Chapter 16, Problem 18PS

Repurchases and the DCF model House of Haddock has 5,000 shares outstanding and the stock price is $140. The company is expected to pay a dividend of $20 per share next year and thereafter the dividend is expected to grow indefinitely by 5% a year. The president, George Mullet, now makes a surprise announcement: He says that the company will henceforth distribute half the cash in the form of dividends and the remainder will be used to repurchase stock. The repurchased stock will not be entitled to the dividend.

1. a. What is the total value of the company before and after the announcement? What is the value of one share?
2. b. What is the expected stream of dividends per share for an investor who plans to retain his shares rather than sell them back to the company? Check your estimate of share value by discounting this stream of dividends per share.

Summary Introduction

To determine: The total value of the company before and after the announcement and also the value of one share.

Explanation of Solution

Computation of value of company before and after the announcement:

$\begin{array}{c}\text{Pre-announcemnt company value}\text{=}\text{outstanding share}\times \text{share price}\\ \text{=5,000}\times \text{\$140}\\ \text{=\$700,000}\end{array}$

The company value and stock price will be same after the announcement.

Therefore, the value of one share is $140.

Summary Introduction

To determine: The expected stream of dividends per share for an investor who plans to retrain the shares rather than sell them back to the company.

Explanation of Solution

For year 1:

$\begin{array}{c}{\text{Annual repurchase amount of year}}_1=\left[\begin{array}{l}\text{number of shares}\times \text{original dividend per share}\times \\ \text{repurchase percent}\end{array}\right]\\ \text{=5,000}\times \text{\$20}\times \text{0}\text{.50}\\ \text{=\$50,000}\end{array}$

$\begin{array}{c}\text{Annual dividend amount }\left({\text{year}}_1\right)=\left[\begin{array}{l}\text{number of shares }\times \text{original dividend per share}\\ \times \text{dividend percent}\end{array}\right]\\ \text{=5,000}\times \text{\$20}\times \text{0}\text{.50}\\ \text{=\$50,000}\end{array}$

$\begin{array}{c}\text{Discount rate}\left(\text{r}\right)\text{=}\frac{{\text{D}}_1}{{\text{P}}_0}+\text{g}\\ \text{=}\frac{\$20}{\$140}+0.05\\ =0.1929\text{or 19}\text{.29\%}\end{array}$

$\begin{array}{c}\text{Stock price}\left({\text{p}}_1\right)\text{=}{\text{p}}_0\times \left(1+\text{r}\right)\\ =\$140\times \left(1+0.1929\right)\\ =\$140\times 1.1929\\ =\$167\end{array}$

$\begin{array}{c}{\text{Share repurchased year}}_1=\frac{\text{Repurchase amount}}{\text{price}}\\ =\frac{\$50,000}{\$167}\\ =299\end{array}$

$\begin{array}{c}\text{New outstanding shares}\text{=}\text{Prior shares outstanding}-\text{shares repurchased}\\ \text{=5,000}-299\\ =4,701\end{array}$

$\begin{array}{c}{\text{D}}_1\text{per share}\text{=}\frac{{\text{D}}_1}{\text{number of shares}}\\ =\frac{\$50,000}{4,701}\\ =\$10.64\end{array}$

For year 2:

$\begin{array}{c}{\text{Annual repurchase amount of year}}_2={\text{Annual repurcahse amount of year}}_1\times \left(1+\text{g}\right)\\ \text{=\$50,000}\times \text{1}\text{.05}\\ \text{=\$52,500}\end{array}$

$\begin{array}{c}\text{Annual dividend amount }\left({\text{year}}_2\right)=\text{Annual dividend amount }\left({\text{year}}_1\right)\times \left(1+\text{g}\right)\\ \text{=\$50,000}\times \left(1+0.05\right)\\ =\$50,000\times 1.05\\ \text{=\$52,500}\end{array}$

$\begin{array}{c}\text{Discount rate}\left(\text{r}\right)\text{=}\frac{{\text{D}}_1\left(1+\text{g}\right)}{{\text{P}}_1}+\text{g}\\ \text{=}\frac{\$10.64\left(1+0.05\right)}{\$167}+0.05\\ =0.1169\text{or 11}\text{.69\%}\end{array}$

$\begin{array}{c}\text{Stock price}\left({\text{p}}_2\right)\text{=}{\text{p}}_1\times \left(1+\text{r}\right)\\ =\$167\times \left(1+0.1169\right)\\ =\$167\times 1.1169\\ =\$186.52\end{array}$

$\begin{array}{c}{\text{Share repurchased year}}_2=\frac{\text{Repurchase amount}}{{\text{price}}_2}\\ =\frac{\$52,500}{\$186.52}\\ =281\end{array}$

$\begin{array}{c}\text{New outstanding shares}\text{=}\text{Prior shares outstanding}-\text{shares repurchased}\\ \text{=4,701}-281\\ =4,419\end{array}$

$\begin{array}{c}{\text{D}}_2\text{per share}\text{=}\frac{{\text{D}}_2}{\text{number of shares}}\\ =\frac{\$52,500}{4,419}\\ =\$11.88\end{array}$

Following table shows the overall scenario of two years based on above calculations:

	Year 1	Year 2
Annual repurchase amount($)	50,000	52,500
Annual dividend amount ($)	50,000	52,500
Discount rate (%)	19.29	11.69
Stock price ($)	167	186.52
Shares repurchased	299	281
New outstanding shares	4,701	4,419
Dividend per share ($)	10.64	11.88

Table no.1

In all subsequent years, the total dividend and repurchase amount will increase by 5% and the number of shares decreasing by:

$\begin{array}{c}\text{Percentage decrease in number of shares}\text{=}\frac{\left(4,701-4,419\right)}{4,419}\\ =\frac{282}{4,419}\\ =0.0638\text{ or}\text{6}\text{.38\%}\end{array}$

$\begin{array}{c}\text{Increase in dividend per share}\text{=}\frac{\left(\$11.88-\$10.64\right)}{\$10.64}\\ =\frac{\$1.24}{\$10.64}\\ =0.1169\text{or 11}\text{.69\%}\end{array}$

Therefore, the share price is as follows:

$\begin{array}{c}{\text{P}}_0=\frac{{\text{D}}_1\text{per share}}{\left(\text{r}-\text{g}\right)}\\ =\frac{\$10.64}{\left(0.1929-0.1169\right)}\\ =\$140\end{array}$