# 1.32 Multiple IRRs

As briefly noted earlier, when one (or more) of the interim cash flows is negative, there may be more than one mathematically correct IRR. If in such cases, there are two future cash flows, one of which is negative, we will get a “quadratic” solution. By definition, a quadratic equation is one that contains three exponents: zero, one, and two.

 Cash Flows 0 (\$1,600) 1 \$10,000 2 (\$10,000)

As we know, the IRR is the rate at which the NPV will be equal to zero. Therefore:

NPV = 0 = [(\$1,600) ÷ (1 + IRR)0] + [\$10,000 ÷ (1 + IRR)1] + [(\$10,000) ÷ (1 + IRR)2]

Where IRR = 25% and 400%!

If we just used the NPV approach, we would not have a problem because the discount rate would be given.

Solutions/Proofs:

25%: 0 = [(1,600) ÷ 1] + [10,000 ÷ 1.25] + [(10,000) ÷ 1.252]

0 = (1,600) + 8,000 + (6,400) = 0

400% 0 = [(1,600) ÷ 1] + [10,000 ÷ 5] + [(10,000) ÷ 25]

0 = (1,600) + 2,000 + (400) = 0

What if the capital costs were just 10% (k = 0.10)? Wouldn’t we accept this project since the costs are less than each quadratic solution? No! If you take the time to calculate the NPV, you will find that it is negative, i.e., (\$773)! If you also calculate the NPV using other discount rates, e.g., 20%, 30%, 40%, and 50%, you will find that the NPV goes back and forth from positive to negative. In short, we cannot use the IRR at all when an interim future cash flow is negative! In other words, we reject the … IRR method!