Chapter 4, Problem 31PS

Valuing a business* Phoenix Corp. faltered in the recent recession but is recovering. Free cash flow has grown rapidly. Forecasts made in 2019 are as follows:

Phoenix’s recovery will be complete by 2024, and there will be no further growth in net income or free cash flow.

1. a. Calculate the PV of free cash flow, assuming a cost of equity of 9%.
2. b. Assume that Phoenix has 12 million shares outstanding. What is the price per share?
3. c. Confirm that the expected rate of return on Phoenix stock is exactly 9% in each of the years from 2020 to 2024.

Summary Introduction

To determine: Present value of free cash flow

Explanation of Solution

Compute the present value of free cash flow:

$\begin{array}{c}{\text{PV}}_{\text{2019}}=\frac{{\text{DIV}}_{\text{2020}}}{1+r}+\frac{{\text{DIV}}_{\text{2021}}}{{\left(1+r\right)}^2}+\frac{{\text{DIV}}_{2022}}{{\left(1+r\right)}^3}+\frac{{\text{DIV}}_{2023}}{{\left(1+r\right)}^4}+\frac{{\text{DIV}}_{2024}}{{\left(1+r\right)}^5}\text{+}\frac{\frac{{\text{DIV}}_{\text{2025}}}{r}}{{\left(1+r\right)}^5}\\ \text{=}\frac{\$0}{1.09}\text{+}\frac{\$1}{{1.09}^2}+\frac{\$2}{{1.09}^3}+\frac{\$2.3}{{1.09}^4}+\frac{\$2.6}{{1.09}^5}+\frac{\left(\frac{\$2.6}{0.09}\right)}{{1.09}^5}\\ =\$24.48\text{ million}\end{array}$

Hence, the present value is $24.8 million.

Summary Introduction

To determine: Price per share

Explanation of Solution

 Note:

 Assume no debt, the share price are as follows,

 $\begin{array}{c}{\text{Price per share}}_{2019}=\frac{{\text{PV}}_{2019}}{\text{Number of shares}}\\ =\frac{\$24.48}{12}\\ =\$2.04\end{array}$

 Hence, the price per share is $2.04.

Summary Introduction

To confirm: The expected rate of return is 9%.

Explanation of Solution

 Compute PV of the cash flows at various points in time:

 $\begin{array}{c}{\text{PV}}_{2020}=\frac{\$1}{1.09}+\frac{\$2}{{1.09}^2}+\frac{\$2.3}{{1.09}^3}+\frac{\$2.6}{{1.09}^4}+\frac{\left(\frac{\$2.6}{0.09}\right)}{{1.09}^4}\\ =\$26.68\end{array}$

 $\begin{array}{c}{\text{PV}}_{2021}=\frac{\$2}{1.09}+\frac{\$2.3}{{1.09}^2}+\frac{\$2.6}{{1.09}^3}+\frac{\left(\frac{\$2.6}{.09}\right)}{{1.09}^3}\\ =\$28.09\end{array}$

 $\begin{array}{c}{\text{PV}}_{2022}=\frac{\$2.3}{1.09}+\frac{\$2.6}{{1.09}^2}+\frac{\left(\frac{\$2.6}{.09}\right)}{{1.09}^2}\\ =\$28.61\end{array}$

 $\begin{array}{c}{\text{PV}}_{2023}=\frac{\$2.6}{.09}\\ =\$28.89\end{array}$

 $\begin{array}{c}{\text{PV}}_{2024}=\frac{\$2.6}{.09}\\ =\$28.89\end{array}$

 Compute rate of return using the formula $r_0=\frac{\left({\text{DIV}}_1+{\text{P}}_1–{\text{P}}_0\right)}{{\text{P}}_0}$

 $\begin{array}{c}{\text{Rate of return}}_{2021}=\frac{\left(\$1+\$28.09-\$26.68\right)}{\$26.68}\\ =0.09,\text{ or }9\%\end{array}$

 $\begin{array}{c}{\text{Rate of return}}_{2022}=\frac{\left(\$2+\$28.61-\$28.09\right)}{\$28.09}\\ =0.09,\text{ or }9\%\end{array}$

 $\begin{array}{c}{\text{Rate of return}}_{2023}=\frac{\left(\$2.3+\$28.89-\$28.61\right)}{\$28.61}\\ =0.09,\text{ or }9\%\end{array}$

 $\begin{array}{c}{\text{Rate of return}}_{2024}=\frac{\left(\$2.6+\$28.89-\$28.89\right)}{\$28.89}\\ =0.09,\text{ or }9\%\end{array}$

 Thus, the above calculation shows that the rate of return on Company P is exactly 9%.