The decision rule for IRR really becomes whether you are making more money than the others. So I'm going to spend a little bit of time on graphics. I think graphics can help a lot, especially by something so complicated. So let me draw, and I hope you follow with me, 0. And let me draw this, NPV. And let me call this, r. Okay? Why am I drawing this? For two reasons. I have a value of 0 here. Why am I making it 0 NPV? Because I know that if I'm going on the south side of this, it's not good. Why? Because I'm actually destroying value, right? So, remember, value creation means positive net NPV. Why am I taking r to 0, up to 0? Because we have assumed, for the purposes of our whole course, that the r cannot be negative, or will not be negative. I shouldn't say cannot, it can. But let's stop there for a second. So now I know what is my project? (100) x 0 + 110, x 1. Got it? Okay, now let me show you what the relationship is. Suppose I don't know the IRR of this, right? I know, because I can calculate 10%. But suppose I don't know. This is what a calculator will do, it'll start off with 0. So if the IRR is 0, if the discount rate is 0, what is the NPV of the project? This is the easiest example I could ask. If in time, value of money is 0, what can you do? [LAUGH] You can add. So the NPV will be 10. Right? But if the IRR, if the discount rate is 10%, what do we know about the NPV of this project? It's 0. Because that's the definition. If I use 10%, what is the NPV? 0. Because 110 divided by 1.1 minus 100. So draw a line. And this is a little bit, just pay attention a little bit to this. How difficult, in this example, is it to calculate the IRR of my project? Very easy. In the graph, 10% is the IRR. Why? Because I know at 10%, the NPV is 0. What's true now? What has this told me? One simple fact, that my project is going to make 10% rate of return. However, doesn't mean anything. Now, I know that I have to compare it to whom? The cost of capital, r. What are other people making? So look what happens. If other people are making less, is this project valuable? Answer is yes. This is +NPV. However, if other people are making more, which direction [LAUGH] am I going with my idea? -NPV. So, the rule of thumb is very obvious here. If IRR > r, yes. If IRR < r, no. But the tragedy of this rule is what? I'm choosing this to be a yes only because NPV is positive. Why am I saying no? Because here, NPV is negative. So the tragedy of IRR is, IRR cannot work by itself. 10% by itself doesn't mean anything. And I, please, encourage you to internalize this because this is so important. And popular press says only report returns. They don't mean anything in isolation, we'll see. But in order to make decisions, if you calculate your IRR, what do you have to compare it to? How much are other people making? That's your benchmark, okay? So if you use that benchmark, you come up with a decision rule that you do things, if you are doing them better than other people. Another example. And this will show you why we use formulas. Tell me, what is the IRR of this idea? I'm going to pause for a second and let you think about it. You see, this is going to make your mind go nuts. So tell me, let's draw the timeline. Let's draw the timeline of this, okay? So what has happened? 0, 1, 2. Does it look like the same problem we had before? Yes. I've thrown one curve ball at you. I've said, you spend $100 today, same as last time, and let that be $1 million again. But, I said, you know, your idea is such that in the first year, you're more likely not to do anything, make any money. Is that possible? Of course it's possible. What do the best ideas of the world do? Not make money for a long period of time [LAUGH] initially, and then, boom, right? So $110 in year 2. Do the numbers. Are the numbers the same? Yep. But for what I have done, I have oranges x 0, apples x 1, and now I've thrown in bananas x 2. So by shifting time by 1, what have I done? I've made life a little bit miserable, and that's why you have formulas. Okay. So what is the IRR over 2 years? So if I want to say, suddenly okay, I'll solve this problem very easily. I'll just make my period 2 years. What is the IRR over 2 years? So you have (100), you have 110, 10%, right? Same answer. But is it comparable to the previous one? No. Because 2 years is not the same as 1 year. I mean, you have to remember time value of money. So the question is, what the heck do I do? What is the IRR of this per year? So that's why things have to have the same periodicity to be compared. So what is the IRR per year? That's a tough one, right? [LAUGH] Because, what have I done? I've thrown in an extra year where nothing is happening. So how will we solve this problem? Very easy to think about, very tough to do. Make NPV 0. What would that do? (100) + 0. How much of discounting do we do to be 0? In year 1, 1+IRR + in year 2, what do you have? 110. 1+IRR square. Quick question. This is an R. Quick question. How many unknowns in this equation? 0 equals 100 negative plus 0 over 1 plus IRR plus 110 over 1 plus IRR square. How many unknowns? One. Which is IRR. What's the problem? It's not easy to calculate. Why? Because of, pause again, compounding. So IRR is tough to do mentally because of compounding when the number of periods increase. If it's one period, it's relatively easy. And that's why, who do we go to? To the computer. And I'm going to do it in a second. Before I do it, I have two things to do. One, I'll show the formula, generically which shouldn't surprise you, it's how do you make NPV 0, in a second. But number two, I'll go to the calculator. But before I do that, the second thing I wanted to say is, can you guess what it is? So, with 2 years if I asked you, suddenly the world is 2 years is 1 year you know, 1 period of time, you know what the answer is,10%. What will it be per year? I think many of you will be tempted to choose 5%. But then again, you are kind of stabbing me. You know, you're forgetting compounding. So if money earns no interest on the money, you're on the right track, but then life is very easy. We don't need to do most of this class. Turns out, the IRR will be < 5%. I can guess that simply because I know there's compounding, and the actual answer is probably 4.9 or something like that. It'll be slightly less than 10%. So I want to do this in a calculator, but before I do that, let's just stare at the generic formula. IRR is the rate that solves the following equation. Where I0 = C1 plus C2. I would rewrite it, if I may, in a slightly different way, where NPV = (I0) + all this junk = 0. So equating (I0) to the right-hand side, if I take (I0) to the right-hand side, it becomes NPV formula, and then you force it to be equal to 0. So let me ask you this. In that period, how many cash flows were there? Nothing here, 110 here. Now you could have many more cash flows. The problem is, from being a quadratic problem, it becomes a problem like which E=MC2 was cool for Einstein because he stopped at square, but when he saw N, he said, man, this is too cool, this is just too much mind-boggling. And it is. The power of compounding now is in reverse, it's in the denominator. We did future value and now trying to figure out present value is a tough thing to do. So what I want to do now is take that problem, simple problem, and do some calculations on the calculator. So, let's go on a tab and let's keep those numbers there. And, actually no. Why don't we just delete that number. And, what was our problem? I'm spending -100, right? This was, what? 0. This was, what? 110. Right? Everybody okay? I think you're okay. Let's do it. So what is the function IRR? Open up the brackets (, what do you know? You want to just throw in values. Now remember, in IRR you have to throw in all the values, because if you don't throw in the a1 IRR, they'll laugh at you. In fact, Excel will say, come on, get real, you're getting 110 for not doing anything. So IRR has to have a1:c1. And you don't want to look stupid, even to Excel you know, because there's no point. Now you can throw in a guess, I'm not going to. And the reason is, the answer is pretty straightforward, and the reason I'm getting 5% is because the number of decimals in this is not big enough. So, the answer here should be actually, if you increase the number of decimals, it should be 4., 4.88%, okay? And the way to double check this is, what? If I use 4.88 to discount 110, what answer will I get? I'll get exactly 100 bucks. So let's do that. Let's take =pv(.0488, right? Am I okay here? Yep. ,2,0,110). Exactly $100. So, by this I know that 4.88 is the answer, and that the decimals are not showing up in the 5% the way it's set up right now. So, what you want to do is you want to make sure, always, that the decimals are showing. So let me just go here and do that for you. One, two. So now I'm showing not just 0 decimals, I'm showing all. So you see, my answer was right because this simple problem, that number was in my head. Now, when you come back, let's take a break. When you come back, we'll do more complicated examples. We'll talk about IRR as a principle, and we'll take it to the last piece of today's session. But I think you need to understand, right now, how to calculate IRR, and what does it mean? Two things, calculation requires making NPV of your project 0, simply because it's the easiest way to figure out IRR in a most updated context. Quadratic, high order numbers throwing in because of compounding. The second, getting 4.88 by itself doesn't mean anything. It doesn't mean anything. 4.88 is better than 0, yep. But how do you judge whether this is a value creating idea? You cannot judge something just by your own cash flows. You have to go figure out what other people are making. In the banana business, this is awesome. Why am I saying that? Because if you earn 5%, 4.88% per year in banana business, you're doing great. But if you are talking about iPads or technology, probably not such a good idea, because others may be making more and money won't come to you to create a new project. So take a break. We'll come back, we'll keep plowing through this stuff. This is actually both intuitive and practical. Take care, see you soon.