Question

A firm can be worth $100 million (with 20% probability), $200 million (with 60% probability), or $300 million (with 20% probability). The firm has one senior bond outstanding, promising to pay $80 million. It also has one junior bond outstanding, promising to pay $70 million. The senior bond promises an interest rate of 5%. The junior bond promises an interest rate of 26%. If the firm’s projects require an appropriate cost of capital of 10%, then what is the firm’s levered equity cost of capital?

Answer 1

First in this question, we need to calculate the expected value of firm.

Hence, expected value of firm = (20% * $100 mil) + (60% * $200 mil) + (20% * $300 mil)

Expected Value of firm = $20mil + $120 mil + $60mil = $200 mil

But this expected value is for one year from today, so the (expected) value of firm today would be:

Expected value of firm today = $200/(1 + 0.1) {Since 10% is the cost of capital}

Expected Value of firm today = $181.82

At a 5% interest rate, Senior bond holders after 1 year would be paid $80 mil * (1 + 5%) = $84 mil

At a 26% interest rate, junior bond holders after 1 year would get $70 mil * (1 + 26%) = $88.2 mil.

Senior bondholders would have a priority of receiving the interest and principal amount of $84 mil. This means, junior bondholders will be paid only in scenarios where the firm is worth $200 mil or $300 mil. Since, in scenario where the firm is worth $100mil, after $84 mil are paid to senior holders only $16 mil would remain, which is less than the amount required to payback junior shareholders.

This means, the firm's equity value is 0, when firm's value is $100 mil (since $16 that remained will be paid proportionately to junior bonholders and nothing would remain for equity holders).

When firm's value is $200, value of equity (=firm value - value payable to senior and junior bond holders) would be equal to $200 - $84 - $88.2 = $27.8 mil

When firm's value is $300, value of equity (=firm value - value payable to senior and junior bond holders) would be equal to $300 - $84 - $88.2 = $127.8 mil

Therefore, expected value of future equity = (20% * $0) + (60% * $27.8 mil) + (20% * $127.8 mil) = $42.24 mil

Value of Equity today = $181.82 mil - $80 mil - $70 mil = $31.82 mil

Hence, firm's levered cost of capital could be found using Time Value of Money relation between PV and FV.

FV = PV * (1 + r)

Here, substituting the values, to calculate r, we get

42.24 = 31.82 * (1 + r)

(1 + r) = 1.3275

r = 0.3275

=> r = 32.75%. Cost of Levered Equity. Answer.