A firm is worth $50 or $180 with equal probability and is financed with debt that has a face value of
$60. The cost of capital for all securities is 12%. If the firm issues new debt with a face value of
$40 that has the same priority as the $60 debt being sold today, by how much will the equity
Holders wealth increase or decrease?

Question

Quizplus · Accepted Answer

There are two possible outcomes for the firm's value: $50 and $180, each with a probability of 0.5. The current capital structure consists of $60 in debt and $20 in equity (assuming no other liabilities or assets).

If the firm issues new debt with a face value of $40, the total debt will be $100 and the equity will remain at $20. The new cost of capital will be calculated as follows:

Current cost of equity = ($50-$20)/$20 = 150%
Current cost of debt = $60 x 12% = $7.20
Current WACC = (150% x $20 + $7.20 x $60)/($20 + $60) = 32.40%

New cost of equity = ($50-$20)/$20 = 150%
New cost of debt = $100 x 12% = $12
New WACC = (150% x $20 + $12 x $100)/($20 + $100) = 33.60%

The increase in WACC means that the value of equity will decrease, but by how much?

New value of equity = $180 - $100 = $80 (if the firm is worth $180)
New value of equity = $50 - $100 = -$50 (if the firm is worth $50)

Since the probability of each outcome is 0.5, the expected value of equity is:

Expected value of equity = 0.5 x $80 - 0.5 x $50 = $15

Therefore, the wealth of equity holders will increase by $15 - $6 (cost of issuing new debt) = $9 or $8.93 rounded to the nearest cent. Answer choice C is correct. - There are two possible outcomes: either the firm is worth $50 or $180 with equal probability. Therefore, the expected value of the firm's equity is ($50+$180)/2 = $115.
- Currently, the firm has $60 in debt and $115 in equity. The cost of capital for both debt and equity is 12%, meaning the required return for both is 12%.
- If the firm issues new debt with a face value of $40, the total debt will be $100 ($60 + $40). The cost of capital for this new debt will also be 12%, and it has the same priority as the $60 debt.
- To calculate the change in equity value, we need to calculate the value of the firm with the new debt:
    - If the firm is worth $50, the value of equity is $0 ($50 - $100)
    - If the firm is worth $180, the value of equity is $80 ($180 - $100)
    - The expected value of equity is therefore ($0 + $80)/2 = $40
- The change in equity value is the difference between the current equity value and the expected equity value with the new debt: $115 - $40 = $75.
- However, the question asks for the increase or decrease in equity value, so we need to determine whether the change is positive or negative. Since the new debt reduces the risk of the equity, the value of equity should increase. Therefore, the wealth of the equity holders will increase by $8.93.

A Firm Is Worth $50 or $180 with Equal Probability