Now let's study these IRR challenges in somewhat more detail. Well, the first challenge is investing versus let's say, borrowing, for simplicity. If we study the project in which one party borrows money and the other lends money, then clearly- and we- the IRR for both actions is the same. But clearly for the borrower, the good project will be to borrow money at a rate lower than this found IRR, while for the lender, it will be the opposite. So, we have to be careful which side of this transaction you're on. And this is luckily an easy thing to overcome because that's a very much, not only clear, but this is very natural to identifying on which side you are, but that still must be kept into account. Now, the next thing is much worse and this is multiple IRRs. I told you in the previous episode when I calculated this 28 percent, I said that if we sold this quadratic equation, then it actually gives us two solutions. One solution is positive 20 percent, and the other is negative. And then I easily put aside this negative thing saying, well, negative IRRs don't make sense. Well, as we all know in recent days, this is not such a clear statement because some very low risk projects do have negative rates. So, that requires some specific treatment. But what if the solution provides for two positive results, what we will do then, let me give you an example. Again, the example would go like this, we have cash flows of minus 800, then, plus 5,000, and then minus 5,000, and these are years, zero, one and two. So, we write the formula for NPV, which NPV is equal to minus 800, plus 501, plus IRR, minus 5,000, divided by one plus IRR squared, but I will not write the formula, instead, I will draw a chart. And the chart again, this is R, and this is NPV. Now, this will be 500 as a benchmark, this is zero, and here, I'll put some benchmark things, so that will be 100 percent. So, this is 100 percent, this is 300 percent just. Now, the curve for this NPV goes like this. So, this first solution, this is 25 percent, and this solution is 400 percent. Both of them are positive. So, both these solutions are fine. But now we can see that if we found these two numbers, the question is what do we have to do? And if we looked at this chart, we can say that actually the area where NPV is positive is between them. So, if you go over 400 percent, then of the NPV of this project becomes negative. So, this multiple IRRs, not only produce mathematical difficulty, but also they're required to go back to the idea of NPV and see exactly where we observed positive NPVs. So, we have to be really careful here and that is not the only piece. So, let me proceed with IRR challenges. Now the third thing is mutually exclusive projects. What do I mean by this? Well, let's compare two projects. Again, I will put some numbers as an example, this is project A, for which cash flows are minus 5,000, plus 10,000 and that's it. And then project B, has cash flows of minus 10,000 and then plus 17,000, and let's say that we analyze them at the rate of return of 10 percent. So, let's see what we have here. The internal rate of return for this project, so this will be IRR, and this will be, NPV at 10 percent. Here, we have 100 percent, for project B, we have 70 percent. However, when it comes to NPV at 10 percent, here this is plus 4,091, and here it's plus 5,454. The numbers exactly are sort of immaterial, but see what happens. Based on the criterion of the internal rate of return, we have to take project A, because it has a higher internal rate of return. Now, beware that these projects are one period. So, if we go back a couple of episodes, we expect to have the same conclusion from the use of NPV and the use of IRR, but it's not quite because indeed the IRR criterion tells us take A. But then, if we study that at the rate of 10 percent, we can see that the NPV of B is higher. Well, we know that the general criterion of NPV maximization will force us to take B. Now, we can go further and say here for the first time, we introduced the idea of incremental projects. What if both projects A and B require let's say, a piece of land, then we can take either A or B, but we can say well, we can start with taking an A, because it requires not only this piece of land but a much smaller amount of money. Let's say that we have $10,000, then we can easily take A that promises the internal rate of return of100 percent, and then the NPV at 10 at 4,000 plus. And then, we still have close to the spare $5,000. If we could find another project that would not require the same piece of land, but that would produce an incremental cash, incremental PV, which is the difference between those which is about, a little bit less than $1,400, then we could be better off by taking these two, A plus another project C, that is not on the slip chart. And that is key. So, oftentimes the idea of NPV here again becomes a little bit but still better thing. So, we have to be really careful about these mutually exclusive projects and I specifically said that this whole approach of using incremental projects, and take into account incremental cash flows, this is actually extremely fruitful and important in the real use of the NPV approach. Now, what else? I would specify just one more problem to IRR or challenge. What if Rs change? As was the case with ones. So we can see that to this end, the use of IRR becomes much less fruitful, because you can hardly use one and the same R, for all these periods of time and you have to analyze the project deeper. However, you can see that the NPV approach still holds perfectly. You just have to feed the NPV formula with different Rs. So, we can see that here again, NPV is better. Now you can say, "Well, you are sort of pushing us towards say, well NPV is always better." Well, it indeed is, and we will in the next episode, where we'll put things together, we will repeat once again why. But now, I just wanted to emphasize that the difference and the areas in which the NPV is a little bit or sometimes much better criterion when compared to some other ones. And, the big question occurs, why is that that everyone knows that NPV is better? And why not drop all others? Well, as oftentimes happens, that is because people, especially people in corporations where they have to make many decisions, they feel tempted to use a simpler proxy, something that is basically not that much worse but really allows them not to worry about many things, not to worry about proper treatments of inputs, not to worry about proper calculations, and not to worry about thinking the way we did in this episode than in the previous one. So, now you can see that oftentimes this push towards decisions that are easier to implement, that we once in the first week put as there, double quoted ignorance. That plays an important role in corporate management and that is why we have to pay attention to that because it's not worthwhile saying that this is all just wrong. It's more important to explain when these proxies can be safely used, because that saves time and saves managerial resources. But when this is actually potentially damaging, and you are better off if you really worried about the inputs to the NPV, and if you really spend time and resources to calculate that and sort of would make your decision based on a more advanced criterion. In what follows, I will briefly wrap this up and then we will move to the other more important piece. How exactly we will use the NPV criterion.