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# 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.

The left-hand side of the expression for the IRR has exactly what we had before on the right-hand side of the N, of the NPV expression. In other words, what you have on the left-hand side of the IRR expression is basically an NPV and what you had on the right-hand side is zero. Therefore, what we are doing here is we need to input the cash flows of the project in exactly the same way as we have done them before. That is, I need to know what my initial investment is going to be.

I need to make a forecast of what the cash flows that I expect for this project, from this project are going to be. Now, here comes the big difference. Now we do not discount those cash flows at the discount rate, at the cost of capital, whatever might be the discount rate? Now we equate this to zero and what we solve for are those IRRs that you are seeing on the screen. Now as we said before if you remember when we calculated in session, three, and the return that we would get from a bond by buying it at the market price and holding it until maturity that was more or less a number we calculated.

We did not actually give it the name internal rate of return when we discussed bonds, but one thing, an interesting thing to relate here that is something that we are doing now with something that we have done before is that the yield to maturity of a bond is exactly the internal rate of return of the bond. That is, given the price and given the expected cash flows of a bond, there is a mean annual return that you are going to get by solving an expression that is identical to the expression that you have in front of you for the internal rate of return of evaluating a project. So now, we are solving for those IRR’s, and the same thing that we said before for bonds applies here. That is not an easy thing to solve. Certainly, this is not something that you can do by hand. You need a scientific calculator, you need Excel, and you need some tool to help you solve that equation. It may get very complicated. If you have only a couple of periods, then it may not be all that complicated, but whenever you have three, four, five periods, not only it gets complicated to actually solve for the IRR, but you may encounter problems, some of which we’re going to discuss a couple of minutes from now.

So, point number one that is the expression of the internal rate of return. Mathematically it is more difficult to solve than an impressing value, but that is what we have to deal with now. So a few things to keep in mind about this expression for the internal rate of return. As we said before, and this is important that you keep in mind, do not minimize this. This is not easy to solve, so do not try to do this by hand. You need some sort of software with this. Excel would do but you need some sort of software to, to do this. In addition, this second thing why I want to emphasize that it’s mathematically complex is because, you know, we typically think of what is the IRR of the project, and what is the IRR of the project. Well, it does not have to be the IRR. There may be more than one, and again, that is one thing that we actually will discuss in a minute.

Of course, it is going to be problematic if we get more than one but it is possible with the equation that you have in front of you. Nothing guarantees that that equation is going to have only one solution. It may have more than one. It may have no solution at all. Therefore, we will get back to these issues in just a second, but for, for now, keep in mind that this is more difficult to solve mathematically speaking than simply calculating a net present value. The rule is actually simple because the intuition once again is simple. Once we solve them for, for that IRR, once we solve for that internal rate of return, that is some sort of the mean annual return that you can expect from this particular project. Exactly as we discussed from bonds before.

Remember, for in the case of bonds we took some money out of our pocket to buy the bond at the market price. We held the bond until maturity and expected to receive those cash flows, and then we backed out. We calculated beginning from the market price and the expected cash flows, our mean annual return. Well, this is identical, identical to that. So when we’re solving the expression, for the IRR, what we’re saying is we’re comparing the initial investment that I have to make in this project with the cash flows I expect to get out of this project, and what we’re basically calculating is the return that I get from this project. Now this return, if those cash flows are annual cash flows, is going to be expressed in annual terms. Therefore, it will be a mean annual return that you expect from investing in this project. Now, whether or not you are going to go ahead with this project or not, goes back to first principles that we mentioned before. That is, if you are calculating here the return of this project, you do not want to invest in anything that does not give you at least the cost of raising funds to invest in the project, and that is exactly what we call the cost of capital before.

So the rule is very simple, if the IRR is higher than the discount rate and for now we still thinking that, that discount rate is the cost of capital then you invest in this project. If the IRR is lower than the discount rate, you do not want to invest in this project because this is basically like burning money up. It is like borrowing money at 5% and then investing money at 3%. Well, that is something you do not want to do. Companies do not want to do that, either. So if you borrow at 5%, you want to invest at something that gives you more than 5%. That is exactly what this rule tells you. All right, so at the end of the day, it is a very intuitive rule. It tells you if the return of the project is higher than the cost of raising funds to invest in the project, go for it.

Otherwise, do not. So the intuition is getting simple. The devil again is, is in the details. So in terms of competing projects is when it gets a little tricky. In addition, it gets a little tricky, because it would seem to make sense to think that the higher the IRR, that is, the higher the mean annual return of the project, the more you would want to invest in it. In addition, sometimes that is true, and sometimes that is not true. In addition, if you are confused as to why that may not be true, we are going to see an example in a minute why that may be the case. However, for now, let me say that in principle it seems to be the case that the higher the return on the project the better. In other words, if we are comparing project A and project B, and the IRR of project A is higher than the IRR of project B, then we should go for project A. In addition, in many cases, that is true, but there are cases in which that is not true, and we’ll get to a specific example about that a couple minutes from now. That has why for now let me just say that, the rule has some loopholes. So again in principle it looks like it makes sense that the higher the return on the project, the more I want to go for it, but we’re going to see a couple of counterexamples, to that.

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 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 a cumbersome calculation, but it is simply a 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 really, 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 to 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 at 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 particular, 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. A 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 are 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 are 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 that IRR is. 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 are 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.

## 3 thoughts on “Internal Rate of Return”

1. [email protected]

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2. realbench

I personally used the internal rate of return mainly to rank several prospective real estate investments. Help me flush out bad investments and pick the most desirable when considering multiple ones. Your post gives me another angles on how to employ it in other areas I had overseen.

Good post

3. Debt Consolidation

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