In this third lecture, we're going to talk about other evaluation techniques, and we're also going to talk a little bit more about the cost of capital. So another very common technique for evaluating projects is call the internal rate of return. And the definition of the internal rate of return is simply this. It's the discount rate that makes the net present value of the project exactly zero. So in that equation down there, what you see is we have a series of cash flows starting at C0 and going through Cn. And what we're going to do is we're going to solve for the internal rate of return, the IRR there, that will make that sequence of cash flows have exactly a zero net present value. So again, all of the internal rate of return is is the discount rate that makes the net present value of a sequence of cash flows exactly zero. So how do you then use this technique? Well, let's suppose that you had an investment of $200 million that you made today. And you thought that at the end of the first year, that was going to yield $120 million, and at the end of the second year, it was going to yield $144 million. And you're interested in whether or not you should accept this project. Well, if you were to calculate the NPV of that project at various discount rates, you would ultimately find that the internal rate of return of that project is 20%. And there are functions in spreadsheets like Excel that will calculate the internal rate of return for you. If our cost of capital, in this case, was below 20%, and again, 20% is the internal rate of return. If our cost of capital was below 20%, we would know that this project has a positive net present value, and we would want to take it. If our cost of capital, on the other hand, is above 20%, this project would have a negative net present value, and you would not want to take it. So the way that the internal rate of return criterion is used is you simply calculate what the internal rate of return is associated with the project, and you then ask whether that internal rate of return is above or below your cost of capital. Again, if it is above your cost of capital, it's a project you would want to take because it has a positive net present value. If it was below your cost of capital, you would not want to take that project because it has a negative net present value. And this is the example that we talked about earlier where we had an initial investment of 200. Then cash flows at year one came in of 120. And then at year two, the cash flows were 144. So in order to calculate the internal rate of return, we're going to use the IRR function in Excel. Now the nice thing about having Excel to calculate an IRR is there's actually no closed form solution for calculating an IRR, but Excel is so fast, it will do this for us pretty quickly. What you enter into the IRR function is the cash flows that you wanted to calculate the IRR of. So in this case, we're going to enter the cash flows b15 to d15. Then the other thing that you have to tell Excel is you just have to give it a starting point, a guess that it's going to iterate from. So let's just guess that the internal rate of return of this project is 15%, so it has something to iterate from. And when we do that, you can see how quickly that happens. Excel iterated and found the internal rate of return right away, which is 20%, which is exactly what we showed earlier when we discussed this example. Here's a simple graph of the net present value of a project as a function of the discount rate. So on the vertical axis, we have the net present value of the project. On the horizontal axis, we're showing different discount rates. And you can see that that line goes through 0 at 20%, and that is the internal rate of return for this project. The internal rate of return is often an appropriate technique for evaluating projects and will often lead you to the right answer, but it doesn't always. It can lead you to select the wrong project in some circumstances. One of those circumstances is mutually exclusive projects. And again, mutually exclusive projects are where you have, let's say, two projects, but you can't take both of them. That could be because there are two solutions to the same problem, and you're only going to solve the problem once. Or it could be you're capital constrained, and you could only take one of the projects. If you're faced with a situation like that, you do not want to rank the projects based upon their internal rate of return and take the project with the highest internal rate of return. What you want to do instead is take the project with the highest net present value. You can also get into situations where the cost of capital varies over the life of the project. So this would be a circumstance where maybe, for example, for the first three or four years of the project, the cost of capital is say 5%. And then for the rest of the project's life, let's say the cost of capital is 10%. Well, you can get into situations where the internal rate of return is right between those two costs of capital. And of course, if that happens, you can't use the internal rate of return criteria because the internal rate of return is above one of the cost of capital and below the other. So you don't know what to do. What would you do in that circumstance? You would calculator the net present value of the project, and if the net present value's positive, you would take it. The other thing that can happen with the internal rate of return is sometimes there's no definitive answer because there are multiple internal rates of return. And I won't get into why that occurs, but sometimes you can have a project that'll have two or three different internal rates of return. And again, here you're going to not really know whether you should take a project or not because some of those internal rates of return could be above your cost of capital, and some could be below your cost of capital. There's a simple rule to follow. If the internal rate of return criteria does not provide a definitive answer regarding whether a project is good or bad, or if the net present value criterion and the internal rate of return criterion do not agree on which project to select, just use the net present value criterion. It will always give you the right answer even though the internal rate of return criteria may not. In general, I would urge you to just calculate the net present value and use the net present value for making your decisions. Some companies, however, like to calculate the internal rate of return, and there's nothing wrong with calculating the internal rate of return, but it's useful to know that the internal rate of return as a criterion for deciding whether to take a project or not does have some problems that net present criterion does not have. Another evaluation technique is called the payback. And all the payback is is how many years will it take for us to recover the initial outlay? So we just sum up the cash flows and figure out how many years it is until the cumulative non-discounted cash flows are equal to 0. So it's a very, very simple calculation. Unfortunately, it has lots of pitfalls. It gives equal weight to all the cash flows before the payback period because there's no discounting associated with the cash flows, and it ignores all the cash flows after the payback period. So you could have a situation where you have a project, and maybe its payback is five years, and you're comparing that against another project that has a payback of three years, and your tendency will be to take the project with a shorter payback. But what if the project that had a five year payback had tremendous cash flows that started after the fifth year? The payback method is going to totally ignore the value of those cash flows. Another thing about the payback method is there is no correct decision criteria. You can't say I'm only going to take projects with a one year payback or two year payback or with a three year payback. There is no correct way to determine whether a shorter or longer payback is better or worse. It's not like the net present value criteria where we have a true objective, take projects that have a positive NPV, because those are going to increase the value of the firm. Firms that use the payback tend to accept too many projects, which are short-lived and reject good projects which were longer-lived. And so it tends to focus the company on shorter term projects which may not be the projects that create the most value. So here is project A and project B that we looked at before. And you can see that with project A, the payback occurs in the third period because it isn't until period three that the sum of the cash flows turns to zero. Whereas with project B, the payback occurs at the end of the first period, right? Because by the time you get to the end of the first period, the sum of the cash flows is zero. So the payback for project A is three years. The payback for project B is one year. But that doesn't mean we should take project B simply because it has a shorter payback. You already saw, previously, that whether or not project A or project B was better depended upon what the discount rate was. When we calculated the net present value at 10%, we did see that project B added more value than project A. But we also saw that if the discount rate fell below 6.5 to 5%, that meant that project A had a higher present value than project B, even though its payback is longer. So the point is payback doesn't really tell you anything about value creation, and we think that's the appropriate objective function for picking projects. Return on investment. This is another common evaluation technique that people will use, and there are many different forms of return on investment. They all tend to be some measure of accounting profits divided by some measure of investment. And the accounting profit measure can vary. It can be before tax, it can be after tax. It's usually not a measure of cash flows, it's usually an accounting-based measure. But what people will do, is they will calculate the return on investment, by taking, again, some measure of accounting profitability, and dividing by the capital that has been invested in that project. Firms usually have some specified minimum required return on investment before a project is accepted. One of the issues that you run into, however, is that the return on investment on a project can vary a lot from year to year. So you could have a situation where the return on investment is negative for a few years, and then it turns positive, or you could have a situation where the return on investment was zero for a few years and then became positive. And one of the problems with the return on investment is there's no correct procedure for aggregating those varying ROIs over time into some single point estimate of what the return on estimate is. And using the return on investment criterion usually causes firms to reject projects which are unprofitable in their early years. Going back to our two projects, A and B. And here I'll use cash flows. I won't use accounting measures because we haven't created any accounting statements, so I'll just calculate these based upon what the cash flows are. You can see for project A that the return on investment in year one would be 0%, and in year two, it would be 0%, and in year three, it would 225%. For project B, the return on investment in the first year would be 100%, it would be 100% in year two, and it would be 5% in year three. But now the question is okay, now you've got those numbers, what are you going to do with them? How do you aggregate that? How do you decide which of these is better? You might be tempted to take project B because, wow, it's got a return on investment of 100% in both year one and year two, right. And you don't get any return on investment from project A in year one and two. But again, there's no way to aggregate those ROIs into some number that tells you whether to take the project of not. And as we saw before, whether you would want to take project A or project B is going to depend upon what the discount rate is. As we saw before, at discount rates below that 6.525%, you would want to take project A. At discount rates above that 6.525%, you would want to take project B. So the problem here again is, when you have varying ROI through time, again, there's no way to aggregate that into a single number to make a decision about whether a project is a good project or a bad project, or whether one project is superior to another project.