# What is the economic meaning of multiple internal rates of return?

Recall that the internal rate of return (IRR) is the discount rate such that the net present value (NPV) of a project is 0.

One interesting complexity of the internal rate of return (IRR) is that it does not have to be unique. For example, consider the following project with cash flows spanning 5 years:

What is the economic meaning of this phenomenon? Beyond the economic meaning, what are the project financing implications? Say your IRR hurdle rate is 10%. Should you do this project or not?

I know there is a related issue, where the IRR doesn't exist. That doesn't seem like as big a deal, because if the NPV of a project is positive for all discount rates then that seems like a very good project (and if always negative then it is a very bad one). Is there a modified IRR decision rule that handles the situations where the IRR is non-unique or non-existent?

Some claim this is a reason to use NPV instead (Jan (2016)). But I don't see how that helps. What do we make of a project like this, one that has a positive NPV with a discount rate of 10% (\$1,429.94) and a negative NPV for a discount rate of 5% ( -\$1,008.15)? Are we really saying that this is a project that if profitable if your cost of capital for the project is ten percent but unprofitable if your cost of capital in five percent? Doesn't that seem off and maybe even wrong? Normally, having a lower cost of capital is an advantage!

• The possibility of multiple IRR's was one issue discussed in the so-called Cambridge capital controversy c 1960. See here, especially the section on Reswitching. – Adam Bailey Jan 24 at 20:30
• You typically get these multiple IRRs when there are flows in one direction at the beginning and the end and flows in the other direction in the middle. One way to assess them is to use the actual cost of finance in the calculations (rather than a conservative threshold) and see if there is an arbitrage which makes the project profitable. If there is, then undertake the project and the arbitrage; if there is not, then do neither. – Henry Jan 24 at 23:48