Now in this episode, the central episode of the first week, we will introduce something that most of you have heard about and that is very famous in corporate finance. This is the approach that deals with the concepts of present value and net present value. Now first of all, let's start with an elementary project like this. Again, my jumbo jet taking off metaphor was specifically pronounced in order to show to the people that although I start from the things that seem to be offensively easy to understand. But, again, our jump from that to something much more advanced and realistic will be very quick. So let's take a look at the elementary project. So we have one period, this is 0 when we make a decision. This is point 1. And we expect To receive $1 at point 1 here. And we pose the question, what is the equivalent to the receipt of that $1 at point 1 at point 0? So that may seem like an answer to the following question. If I invest in this project, What is the equivalent of putting some money in my pocket right now, or maybe taking some money out of my pocket? Well, in order to tackle that, we have to realize how would we treat this time flow, and how would we treat this project? Because something happens from 0 to point 1. And we say that this project has a rate of return of r, what does that mean? That means that for any amount of money that you invest here, you will get this amount multiplied by 1 plus r at point 1. If this is the case, then we can easily get an answer to our question. We'll say, if x is the equivalent of that at point 0, then x multiplied by 1+r should be equal to our expected $1. And we get the first famous formula that x is equal to present value of this $1, and this 1 over 1+r, where r is the return for this project. Now projects have different returns, projects have different lengths. And as we understand projects have also different risks. But for the first part of this course, in weeks one, two, and three we will take risk and return as a given, and we'll study that all the later. But for now, this is like I said an elementary project. We can have clearly other projects, and now we can make some generalizations. Now generalization one will be, what if we expect to receive C dollars? Well, then free NPV will be equal to C over 1 + r, great. But look, what if by the way here we assume that r is positive and C is positive, too. If this is the case, then we can say that PV is positive which is kind of cool. So the question is in order to receive that, I have to enter this project. I have to pay something to enter into this project. Because I receive a positive amount right now. Remember, I said that PV answers the following question. How much money do I put in my pocket right now that is equivalent to my investing in this project? So if I put a positive amount in my pocket right now, that seems that I got a gift. Well, again, this is a very clear idea that in general we have to pay something for these projects. And that leads us to the next formula. If you paid, I put -C sub 0 for this project, and then you receive this PV whatever it is, then this is called net present value. So that basically answers the following question, how much do I put in my pocket or do I take out of my pocket after I entered in this project, after having paid C sub 0? Well, all that is great. But now if I know how to calculate NPVs, then I have a fundamental criterion for valuation. I can say remember I started out in the very first episode saying that we have to find the criterion of affixing labels, what is good and what is bad. And now we can say that clearly a black box is good. When? If its NPV is positive. So this is the most fundamental valuation criteria. So that basically means that I found a project by having invested in which I put some money in my pocket right now. Sort of from nothing. Well, not exactly from nothing. And we understand that if all projects in all this world they would have been let's say absolutely clear to everyone, then as a result of competition, it's unlikely that people would be able to find some some positive NPV projects. We will indeed see that in some markets that are very liquid, very close to being perfect, this is indeed the case. For example, this is NPVs of most investments in riskless bonds are very close to zero NPV projects. But if you take up a lot risk, if you engage in something uncertain, then it's very likely that if you are lucky, or if you're successful, then you do indeed get a positive NPV. So basically, this is the general thing. And this process, as you know, is called discounting. And from here we can proceed with some more generalizations, because here I used a one period project. Well, first of all, I will show you some simplistic examples how business this discounting works. Let's say that we analyze the following project. So this is, again, 0 and 1. I invest $1,000. And then in the blue project, I expect to receive $1,200. And the return of that is 10%. And in the black project, for which cash flows are exactly the same, so this is minus 1,000 here and plus 12,000 here. But the rate of return on black project is 30%. Now you don't have to be a wiz to calculate that for a blue project the NPV is equal to +91, while for a black project the NPV is equal to -77. So you can see that although in both projects you are expected to make more money than you invested. But the question is not only more but how much more. How much more is enough to make the project good. And now we can see that the threshold point is clearly 20%. So if, for example, this project would require a 20% return, then its NPV would be equal to 0. All these things will happen very often in our future examples and studies, but for now this is kind of thing that you have to keep in mind. So this idea of how much more makes it good. But then we can make generalization two. We can say, well, projects Have different lengths. And we know the different rs for all these lengths. And so we can say that if, for example, this is 0, 1, and 2. And this is C1, this is C2. And then let's say that r1 is the correct expected rate of return for this kind of investment. And then r2, if we invest for two years, then clearly this in general is going to be a different rate of return. Not because of the length, but also because of the fact that maybe the expectations for a two year investment are different. We have to compare returns that are sort of normalized to the same period. So here r2, in order to get the PV of that, you have to square that. Then the NPV of this will be equal to -C sub 0 plus C1 over 1 + r1 + C2 over 1 + r2 squared. So we generalized for two periods. But, again, now we can make the general formula. We can say that the general NPV is equal to -C sub 0 + C1 over 1+r1 + the general term Ck over (1+rk) to the k-th power. And then plus because that doesn't have to be finite. So this is the general formula which we reached very quickly. But the problem is that, in order to find MPV, you have to know all the C sub 0, C1, Ck, and so forth. And also you have to know r1 rk, and so on and so forth. So, well, you can always say that In reality you can make investments not only at point 0 but then later on 2. But for the sake of simplicity, we can always think that these investments can be treated sort of like costs at these points. So we will not delve deeper into that for now. We discuss these differences between capital expenditures and some kinds of costs. Not only in what follows in this course, but also in some greater detail in our third course. But for now, we've reached this formula and that all sounds great. We know that unfortunately we posed more extensions than we have answers. Because you say, great, if I only knew all Cs and all Rs, I would be great. However, we do not always know them. And starting from the next episode, we'll start to see how this formula works if we do know some of these Cs, or we have a good forecast of them, or if we make some special cases that will simplify the use of this formula. Let me tell you right away that although some of the examples are, not only widely known but seem to be very much made up to make the formula look simple. But the key story is that actually people do use these simple results in actual corporate evaluations and evaluations of some real assets. So our examples from the next episodes will not be so irrelevant, if you will. So starting the next episode, we will study these short cuts.