Chapter 9, Problem 23IC

a.

Summary Introduction

To describe: The legal rights of and privileges of common stockholders.

Introduction:

Stock Valuation:

The process to find the worth of a company’s share is called stock valuation. It can be computed based on the estimated future cash flows by company or on the basis of dividends payment by the company.

Answer to Problem 23IC

• The control of the company.
• The preventative right.
• Share in profitability.

Explanation of Solution

• The shareholders have the right to control the decision with regards to the election of directors.
• The preemptive right makes them first candidates for any new issue of shares before those shares are offered to the general public.
• Also, the shareholders are entitled to any dividend that is declared by the company.

Conclusion

So, the shareholders of a company have above rights and privileges.

b.

1.

Summary Introduction

To identify: The formula that can be used to value any stock irrespective of its dividend pattern.

Answer to Problem 23IC

The formula used to value a stock,

${\text{P}}_0=\frac{{\text{D}}_1}{\left(1+{\text{r}}_1\right)}+\frac{{\text{D}}_2}{{\left(1+{\text{r}}_2\right)}^2}+\frac{{\text{D}}_3}{{\left(1+{\text{r}}_3\right)}^3}+\mathrm{.......}\frac{{\text{D}}_{\text{n}}}{{\left(1+{\text{r}}_{\text{n}}\right)}^{\text{n}}}$

Where,

• ${\text{P}}_0$ is value of stock.
• $\text{D}$ is dividend at different time period.
• r is required rate of return for each period.
• n is number of period.

Explanation of Solution

• The dividend discount model is based on the expected dividend in future.
• It can be used to value a stock regardless of its dividend payment as it considered the dividend for each period separately.
• It computes the present value of each dividend payment regardless of its amount.
• The total value of dividends at current date is declared as the value of that stock.

Conclusion

So, the dividend discount model can be used to value a stock regardless of its dividend pattern.

2.

Summary Introduction

To explain: The constant growth stock and its valuation method.

Answer to Problem 23IC

• Constant growth stock is a stock that provides identical increase in dividend during each time period.
• The valuation of such stocks are as per the estimated dividend for the next period, discounted based on the difference of the expected rate of return and growth rate.

Explanation of Solution

• The constant growth stock has the same growth rate for each time period, depicted in percentage.
• The constant growth rate is deducted from the expected rate of return and the resulted rate is used to compute the present value of future dividend.

Conclusion

So, the constant growth stock is one with the constant growth in dividend and is valued based on the dividend discounted by expected rate minus growth rate.

3.

Summary Introduction

To identify: The implications of growth rate higher than expected rate of return. The possibilities of such scenario in short run and long run.

Answer to Problem 23IC

• The constant growth higher than expected rate of return is not possible in the long run.
• The value of a stock with a constant growth rate higher than expected rate of return is not determined based on dividend discount model.

Explanation of Solution

• In short run, the stock may have supernormal growth rate, but that stock’s value can’t be computed based on the dividend discount model as it requires expected rate higher than the constant growth rate.
• The long-run implies that the expected rate is based on the growth of the company and its percentage.
• So, the expected rate includes the effect of growth rate and can’t be lower than the constant growth in the long run.

Conclusion

So, the implication is the computation of stock value becomes tough and it is not possible for long run.

c.

Summary Introduction

To compute: The required rate of return of B.

Explanation of Solution

Given information:

Risk free rate is 3%.

Beta is 1.2.

Market risk premium is 5% $\left(8\%-3\%\right)$.

Formula to compute the required rate of return,

$\text{Required Rate of Retun}=\text{Risk-free Rate}+\text{Beta}\times \left(\text{Market Risk Premium}\right)$

Substitute 3% for risk free rate, 1.2 for beta and 5% for market risk premium.

$\begin{array}{c}\text{Required Rate of Retun}=3\%+1.2\times \left(5\%\right)\\ =3\%+6\%\\ =9\%\end{array}$

Conclusion

So, the required rate of return is 9%.

d.

1.

Summary Introduction

To compute: The expected dividend stream for next year.

Explanation of Solution

The spreadsheet to compute the expected dividend stream,

Table (1)

Conclusion

So, the dividend stream during next three years is $2.08, $2.16 and $2.25.

2.

Summary Introduction

To compute: The current stock price.

Explanation of Solution

Given information:

The current dividend is $2.

Growth rate is 4%.

Required rate of return is 9% (computed in part 1).

Formula to compute the current stock price,

$\text{Current Stock Price}=\frac{\text{Current Dividend}\left(1+\text{Growth Rate}\right)}{\text{Required Rate of Retun}-\text{Growth Rate}}$

Substitute $2 for dividend and 9% for required rate of return and 4% for growth rate.

$\begin{array}{c}\text{Current Stock Price}=\frac{\$2\left(1+0.04\right)}{0.09-0.04}\\ =\frac{2.08}{0.05}\\ =\$41.6\end{array}$

Conclusion

So, the current stock price is $41.6.

3.

Summary Introduction

To compute: The expected value of stock after one year.

Explanation of Solution

Given information:

The current dividend is $2.

Growth rate is 4%.

Required rate of return is 9% (computed in part 1).

Formula to compute the expected value of stock,

$\text{Value of Stock}=\frac{\text{Current Dividend}\times {\left(1+\text{Growth Rate}\right)}^2}{{\left(\text{Required Rate of Retun}-\text{Growth Rate}\right)}^1}$

Substitute $2 for dividend and 9% for required rate of return and 4% for growth.

$\begin{array}{c}\text{Value of Stock}=\frac{\text{\$2}\times {\left(1+\text{0}\text{.04}\right)}^2}{{\left(\text{0}\text{.09}-0.04\right)}^1}\\ =\frac{\$2.1632}{0.05}\\ =\$43.26\end{array}$

Conclusion

So, the value of stock after one year is $43.26.

4.

Summary Introduction

To compute: The dividend yield, capital gains yield and total return during first year.

Explanation of Solution

Given information:

Dividend after one year is $2.08 (refer part d-1).

Current stock price is $41.6 (refer part d-3).

Stock price at year1 is $43.26 (refer part d-3).

Formula to compute the dividend yield,

$\text{Dividend Yield}=\frac{\text{Dividend at Year1}}{\text{Current Stock Price}}$

Substitute $2.08 for dividend at year1 and $41.6 for current stock price.

$\begin{array}{c}\text{Dividend Yield}=\frac{\$2.08}{\$41.6}\\ =0.05\text{ or 5\%}\end{array}$

Formula to compute the capital gain yield,

$\text{Capital Gain Yield}=\frac{\text{Stock Price at Year1}-\text{Current Stock Price}}{\text{Current Stock Price}}$

Substitute $43.26 for stock price at year1 and $41.6 for current stock price.

$\begin{array}{c}\text{Capital Gain Yield}=\frac{\text{\$43}\text{.26}-\$41.6}{\$41.6}\\ =\frac{\$1.66}{\$41.6}\\ =0.00399\text{ or 3}\text{.99\%}\end{array}$

Calculated values,

Dividend yield is 5%.

Capital gain yield is 3.99%.

Formula to compute total return during year1,

$\begin{array}{c}\text{Total Return}=\text{Dividend Yield}+\text{Capital Gain Yield}\\ =5\%+3.99\%\\ =8.99\%\end{array}$

Conclusion

So, the dividend yield is 5%, capital gain yield is 3.99%and total return is 8.99% during year1.

e.

Summary Introduction

To compute: The expected rate of return if current stock price is $30.29.

Explanation of Solution

Given information:

Current stock price is $40.

Dividend for year1 is $2.08.

Growth rate is 4%.

Formula to compute expected rate of return,

$\text{Expected Rate of Return}=\frac{\text{Dividend at Year1}}{\text{Current Stock Price}}+\text{Growth Rate}$

Substitute $2.08 for dividend at year1, $40 for current stock price and 4% for growth rate.

$\begin{array}{c}\text{Expected Rate of Return}=\frac{\$2.08}{\$40}+0.04\\ =0.052+0.04\\ =0.092\text{ or 9}\text{.2\%}\end{array}$

Conclusion

So, the expected rate of return is 9.2%.

f.

Summary Introduction

To compute: The stock price if dividend growth rate is zero.

Explanation of Solution

Given information:

Required rate of return is 8%.

Dividend is $2.

Formula to compute the stock price,

$\text{Stock Price}=\frac{\text{Dividend}}{\text{Required Rate of Rate}}$

Substitute $2 for dividend and 8% for required rate of return.

$\begin{array}{r}\text{Stock Price}=\frac{\$2}{0.08}\\ =\$25\end{array}$

Conclusion

So, the stock price is $25.

g.

Summary Introduction

To compute: The stock value, expected dividend yield and capital gains yield in year1 and year4 if initial growth is 30%.

Explanation of Solution

Given information:

Growth rate for first three years is 30%.

Current dividend is $2.

Table (2)

Now, compute the terminal value for dividends after year3.

Given,

Dividend at year3 is $4.39.

Constant growth rate is 4%.

Required rate of return is 9%.

Formula to compute the horizon value,

$\text{Horizon Value}=\frac{\text{Dividend}\left(1+\text{g}\right)}{\text{Required Rate of Return}-\text{Growth Rate}}$

Substitute $4.39 for dividend, 9% for required rate of return and 4% for growth rate.

$\begin{array}{c}\text{Horizon Value}=\frac{\$4.39\left(1+0.04\right)}{0.09-0.04}\\ =\frac{\$4.5656}{0.05}\\ =\$91.31\end{array}$

Compute the present value of this horizon value,

$\begin{array}{c}\text{PV of Horizon Value}=\frac{\text{Horizon Value}}{{\left(1+0.09\right)}^3}\\ =\frac{\$91.31}{1.295029}\\ =\$70.51\end{array}$

The sum of total present value of dividends during non-constant growth and the present value of horizon value will the value of stock today.

$\begin{array}{c}\text{Stock Value}=\$9.52+\$70.51\\ =\$80.03\end{array}$

Compute dividend yield,

Given,

Dividend at year1 is $2.60.

Current stock price is $80.03.

Formula to compute the dividend yield,

$\text{Dividend Yield}=\frac{\text{Dividend at Year1}}{\text{Current Stock Price}}$

Substitute $2.60 for dividend at year1 and $80.03 for current stock price.

$\begin{array}{c}\text{Dividend Yield}=\frac{\$2.60}{\$80.03}\\ =0.032\text{ or 3}\text{.2\%}\end{array}$

Compute capital gain yield,

Given,

Current price of stock is 80.03.

Stock price at year1 is 77.64.

Formula to compute the capital gain yield,

$\text{Capital Gain Yield}=\frac{\text{Stock Price at Year1}-\text{Current Stock Price}}{\text{Current Stock Price}}$

Substitute $77.64 for stock price at year1 and $80.03 for current stock price.

$\begin{array}{c}\text{Capital Gain Yield}=\frac{\text{\$77}\text{.64}-\$80.03}{80.03}\\ =\frac{-\$2.39}{\$80.03}\\ =-0.0298\text{ or }-2.98\text{\%}\end{array}$

Working note:

Table (3)

Calculation of stock price at year 1,

$\begin{array}{c}\text{Stock Price at Year1}=\$7.13+\$70.51\\ =\$77.64\end{array}$

Conclusion

So, the stock value is $80.03, dividend yield is 3.2% and capital gain yield is -2.98%.

h.

Summary Introduction

To compute: The stock value, expected dividend yield and capital gains yield in year1 and year4 if initial growth is 30%.

Explanation of Solution

Given information:

Growth rate for first three years is zero.

Current dividend is $2.

Table (4)

Now, compute the terminal value for dividends after year3.

Given,

Dividend at year3 is $2.

Constant growth rate is 4%.

Required rate of return is 9%.

Formula to compute the horizon value,

$\text{Horizon Value}=\frac{\text{Dividend}\left(1+\text{g}\right)}{\text{Required Rate of Return}-\text{Growth Rate}}$

Substitute $2 for dividend, 9% for required rate of return and 4% for growth rate.

$\begin{array}{c}\text{Horizon Value}=\frac{\$2\left(1+0.04\right)}{0.09-0.04}\\ =\frac{\$2.08}{0.05}\\ =\$41.6\end{array}$

Compute the present value of this horizon value,

$\begin{array}{c}\text{PV of Horizon Value}=\frac{\text{Horizon Value}}{{\left(1+0.09\right)}^3}\\ =\frac{\$41.6}{1.295029}\\ =\$32.12\end{array}$

The sum of total present value of dividends during non-constant growth and the present value of horizon value will the value of stock today.

$\begin{array}{c}\text{Stock Value}=\$5.50+\$32.12\\ =\$37.62\end{array}$

Compute dividend yield,

Given,

Dividend at year1 is $2.

Current stock price is $37.62.

Formula to compute the dividend yield,

$\text{Dividend Yield}=\frac{\text{Dividend at Year1}}{\text{Current Stock Price}}$

Substitute $2 for dividend at year1 and $37.62 for current stock price.

$\begin{array}{c}\text{Dividend Yield}=\frac{\$2}{\$37.62}\\ =0.05316\text{ or 5}\text{.32\%}\end{array}$

Compute capital gain yield,

Given,

Current price of stock is $37.62.

Stock price at year1 is $35.64.

Formula to compute the capital gain yield,

$\text{Capital Gain Yield}=\frac{\text{Stock Price at Year1}-\text{Current Stock Price}}{\text{Current Stock Price}}$

Substitute $35.64 for stock price at year1 and $37.62 for current stock price.

$\begin{array}{c}\text{Capital Gain Yield}=\frac{\$35.64-\$37.62}{37.62}\\ =\frac{-\$1.98}{\$37.62}\\ =-0.05263\text{ or }-5.27\text{\%}\end{array}$

Working note:

Table (5)

Calculation of stock price at year 1,

$\begin{array}{c}\text{Stock Price at Year1}=\$3.52+\$32.12\\ =\$35.64\end{array}$

Conclusion

So, the stock value is $37.62, dividend yield is 5.32% and capital gain yield is -5.27%.

i.

Summary Introduction

To compute: The price of share, also calculate the expected dividend and capital gain yields.

Explanation of Solution

Given information:

The current dividend is $2.

Growth rate is $-$4%.

Required rate of return is 9% (computed in part 1 of d).

Formula to compute the current stock price,

$\text{Current Stock Price}=\frac{\text{Current Dividend}\left(1+\text{Growth Rate}\right)}{\text{Required Rate of Retun}-\text{Growth Rate}}$

Substitute $2 for dividend and 9% for required rate of return and $-$4% for growth rate.

$\begin{array}{c}\text{Current Stock Price}=\frac{\$2\left(1-0.04\right)}{0.09+0.04}\\ =\frac{1.92}{0.13}\\ =\$14.77\end{array}$

Calculate expected dividend.

Given,

The current dividend is $2.

Growth rate is $-$4%.

Formula to calculate expected dividend is,

$\text{Expected dividend}=\text{Current Dividend}\left(1+\text{Growth Rate}\right)$

Substitute $2 for current dividend and $-$4% for growth rate in the above equation.

$\begin{array}{c}\text{Expected dividend}=\$2\left(1-0.04\right)\\ =\$2\times 0.96\\ =\$1.92\end{array}$

Calculate capital gain yield.

Given,

Stock price at year 1 is $43.26.

Current stock price is $14.77 as calculated above when growth rate is negative 6%.

Formula to compute the capital gain yield,

$\text{Capital Gain Yield}=\frac{\text{Stock Price at Year1}-\text{Current Stock Price}}{\text{Current Stock Price}}$

Substitute $43.26 for stock price at year1 and $14.77 for current stock price.

$\begin{array}{c}\text{Capital Gain Yield}=\frac{\text{\$43}\text{.26}-\$14.77}{\$14.77}\\ =\frac{\$28.49}{\$14.77}\\ =1.928\end{array}$

Conclusion

Thus, if the growth is negative the stock price is reduces and this is the reason the investor not want to purchase this stock, the current stock price is $14.77. The expected dividend for the next year is $1.92 and the capital gain yield is 1.928 respectively.

j.

Summary Introduction

To compute: The total value of B. and the price per share.

Explanation of Solution

Calculated values,

Cash flow of year 3 is $20 million.

WACC is 7%.

Growth rate is 5%.

Present value total is $20.38 million.

Formula to calculate the total value of B stock is,

$\text{Total value}=\left[\begin{array}{l}\left(\frac{\text{Year}3\text{ cash flow}\left(1+\text{Growth rate}\right)}{\left(\text{WACC}-\text{Growth rate}\right)}\times \text{Year}3\text{ discount factor}\right)\\ \text{+Total of present values}\end{array}\right]$

Substitute $20 for year 3 cash flows, growth rate is 4%, WACC is 7% and total of present value is $20.38 million in the above equation.

$\begin{array}{c}\text{Total value}=\left[\frac{\$20\text{ million}\left(1+0.05\right)}{\left(0.07-0.05\right)}\times 0.816+\$20.38\text{ million}\right]\\ =\left[\frac{\$21\text{ million}}{0.02}\times 0.816+\$20.38\text{ million}\right]\\ =\$877.18\text{ million}\end{array}$

Calculate share price.

Table (6)

Working Notes:

Calculate the present value of 3 year’s cash flow.

Table (7)

Conclusion

Thus, total value of B is $877.18 million and the price per share is $83.72 respectively.

k.

Summary Introduction

To determine: The preferred stock expected return and the expected return if the stock was a perpetual issue or if it had 20-year maturity.

Explanation of Solution

Given information:

Annual dividend is of $5.

Issue price is $100 per share

Formula to calculate the expected return is,

$\text{Expected return}=\frac{\text{Dividend}}{\text{Issue price}}$

Substitute $5 for dividend and $100 per share for issue price in the above equation

$\begin{array}{c}\text{Expected return}=\frac{\$5}{\$100}\\ =0.05\text{ or 5\%}\end{array}$

Conclusion

Thus, the expected return of preferred stock is 10% and it will not change if the preferred stock was a perpetual issue or if it had 20-year maturity.