Welcome back. I'm sorry I'm being a little painful, but I'm forcing you to do all, 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? I NOT is equal to 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 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 is 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 in the 50 percent probability? Half multiplied by seven million, remember that? So, if you have null, remember the context. You've written out the data that the 50 percent probability you get seven million. What was the next? The 40 percent probability, you get how much? You get five million. So, what I'm going to do is I'm just simple going to write 0.40 times five million. And this can be written as what? 0.5. And what's happening in the last possible state of the word? With the ten percent probability you will get nothing. Right? So, think about it. The properties should always add up to one. In the real world, how many such possi, possibilities are there. Numerous, again that's why you need a spreadsheet. Here, I am saying suppose there are only three possibilities. By the way, this approximation turns out is not that bad. You can do very accurate calculation about the future, but you're probably going to be wrong anyways. So, so might as well use rules of thumb. You know, that's how people think. Thinking is more important than the final number. Right? So, how much is this? Half time seven is what? 3.5. 40 percent of five is what? Two plus zero, it's 5.5, no. The important thing to remember is its never 5.5 in any particular year, is 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 I NOT is IRR for a perpetuity. Okay? And that is equal to 5.5 divided by 50 is, I believe, equal to how much? So, 5.5 divided by 50 million is equal to eleven%. Okay? Very straightforward. Right? So, take the zero fifty. Five goes eleven times into 55, and you get eleven%, right? So, another way of thinking about this. At eleven%, what would be the NPV of this? So, if I take eleven 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'll double check that the eleven percent 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 zero NPV. I write everywhere and that's one of my weaknesses, it's 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, quick, so in order to do a decision, now what do I have to do? Given that this is well-behaved cash flows, there's only one project, IRR is well-behaved meaning, this change of sign only once and you don't have multiple IRR's, 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 are enjoying this.