Chapter 7, Problem 31QP

Summary Introduction

To determine: The dividend yield for each four stocks.

Introduction:

Dividend is a sum of money that is 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 firm pays as dividends every year compared with its share price. It is considered as the stock’s return on investment.

Answer to Problem 31QP

The dividend yield of Stock W is 8%.

The dividend yield of Stock X is 16%.

The dividend yield of Stock Y is 21%.

The dividend yield of Stock Z is 3.7%.

Explanation of Solution

Given information:

Four different stocks have a required rate of return of 16% and the recent dividend is $2.80 per share. Stock Z has a growth rate of 20%.

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%.

Notes:

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 16%, which is the sum of dividend yield and capital gains yield.
• After determining the price for 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.

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:

$\begin{array}{c}{\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}\\ =\$2.80\times \frac{\left(1+\frac{8}{100}\right)}{\left(\frac{16}{100}-\frac{8}{100}\right)}\\ =\$2.80\times \frac{1.08}{\left(0.16-0.08\right)}\\ =\$2.80\times \left(\frac{1.08}{\text{0}\text{.08}}\right)\end{array}$

$\begin{array}{c}=\$2.80\times 13.5\\ =\$37.8\end{array}$

Hence, the stock price of Stock W is $37.8.

Compute the dividend yield of Stock W:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$2.80\times \left(1+\frac{8}{100}\right)}{\$37.8}\\ =\frac{\left(\$2.80\times 1.08\right)}{\$37.8}\end{array}$

$\begin{array}{c}=\frac{\$3.024}{\$37.8}\\ =0.08\end{array}$

Hence, the dividend yield of Stock W is 0.08 or 8%.

Compute the stock price of Stock X:

$\begin{array}{c}{\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}\\ =\$2.80\times \frac{\left(1+\frac{0}{100}\right)}{\left(\frac{16}{100}-\frac{0}{100}\right)}\\ =\$2.80\times \frac{1}{\left(0.16-0.00\right)}\end{array}$

$\begin{array}{l}=\$2.80\times \left(\frac{1}{\text{0}\text{.16}}\right)\\ =\$17.5\end{array}$

Hence, the stock price of Stock X is $17.5.

Compute the dividend yield of the Stock X:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$2.80}{\$17.5}\\ =0.16\end{array}$

Hence, the dividend yield of Stock X is 0.16 or 16%.

Compute the stock price of Stock Y:

$\begin{array}{c}{\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}\\ =\$2.80\times \frac{\left(1+\left(-\frac{5}{100}\right)\right)}{\left(\frac{16}{100}-\left(-\frac{5}{100}\right)\right)}\\ =\$2.80\times \frac{\left(1-0.05\right)}{\left(0.16+0.05\right)}\\ =\$2.80\times \left(\frac{0.95}{0.21}\right)\end{array}$

$\begin{array}{c}=\$2.80\times 4.52380\\ =\$12.67\end{array}$

Hence, the stock price of Stock Y is $12.67.

Compute the dividend yield of Stock Y:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$2.80\times \left(1+\left(\frac{-5}{100}\right)\right)}{\$12.67}\\ =\frac{\$2.80\times \left(1-0.05\right)}{\$12.67}\\ =\frac{\left(\$2.80\times 0.95\right)}{\$12.67}\end{array}$

$\begin{array}{c}=\frac{\$2.66}{\$12.67}\\ =0.21\end{array}$

Hence, the dividend yield of Stock Y is 0.21 or 21%.

Compute the stock price in Year 2 of Stock Z:

$\begin{array}{c}{\text{P}}_2={\text{D}}_o\frac{{\left(1+\text{g}\right)}^2\left(1+{\text{g}}_2\right)}{\left(\text{R}-\text{g}\right)}\\ =\$2.80\times \frac{{\left(1+\frac{20}{100}\right)}^2\times \left(1+\frac{12}{100}\right)}{\left(\frac{16}{100}-\frac{12}{100}\right)}\\ =\$2.80\times \left(\frac{{\left(1+0.20\right)}^2\times \left(1+0.12\right)}{\left(0.16-0.12\right)}\right)\\ =\$2.80\times \left(\frac{\left(1.44\times 1.12\right)}{\text{0}\text{.04}}\right)\end{array}$

$\begin{array}{c}=\$2.80\times \frac{1.6128}{0.04}\\ =\$2.80\times 40.32\\ =\$112.90\end{array}$

Hence, the stock price in Year 2 of Stock Z is $112.90.

Compute the current stock price of Stock Z:

$\begin{array}{c}{\text{P}}_o=\frac{{\text{D}}_1}{{\left(\text{1+R}\right)}^1}+\frac{{\text{D}}_2}{{\left(\text{1+R}\right)}^2}+\frac{{\text{P}}_2}{{\left(\text{1+R}\right)}^2}\\ =\frac{\$2.80\times \left(1+\frac{20}{100}\right)}{\left(1+\frac{16}{100}\right)}+\frac{\$2.80\times {\left(1+\frac{20}{100}\right)}^2}{{\left(1+\frac{16}{100}\right)}^2}+\frac{\$112.90}{{\left(1+\frac{16}{100}\right)}^2}\\ =\frac{\$2.80\times \left(1.20\right)}{1.16}+\frac{\$2.80\times {\left(1.20\right)}^2}{{\left(1.16\right)}^2}+\frac{\$112.90}{{\left(1.16\right)}^2}\\ =\frac{\$3.36}{1.16}+\frac{\left(\$2.80\times 1.44\right)}{1.3456}+\frac{\$112.90}{1.3456}\end{array}$

$\begin{array}{c}=\$2.8966+\$3.9964+\$83.9031\\ =\$90.7961\end{array}$

Hence, the stock price of Stock Z is $90.7961.

Compute the dividend yield of Stock Z:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$2.80\times \left(1+\frac{20}{100}\right)}{\$90.7961}\\ =\frac{\left(\$2.80\times 1.20\right)}{\$90.7961}\end{array}$

$\begin{array}{c}=\frac{\$3.36}{\$90.7961}\\ =0.0370\end{array}$

Hence, the dividend yield of Stock Z is 0.0370 or 3.7.

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%.

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

The capital gains yield of Stock Z is 12.3%.

Explanation of Solution

Given information:

Four different stocks have a required rate of return of 16%. The computed dividend yield of each of the four stocks is as follows:

The dividend yield of Stock W is 8%.

The dividend yield of Stock X is 16%.

The dividend yield of Stock Y is 21%.

The dividend yield of Stock Z is 3.7%.

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:

$\begin{array}{c}\text{Capital gains yield}=\text{Total required return}-\text{Dividend yield}\\ =\frac{16}{100}-\frac{8}{100}\\ =0.16-0.08\\ =0.08\end{array}$

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

Compute the capital gains yield of Stock X:

$\begin{array}{c}\text{Capital gains yield}=\text{Total required return}-\text{Dividend yield}\\ =\frac{16}{100}-\frac{16}{100}\\ =0.16-\text{0}\text{.16}\\ =\text{0}\text{.00}\end{array}$

Hence, the capital gains yield of Stock X is 0%.

Compute the capital gains yield of Stock Y:

$\begin{array}{c}\text{Capital gains yield}=\text{Total required return}-\text{Dividend yield}\\ =\frac{16}{100}-\frac{21}{100}\\ =0.16-0.21\\ =-0.05\end{array}$

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

Compute the capital gains yield of Stock Z:

$\begin{array}{c}\text{Capital gains yield}=\text{Total required return}-\text{Dividend yield}\\ =\frac{16}{100}-\frac{3.7}{100}\\ =0.16-0.037\\ =0.123\end{array}$

Hence, the capital gains yield of Stock Z is 0.123 or 12.3%.

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

The entire four stocks have a required rate of return of 16%. 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.