MC algo 8-17 Calculating Crossover Rate

You are considering the following two mutually exclusive projects. The crossover rate between these two projects is _________
blank
percent and Project _________
blank
should be accepted if the required return is greater than the crossover rate.

Year Project A Project B
0 −$ 30,000 −$ 30,000
1 11,500 19,630
2 11,500 9,500
3 19,500 11,650

Question

MC algo 8-17 Calculating Crossover Rate

You are considering the following two mutually exclusive projects. The crossover rate between these two projects is _________
blank
 percent and Project _________
blank
 should be accepted if the required return is greater than the crossover rate.

Year	Project A	Project B
0	−$ 30,000	−$ 30,000
1	11,500	19,630
2	11,500	9,500
3	19,500	11,650

BlackTom AI · Accepted Answer

We start by restating the problem: we have two mutually exclusive projects, A and B, with cash flows given for years 0 to 3, and we are asked for the crossover rate and which project to accept if the required return is greater than the crossover rate.

Option-by-option analysis:

Option 1: 11.33; B
- This option proposes a crossover rate of 11.33% and that Project B should be accepted when the required return exceeds the crossover rate. If we compute the crossover rate, we compare the two projects by constructing the incremental cash flows of A minus B: Year1: 11,500 − 19,630 = −8,130; Year2: 11,500 − 9,500 = 2,000; Year3: 19,500 − 11,650 = 7,850. The initial outlays are equal (both −30,000). Setting the NPV of the incremental project (A − B) to zero gives the equation −8,130/(1+r) + 2,000/(1+r)^2 + 7,850/(1+r)^3 = 0. Solving this yields a positive root around r ≈ 11.33%, which matches the stated crossover rate. For r above the crossover, the incremental NPV becomes negative, meaning Project B has the higher NPV and should be chosen. Therefore, this option aligns with the standard crossover-rate reasoning and the rule “accept the project with the higher NPV when r is greater than the crossover.”

Option 2: 19.28; A
- This would imply a crossover rate much higher than the calculated value; since the true crossover is around 11.33%, 19.28% is inconsistent with the computed intersection of the two projects’ NPVs. If r were 19.28%, the sign of the incremental NPV would be negative (A would be less favorable than B), so claiming A should be chosen is incorrect.

Option 3: 11.33; A
- While the numeric crossover rate 11.33% is correct, the recommendation that A should be chosen if r is greater than the crossover is incorrect. As explained, for r above the crossover, the incremental NPV is negative, so B would be preferred, not A. This option errs in the project selection rule.

Option 4: 17.76; A
- The crossover rate value 17.76% is not supported by the cash-flow comparison. The incremental cash-flow equation would yield a root near 11.3%, not 17.76%. Moreover, using 17.76% would falsely indicate a different project preference under the standard rule, and selecting A would be inconsistent with the sign of the incremental NPV at that rate.

Option 5: 17.76; B
- This option incorrectly pairs an incorrect crossover rate with the correct project choice. Even if B is the objectively correct choice at rates above the crossover, the stated crossover rate is wrong here, so the recommendation cannot be trusted. The mismatch shows the reasoning is flawed.

Summary of conclusions:
- The mathematically derived crossover rate from the incremental cash flows is about 11.33%. 
- For required returns greater than the crossover rate, Project B yields the higher NPV and should be selected. 
- Among the options, only 11.33; B is consistent with both the computed crossover rate and the selection rule for r > crossover, while the other options either give an incorrect rate or the wrong project choice.

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