I am having some trouble figuring out how to approach this problem. I understand the concept that the change in sign of the cash flows crates multiple NPVs, I’m not sure how to calculate them. The answer explanation does not show how the NPVs were calculated.

A project has the following annual cash flows:

Year 0** Year 1 Year 2 Year 3 **Year 4 ‒$4,662,005 $22,610,723 ‒$41,072,261 $33,116,550 ‒$10,000,000

Which of the following discount rates most likely produces the highest net present value (NPV)?

8%

15%

10%

See below for answer:

B is correct. The NPV at 15% is $99.93. The NPV at 10% is −$0.01. The NPV at 8% is −$307.59.

If it helps to distinguish between IRR and NPV (and to really understand what is going on mathematically), try remembering that IRR is what you need to solve for in order to make the equality = 0 (the NPV = 0). And so like any polynomial, you just solve for when x = 0. There are always solutions, but they may not be over the real number system (they may be complex number system), and so, an IRR simply makes no sense. This is not the case for NPV since you are not solving for r to balance the equality, but rather just summing discounted cash flows up.

Take the CFA Institute Books example: 100+ −300/(1+IRR)^1 +250(/1+IRR)^2 = 0. With algebra, you will find this simplifies to 0 = -1 + 2(IRR) - 2(IRR)^2. To solve simply add 2(IRR)^2 to both sides, meaning 2(IRR)^2 = -1 + 2(IRR), then square root both sides. However, you’ll find you have square rooted a negative number which is only possible over the complex number system. If that complicates things, then just understand the first paragraph to understand what is going on.