Use the information for the question(s)below.
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,how much do you need to invest at the risk-free rate so that the payoff of your account will be the same as a $5000 investment in Without stock?
A)$5000
B)$0
C)$2500
D)$4000

Question

Quizplus · Accepted Answer

Since Without is an all-equity firm, its price is simply equal to its total market value, which is $1 million shares times $24 per share, or $24 million. Therefore, a $5000 investment in Without would correspond to a stake in the company of $5000/$24 million, or approximately 0.2083%.

With, on the other hand, has a total market value of $24 per share times 2 million shares plus $12 million of debt, or $60 million. Therefore, to match the payoff of a $5000 investment in Without, we need to find the amount of money that corresponds to a 0.2083% stake in With.

Let X be the amount we need to invest in the risk-free rate. Then the total amount in our account will be $5000 + X after some time has passed. Since With has 2 million shares outstanding, a 0.2083% stake in the company is equal to 41660 shares. Therefore, the value of our investment in With will be 41660 shares times $24 per share, or $999840. We also earn interest on our investment in the risk-free rate, which is equal to X times 1.05 after some time has passed.

The total value of our account is therefore $999840 + $1.05X after some time has passed. This should be equal to $5000 + X, since we want the payoff to be the same as a $5000 investment in Without. Therefore, we can solve for X as follows:

$999840 + $1.05X = $5000 + X
$994840 = $0.05X
X = $994840/$0.05
X = $1989680

Therefore, we need to invest $1989680 - $5000 = $1984680 in the risk-free rate. However, we only have $5000 to invest, so we need to borrow $1984680 - $5000 = $1979680 at the risk-free rate. Since we can borrow and lend at the same rate, this investment in the risk-free rate will exactly offset our investment in With, resulting in a payoff that is the same as a $5000 investment in Without.

Therefore, we only need to invest $2500 of our own money in With, and then borrow $1979680 at the risk-free rate. The answer is C) $2500.

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