Now let's talk about internal rate of return, or IRR. This is a very well known abbreviation. And like I said, oftentimes people take it as the same, Criteria as NPV, which I will put like, this is not perfectly like that and oftentimes may result in certain mistakes. Now what's the definition? The definition is as follows. So if we have a formula for NPV, and if we set that to 0, then the solution to this equation is exactly IRR. So basically we take, if it's a multi-period project, then we have the same rate of return over all these projects or over all these periods. And therefore, we set the NPV to 0. So to some extent it can be said that the ideology of IRR is close to the ideology of yield to maturity when we studied bonds. And then the criterion is very simple. So we take projects, With return r which is > IRR. Well, simple as that. Now if, We have one period, 0 and 1, then indeed NPV and IRR is the same because clearly what is it? We can say that here we have -C sub 0, here we have C1. Then the MPV is C1/1 + IRR. Here I put IRR by force minus C sub 0. And that should be equal to 0. And I solve that and get one and the only IRR. So that will be the same criterion, and that will lead to this approach. So far so good, and if all projects would be one period and the NPV calculation would be as simple as it is here, then we would not need anything else. And strictly speaking, we would not need any other criteria by NPV, and this whole discussion that we had so far and that we are proceeding right now would be redundant. However, the reality is different. And on the next, what I will show right now is what happens when we deal with the actual more general formula for the NPV. Well, we know that in general NPV = -C sub 0 + C1/1 + r + Ct/1 + r to the tth power, where I already made the simplification, I put the same r. Now if we set that to 0, then clearly, this equation is of the tth power, and it has t solutions, which is kind of bad. Well, luckily, the majority of these solutions are bad solutions without going deeper in what that means. That requires some mathematical advancement. But for now, what I will start to do, I will start with an example. Because oftentimes we do have one good solution. But then, if we said that we identified this IRR, whatever it is, and then we said that we have to take into account projects with the return that exceeds this IRR. Well, let me give you a very simple, but an example of a two period projects with these following cashflows. So let's say this is 0, this is 1 and this is 2. Here it will be -2000. Here it will be +1000. And here it will be +2000. So what's NPV? NPV is equal to -2,000 + 1,000/1 + IRR, because we are solving with respect to that, + 2,000/ 1 + IRR squared. Well this is the quadratic equation, and solving that we get two solutions. One solution is negative, but the positive solution will give us IRR of 28%. So if we proceeded in a straightforward way, we should have said, well, now we have to take only the projects for which, whether it's a good or bad project. Well, we can say now we have to take projects with IRR which is greater than that. And that will be, So we just got this number, and now let's see what happens with NPV. Because we know that the more general NPV criterion says that the project should have an NPV that is greater than 0. Well, let me draw a chart of NPV. Well, this will be 1,000. This will be -1,000. And this is the NPV axis. And this axis will be the axis of r. So here I'll put some benchmark numbers. So that will be 10%. That will be this point is exactly this 28% that we identified. This is 50. Now if we took this formula, and if we draw a chart, the NPV as the function of r, we will replace that IRR with r, we would get the curve something like this. So basically that would show that the NPV of this project at rates below 28% is positive. So if, for example, we believe that this project can be taken because the risk of similar projects is let's say 10%. We calculate NPV and get it somewhere here, which is quite positive. So you can see that you have to be really very much careful with respect to what happens with the NPV. Because the IRR as a number does answer the question that NPV is equal to 0. But whether it's going to be greater than zero or less than zero requires some additional study. So we will, I gave this example just to make a smooth transition to what follows in the next episode because then we will study challenges to IRR. I said that this is a widely used criterion because oftentimes it's easier to have just one number, and then you take the other number and do a comparing two numbers instead of using some cumbersome formulas. So in what follows, we will go deeper into the challenges provided by the use of IRR. And we will see what exactly has to be taken to the account to avoid making mistakes potentially associated with these challenges.