A firm has equity market capitalization of $200 and $100 in debt. The beta is 2. The swaps curve is flat at 3%. The credit spread on the firm's debt is 2%. It plans to finance a $10 project with debt. You can ignore taxes. The risk premium on the market portfolio is 4%. In theory, this project makes sense if the expected IRR

Question

A firm has equity market capitalization of $200 and $100 in debt. The beta is 2. The swaps curve is flat at 3%. The credit spread on the firm's debt is 2%.  It plans to finance a $10 project with debt. You can ignore taxes. The risk premium on the market portfolio is 4%. In theory, this project makes sense if the expected IRR >

BlackTom AI · Accepted Answer

We start by restating the situation and the options to ensure clarity about what’s being evaluated.
Question and data: A firm with equity market value 200 and debt 100 has a levered beta of 2. It finances a $10 project with debt. Taxes are ignored. The swaps curve is 3% (assumed risk-free rate), the firm’s debt carries a credit spread of 2%, the market risk premium is 4%. The available answer choices for the project’s required return are 5%, 6.33%, 9%, and 7%.
First, determine the asset beta (unlevered beta) to capture the project’s inherent risk independent of financing. With D/E = 100/200 = 0.5 and ignoring taxes, the relation between levered beta (beta_L) and unlevered beta (beta_U) is beta_L = beta_U * (1 + D/E). Solving for beta_U gives beta_U = beta_L / (1 + D/E) = 2 / (1 + 0.5) = 2 / 1.5 = 1.333...
Next, compute the asset return required by CAPM using the asset beta. The risk-free rate from the swaps curve is 3%, and the market risk premium is 4%. Therefore, the asset return (cost of capital for the project assets) is r_A = r_f + beta_U * (MRP) = 3% + 1.333... * 4% = 3% + 5.333...% ≈ 8.333...%
This r_A represents the required return for projects with the same asset risk profile, i.e., the discount rate that should be applied to the project’s cash flows to determine its value, given the firm’s overall risk independent of financing choices.
Now evaluate each answer option against this threshold of about 8.33%:
- Option 5%: This is well below the computed asset-cost-of-capital threshold. A project yielding 5% would be unattractive because it fails to cover the risk-adjusted return demanded by the asset risk; it would destroy value under the CAPM framework.
- Option 6.33%: This remains below the 8.33% threshold. Although closer than 5%, it still does not compensate for the asset risk as captured by beta_U, so the project would be considered too risky relative to the required return.
- Option 9%: This exceeds the 8.33% threshold. Since 9% surpasses the required return for the asset risk class, the project would be attractive on a risk-adjusted basis and could be justified economically given the stated assumptions.
- Option 7%: Like the previous low candidates, 7% is below the calculated 8.33% hurdle. It would not meet the required return to compensate for the project’s risk.
In summary, among the provided options, the only one that clears the risk-adjusted hurdle derived from CAPM with the given betas and market data is the 9% figure. The others fail to compensate for the asset risk and would be insufficient to justify undertaking the project under the stated conditions.

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