###### Binomial Option Valuation Model

One-Period Binomial Option Valuation Model In the one-period binomial model, we start today... **Read More**

When the market price per share is greater than its book value, the book value per share will reduce after share repurchases. The book value per share will increase after repurchases when the market price per share is less than the book value per share. Let us illustrate this using an example.

$$\small{\begin{array}{l|c|c} {}& \textbf{Delta} & \textbf{Omega}\\ \hline\text{Market Price per share} & \$30 & \$30 \\ \hline\text{Outstanding shares} & \text{12 million} & \text{12 million}\\ \hline\text{Book value of equity} & \$400\ \text{million} & \$300\ \text{million}\\ \hline\text{Amount of buyback} & \$6\ \text{million} & \$6\ \text{million}\\ \hline\text{Book value per share}^{1} & \$33.33 & \$25\\ \hline\text{Equity at book value after share buyback}^{2} & \$394\ \text{million} & \$294\ \text{million}\\ \hline\text{Book value per share after share buyback}^{3} & \$33.38 & \$24.91\\ \end{array}}$$

\(\textbf{Book value per share}^{1}\) is calculated as follows:

For Delta, \(=\frac{\text{Book value of equity}}{\text{Outstanding shares}}=\frac{$400\ \text{million}}{12\ \text{million}}=$33.33\)

For Omega, \(=\frac{\text{Book value of equity}}{\text{outstanding shares}}=\frac{$300\ \text{million}}{12\ \text{million}}=$25\)

\(\textbf{Equity at book value after share buyback}^{2}\) is calculated as follows:

$$\begin{align*}\text{For Delta}&=\text{Equity at book value}–\text{Amount of buyback}\\&=$400\ \text{million}-$6\ \text{million}\\&=$394\ \text{million}\end{align*}$$

$$\begin{align*}\text{For Omega}&=\text{Equity at book value}-\text{Amount of buyback}\\&=$300\ \text{million}-$6\ \text{million}\\&=$294\ \text{million}\end{align*}$$

\(\textbf{Book value per share after share buyback}^{3}\) is calculated as follows:

\($6\ \text{million worth of shares to be repurchased} = 200,000\ \text{shares}= (\frac{$6\ \text{million}}{$30})\)

We will be left with 11.8 million = (12 million – 0.2 million).

For Delta, book value per share after the the buyback:

$$\frac{\text{Book value of equity}}{\text{outstanding shares}}=\frac{$394\ \text{million}}{11.8\ \text{million}}=$33.38$$

For Omega, book value per share after the buyback:

$$\frac{\text{Book value of equity}}{\text{outstanding shares}}=\frac{$294\ \text{million}}{11.8\ \text{million}}=$24.91$$

As we can see, when the market price is greater than the book value per share, the book value per share will increase, as is the case with Delta.

## Question

Apex Ltd. has a total of 30 million outstanding shares. The company decides to buy back 3 million shares at a market price of $13.33. At the time of the repurchase, the book value of equity is $400 million.

As a result of the buyback, the company’s book value per share will

most likely:

- Increase.
- Remain unchanged.
- Decrease.
## Solution

The correct answer is C:The company’s book value per share before buyback of shares is:

$$\text{BVPS}_{\text{Before}}=\frac{$400\ \text{million}}{30\ \text{million shares}}=$13.33 / \text{share}$$

$$\text{Value of repurchased shares in dollar amount}=3\ \text{million shares}\times$13.33 / \text{share}=$39,990,000$$

The book value of equity will be reduced to $360.01 million ($400 million – $39.99 million), and the number of shares will be reduced to 27 million.

The company’s book value per share after the share repurchase is:

$$\frac{$360,010,000}{27,000,000 \text{ shares}}=$13.33 / \text{share}$$

A is incorrect:The book value per share will increase after repurchases when the market price per share is less than the book value per share.

B is incorrect:The book value per share will remain the same after repurchases when the market price per share is equal to the book value per share.

Reading 16: Analysis of Dividends and Repurchases

*LOS 16 (j). Calculate the effect of a share repurchase on book value per share*