# Internal Rate of Return

Internal Rate of Return is another important technique used in Capital Budgeting Analysis to access the viability of an investment proposal. This is considered to be the most important alternative to Net Present Value (NPV). IRR is “The Discount rate at which the costs of investment equal to the benefits of the investment. Or in other words, IRR is the Required Rate that equates the NPV of an investment zero.

NPV and IRR methods will always result identical accept/reject decisions for independent projects. The reason is that whenever NPV is positive, IRR must exceed Cost of Capital. However, this is not true in the case of mutually exclusive projects.

If it ever happens, and, and it might happen actually more than once, that you have a conflict between what the NPV rule recommends and what the IRR rule recommends, you always want to fall back on the NPV rule. Moreover, the reason you want to fall back on the NPV rule is, is that mathematical reason that we actually mentioned in passing. We did not get into the details, but that mathematical reason that we mentioned before. When you are calculating it at present value that is not a tricky calculation. It may be a messy calculation. It may be cumbersome calculation, but it is simply calculation of a present value, there is no mathematical complexity involved there. When you are trying to solve the expression for the IRR, you may run into trouble. You may run into situations where you have no solutions, where you have many solutions. And therefore it gets a little bit tricky, and again we’re going to discuss a couple of examples very soon but as far as what matters real, or right now is concerned do remember that. If you calculate a project’s NPD, if you calculate a project, a, a project’s IRR, and it happens to be the case, because it doesn’t have to be the case, but if it happens to be the case that the NPD tells you go for this project, and the IRR tells you don’t go for this project, or the other way around, always fall back, always rely on the recommendation of the NPV. So that’s important that you keep in mind. In any case, of conflict, you want to fall back; you want to follow the NPV rule.

Why are we saying all this? Let me start from the end of what I want to cover in the next few minutes. I do not want to tell you that you should not use the IRR. I do not want to tell you that the IRR is very problematic and very cumbersome and therefore you should actually forget about it. What I want to tell you is you should be careful when you use the IRR. Moreover, in particularly, I want to tell you that you should know the structure of the cash flows that you are dealing with. Moreover, this will be clearer in a minute. However, let me start with an example that actually illustrates why you have to be careful with the IRR. Let us look at these numbers. Very simple project that you see. Only three cash flow. We have to put down 100 million today. We expect to get 260 million a year from now. Moreover, 165 million a year, two years from now. Moreover, we are dealing with a company whose discount rate is 12%. So if we have those cash flows and a discount rate of 12%, well, we can do two things. We can calculate the net present value, but because we are talking about shortcomings, or problems with the IRR let, let us focus on the IRR. In addition, here is a picture for you to take a look at. Remember the definition of the IRR is the solution of that expression; the IRR is the solution that gives you an NPV equal to zero. That means that if I am plotting different discount rates on the horizontal axis, as the picture shows you, and I look at the NPV on the vertical axis, for each discount rate I can calculate what the NPV would be. Moreover, for some of those discount rates the NPV will be equal to zero. Well in those cases, that’s exactly the NPV that tells me that that is the IRR, because remember that by definition, the IRR is the rate, the discount rate that makes the net present value equal to zero. Now you can see what the problem is with the picture. That there is two times in which the line, the blue line crosses the horizontal axis. And that means that there are two solutions for that equation. And that means that there’s two instances in which the net present value is equal to zero. Not just one, but two. Now let me put down, if in case you cannot see the picture very clearly, what those IRR are, are. One is an IRR of 10%, which is the one on the left, and the other is an IRR of 50%, which is the one on the right. And here comes the problem. We are dealing with a company whose discount rate is 12%. So what do we do? We do not invest in this project, because the IRR is 10% and therefore lower than 12%, or we do invest in this project because the IRR is 50% and the discount rate is 12%. That is the problem, and we cannot tell one or the other. We cannot tell that one IRR is better, more accurate, or superior to the other. There is no mathematical or intuitive way of doing that. So here you see what the problem is. We may be facing a situation in which the expression that we need to solve does not have one solution but more than one. In our extremely simple case, that solution is actually, there is two solutions, and they happen to be one below and one above our discount rate. So what do we do in situations like that? Exactly what we said before. We need to fall back on the NPV. And if you calculate the NPV of those cash flows and you discount them at the rate of 12%, then you’re going to get an NPV of 0.6 million, or, or \$600,000, and that number is positive, and that tells you that you should ho, go ahead with this project. So problem number one of the internal rate of return, we can have more than one. In this very, very simple case, we actually have two, but you can actually have many more, depending on how complex, how long is the expression for the, for the NPV. Just as an aside and you know, this, this doesn’t really go to the heart of what we are discussing here, but if you are wondering, what makes you know, what is the problem here? Why do we have two solutions? Well, you know, if you have a clean case, in which you have a whole bunch of negative signs and then a whole bunch of positive signs. In other words, you have changes in the signs of the cash flows only once from negative to positive and from positive to negative, then that guarantees that you’re going to have only one IRR. But of course you know, suppose that you’re putting down some money to start a project, you start receiving money out of the project. But eventually three, four, five years down the road you need to actually put down some money to add fresh capital to the project, to maybe invest in buildings that have depreciated or the machine that has depreciated and now you have another negative cash flow.

Once that additional negative cash flow appears then you, go from negative to positive to negative, and then it presumably to positive again. Now once you start having those changes in the signs of the cash flows, then you know, all bets are off. Then you can have a number of different discount rates or different solutions to the expression that, that we actually set up. In fact, the mathematical rule says that you can have up to as many solutions as changes in the sign we have in the equation. Again, this is just a passing comment if you are curious about it, or if you are mathematically inclined, but what is important is that nothing guarantees that you know, its anti without knowing the cash flows of the project that we are going to have only one IRR. All right, if that were bad enough, here comes sort of the opposite side of the coin. The opposite, the one side of the coin is look, there is a problem with the IRR and depending on the structure of the cash flows, and we can have more than one solution. Then it gets problematic to decide whether we should go ahead with the project or not. As we said before, if that is the case, go for the NPV rule.

The problem with IRR

The problem with IRR comes about when Cash Flows are non-conventional or when we are looking for two projects which are mutually exclusive. Under such circumstances, IRR can be misleading.

Suppose we have to evaluate two mutually exclusive projects. One of the projects requires a higher initial investment than the second project; the first project may have a lower IRR value, but a higher NPV and should thus be accepted over the second project (assuming no capital rationing constraint).

Decision Rule of Internal Rate of Return:

If Internal Rate of Return exceeds the required rate of Return, the investment should be accepted or should be rejected otherwise.

Internal rate of return (IRR)NPVPVcapital budgeting techniquesInternal Rate of Return Internal Rate of Return is another important technique used in Capital Budgeting Analysis to access the viability of an investment proposal. This is considered to be the most important alternative to Net Present Value (NPV). IRR is “The Discount rate at which the costs of investment...