Chapter 9, Problem 12P

VALUATION OF A CONSTANT GROWTH STOCK Investors require a 15% rate of return on Levine Company’s stock that is, r_s = 15%).

1. a. What is its value if the previous dividend was D₀ = $2 and investors expect dividends to grow at a constant annual rate of (1) −5%, (2) 0%, (3) 5%, or (4) 10%?
2. b. Using data from Part a, what would the Gordon (constant growth) model value be if the required rate of return was 15% and the expected growth rate was (1) 15% or (2) 20%? Are these reasonable results? Explain.
3. c. Is it reasonable to think that a constant growth stock could have g > r_s? Why or why not?

Summary Introduction

To compute: The value of stock for company L with constant growth in dividends.

Value of Stock:

Value of stock is an amount computed to evaluate the stock of a company for investment purposes. It determines the dividend payout at the present value at required rate of return less growth rate or plus growth rate for stock with declining growth in dividends.

Explanation of Solution

Given,

Dividend is $2.

Required rate of return is 15%.

Growth rate is $-5\%$.

Formula to compute stock value,

${\text{P}}_0=\frac{{\text{D}}_0\times \left(1+\text{g}\right)}{{\text{r}}_{\text{s}}-\text{g}}$

Where,

• ${\text{D}}_0$ is dividend.
• ${\text{P}}_{\text{0}}$ is current value of stock.
• ${\text{r}}_{\text{s}}$ is required rate of return.
• $\text{g}$ is growth rate.

Substitute $2 for ${\text{D}}_0$ and 15% for ${\text{r}}_{\text{s}}$ and –5% for $\text{g}$.

$\begin{array}{c}{\text{P}}_0=\frac{\$2\times \left(1-0.05\right)}{0.15+0.05}\\ =\frac{\$1.9}{0.20}\\ =\$9.5\end{array}$

Conclusion

So, the current value of L Company’s stock is $9.5.

Summary Introduction

To compute: The value of stock for company L with constant growth in dividends.

Explanation of Solution

Given,

Dividend is $2.

Required rate of return is 15%.

Growth rate is 0%.

Formula to compute stock value,

${\text{P}}_0=\frac{{\text{D}}_0\times \left(1+\text{g}\right)}{{\text{r}}_{\text{s}}-\text{g}}$

Where,

• ${\text{D}}_0$ is dividend.
• ${\text{P}}_{\text{0}}$ is current value of stock.
• ${\text{r}}_{\text{s}}$ is required rate of return.
• $\text{g}$ is growth rate.

Substitute $2 for ${\text{D}}_0$ and 15% for ${\text{r}}_{\text{s}}$ and 0% for $\text{g}$.

$\begin{array}{c}{\text{P}}_0=\frac{\$2\times \left(1+0\right)}{0.15-0}\\ =\frac{\$2}{0.15}\\ =\$13.33\end{array}$

Conclusion

So, the current value of L Company’s stock is $13.33.

Summary Introduction

To compute: The value of stock for company L with constant growth in dividends.

Explanation of Solution

Given,

Dividend is $2.

Required rate of return is 15%.

Growth rate is 5%.

Formula to compute stock value,

${\text{P}}_0=\frac{{\text{D}}_0\times \left(1+\text{g}\right)}{{\text{r}}_{\text{s}}-\text{g}}$

Where,

• ${\text{D}}_0$ is dividend.
• ${\text{P}}_{\text{0}}$ is current value of stock.
• ${\text{r}}_{\text{s}}$ is required rate of return.
• $\text{g}$ is growth rate.

Substitute $2 for ${\text{D}}_0$ and 15% for ${\text{r}}_{\text{s}}$ and 5% for $\text{g}$.

$\begin{array}{c}{\text{P}}_0=\frac{\$2\times \left(1+0.05\right)}{0.15-0.05}\\ =\frac{\$2.1}{0.10}\\ =\$21\end{array}$

Conclusion

So, the current value of L Company’s stock is $21.

Summary Introduction

To compute: The value of stock for company L with constant growth in dividends.

Explanation of Solution

Given,

Dividend is $2.

Required rate of return is 15%.

Growth rate is 10%.

Formula to compute stock value,

${\text{P}}_0=\frac{{\text{D}}_0\times \left(1+\text{g}\right)}{{\text{r}}_{\text{s}}-\text{g}}$

Where,

• ${\text{D}}_0$ is dividend.
• ${\text{P}}_{\text{0}}$ is current value of stock.
• ${\text{r}}_{\text{s}}$ is required rate of return.
• $\text{g}$ is growth rate.

Substitute $2 for ${\text{D}}_0$ and 15% for ${\text{r}}_{\text{s}}$ and 10% for $\text{g}$.

$\begin{array}{c}{\text{P}}_0=\frac{\$2\times \left(1+0.10\right)}{0.15-0.10}\\ =\frac{\$2.2}{0.05}\\ =\$44\end{array}$

Conclusion

So, the current value of L Company’s stock is $44.

Summary Introduction

To compute: The value of stock for company L with constant growth in dividends.

Explanation of Solution

Given,

Dividend is $2.

Required rate of return is 15%.

Growth rate is 15%.

Formula to compute stock value,

${\text{P}}_0=\frac{{\text{D}}_0\times \left(1+\text{g}\right)}{{\text{r}}_{\text{s}}-\text{g}}$

Where,

• ${\text{D}}_0$ is dividend.
• ${\text{P}}_{\text{0}}$ is current value of stock.
• ${\text{r}}_{\text{s}}$ is required rate of return.
• $\text{g}$ is growth rate.

Substitute $2 for ${\text{D}}_0$ and 15% for ${\text{r}}_{\text{s}}$ and 15% for $\text{g}$.

$\begin{array}{c}{\text{P}}_0=\frac{\$2\times \left(1+0.15\right)}{0.15-0.15}\\ =\frac{\$2.3}{0}\\ =\text{ Can not be computed}\end{array}$

Conclusion

So, the current value of L Company’s stock can’t be computed as this is not reasonable to have growth rate equal to or greater than required rate of return.

Summary Introduction

To compute: The value of stock for company L with constant growth in dividends.

Explanation of Solution

Given,

Dividend is $2.

Required rate of return is 15%.

Growth rate is 20%.

Formula to compute stock value,

${\text{P}}_0=\frac{{\text{D}}_0\times \left(1+\text{g}\right)}{{\text{r}}_{\text{s}}-\text{g}}$

Where,

• ${\text{D}}_0$ is dividend.
• ${\text{P}}_{\text{0}}$ is current value of stock.
• ${\text{r}}_{\text{s}}$ is required rate of return.
• $\text{g}$ is growth rate.

Substitute $2 for ${\text{D}}_0$ and 15% for ${\text{r}}_{\text{s}}$ and 20% for $\text{g}$.

$\begin{array}{c}{\text{P}}_0=\frac{\$2\times \left(1+0.20\right)}{0.15-0.20}\\ =\frac{\$2.4}{-0.05}\\ =-\$48\end{array}$

Conclusion

So, the current value of L Company’s stock is -$48 and this is not reasonable to have growth rate equal to or greater than required rate of return.

Summary Introduction

To explain: If a constant growth stock can have growth rate more than required rate of return.

Answer to Problem 12P

No, it is not reasonable because a company is not supposed to give back the stock itself along with the periodic dividends.

Explanation of Solution

• The growth rate reflects the change in periodic dividends payments.
• The growth rate can’t be more than the required return as it will result in negative value of stock (refer part 2 of b).

Conclusion

So, it is not reasonable to have growth rate more than required rate of return for a stock.