Welcome back. I'm sorry I'm being a little painful, but I'm forcing you to do everything at the same time. So let's see what IRR is. Question number one, what do you need to know to figure out the IRR of the project? You just need to know the cash flows. So let's see what's happening. At time zero, You're making an investment. How much is the investment? I0 = 50 million, right? Bunch of zeros I'm leaving out. So you need to know your investment, what else do you need to know? You need to know your cash flows. So your cash flow at time C1, C2. I've kept them the same, and go on forever. However, I made the problem interesting, it's expected cash flow. Why, because it's not known for sure. So let's figure out expected cash flow, and by the way, this helps us do the problem together. In real life the expected cash flow will be what? Expected cash flow one, two, and so on. Those numbers will be changing. But that's not that important here. So let's do it. How will you figure it out? How much is the cash flow with a 50% probability? 1/2(7 million) remember that? So if you remember the context, you've written out the data, with a 50% probability, you get 7 million. What was the next? With a 40% probability, you get how much? You get 5 million. So what I'm going to do is, I'm just simply going to write (0.40)(5 million). And this can be written as what? 0.5, and what's happening in the last possible state of the work? With a 10% probability, you get nothing, right? So think about it. The probabilities should always add up to one. In the real world how many such possibilities are there? Numerous, again, that's why you need a spreadsheet. Here I'm saying, suppose there are only three possibilities. By the way, this approximation, turns out is not that bad. You can do very accurate calculations about the future, but you're probably going to be wrong anyways. [LAUGH] So might as well use rules of thumb. That's how people think. Thinking is more important than the final number, right? So how much is this? 1/2 times 7 is what? 3.5, 40% of 5 is what? 2, plus 0, it's 5.5 million. The important thing to remember is it's never 5.5 in any particular year. It's bouncing up and down around 5.5 million, okay? So let me just wrap this up and ask you, what is the IRR? The beauty about IRR of a perpetuity is what? All you need to know is, how much cash flows are you getting per period divided by the investment. So the C over I0 is IRR for perpetuity, okay, and that's = 5.5 divided by 50, is I believe, equal to how much? So, 5.5 divided by 50 million = 11%, okay? Very straightforward, right? So, take the 0, 50. 5 goes 11 times into 55, and you get 11%, right? So another way of thinking about it is, at 11%, what would be the NPV of this? So, if I take 11 and I figure out the PV of 5.5, what is it? Think about it, what is the PV of 5.5 perpetuity? It's 5.5 divided by, PV is 5.5 divided by 0.11, if I use the IRR. If I use the IRR, and it's equal to 50 million. But that's exactly equal to my investment. So now you double check that the 11% is actually the IRR. Because remember, what is the IRR? IRR is that rate of return that makes NPV zero. So this minus 50 is equal to 0 NPV. I write everywhere, and that's one of my weaknesses, is that I'm messy. But hopefully you understood what I'm doing here. Quick question, did I use any cost of capital in making this calculation? Answer is, no. So in order to do a decision, now what do I have to do? Given that this is well-behaved cash flow, there's only one project, the IRR is well-behaved, meaning there's change of sign only once, and do you don't have multiple IRR, we can use IRR. But in isolation, it doesn't make any sense. Let's take a break. We'll come back to question number three and then take a break, and so on and so forth, okay? Hopefully you’re enjoying this.