A firm has 60% probability of being worth $100 million and a 40% probability of being worth $150 million. There is one bond outstanding that promises to pay $100 million at an interest rate of 4%. The cost of capital for the firm's projects is 8%.
-Refer to the information above. What is the expected return on the levered equity?

Question

Quizplus · Accepted Answer

The expected value of the firm is calculated as (0.6 * $100 million) + (0.4 * $150 million) = $60 million + $60 million = $120 million. The bond promises to pay $100 million. The equity value is the difference between the firm value and the bond value, which is $120 million - $100 million = $20 million. The expected return on the levered equity can be calculated using the formula for levered equity return: [(Equity Value / Firm Value) * Cost of Capital] + [(Debt Value / Firm Value) * Interest Rate] - Interest Rate. Plugging in the values: [(20/120) * 8%] + [(100/120) * 4%] - 4% = 1.33% + 3.33% - 4% = 0.67%. However, this calculation seems to have gone awry. The correct approach to find the expected return on equity is to consider the payoff to equity holders under each scenario and then calculate the expected return based on those payoffs. Given the bond pays $100 million, if the firm is worth $100 million, equity holders get nothing (since all value goes to bondholders). If the firm is worth $150 million, equity holders get $50 million (since bondholders are paid first, and the rest goes to equity). The expected equity value is thus (0.6 * $0) + (0.4 * $50 million) = $20 million. Given the initial equity value is $20 million (from the difference between the expected firm value and the bond value), and the payoff to equity in the good state is $50 million, the return on equity in the good state is ($50 million - $20 million) / $20 million = 150%. The expected return, accounting for the 40% probability of this happening, is 0.4 * 150% = 60%. This calculation, however, simplifies the scenario and does not directly apply the correct formula or method for calculating expected return on levered equity, which involves more nuanced financial theory and may consider the cost of debt and equity in a leveraged firm to determine the overall expected return to equity holders. The correct answer involves calculating the expected payoff to equity holders and then determining the return based on the initial equity value. Given the probabilities and outcomes, the expected return on levered equity is significantly higher than the cost of debt or the weighted average cost of capital, reflecting the higher risk to equity holders in a leveraged firm. The correct calculation should consider the leveraged return formula, which accounts for the risk and return to equity holders in scenarios where the firm's value exceeds the debt obligation. Without the precise calculation steps provided in the question, the correct answer hints at a higher expected return on equity due to leverage, which is consistent with financial theory that leveraged equity carries higher risk and potentially higher returns than unleveraged equity or debt.

A Firm Has 60% Probability of Being Worth $100 Million