Spam Corp. is financed entirely by common stock and has a beta of 1.0. The firm is expected to generate a level, perpetual stream of earnings and dividends. The stock has a price-earnings ratio of 8 and a cost of equity of 12.5%. The company's stock is selling for $50. Now the firm decides to repurchase half of its shares and substitute an equal value of debt. The debt is risk-free, with a 5% interest rate. The company is exempt from corporate income taxes. Assuming MM are correct, calculate the following items after the refinancing:

a. cost of equity
b. overall cost of capital (WACC)
c. price-earnings ratio
d. Stock price
e. stock's beta

To calculate the items after the refinancing, we will use the Modigliani-Miller (MM) theorem, which states that in the absence of taxes and bankruptcy costs, the value of a firm is unaffected by its capital structure.

a. Cost of Equity (Re)

The cost of equity remains the same as before at 12.5%. The change in capital structure does not affect the risk associated with the equity, so the cost of equity remains unchanged.

b. Overall Cost of Capital (WACC)

The overall cost of capital (WACC) is calculated as the weighted average of the cost of equity and the after-tax cost of debt. Since the company is exempt from corporate income taxes, the after-tax cost of debt is equal to the interest rate.

WACC = (E / V) * Re + (D / V) * Rd, where:
E = Market value of equity after the refinancing
V = Total market value of the firm after the refinancing
D = Market value of debt after the refinancing
Re = Cost of equity
Rd = After-tax cost of debt

First, let's calculate the new market values of equity and debt:
Market value of equity after refinancing = 0.5 * Stock price * Number of shares repurchased
= 0.5 * $50 * (0.5 * Number of original shares)

Since the firm repurchases half of its shares, it is left with half of its original number of shares.

Market value of debt after refinancing = 0.5 * Stock price * Number of shares repurchased
= 0.5 * $50 * (0.5 * Number of original shares) * (Debt / Equity ratio)

Assuming the debt-to-equity ratio is equal to the equity-to-debt substitution ratio, which is 1:1, the debt value will be equal to the equity value.

Now, calculate the WACC:
WACC = (E / V) * Re + (D / V) * Rd
= (Market value of equity after refinancing / (Market value of equity after refinancing + Market value of debt after refinancing)) * Re + (Market value of debt after refinancing / (Market value of equity after refinancing + Market value of debt after refinancing)) * Rd
= (E / (E + D)) * Re + (D / (E + D)) * Rd

Substituting the calculated values will give us the overall cost of capital (WACC).

c. Price-earnings Ratio

The price-earnings ratio remains the same as before at 8. The change in capital structure does not affect the relationship between the stock price and earnings.

d. Stock Price

To calculate the new stock price, divide the market value of equity after refinancing by the number of remaining shares.

New stock price = Market value of equity after refinancing / (0.5 * Number of original shares)

e. Stock's Beta

The stock's beta remains the same as before at 1.0. The change in capital structure does not affect the risk associated with the stock.

Please note that the calculations heavily depend on the assumed values for the debt-to-equity ratio and the equity-to-debt substitution ratio.

To calculate the various items after the refinancing, we need to consider the concepts of the Modigliani-Miller (MM) theorem.

a. Cost of Equity:
The cost of equity represents the return required by the shareholders of the company. According to MM theorem, the cost of equity does not change with the change in capital structure. Therefore, the cost of equity remains the same. In this case, the cost of equity is 12.5%.

b. Overall Cost of Capital (WACC):
WACC stands for Weighted Average Cost of Capital, which represents the average rate of return required by both equity and debt holders. Since the company has repurchased half of its shares and substituted equal value of debt, the new capital structure will have equal proportions of equity and debt. Therefore, the weight of equity and debt will be 50% each.
To calculate the WACC, we need to find the cost of debt. Since the debt is risk-free with a 5% interest rate and the company is exempt from taxes, the cost of debt is simply 5%.
Now, using the formula:
WACC = (Weight of Equity * Cost of Equity) + (Weight of Debt * Cost of Debt)
WACC = (0.50 * 12.5%) + (0.50 * 5%)
WACC = 6.25% + 2.5%
WACC = 8.75%

c. Price-Earnings Ratio:
The price-earnings ratio (P/E ratio) compares the price of a company's stock to its earnings per share (EPS). Here, we are given that the original P/E ratio is 8. However, after the refinancing, the cost of equity and the cost of debt remain the same. As a result, the P/E ratio does not change. Therefore, the P/E ratio remains 8.

d. Stock Price:
To calculate the new stock price after the refinancing, we need to consider the change in the capital structure and the new WACC. The formula to calculate the stock price is:
Stock Price = Earnings per Share (EPS) * Price-Earnings Ratio (P/E Ratio)
The EPS is not given in the information provided, but we know that the firm is expected to generate a level, perpetual stream of earnings and dividends. Therefore, we can assume that the EPS remains constant.
Using the original stock price of $50 and the P/E ratio of 8, we can calculate the EPS as follows:
EPS = Stock Price / P/E Ratio
EPS = $50 / 8
EPS = $6.25
Now, using the new WACC of 8.75%, we can calculate the new stock price:
New Stock Price = EPS * P/E Ratio
New Stock Price = $6.25 * 8
New Stock Price = $50

e. Stock's Beta:
According to MM theorem, the change in capital structure, through the substitution of debt for equity, does not affect the firm's beta. Therefore, the stock's beta remains the same as before, which is 1.0.

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