Alpha Manufacturing has debt with both a face value and market value of $8,000. The debt has a coupon rate of 5 percent and pays interest annually. The expected earnings before interest and taxes are $1,800, the tax rate is 35 percent, and the unlevered cost of capital is 10.5 percent. What is the firm's cost of equity?

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BlackTom AI · Accepted Answer

We start by restating the problem parameters to ensure clarity: Alpha Manufacturing has debt with face value and market value both at 8,000. Coupon rate is 5% (paid annually). EBIT is 1,800, tax rate is 35%, and the unlevered cost of capital Ru is 10.5%. The question asks for the firm’s cost of equity Re.

Option-by-option analysis:

Option A: 14.7%
- This value is noticeably lower than Ru and would imply that debt reduces equity risk or that leverage somehow lowers required return on equity, which contradicts the basic intuition of the levered beta effect: debt increases financial risk to equity, generally pushing Re up, not down. Given the data that there is debt at 5% coupon, we should expect Re to be higher than Ru, not lower. Therefore this option seems implausible.

Option B: 12.89%
- A cost of equity around 12.9% is only modestly above Ru (10.5%). In a levered situation, you typically see a larger increase in Re due to the debt burden amplifying equity risk, assuming a nonzero debt level. Unless the debt is very small or tax effects produce an unusually large tax shield relative to risk, 12.89% would understate the leverage effect. This option is suspect given the debt level presented.

Option C: 15.31%
- A rise from 10.5% to about 15.3% puts Re well above Ru, aligning with the general effect of debt on equity risk. This value is plausible if the debt-to-equity ratio is moderate and the after-tax cost of debt and tax shield are accounted for in the standard levered cost of equity formula Re = Ru + (Ru − Rd)(D/E). It fits within a reasonable range for a firm with $8,000 of debt and a sizable equity base.

Option D: 19.6%
- This option represents a substantially higher equity cost, consistent with a high leverage impact. If we compute using the standard levered cost of equity formula with Rd as the cost of debt (5%), Ru as 10.5%, and the debt-to-equity relation inferred from NI and Re (since NI = Re × E), the resulting Re tends to be around the upper end of the provided options. In fact, a more exact calculation yields a value close to 20% (approximately 20.3%), which is closest to 19.6% among the listed choices. This makes D a strong candidate as the correct answer.

Deeper calculation outline (without final single-step assertion):
- Let E be equity value. Net income NI = Re × E, so E = NI / Re. Here NI = 910 (calculated from EBIT 1800 minus interest 400 times (1 − Tc) = 900? Actually it’s 1800 × (1 − 0.35) − 400 × 0.35? For this approach, we use NI = Re × E as the fundamental link for solving Re via the relationship D/E with the levered cost of equity formula Re = Ru + (Ru − Rd) × (D/E).
- The debt-to-equity ratio is D/E = 8000 / E. Substituting E = 910 / Re into D/E gives D/E = 8000 × Re / 910.
- Plug into the levered formula: Re = Ru + (Ru − Rd) × (D/E) = 0.105 + (0.105 − 0.05) × (8000 × Re / 910).
- Compute constants: (0.105 − 0.05) = 0.055; 8000/910 ≈ 8.7912; product 0.055 × 8.7912 ≈ 0.4835. So Re ≈ 0.105 + 0.4835 × Re.
- Solve for Re: Re − 0.4835 × Re ≈ 0.105 → 0.5165 × Re ≈ 0.105 → Re ≈ 0.2033, i.e., about 20.3%.
- This computed Re is closest to 19.6% among the given options, supporting option D as the most accurate choice.

Summary takeaway: While the exact algebra points to a value around 20.3%, the option that best matches the levered equity cost given the inputs is 19.6%, making option D the most consistent choice with the standard levered cost-of-equity framework.

FA25-BL-BUS-F305-1130 Final Exam

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