Chapter 8, Problem 32QP

Summary Introduction

To determine: The dividend yield for each of the four stocks.

Introduction:

Dividend is a sum of money paid to the shareholders of the company. It is distributed among the investors from a portion of the company’s earnings.

Dividend yield is a ratio that specifies how much a company pays as dividends every year, on comparing with its share price. It is considered as the return on investment for a stock.

Answer to Problem 32QP

The dividend yield of Stock W is 7%.

The dividend yield of Stock X is 17%.

The dividend yield of Stock Y is 22%.

The dividend yield of Stock Z is 10.34%.

Explanation of Solution

Given information:

Four different stocks have a required rate of return of 17% and the recent dividend is $4.50 per share.

The constant growth rate in dividends of Stock W is 10%.

The constant growth rate in dividends of Stock X is 0%.

The constant growth rate in dividends of Stock Y is -5%.

The constant growth rate in dividends of Stock Z is 20%.

Steps to determine the dividend yields and capital gains yield for each of the stocks:

• Firstly, determine the stock price for each stock. All the stocks have a required return of 17%, which is the sum of dividend yield and capital gains yield.
• After determining the stock price of each stock, use the stock price and dividend to compute the dividend yield of the four stocks.
• Finally, determine the capital gains yield for each stock by subtracting the dividend yield from the total return.

Formula:

The formula to calculate the price of a stock:

${\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}$

Where,

P_o refers to the present value of a share of stock,

D_o refers to the current year dividend paid,

R refers to the discount rate,

g refers to the constant growth of dividends.

The formula to calculate the dividend yield:

$\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}$

Where,

D₁ refers to the next period dividend per share,

P_o refers to the present value of a share of stock.

Compute the stock price of Stock W:

$\begin{array}{c}{\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}\\ =\$4.50\times \frac{\left(1+\frac{10}{100}\right)}{\left(\frac{17}{100}-\frac{10}{100}\right)}\\ =\$4.50\times \left(\frac{\left(1.10\right)}{\left(0.17-0.10\right)}\right)\\ =\$4.50\times \left(\frac{1.10}{0.07}\right)\\ =\$4.50\times 15.71428\\ =\$70.71\end{array}$

Hence, the stock price of the Stock W is $70.71.

Compute the dividend yield of the Stock W:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$4.50\times \left(1+\frac{10}{100}\right)}{\$70.71}\\ =\frac{\left(\$4.50\times 1.10\right)}{\$70.71}\\ =\frac{\$4.95}{\$70.71}\\ =0.07\end{array}$

Hence, the dividend yield of the Stock W is 0.07 or 7%.

Compute the stock price of Stock X:

$\begin{array}{c}{\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}\\ =\$4.50\times \frac{\left(1+\frac{0}{100}\right)}{\left(\frac{17}{100}-\frac{0}{100}\right)}\\ =\$4.50\times \frac{1}{\left(0.17-0.00\right)}\\ =\$4.50\times \left(\frac{1}{\text{0}\text{.17}}\right)\\ =\$4.50\times 5.88235\\ =\$26.47\end{array}$

Hence, the stock price of Stock X is $26.47.

Compute the dividend yield of Stock X:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$4.50}{\$26.47}\\ =0.17\end{array}$

Hence, the dividend yield of Stock X is 0.17 or 17%.

Compute the stock price of Stock Y:

$\begin{array}{c}{\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}\\ =\$4.50\times \frac{\left(1-\frac{5}{100}\right)}{\left(\frac{17}{100}+\frac{5}{100}\right)}\\ =\$4.50\times \frac{0.95}{\left(0.17+0.05\right)}\\ =\$4.50\times \left(\frac{0.95}{0.22}\right)\\ =\$4.50\times 4.318\\ =\$19.43\end{array}$

Hence, the stock price of Stock Y is $19.43.

Compute the dividend yield of Stock Y:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$4.50\times \left(1+\left(\frac{-5}{100}\right)\right)}{\$19.43}\\ =\frac{\left(\$4.50\times 0.95\right)}{\$19.43}\\ =\frac{\$4.275}{\$19.43}\\ =0.22\end{array}$

Hence, the dividend yield of Stock Y is 0.22 or 22%.

Compute the stock price in Year 2 of Stock Z:

$\begin{array}{c}{\text{P}}_2={\text{D}}_o\times \frac{{\left(1+\text{g}\right)}^2\left(1+{\text{g}}_2\right)}{\left(\text{R}-\text{g}\right)}\\ =\$4.50\times \frac{{\left(1+\frac{20}{100}\right)}^2\times \left(1+\frac{12}{100}\right)}{\left(\frac{17}{100}-\frac{12}{100}\right)}\\ =\$4.50\times \left(\frac{{\left(1+0.20\right)}^2\times \left(1+0.12\right)}{\left(0.17-0.12\right)}\right)\\ =\$4.50\times \left(\frac{\left(1.44\times 1.12\right)}{\text{0}\text{.05}}\right)\\ =\$4.50\times \left(\frac{1.6128}{0.05}\right)\\ =\$4.50\times 32.256\\ =\$145.152\end{array}$

Hence, the stock price in Year 2 of Stock Z is $145.152.

Compute the current stock price of Stock Z:

$\begin{array}{c}{\text{P}}_o=\frac{{\text{D}}_1}{{\left(\text{1+R}\right)}^1}+\frac{{\text{D}}_2}{{\left(\text{1+R}\right)}^2}+\frac{{\text{P}}_2}{{\left(\text{1+R}\right)}^2}\\ =\frac{\$4.50\times \left(1+\frac{20}{100}\right)}{\left(1+\frac{17}{100}\right)}+\frac{\$4.50\times {\left(1+\frac{20}{100}\right)}^2}{{\left(1+\frac{17}{100}\right)}^2}+\frac{\$145.152}{{\left(1+\frac{17}{100}\right)}^2}\\ =\frac{\$4.50\left(1.20\right)}{1.17}+\frac{\$4.50\times {\left(1.20\right)}^2}{{\left(1.17\right)}^2}+\frac{\$145.152}{{\left(1.17\right)}^2}\\ =\frac{\$\text{5}\text{.4}}{1.17}+\frac{\left(\$4.50\times 1.44\right)}{\text{1}\text{.3689}}+\frac{\$145.152}{\text{1}\text{.3689}}\\ =\$\text{4}\text{.61538}+\left(\frac{\$\text{6}\text{.48}}{\text{1}\text{.3689}}\right)+\$\text{106}\text{.03550}\\ =\$\text{4}\text{.61538}+\$\text{4}\text{.73372}+\$42.87334\\ =\$\text{52}\text{.22}\end{array}$

Hence, the stock price of Stock Z is $52.22.

Compute the dividend yield of Stock Z:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$4.50\times \left(1+\frac{20}{100}\right)}{\$52.22}\\ =\frac{\left(\$4.50\times 1.20\right)}{\$52.22}\\ =\frac{\$5.4}{\$52.22}\\ =0.1034\end{array}$

Hence, the dividend yield of the Stock Z is 0.1034 or 10.34%.