. The dividends are expected to grow at 10 percent over the next five years. The company has a payout ratio of 40 percent and a benchmark PE of 21. What is the target stock price in five years? What is the stock price today assuming a required return of 11 percent on this stock?

Summary Introduction

To determine: The target stock price in five years.

Introduction:

Target stock price is a price in which the investor wants to exit from the current position to attain maximum earnings.

Answer to Problem 30QP

The target stock prices in five years are $97.23.

Explanation of Solution

Given information:

The company currently paid dividends of $1.15. The constant growth rate is 10% over the next five years. The benchmark price to earnings ratio is 21 and the estimated payout ratio is 40%.

The formula to calculate the dividend for each year:

${\text{D}}_1={\text{D}}_o{\left(1+\text{g}\right)}^n$

Where,

D₁refers to the next periods dividends per share

D_orefers to the currently dividend paid

g refers to the expected constant growth rate

n refers to the number of years

The formula to calculate the earnings per share:

$\text{Earnings per share}=\frac{\text{Dividend of the corresponding year }}{\text{Payout ratio}}$

The formula to calculate the target stock price of the year:

${\text{P}}_o=\text{Benchmark price to earnings ratio }\times {\text{EPS}}_1$

Where,

EPS₁ refers to the earnings per share of next year,

P₁ refers to the price of stock per share.

Note:

The current stock price is often called as target stock prices over the next year.

Compute the dividend of Year 1:

$\begin{array}{c}{\text{D}}_1={\text{D}}_o{\left(1+\text{g}\right)}^n\\ =\$1.15\times {\left(1+\frac{10}{100}\right)}^1\\ =\$1.15\times 1.10\\ =\$1.27\end{array}$

Hence, the dividend of Year 1 is $1.27.

Compute the dividend of Year 2:

$\begin{array}{c}{\text{D}}_1={\text{D}}_o{\left(1+\text{g}\right)}^n\\ =\$1.15\times {\left(1+\frac{10}{100}\right)}^2\\ =\$1.15\times {\left(1+0.10\right)}^2\\ =\$1.15\times {\left(1.10\right)}^2\end{array}$

$\begin{array}{c}=\$1.15\times 1.21\\ =\$1.39\end{array}$

Hence, the dividend of Year 2 is $1.39.

Compute the dividend of Year 3:

$\begin{array}{c}{\text{D}}_1={\text{D}}_o{\left(1+\text{g}\right)}^n\\ =\$1.15\times {\left(1+\frac{10}{100}\right)}^3\\ =\$1.15\times {\left(1+0.10\right)}^3\\ =\$1.15\times {\left(1.10\right)}^3\end{array}$

$\begin{array}{c}=\$1.15\times 1.331\\ =\$1.53\end{array}$

Hence, the dividend of Year 3 is $1.53.

Compute the dividend of Year 4:

$\begin{array}{c}{\text{D}}_1={\text{D}}_o{\left(1+\text{g}\right)}^n\\ =\$1.15\times {\left(1+\frac{10}{100}\right)}^4\\ =\$1.15\times {\left(1+0.10\right)}^4\\ =\$1.15\times {\left(1.10\right)}^4\end{array}$

$\begin{array}{c}=\$1.15\times 1.4641\\ =\$1.68\end{array}$

Hence, the dividend of Year 4 is $1.68.

Compute the dividend of Year 5:

$\begin{array}{c}{\text{D}}_1={\text{D}}_o{\left(1+\text{g}\right)}^n\\ =\$1.15\times {\left(1+\frac{10}{100}\right)}^5\\ =\$1.15\times {\left(1+0.10\right)}^5\\ =\$1.15\times {\left(1.10\right)}^5\end{array}$

$\begin{array}{c}=\$1.15\times 1.61051\\ =\$1.85\end{array}$

Hence, the dividend of Year 5 is $1.85.

Compute the earnings per share in Year 5:

$\begin{array}{c}\text{Earnings per share in Year 5}=\frac{\text{Dividend of Year 5}}{\text{Payout ratio}}\\ =\frac{\$1.85}{\frac{40}{100}}\\ =\frac{\$1.85}{40}\times 100\\ =\$4.63\end{array}$

Hence, the earnings per share in Year 5 are $4.63.

Compute the target stock price in Year 5:

$\begin{array}{c}{\text{P}}_5=\text{Benchmark price to earnings ratio }\times {\text{EPS}}_5\\ =21\times \$4.63\\ =\$\text{97}\text{.23}\end{array}$

Hence, the target stock price in Year 5 is $97.23.

Summary Introduction

To determine: The current stock price when the required rate of return is 11%.

Answer to Problem 30QP

The current stock price is $63.30.

Explanation of Solution

Given information:

The required return on its stock is 11%.

The formula to calculate the current stock price:

${\text{P}}_o=\frac{{\text{D}}_1}{{\left(1+\text{R}\right)}^1}+\frac{{\text{D}}_2}{{\left(1+\text{R}\right)}^2}+\frac{{\text{D}}_3}{{\left(1+\text{R}\right)}^3}\mathrm{...}+\mathrm{...}\frac{\left({\text{D}}_{\text{n}}+{\text{P}}_1\right)}{{\left(1+\text{R}\right)}^{\text{n}}}$

Where,

${\text{P}}_o$ refers to the present price of stock

P₁ refers to the next period’s price of the stock

${\text{D}}_1$… ${\text{D}}_n$ refers to the next period dividend per share

$\text{R}$ refers to the required return

nrefers to the number of years

Compute the price of the stock:

$\begin{array}{c}{\text{P}}_o=\frac{{\text{D}}_1}{{\left(1+\text{R}\right)}^1}+\frac{{\text{D}}_2}{{\left(1+\text{R}\right)}^2}+\frac{{\text{D}}_3}{{\left(1+\text{R}\right)}^3}+\frac{{\text{D}}_{\text{4}}}{{\left(1+\text{R}\right)}^{\text{4}}}+\frac{\left({\text{D}}_5+{\text{P}}_5\right)}{{\left(1+\text{R}\right)}^5}\\ =\frac{\$1.27}{{\left(1+\frac{11}{100}\right)}^1}+\frac{\$1.39}{{\left(1+\frac{11}{100}\right)}^2}+\frac{\$1.53}{{\left(1+\frac{11}{100}\right)}^3}+\frac{\$1.68}{{\left(1+\frac{11}{100}\right)}^4}+\frac{\left(\$1.85+\$97.23\right)}{{\left(1+\frac{11}{100}\right)}^5}\\ =\frac{\$1.27}{1.11}+\frac{\$1.39}{{\left(1.11\right)}^2}+\frac{\$1.53}{{\left(1.11\right)}^3}+\frac{\$1.68}{{\left(1.11\right)}^4}+\frac{\$\text{99}\text{.08}}{{\left(1.11\right)}^5}\\ =\$\text{1}\text{.1441}+\frac{\$1.39}{1.2321}+\frac{\$1.53}{1.36763}+\frac{\$1.68}{1.51807}+\frac{\$\text{99}\text{.08}}{1.68505}\end{array}$

$\begin{array}{c}=\$1.1441+\$1.12815+\$1.11872+\$1.10666+\$58.79944\\ =\$63.30\end{array}$

Hence, the price of a stock is $63.30.