Chapter 9, Problem 29QP

Capital Gains versos Income Consider four different stocks, all of which have a required return of 14 percent and a most recent dividend of $3.50 per share. Stocks W, X, and Y are expected to maintain constant growth rates in dividends for the foreseeable future of 7 percent, 0 percent, and –5 percent per year, respectively. Stock Z is a growth stock that will increase its dividend by 30 percent for the next two year and then maintain a constant 8 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 yields for each stock and also computes the capital gain yield also express the relationship between dividend yield and capital gain yield.

Dividend Yield:

It is a relation of dividend and its share price, it interpret that the investment in share gives the appropriate amount of dividend to the investor.

Capital Gain Yield:

It is a method used to calculate the capital gain in the share price of a stock. The capital gain is computed by dividing the rise in share by its original price of the stock.

Answer to Problem 29QP

Solution: The dividend yield of W, X, Y and Z is 7%, 14%, 19% and 5.03%. The capital gain yield of W, X, Y and Z is 7%, 0%, -5% and 8.97%.

Explanation of Solution

Formula to calculate dividend yield is,

$\text{Dividend yeild}=\frac{\text{Dividend}\left(1+\text{Growth rate}\right)}{\text{Share price}}$ (1)

Formula to calculate the capital gain yield is,

$\text{Capital gain yeild}=\text{Rate of return}-\text{Dividend yeild}$ (2)

For stock W

Given,

Dividend is $3.50.

Growth rate is 7%.

Share price is $53.50.

Calculate dividend yield.

Substitute $3.50 for dividend, 7% for growth rate and $53.50 for share price in equation (1).

$\begin{array}{c}\text{Dividend yeild}=\frac{\$3.50\left(1+0.07\right)}{\$53.50}\\ =\frac{\$3.745}{\$53.50}\\ =0.07\text{ or 7\%}\end{array}$

Calculate capital gain yield.

Substitute 14% for rate of return and 7% for dividend yield in equation (2).

$\begin{array}{c}\text{Capital gain yeild}=0.14-0.07\\ =0.07\text{ or }7\%\end{array}$

For stock X

Given,

Dividend is $3.50.

Growth rate is 0%.

Share price is $25.

Calculate dividend yield.

Substitute $3.50 for dividend, 0% for growth rate and $25 for share price in equation (1).

$\begin{array}{c}\text{Dividend yeild}=\frac{\$3.50\left(1+0\right)}{\$25}\\ =\frac{\$3.50}{\$25}\\ =0.14\text{ or 14\%}\end{array}$

Calculate capital gain yield.

Substitute 14% for rate of return and 14% for dividend yield in equation (2).

$\begin{array}{c}\text{Capital gain yeild}=0.14-0.14\\ =0\%\end{array}$

For stock Y

Given,

Dividend is $3.50.

Growth rate is 5%.

Share price is $17.50.

Calculate dividend yield.

Substitute $3.50 for dividend, 5% for growth rate and $17.50 for share price in equation (1).

$\begin{array}{c}\text{Dividend yeild}=\frac{\$3.50\left(1+\left(-0.05\right)\right)}{\$17.50}\\ =\frac{\$3.325}{\$17.50}\\ =0.19\text{ or 19\%}\end{array}$

Calculate capital gain yield.

Substitute 14% for rate of return and 19% for dividend yield in equation (2).

$\begin{array}{c}\text{Capital gain yeild}=0.14-0.19\\ =-0.05\text{ or }-5\%\end{array}$

For stock Z

Given,

Dividend is $3.50.

Growth rate is 30%.

Share price is $90.47.

Calculate dividend yield.

Substitute $3.50 for dividend, 30% for growth rate and $90.47 for share price in equation (1).

$\begin{array}{c}\text{Dividend yeild}=\frac{\$3.50\left(1+0.30\right)}{\$90.47}\\ =\frac{\$4.55}{\$90.47}\\ =0.0503\text{ or 5}\text{.03\%}\end{array}$

Calculate capital gain yield.

Substitute 14% for rate of return and 5.03% for dividend yield in equation (2).

$\begin{array}{c}\text{Capital gain yeild}=0.14-0.0503\\ =0.0897\text{ or }8.97\%\end{array}$

Working notes:

Calculate the stock price of stock W,

$\begin{array}{c}\text{Share price}=\frac{\text{Dividend}\left(1+\text{Growth rate}\right)}{\left(\text{Required rate of return}-\text{Growth rate}\right)}\\ =\frac{\$3.50\left(1+0.07\right)}{\left(0.14-0.07\right)}\\ =\frac{\$3.745}{0.07}\\ =\$53.50\end{array}$

Calculate the stock price of stock X,

$\begin{array}{c}\text{Share price}=\frac{\text{Dividend}\left(1+\text{Growth rate}\right)}{\left(\text{Required rate of return}-\text{Growth rate}\right)}\\ =\frac{\$3.50\left(1+0\right)}{\left(0.14-0\right)}\\ =\frac{\$3.50}{0.14}\\ =\$25\end{array}$

Calculate the stock price of stock Y.

$\begin{array}{c}\text{Share price}=\frac{\text{Dividend}\left(1+\text{Growth rate}\right)}{\left(\text{Required rate of return}-\text{Growth rate}\right)}\\ =\frac{\$3.50\left(1+\left(-0.05\right)\right)}{\left(0.14-\left(-0.05\right)\right)}\\ =\frac{\$3.325}{0.19}\\ =\$17.5\end{array}$

Calculate the stock price of stock Z for next 2 years,

$\begin{array}{c}\text{Price}=\frac{\text{Dividend }{\left(1+{\text{Growth rate}}_1\right)}^2\left(1+{\text{Growth rate}}_2\right)}{\left(\text{Rate of return}-\text{Growth rate}\right)}\\ =\frac{\$3.50{\left(1+0.30\right)}^2\left(1+0.08\right)}{\left(0.14-0.08\right)}\\ =\frac{\left(\$3.50\times 1.69\times 1.08\right)}{0.06}\\ =\$106.47\end{array}$

Calculate current price of Z,

$\begin{array}{c}\text{Price}=\frac{\$3.50\left(1+0.30\right)}{\left(1.14\right)}+\frac{\$3.50{\left(1+0.30\right)}^2}{{\left(1.14\right)}^2}+\frac{\$106.47}{{\left(1.14\right)}^2}\\ =\$90.47\end{array}$

The relation between capital gain yield and dividend yield is that the stock which has high growth rate having high capital gain yield and dividend yield will be low.

Conclusion

Thus, the dividend yield of W, X, Y and Z is 7%, 14%, 19% and 5.03%. The capital gain yield of W, X, Y and Z is 7%, 0%, -5% and 8.97%.