[MUSIC] So now, let's develop the concept of net present value. This is a fundamental concept in corporate finance, because that's what's going to allow us to make decisions about projects, about collection systems, about acquisitions and pretty much everything else we're gonna talk about in this course. What is the net present value? The net present value is the sum of all the incremental cash flows. Remember, we already defined incremental cash flows, new minus old. All the incremental cash flows discounted from the future to the current period. The important thing here is that to get your NPV, you have to take all the incremental cash flows into account. Everything. You take all the consequence, any decision you're thinking about, you have to model all the consequence of taking that decision and then discount to the current period. That is NPV. Notice that there is the word discounting here. So we have to discount. And so we are going to need to use a couple formulas. That I'm pretty sure you've seen those already at some point. If you didn't, it really is very easy to learn and to apply these formulas. All you need really are these two formulas, the computer and the calculator. And in some cases, pencil and paper really works. It might be even the best way to solve a problem. So these are the two formulas that we need. So r, there is this term r which is the discount rate. So that's the rate that we're going to use to discount the cash flows. So here you have a situation where you have a cash flow, Ct that is happening t periods ahead. Remember this is a timeline, so t here could be anything. T here could be anything, it could be a year, it could be a month whatever you want. And to discount that cash flow, what you do is you divide it by 1 plus r to the power t. So if the cash flow happens two periods ahead, you do 1 plus r to the power of 2. Very simple stuff. I hope. [LAUGH] This is a little bit more complicated, but its very important that we learn that. We cannot work, we cannot do corporate finance without a growing perpetuity formula. That this is what it is. Now, we have a cash flow that happens every period. So here, there is a cash flow C with period one, there is a cash flow C another cash flow in period two. And notice that now the cash flow is going to grow at a certain rate. G here is the growth rate of cash flow. So if you look at this. If g is 5%, then the cash flow in period two, remember it's not necessarily a year. The period two is going to be 5% larger than the cashflow in period one. If you go to period three again, you're growing cash flow at 5%. If you have this pattern which as we're going to see is gonna be a very common pattern. If you have this pattern, the present value is just gonna deceit the cash flow divided by r minus g. So I know this is math, it's a formula. Many people get scared with math, but it really is very simple. All you need to do is to remember these two formulas or even look them up if you forgot. And apply them to solve some different problems and you're gonna get this very quickly I think. All calculations can be done with these two formulas. All of them. So I like to really narrow down and just tell you about those two formulas, cuz they are sufficient. To allow you to practice this, I want you to try to answer these two questions on your own. So very simple present value problems. So first one, we have $1 million in one year, the discount rate is 6% a year, the second one is a perpetuity. So it's a payment of 1 million every year that's gonna last forever and the discount rate is 6%. So please work on these two questions. Answer should be really simple. The first problem, all you need to do is to discount the cash flow of 1 million by 6%, so you divide 1 million by 1 plus 6%. And then you should get a present value of 943,396. If you're dealing with a stream of payments. $1 million every year lasting forever, then you have to use the perpetuity form. Like I said, that's gonna come over and over again. Perpetuity formula. Now, we have the discount rate is 6%, what is the growth rate? The growth rate in this case is 0. Cuz the payment is not growing, so all you have to do is to divide 1 million by 6%. And you should get a present value of 16.666 million this is one of those numbers that are annoying us. You'll get six forever. So 16,666,000. It's a tongue twister. So it's really, really, very easy to use this form. Let's go back to the accounts receivable example. Remember, we have the incremental cash flows. $82 million today, and then a loss, a negative cash flow of minus 20 million every year, starting next year. And now the time period here is years. So we're expressing this in years. This is the trade off that we've already derived. Now, we have to think about, how many years into the future are we going to consider? Are we going to consider 10, 20 years, what's the timeline? What you are going to see and once you think about this, it might sound like a not very natural idea but for most corporate finance problems like acquisitions, valuation, project analysis, we have to end up considering an infinite horizon. This sounds strange, so I really want to talk about this now, but the issue is that there is no natural date for a company to end. So suppose that we're thinking about a company and we think that in the future the company's gonna be sold to another buyer. That's an example you might think about. So if you're going to value this company today, what you need to do is you need to think about what's the sale price. The sale price if it's not zero, is gonna be part of the value. And now what? The sale price, how are you going to determine that sale price? Again, it's the same idea, net present value. The sale price if going to depend on the future cash flows after the sale, that are going to go to the buyer. So it doesn't stop. Even if you sell the company it doesn't stop. The cash flows are always going to go on. So really this is the infinite, the sign, the mathematical sign for infinity which is something that is going to show up in corporate finance over and over again. There's no way to get around that. And you might be saying yeah, infinity, no one lives forever. And a cash flow that happen, let's say a thousand years in the future shouldn't matter. This intuition is right. And it is actually taken into account in our formula. This is what the present value formula is doing for us. Think about this, you don't have to do this calculation, I've done it here for you. But if you want to check it, do it so you believe me. Supposed you're going to get a billion dollars sounds a lot. If I had a billion dollars I would be I would be a billionaire. Yeah, of course. [LAUGH] But today, okay, what if I get a billion dollars in a thousand years? How much is that worth? Do the math. You can discount a billion by 1 plus 6% to the thousandth power. Guess what. What answer you get, here's the number. I can't even count the number of zeros. Really, let me just go here to emphasize this. The answer is zero. A billion dollars in a thousand years is worth nothing. Unless the discount rate is zero, which it's not. As we're going to see in corporate finance discount rates are not going to be zero. So yeah, a thousand years ahead, it really not going to affect our calculation. But this is taken into account. All you have to do is to use the present value formula. And you're gonna get that. So let's solve this problem. So finally, we've been working on this for a long time. But like I said, I really like this cuz every concept builds on another concept and now we can finally get a solution. Suppose that the companies we're going to need a discount rate. We just learned we have to discount. We're gonna need a discount rate. Let's say it's 10%. In this model we're really not going to talk about where discount rates come from that's something we're gonna talk about in module four. So if the discount rate is 10%, what is the NPV of the new system? Let's go back here. And do this. So what is NPV? You have to take all cash flows into account and discount them. The $82 million comes today, do we have to discount it? No. And then what we have, we have these infinite cash flows of $20 million. It goes on forever, so all you need to do is to apply the perpetuity formula. It's a magic formula. All you need to do is to divide 20 by 10%, and if you're not mathematically inclined, you might need a calculator to do this but you'll find that the answer is 200. Actually it's 82 here, not 80. So the present value of a $20 million stream of payments happening every year is 200, if the discount rate is 10%. So this means that the NVP is minus 118. The NVP is minus 118. So now, we have the number that is the net present value of this particular decision. So before we talk about, what are we going to do with the NPV, let's talk a little bit about Excel. So I want to show you how to compute net present values in Excel, and do a little exercise for you to talk about the perpetuity formula. So we could set the horizon to 30 years, instead if you had never seen the perpetuity formula, one way you might want to solve this problem is by setting the horizon to a large number of years. So here, very boring to write this down but I wrote all the 30 years here. To compute the NPV in Excel, all you need to do is to use this Excel function. There is an Excel function in Excel that is literally called NPV. So the NPV function you write the discount rate, in this case it's 10% of the first point and then all you do here is you refer to the cells. Cell 1 to cell 30. And then you should be able to computer NPV in Excel. [LAUGH] The problem is that what I end up getting is that answer here. Instead of getting minus 118, I just get a blank. So what's going on? Actually, the reason why Excel does that is because the amount of numbers that Excel reports is too large. That's what Excel is reporting, minus 10,653%. It seems to make no sense. So here's a little letter to Microsoft. NPV is not a percentage. The problem is, in some versions of Excel, the NPV function is programmed to give you an answer in the percentage. Come on, it's the wrong answer. It's a currency. It's either a number or a currency please. And by the way, we do not need to have decimal points. Anyway, so you may need to reformat the answer. I showed you, this is I'm just showing you a tab from Excel. You can figure this out on your own, I'm sure but here is the formatting. And then what happens is you're going to get an answer of minus 106.54. Our answer of course is minus 118. What's going on? So what's going on is that we considered 30 years. We are considering 30 years. We're not considering the entire future. 30 years is not that far, if somebody tells you you're gonna get $20 million in 30 years, you're gonna be happy. That matters, that has value. $20 million in 30 years is a lot. So what you need to do if you really want to get an answer that is closer to the perpetuity value is to increase the horizon. If you have the patience to do this, I recommend you do it once. So do it with 50 years. You're gonna get minus 116.3. Do it with 100 years. You're gonna get 117.99, which is mathematically it's the same. It's really very close to minus 118. So a cash flow of $20 million in 100 years is worth nothing or very close to nothing. So if somebody tells you you're gonna get $20 million in 100 years, you should really think about whether this is a lot of money or not, probably not. [LAUGH] The pattern of course, is the more years we put in the formula, in the calculation in Excel, the closer we're going to get to the perpetuity form.