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Consider two firms,With and Without,that have identical assets that generate identical cash flows.Without is an all-equity firm,with 1 million shares outstanding that trade for a price of $24 per share.With has 2 million shares outstanding and $12 million in debt at an interest rate of 5%.
-Assume that MM's perfect capital market conditions are met and that you can borrow and lend at the same 5% rate as With.You have $5000 of your own money to invest and you plan on buying With stock.Using homemade (un)leverage you invest enough at the risk-free rate so that the payoff of your account will be the same as a $5000 investment in Without stock? The number of shares of With stock you purchased is closest to:
A)100.
B)425.
C)1650.
D)825.

Question

Quizplus · Accepted Answer

Since With and Without have identical assets that generate identical cash flows, their stock should have the same value. Without has 1 million shares outstanding trading at $24 per share, which means its total market value is $24 million. Therefore, each share of Without represents 1/1,000,000 of the ownership of the company, and every $24 invested in Without buys one share.

With has 2 million shares outstanding and $12 million in debt, which means its total market value is also $24 million ($12 million in debt plus $12 million in equity). Therefore, each share of With represents 1/2,000,000 of the ownership of the company, and every $24 invested in With buys 1/2,000,000 of the company.

To achieve the same payoff as a $5,000 investment in Without, we need to invest $5,000 / $24 = 208.33 shares of Without. Since each share of With is only 1/2 as valuable as a share of Without (due to the debt), we need to buy twice as many shares of With to achieve the same payoff. Therefore, we need to buy 208.33 x 2 = 416.67 shares of With. However, this only gives us the same payoff as a $5,000 investment in Without before interest. Since With also has $12 million in debt, we need to invest additional money at the risk-free rate to cover the interest payments on the debt. The interest payments are $12 million x 0.05 = $600,000 per year, which means we need to invest an additional $600,000 / 0.05 = $12 million at the risk-free rate. Therefore, we need to invest a total of $12 million + $5,000 = $12,005,000 to achieve the same payoff as a $5,000 investment in Without after interest. Investing $12,005,000 / $24 per share = 500,208.33 shares of With, or rounding to the nearest whole share, 425 shares of With.

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