All right, now let me get to the one that I always think, that makes people's eyes glaze over, kind of like the net present value discussion we had a few slides back. But again, Internal Rate of Return is not a complicated concept, it's not a complicated procedure, or calculation. By far it's the best methodology of the alternative methodologies we've been talking about in determining the appropriate investment. So, an IRR has become, over the years, more respected. Even their book rate of return and project payback are kind of, they're kind of special purpose measures that people have used in the past. I think people are getting away from that now, and they're looking more at IRR. In fact, 20 years ago when I was in corporate, we used IRR even back in those days. So the IRR is much more respected as a methodology of valuating investment proposal recommended by many, many textbooks. Okay, so I'm going to focus on the negative aspects of IRR. Why am I doing that? Well, [COUGH], excuse me, it's not because IRR is not a good thing, or it's a deficient process, it's just that the deficiencies that can arise in using IRR if you don't understand how this works is less obvious to you than, let me see. It's a less obvious problem and it's a lot more numerous so just less obvious. So let's define what IRR means. Okay, so Internal Rate of Return, IRR, is defined as the discount rate that makes the net present value of the entire cash flows over the life of the project equal to zero. Okay, I'll let that sink in for a moment. The discount rate that makes the net present value of all the project cash flows for the life of the project equal to zero, okay? Okay, so let's talk about the IRR formula, how do you calculate IRR formula, what's the rule that you should follow in using IRR to evaluate investment proposals? Well, so here's the formula and it's not complicated. So, you calculate an NPV of the project, but instead of using the discount rate, the corporate cost of capital in your formula, in the denominator of the formula, you come up with the internal rate of return to use. And so the formula basically says, cash zero, meaning that my investment in year zero + the value of the cash returned in year one. At the internal rate of return for the project + the cash return in year two at the internal rate of return for a project, plus, go on until the end of the project, dash t means how many years the project lasts. And it's basically at the internal rate of return or you discount all of the cash flows back using the internal rate of return of the project. Now, back in the day, people had to calculate this. Now anybody with reasonable Excel spreadsheet skills can go in there and use that function. There's an IRR function built into the program where all you have to do is schedule your cash flows out a couple of columns. The first year's always negative, right, because it's an outflow cash. And the next years are probably positive, although they can be fluctuating, we can talk about that a little later. Let's just say the simple process is a negative number in year zero, positive numbers in years one through whatever. And then, luckily, the program lets you make a guess on what the internal rate of return is. And the program will actually iterate, until it figures out what the IRR is. You don't even have to think about it, you just have to put the calculation in the formula. So, that's how you get the net present value using the IRR. So, the rule that corporations follow is simple. You accept the investment, if the opportunity cost of capital is less than the IRR, less than the IRR, right? And a lot of firms are using the IRR in preference to the net present value. As I said, when I was in corporate, that's what we used. We used the IRR. And generally it works well, okay. But sometimes it doesn't work well, we'll talk about when it doesn't work well. Okay, so the IRR works well in most cases, but there are some pitfalls that you can run into you should be aware of and if you run into those, you should use alternative ways of evaluating the project. So the first pitfall is what they call lending and borrowing. It simply means that the project doesn't have cash flows that decline as the discount rate goes up. And that happens when you have uneven and irregular cash flows. Uneven meaning, not the same flow of cash and irregular means they're not happening year after year after year. So if that happens, if you have a project like that, then you should use NPV. That's a way to analyze that investment proposal. There's another one that's actually funny to watch when you have Excel, get into a calculation mood, that can happen, and you can actually generate more than one internal rate of return on the same project. And that happens because of changes in the timing of when cash goes out, when cash comes in. And essentially, if that happens several times in a project, and I could schematically draw that for you, but essentially if you look at the IRR curve, let's say this is IRR, the cash goes up and it comes back down again. It crosses it more than once. If it crosses up more than once, then that means you got two IRRs to the same project. Simple solution? Use NPV. And the next one is mutually exclusive projects, okay. So depending upon how the cash flows fall, a project can have a lower internal rate of return, but a higher net present value. And it goes back to kind of the way IRR works, right? Because it's discounting everything back to the present time. So, in that case, when you have mutually exclusive projects, you want to use net present value as a tiebreaker. And finally, the fourth pitfall is if you have differences between short and long term interest rates, the rule, the IRR rule, says accept any project when the IRR is higher than the corporate cost of capital they often are. But if there's more than one r, which one should you use, right? There's more than one r when you have, let's say, a short term interest rate for short term projects and the long term interest rates for long term projects. That happens, use net present value. So, the verdict on IRR is, is that it may be more difficult to use than NPV. But if you use it correctly, it should give you the same answer. It's much, much better than using [INAUDIBLE] rate of return payback periods. And firms who encourage their managers to look for projects that have high IRRs can lead to a problem. And that is, number one, they try to, let's say, they try to make their IRRs as high as possible by maybe being a little more loose with their numbers. And as we looked at, in one of our earlier examples, high internal rate of return projects typically come from short lived projects that have a initial investment and then you have the high returns early in the period. But if you have, like another example A, B, C, the two projects, B and C, both had, they were short term. And the IRR would've said to you, invest in those. But if you remember in project A, that was the most reasonable one to use, okay? So that can be a problem. So those are the kind of alternative methods that are often used by corporate finance to evaluate your investment portfolio. Again, I think net present value is the best one to use.