Chapter 4, Problem 18PS

Two-stage DCF model* Consider the following three stocks:

1. a. Stock A is expected to provide a dividend of $10 a share forever.
2. b. Stock B is expected to pay a dividend of $5 next year. Thereafter, dividend growth is expected to be 4% a year forever.
3. c. Stock C is expected to pay a dividend of $5 next year. Thereafter, dividend growth is expected to be 20% a year for five years (i.e., years 2 through 6) and zero thereafter.

If the market capitalization rate for each stock is 10%, which stock is the most valuable? What if the capitalization rate is 7%?

Summary Introduction

To determine: The more valuable stock if market capitalization rate is 10% and 7%.

Explanation of Solution

Compute price of the each stocks:

If market capitalization rate is 10%

$\begin{array}{c}{\text{P}}_{\text{0}}{}_{\text{Stock A}}=\frac{{\text{DIV}}_{\text{1}}}{r}\\ =\frac{\$10}{0.1}\\ =\$100\end{array}$

$\begin{array}{c}{\text{P}}_{\text{0 Stock B}}=\frac{\text{DIV}}{\left(r–g\right)}\\ =\frac{\$5}{0.1–0.04}\\ =\$83.33\end{array}$

$\begin{array}{c}{\text{P}}_{\text{0 Stock C}}= \left(\begin{array}{c}\frac{{\text{DIV}}_1}{1+r}+\frac{{\text{DIV}}_1\times \left(1+g\right)}{{\left(1+r\right)}^2}+\frac{{\text{DIV}}_1\times {\left(1+g\right)}^2}{{\left(1+r\right)}^3}+\frac{{\text{DIV}}_1\times {\left(1+g\right)}^3}{{\left(1+r\right)}^4}+\frac{{\text{DIV}}_1\times {\left(1+g\right)}^4}{{\left(1+r\right)}^5}\\ +\frac{{\text{DIV}}_1\times {\left(1+g\right)}^5}{{\left(1+r\right)}^6}+\frac{\frac{{\text{DIV}}_1\times {\left(1+g\right)}^5\times \left(1+g_2\right)}{r}}{{\left(1+r\right)}^6}\end{array}\right)\\ =\left(\begin{array}{c}\frac{\$5}{1.1}+\frac{\$5\times 1.2}{{1.1}^2}+\frac{\$5\times {1.2}^2}{{1.1}^3}+\frac{\$5\times {1.2}^3}{{1.1}^4}+\frac{\left(\$5\times {1.2}^4\right)}{{1.1}^5}+\\ \frac{\left(\$5\times {1.2}^5\right)}{{1.1}^6}+\frac{\left\{\frac{\left[\$5\times {1.2}^5\times \left(1+0\right)\right]}{0.1}\right\}}{{1.1}^6}\end{array}\right)\\ =\$104.51\end{array}$

Therefore, stock C has the largest present value.

Compute price of the each stocks:

If market capitalization rate is 7%

$\begin{array}{c}{\text{P}}_{\text{0}}{}_{\text{Stock A}}=\frac{{\text{DIV}}_{\text{1}}}{r}\\ =\frac{\$10}{0.07}\\ =\$142.86\end{array}$

$\begin{array}{c}{\text{P}}_{\text{0 Stock B}}=\frac{\text{DIV}}{\left(r–g\right)}\\ =\frac{\$5}{0.07–0.04}\\ =\$166.67\end{array}$

$\begin{array}{c}{\text{P}}_{\text{0 Stock C}}= \left(\begin{array}{c}\frac{{\text{DIV}}_1}{1+r}+\frac{{\text{DIV}}_1\times \left(1+g\right)}{{\left(1+r\right)}^2}+\frac{{\text{DIV}}_1\times {\left(1+g\right)}^2}{{\left(1+r\right)}^3}+\frac{{\text{DIV}}_1\times {\left(1+g\right)}^3}{{\left(1+r\right)}^4}+\frac{{\text{DIV}}_1\times {\left(1+g\right)}^4}{{\left(1+r\right)}^5}\\ +\frac{{\text{DIV}}_1\times {\left(1+g\right)}^5}{{\left(1+r\right)}^6}+\frac{\frac{{\text{DIV}}_1\times {\left(1+g\right)}^5\times \left(1+g_2\right)}{r}}{{\left(1+r\right)}^6}\end{array}\right)\\ =\left(\begin{array}{c}\frac{\$5}{1.07}+\frac{\$5\times 1.2}{{1.07}^2}+\frac{\$5\times {1.2}^2}{{1.07}^3}+\frac{\$5\times {1.2}^3}{{1.07}^4}+\frac{\left(\$5\times {1.2}^4\right)}{{1.07}^5}+\\ \frac{\left(\$5\times {1.2}^5\right)}{{1.07}^6}+\frac{\left\{\frac{\left[\$5\times {1.2}^5\times \left(1+0\right)\right]}{0.07}\right\}}{{1.07}^6}\end{array}\right)\\ =\$156.50\end{array}$

Therefore, stock B has the largest present value at a 7% market capitalization rate.