You are evaluating two projects with the following cash flows:

Year Project X Project Y
0 −$ 541,200 −$ 511,500
1 219,600 209,300
2 229,500 219,100
3 236,700 227,000
4 196,400 187,800

What is the crossover rate for these two projects?

Question

You are evaluating two projects with the following cash flows:

Year	Project X	Project Y
0	−$ 541,200	−$ 511,500
1	219,600	209,300
2	229,500	219,100
3	236,700	227,000
4	196,400	187,800

What is the crossover rate for these two projects?

BlackTom AI · Accepted Answer

We start by restating the problem in my own words to ensure understanding: two projects have identical crisis curves for the difference between their cash flows, and the crossover rate is the rate at which the incremental project (X minus Y) has zero net present value.

First, compute the incremental cash flows: subtract Project Y from Project X year by year.
- Year 0: -541{,}200 - (-511{,}500) = -29{,}700
- Year 1: 219{,}600 - 209{,}300 = 10{,}300
- Year 2: 229{,}500 - 219{,}100 = 10{,}400
- Year 3: 236{,}700 - 227{,}000 = 9{,}700
- Year 4: 196{,}400 - 187{,}800 = 8{,}600

The crossover rate is the internal rate of return (IRR) of this set of incremental cash flows, i.e., the r that makes the net present value (NPV) of these cash flows equal to zero.

Next, evaluate the IRR (roughly) by testing rates and observing how the NPV behaves:
- At a rate around 12.0–13.0%, the discounted inflows (years 1–4) sum to a figure very close to the initial outflow, so the NPV is near zero. When you plug in r ≈ 12.3% (precisely, a bit above 12.2%), the NPV of the incremental series moves toward zero.
- The calculation can be summarized as solving: −29{,}700 + 10{,}300/(1+r) + 10{,}400/(1+r)^2 + 9{,}700/(1+r)^3 + 8{,}600/(1+r)^4 = 0, which yields r ≈ 12.3%.

Now, evaluate each answer option against this approximately computed crossover rate:
- Option A: 0.60% — This is far too low. If r were this small, the present value of the positive incremental cash flows would be much larger, producing a positive NPV well above zero. This does not align with the nearly zero NPV observed at the correct crossover rate.
- Option B: 12.26% — This is extremely close to the calculated ~12.3% IRR for the incremental cash flows. Since NPV crosses zero near this rate, this option aligns with the mathematical result.
- Option C: 23.21% — This is significantly higher than the IRR of the incremental cash flow. At such a rate, the discounted inflows would be much smaller, yielding a negative NPV, which contradicts the zero-NPV condition at crossover.
- Option D: 13.48% — This is somewhat higher than the accurate ~12.3% value. While nearby, small differences in rate can shift NPV from near zero to positive or negative, so this is not the rate that zeroes out the NPV for the incremental cash flow.
- Option E: 23.81% — Like C, this is far from the actual crossover rate and would produce a distinctly negative NPV for the incremental cash flow.

In summary, the crossover rate is the rate that zeros out the NPV of the incremental cash flows, which calculations place around 12.3%. Among the provided choices, the option closest to this value is 12.26%, and it best fits the mathematical result for the crossover rate.

BUSFIN 3220 AU2025 (2110) Exam 2 - Requires Respondus LockDown Browser

You are evaluating two projects with the following cash flows: Year Project X Project Y 0 −$ 541,200 −$ 511,500 1 219,600 209,300 2 229,500 219,100 3 236,700 227,000 4 196,400 187,800 What is the crossover rate for these two projects?

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