# Multiple NPV Problem

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)?

1. 8%
2. 15%
3. 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.

It’s interesting that you understand this . . . because it isn’t true.

Nonstandard cash flows _ may _ lead to multiple IRRs (though not necessarily), but not multiple NPVs; there’s only one NPV.

By the way, you’ll never see a question like this on the real exam: they’ll never make you do the same calculation three times. That’s just stupid.

Where did you get this question?

It is not possible to get multiple NPVs: there is only 1 NPV per discount rate. Don’t confuse IRR with NPV.

Just use your calculator:

CF0 = ‒\$4,662,005 CF1 = \$22,610,723 CF2 = ‒\$41,072,261 CF3 = \$33,116,550 CF4 = ‒\$10,000,000

Press NPV. Enter I. Press NPV. Compute.

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.

I don’t think there’ll be any electrical engineering problems in the LI exam, so we can safely dispense with the complex number system!

This came from the CFA Institute practice question bank

A higher discount rate will result in a higher NPV…Don’t even think about any calculation in such questions…its trick

Um . . . no, it won’t.