A company is choosing between two projects. The larger project has an initial cost of $100,000, annual cash flows of $30,000 for 5 years, and an IRR of 15.24%. The smaller project has an initial cost of $51,600, annual cash flows of $16,000 for 5 years, and an IRR of 16.65%. The projects are equally risky. Which of the following statements is CORRECT?

Question

BlackTom AI · Accepted Answer

First, let's restate the problem to be sure we’re evaluating the right claim: two projects differ in size, cash flows, and IRR. The larger project costs 100,000 with 30,000 per year for 5 years and an IRR of 15.24%. The smaller costs 51,600 with 16,000 per year for 5 years and an IRR of 16.65%. The question asks which statement is correct about their NPV profiles and the impact of the discount rate. Option evaluation: The provided statement is that the larger project will have the higher NPV if the WACC is less than the crossover rate, and that the NPV profiles cross because the smaller project has the higher IRR while the larger project has a higher NPV at zero discount rate. First, reason about the crossover concept: the NPV profiles cross at the discount rate r* where NPV_large(r*) = NPV_small(r*). To find r*, set up the equality using the NPVs: - NPV_large = -100,000 + 30,000 * a5(r) - NPV_small = -51,600 + 16,000 * a5(r) Where a5(r) = (1 - (1+r)^-5)/r. Equating them: -100,000 + 30,000*A = -51,600 + 16,000*A, with A = a5(r). Solving for A gives 14,000*A = 48,400, so A ≈ 3.457. Next, relate A to r: the annuity factor a5(r) decreases as r increases, so there is a unique r* where a5(r*) ≈ 3.457. Numerical checks show a5(10%) ≈ 3.791, a5(12%) ≈ 3.606, and a5(15%) ≈ 3.35, so r* lies between about 12% and 15%. In other words, the crossover rate is around 13–14%. Now test the claim about behavior relative to the crossover rate: the difference in NPV between the two projects is NPV_large - NPV_small = (-100,000 + 30,000*A) - (-51,600 + 16,000*A) = -48,400 + 14,000*A. At rates r < r*, A = a5(r) > 3.457, so -48,400 + 14,000*A > 0, meaning NPV_large > NPV_small. At rates r > r*, A < 3.457, so the difference becomes negative and NPV_small surpasses NPV_large. The statement also correctly notes that the larger project has a higher NPV at a zero discount rate: when r = 0, a5(0) equals 5, so NPV_large = -100,000 + 30,000*5 = 50,000, and NPV_small = -51,600 + 16,000*5 = 28,400, confirming larger has higher NPV at r = 0. Thus, the logic that the profiles cross and that the larger project will have the higher NPV if the WACC is less than the crossover rate is consistent with the standard cross-over (or cross-over rate) analysis in capital budgeting. In summary, the core idea is: there exists a crossover rate around 13–14%; for any WACC below that, the larger project dominates in NPV, and above it, the smaller project does; plus, at r = 0 the larger project already has the higher NPV. The given option accurately captures this relationship, and its reasoning aligns with the standard cross-over-rate framework. Note on completeness: since no alternate answer choices are provided for comparison here, we focused on validating the correctness of the single listed statement and explained why it matches the cross-over-rate behavior and the r = 0 check.

Business Finance Chapter 11 Quiz

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