So when my NPV and IRR not agree? Let's consider this example. So we have the first cash flow at time zero is a $100. The cash flow at time one is negative $130. NPV of this project is 100 plus negative 130 over 1 plus r. The IRR, namely the r that makes NPV equal to zero, is equal to 30 percent. Because if we say zero equals 100 minus 130 over 1 plus r that implies that r is equal to 130 over 100 minus 1. So now suppose that the discount rate is 10 percent. So the IRR rule reasons as follows. The IRR, which is equal to 30 percent is bigger than 10 percent. So we should accept the project. Let's see. NPV is equal to 100 plus negative 130 over, now for 1 plus r, we're going to plug in 1 plus 0.1, so 1.10. That's equal to minus $18.20 less than zero. So the NPV rule says reject the project. So here, the IRR rule and the NPV rule disagree. They come to precisely opposite conclusions. What's going on? Notice what the NPV looks like as a function of r. So NPV equals 100 plus negative 130 over 1 plus r. When r is zero, NPV is negative 130. It looks something like this. It crosses the x-axis at the IRR, which is 0.30. Ten percent is somewhere over here. So notice that indeed 10 percent is less than 30 percent. So the IRR rule apply strictly says accept, but the NPV is negative. This is minus 18 more or less. So that's mathematically what's going on. Instead of having NPV a decreasing function of r, it's an increasing function of r. Intuitively what's going on. So notice in this project, you start by getting an inflow of $100 and then you pay a $130. This is a financing project, not an investing project. In this project, you are in fact borrowing a $100 at an interest rate of 30 percent. For a financing project, you want a low rate of return. Because you are the one borrowing the money, but the IRR rule doesn't know that. The IRR rule just calculates a rate of return. So if you really like the IRR rule, you need to reverse the IRR rule to get it to work for a financing project. If you reverse the IRR rule, meaning you accept if r is greater than the IRR, then IRR and NPV rules agree. Though I would say why fuss with this. In this case, perhaps it's just best to stay away from the IRR rule. Now, I think it's pretty intuitive that you wouldn't want to go around applying the IRR rule, by the books IRR rule, IRR being greater than r in a financing project. There's a few more subtle things that could go wrong. The first is there is no guarantee that you will find a single IRR. So let's consider the following. At time zero, you have a cash flow of minus 100. At time one, you have a cash flow of 230. At time two, you have a cash flow of minus 132. So you can think about this as partially an investing project and partially a financing project. There's a little bit of investing, say between zero and one and a little bit of financing between one and two. So in this case, if you graph the NPV as a function of r. What does it look like? Unfortunately, it looks something like that. The NPV which equals minus 100 plus 230 over 1 plus r plus negative 132 over 1 plus r squared crosses the x-axis twice, once at 0.10 and once at 0.20. So in this example, there is simply no way to apply the IRR rule. If you try, you might make a mistake depending on where the r is relative to these two numbers. So really the correct region is when r is between 0.1 and 0.20. So in this case we have multiple IRR. There's multiple solutions to this equation, so you can't use the IRR rule. So that's one thing that can go wrong. Another is that you might not find any solution at all. So suppose at time zero you have 100. Then at time one you have minus 300. At time two you have 250. So again, this is a mix between an investing and financing project. This is an interesting project because it turns out that for this project, NPV is greater than zero for any r greater than or equal to zero. So there happens to be no IRR. Again, you can't apply the IRR rule. So to summarize, when you are trying to accept or reject a project. You can apply the IRR rule when the project is an investing project. It starts out with negative cash flows and the cash flows become positive. Then the IRR rule and the NPV rule agree, but if the cash flow switch in sign or if they start out positive and become negative, then the IRR rule can give some pretty strange answers.
