Chapter 7, Problem 31QP

Capital Gains versus Income. Consider four different stocks, all of which have a required return of 17 percent and a most recent dividend of $2.40 per share. Stocks W, X, and Y are expected to maintain constant growth rates in dividends for the foreseeable future of 8 percent, o percent, and –5 percent per year, respectively. Stock Z is a growth stock that will increase its dividend by 20 percent for the next two years and then maintain a constant 12 percent growth rate, thereafter. What is the dividend yield for each of these four stocks? What is the expected capital gains yield? Discuss the relationship among the various returns that you find for each of these stocks.

Summary Introduction

To determine: The dividend yield for each 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 company’s earnings. This can be issued or paid as shares of stock or cash payment.

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

Answer to Problem 31QP

The dividend yield of Stock W is 9%.

The dividend yield of Stock X is 16.99%.

The dividend yield of Stock Y is 22%.

The dividend yield of Stock Z is 4.68%.

Explanation of Solution

Given information:

Four different stocks have a required rate of return of 17% and the recent dividend is $2.40 per share. The Stock Z has a growth rate of 12%.

The constant growth rate in dividends of Stock W is 8%.

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 for each of the stocks:

• Firstly, determine the stock price for each stock. It is because 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 the each stock by subtracting the dividend yield from the total return.

Formulae:

The formula to calculate the price of 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:

$\begin{array}{c}{\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}\\ =\$2.40\times \frac{\left(1+\frac{8}{100}\right)}{\left(\frac{17}{100}-\frac{8}{100}\right)}\\ =\$2.40\times \frac{1.08}{\left(0.17-0.08\right)}\\ =\$2.40\times \left(\frac{1.08}{\text{0}\text{.09}}\right)\end{array}$

$\begin{array}{c}=\$2.40\times 12\\ =\$28.80\end{array}$

Hence, the stock price of the Stock W is $28.80.

Compute the dividend yield of the Stock W:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$2.40\times \left(1+\frac{8}{100}\right)}{\$28.80}\\ =\frac{\left(\$2.40\times 1.08\right)}{\$28.80}\end{array}$

$\begin{array}{c}=\frac{\$2.592}{\$28.80}\\ =0.09\end{array}$

Hence, the dividend yield of the Stock W is 0.09 or 9%.

Compute the stock price of Stock X:

$\begin{array}{c}{\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}\\ =\$2.40\times \frac{\left(1+\frac{0}{100}\right)}{\left(\frac{17}{100}-\frac{0}{100}\right)}\\ =\$2.40\times \frac{1}{\left(0.17-0.00\right)}\\ =\$2.40\times \left(\frac{1}{\text{0}\text{.17}}\right)\end{array}$

$\begin{array}{c}=\$2.40\times 5.88235\\ =\$14.12\end{array}$

Hence, the stock price of the Stock X is $14.12.

Compute the dividend yield of the Stock X:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$2.40}{\$14.12}\\ =0.1699\end{array}$

Hence, the dividend yield of the Stock X is 0.1699 or 16.99%.

Compute the stock price of Stock Y:

$\begin{array}{c}{\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}\\ =\$2.40\times \frac{\left(1+\left(-\frac{5}{100}\right)\right)}{\left(\frac{17}{100}+\left(-\frac{5}{100}\right)\right)}\\ =\$2.40\times \frac{\left(1-0.05\right)}{\left(0.17+0.05\right)}\\ =\$2.40\times \left(\frac{0.95}{0.22}\right)\end{array}$

$\begin{array}{c}=\$2.40\times 4.31818\\ =\$10.36\end{array}$

Hence, the stock price of the Stock Y is $10.36.

Compute the dividend yield of the Stock Y:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$2.40\times \left(1+\left(\frac{-5}{100}\right)\right)}{\$10.36}\\ =\frac{\$2.40\times \left(1-0.05\right)}{\$10.36}\\ =\frac{\left(\$2.40\times 0.95\right)}{\$10.36}\end{array}$

$\begin{array}{c}=\frac{\$2.28}{\$10.36}\\ =0.22\end{array}$

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

Compute the stock price in Year 2 of Stock Z:

Note: The stock starts at constant growth rate in the Year 3. As a result, now determine the price of stock in Year 2.

$\begin{array}{c}{\text{P}}_2={\text{D}}_o\frac{{\left(1+\text{g}\right)}^2\left(1+{\text{g}}_2\right)}{\left(\text{R}-\text{g}\right)}\\ =\$2.40\times \frac{{\left(1+\frac{20}{100}\right)}^2\times \left(1+\frac{12}{100}\right)}{\left(\frac{17}{100}-\frac{12}{100}\right)}\\ =\$2.40\times \left(\frac{{\left(1+0.20\right)}^2\times \left(1+0.12\right)}{\left(0.17-0.12\right)}\right)\\ =\$2.40\times \left(\frac{\left(1.44\times 1.12\right)}{\text{0}\text{.05}}\right)\end{array}$

$\begin{array}{c}=\$2.40\times \frac{1.6128}{0.05}\\ =\$2.40\times 32.256\\ =\$77.41\end{array}$

Hence, the stock price in Year 2 of Stock Z is $77.41.

Compute the current stock price of Stock Z:

$\begin{array}{c}{\text{P}}_o=\frac{{\text{D}}_1}{{\left(\text{1+R}\right)}^1}+\frac{{\text{D}}_2}{{\left(\text{1+R}\right)}^2}+\frac{{\text{P}}_2}{{\left(\text{1+R}\right)}^2}\\ =\frac{\$2.40\times \left(1+\frac{20}{100}\right)}{\left(1+\frac{17}{100}\right)}+\frac{\$2.40\times {\left(1+\frac{20}{100}\right)}^2}{{\left(1+\frac{17}{100}\right)}^2}+\frac{\$77.41}{{\left(1+\frac{17}{100}\right)}^2}\\ =\frac{\$2.40\times \left(1.20\right)}{1.17}+\frac{\$2.40\times {\left(1.20\right)}^2}{{\left(1.17\right)}^2}+\frac{\$77.41}{{\left(1.17\right)}^2}\\ =\frac{\$2.88}{1.17}+\frac{\left(\$2.40\times 1.44\right)}{1.3689}+\frac{\$77.41}{1.3689}\end{array}$

$\begin{array}{c}=\$2.46153+\frac{\$3.456}{1.3689}+\$42.87334\\ =\$2.46153+\$2.52465+\$56.54905\\ =\$61.54\end{array}$

Hence, the stock price of the Stock Z is $61.54.

Compute the dividend yield of the Stock Z:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$2.40\times \left(1+\frac{20}{100}\right)}{\$61.54}\\ =\frac{\left(\$2.40\times 1.20\right)}{\$61.54}\end{array}$

$\begin{array}{c}=\frac{\$2.88}{\$61.54}\\ =0.0468\end{array}$

Hence, the dividend yield of the Stock Z is 0.0468 or 4.68%.

Summary Introduction

To determine: The capital gains yield of four stocks.

Introduction:

Capital gains yield is a ratio that indicates the rise in the price of a common stock.

Answer to Problem 31QP

The capital gains yield of Stock W is 8%.

The capital gains yield of Stock X is 0.01%.

The capital gains yield of Stock Y is −5%.

The capital gains yield of Stock Z is 12.32%.

Explanation of Solution

Given information:

Four different stocks have a required rate of return of 17%. The computed dividend yield of each of the four stocks is as follows:

• The dividend yield of Stock W is 9%.
• The dividend yield of Stock X is 16.99%.
• The dividend yield of Stock Y is 22%.
• The dividend yield of Stock Z is 4.68%.

The formula to calculate the capital gains yield:

$\text{Capital gains yield}=\text{Total required return}-\text{Dividend yield}$

Compute the capital gains yield of Stock W:

$\begin{array}{c}\text{Capital gains yield}=\text{Total required return}-\text{Dividend yield}\\ =\frac{17}{100}-\frac{9}{100}\\ =0.17-0.09\\ =0.08\end{array}$

Hence, the capital gains yield of stock W is 0.08 or 8%.

Compute the capital gains yield of Stock X:

$\begin{array}{c}\text{Capital gains yield}=\text{Total required return}-\text{Dividend yield}\\ =\frac{17}{100}-\frac{16.99}{100}\\ =0.17-\text{0}\text{.1699}\\ =\text{0}\text{.0001}\end{array}$

Hence, the capital gains yield of stock X is 0.0001 or 0.01%.

Compute the capital gains yield of Stock Y:

$\begin{array}{c}\text{Capital gains yield}=\text{Total required return}-\text{Dividend yield}\\ =\frac{17}{100}-\frac{22}{100}\\ =0.17-0.22\\ =-0.05\end{array}$

Hence, the capital gains yield of stock Y is −0.05 or −5%.

Compute the capital gains yield of Stock Z:

$\begin{array}{c}\text{Capital gains yield}=\text{Total required return}-\text{Dividend yield}\\ =\frac{17}{100}-\frac{4.68}{100}\\ =0.17-0.0468\\ =0.1232\end{array}$

Hence, the capital gains yield of stock Z is 0.1232 or 12.32%.

Interpretation regarding the relationship among the various returns of each stock:

The entire four stocks have a required rate of return of 17%. However, this return is distributed in different ways between the capital gains and current income. As per the analysis, a higher growth stock has an appreciable capital gains yield but a relatively smaller current income yield. Moreover, a negative growth stock provides a higher current income even when the price decreases over time.