When to Use IRR (Internal Rate of Return)? The Case of a Nonexistent IRR

The internal rate of return is a very useful decision rule in deciding whether or not to accept a project or a firm but in some cases, we don't even have an IRR. It's non-existent, There is no IRR for the cash flows of a given project and in those cases, we have to use the net present value.

So I'm going to walk you through an example where we have a non-existent IRR for a project but first I just want to quickly review the decision rule for IRR and that is we're only going to accept a project if the internal rate of return is greater than the cost of capital and by that I mean the opportunity cost of capital so if IRR is higher than the cost of capital we accept but this decision rule is only guaranteed to work if all of the negative cash flows of that project come before the positive cash flows.

So if the negative cash flows come upfront and then they're followed by the positive cash flows then you can use IRR if that isn't the case if the negative cash flow comes later or there's some kind of issue then you might not be able to use IRR it might break down. 

So in our example let's say that 
You start a construction company and you raise or you get this first project where you get 5million dollars upfront is sort of a down payment and then you get the rest of the money for the project which is 17million at the end of year 5 so there's going to be a five-year project now for the first 4 years of the project you're gonna be incurring expenses as you build this building or whatever it is you're constructing so there's going to be 4million dollars a year in expenses and your cost of capital will be 10%.
So let me walk you through a little timeline here to make sure you understand how the cash flows are working out 

So there's going to be a five-year project so we have zero here then 1st year, 2nd year, 3rd year, year 4 and 5. The zero this is just the beginning of the project you're gonna get 5million dollars now and then in the middle or I should say in between this initial cash flow that you get in year zero and the end at year 5 where you get 17 million you're going to have negative cash flows for four years of 4million dollars. So you get 5million upfront and then 4 years in a row you get it for the next 4 years you get negative 4 million as you're building the building and incurring expenses and at the end of year 5 you don't incur any expenses you just get that final payment of 17million dollars so the cost of capital is 10%
Now we are saying okay we want to solve for the internal rate of return for these cash flows, so I go and I plug it into Excel

I use the equals IRR function and however, you want to calculate it by hand you can do it we have another article about calculating IRR, and I got an error message in Excel because there is no internal rate of return for this project for this particular set of cash flows there is no IRR. So there's nothing where the NPV would be zero there's no rate of return.  So basically we have to solve for NPV. We don't even have an IRR to even compare against our cost of capital, so really IRR is meaningless here.
So here basically I'll just calculate out our net present value 

We're just gonna discount all these cash flows and so forth so we're gonna end up with a net present value of positive $2,876,201. So that means we are going to accept the project. So we used NPV because there is no IRR but just because we're accepting this project and we had a positive NPV does not mean that if you have no IRR that you're automatically gonna accept the project that it means that there's a positive NPV it just as easily could have been that there was a negative NPV. So we could come up with a different set of cash that would end up with a negative NPV and still, no IRR so if there's no IRR you don't have any internal rate of return and it just doesn't exist just forget about the internal rate of return altogether and go ahead and calculate the NPV and then here we've got a positive NPV so you accept the project. That's your decision.

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