This project will offer $200 before-tax profit in year 1. Discounted back at the firm’s cost of capital (don’t worry if this is exact), the NPV without taxes is −$300 + $500/1.2 ≈ $116.67. But, if equity-financed, the IRS will declare taxes due on $200 of profit, or $80. Therefore, the NPV with taxes and all equity-financed is

−$300 + $420/1.2 = $50.

Now, right after the investment, the firm has a value of $420/1.2 = $350. With debt of $50 ($100), the firm carries a debt load of around $50/$350 ≈ 14.3% (28.6%). Let’s round this to 15% (30%). The cost of debt capital formula given in the question suggests that E(˜rDebt) = 15% + 15% . 5% = 15.75% (16.5%).

(Note: The question is a bit ambiguous in that it does not tell you what to use as firm value. The 15% and 30% debt ratios are reasonable values, though.)

Interest payments on $50 ($100) at a cost of capital of 15.75% (16.5%) are $7.88 ($16.50) next year.

Facing a tax rate of 40%, Uncle Sam would thereby subsidize the project to the tune of 40% . $7.88 ≈ $3.15

($6.60), which in today’s value would be worth around $3.15/1.2 ≈ $2.63 ($5.50). Therefore, under APV, if financed with $50 in debt, the project is worth $50 + $2.63 = $52.63. (With $100 in debt, the APV is

$50 + $5.50 = $55.50).

The equity cost of capital, if 15% of the firm is financed by debt at a rate of 15.75%, is the solution to 15% . 15.75% + 85% . E(˜rEquity) = 20% ⇒ E(˜rEquity) = 20.75%. Therefore, the WACC is given by the formula, wEquity . E(˜rEquity) + wDebt . E(˜rDebt) . (1 − τ) = 85% . 20.75% + 15% . 15.75% . (1 −

40%) ≈ 19.06%. Similarly, if $100 is borrowed, E(˜rEquity) = 21.5%, and WACC = wEquity . E(˜rEquity) +

wDebt . E(˜rDebt) . (1 − τ) = 70% . 21.5% + 30% . 16.5% . (1 − 40%) ≈ 18.02%. The WACC-based value with $50 in debt is thus −$300 + $420/1.1906 ≈ $52.76. (With $100 in debt, it is −$300 +

$420/1.1802 ≈ $55.87.) Note that you have made enough little assumptions and approximations that it would make little sense to worry now about being off by a little in the APV and WACC computations

($52.76 and $52.63).