So, that was last week. Going forward this week, I want to emphasize one more thing. Last week, we had between the introduction to the class and the present value, future value stuff that you did, we covered a lot of video. And video watching is very intense, and I want it to be intense, because that's how you're learning. However, again, following my philosophy that the amount of time you spend on something, cannot be the same for every individual. Similarly, when I create videos, I'm going to let the natural process of that day's session topic drive the length of the video. I will not cross two hours, if I do, we'll edit it. But I will not force myself to do two hours. It'll depend on the nature of the beast. So one warning about today, you got to be an active listener. Because we have done some finance. You know a little bit about what's going on. We are going to complicate it, but the complication comes not just because of the formula, but because I'm going to make it more real world. So, you've got to be engaged and you've got to pause the video if you want to. We'll take some breaks, natural breaks and I'll try to take them at times where I think it's more convenient to do so for you rather than for me. But I think today will be quite intense, you need to be an active listener. And have a Excel spreadsheet with you, or a piece of paper with you to follow as we go along. Because I'll do a lot of problems, so just a little bit, it's likely to be much more intense than last time. Okay, so let's get started. Okay. Today, the main issue we'll deal with what vary simply is the notion of multiple payments and what do I mean by that? Last time, if you remember, we did do one period but then we crossed over to multiple periods. But one thing was common to everything we did, whether it was present value or future value. We had one thing to carry for overload back. One payment, one amount of money, one piece of happiness, whatever. Just happening and you're trying to get hold of how to mess around with time. As I said, time travel, if you like it, is what finance is all about. Today, we'll go into multiple payments? Why are we doing this? As I said, there's no necessity to do it except, life is like that. Very seldom will you find that you have to make a decision based on a single payment at a single point in time. Or a single input or output at a single point in time. In fact, life's more interesting issues have something happening today and a lot happening in the future. So for example when Google was started, I doubt if they said we'll put in so much effort in to creating something fantastic which will last only for one period or two periods at best. But, and things will happen only once. No. They were hoping and reality proved them right, is that things will keep happening. It'll expand and time will be a major role for hopefully forever. So, the first element of multiple payments which I saw in textbooks is called an Annuity. This is a special case and annuities we'll call either C, C stands, by the way, for cash flow or PMT which stands for payment and you'll see why I'm using PMT. PMT is the symbol used in calculators and PMT is the symbol used in textbooks, and PMT is the symbol used for an annuity in a spreadsheet. And it derives from the fact that payments are made back for an obligation. You'll understand in a second. So, let me just first show you some terminology. So cash flow which by the way, is being used here simply because I'm going to use a lot of examples where cash is involved. And today another aspect of whatever examples I do, after introducing the concept, is that they're personal examples. They apply to you personally. We will do a lot of corporate applications. Applications of a new started project in a firm, and so on. But this class will start off doing a lot of applications that mean something to you. So if you look over here, there is the timeline. Another way of drawing a timeline and I said timeline's are everything. If you take a word problem and put it on a timeline, that's what we'll do in a second, actually, you've arrived. Finance will just make your life so easy after that. So here's what an annuity is, an annuity pays C dollars three times in this chart and something is very important for you to recognize. If you stare at this chart, you'll recognize that nothing is happening at Time Zero. And when I say Time Zero, I mean a specific point in time. What is one? End of period one going from zero to one. So, what is two? End of period two and the period lasts from one to two. What is three? End of period three. So one aspect of finance which you have to remember is, when you there is some assumptions built in to the formulas that you use. And here the assumption is, the first payment of the annuity occurs one period from now. And the reason for this is very simple. You see in examples that the classic annuity, if you think of an example, what is it? In a classic annuity there's a lot, why? Because you take out some money, the bank gives you some money, and then you pay back. And typically, although you're not required to pay back a fixed amount, you do tend to pay back a fixed amount because the interest rate is fixed and so on, so forth. And we can change all that, but it's very easy to try to understand something that is fixed for three periods of time and then change it to the sea changing over three periods, that becomes easier to do. So what I'm going to do, is I'm going to first explain this concept and I'm going to, as I said, go slow initially and in all the problems, too. So let's fill in the cash flow here, I left it open. So this is zero, nothing is happening at time zero, which is today, right now. So zero, one, two, three, are points in time. Periods of time are zero to one, one to two, two to three. And time value of money is simply the existence of an interest rate per period, okay? So, we'll come to that in a second. So suppose I ask you, how many years till the end? And you should be able to say that there are three years left to the n. How did I figure that out? Very simple. Zero to one is one, one to two is another one and two to three is the third one. Why am I doing this. I'm just giving you a sense of how many periods are left, right? And usually, this is something that should be second nature to you but many times we will start counting. How many periods are there? How many times periods are involved, they get confused? Don't worry. How many periods left here? Two. How many periods left here? One. And how many periods left here? Zero. So the thing to recognize, is that there is no cash flow occurring at time point zero. And there is cash flow occurring at point three, which is the end of the period. So, those are the two things important to recognize. So now, let's do the future value of this guy. And before, as I said, I jump into giving you the formula, which textbooks tend to do, I really don't like that. Because it doesn't take advantage of your learning, that already has happened. What is the future value of the first row? Let's do it one period at a time. One cash flow at a time. Why am I doing that? Because you already know how to do it, right? We did it last time. So what is the future value of this, and I'll do it with you. It has to be zero, not because time is not three years left, but because by convention you do not get a cash flow at time zero. And the notion that you take a loan and you start paying it one period later, right? If you do pay back some today, then the loan is lower, right? Okay, so what's happening in the period two? Clearly, you're taking C but for how many periods are you taking it forward? Well, it helps to have column number three, which says years to the end. So, you'll know it has to be (1 + r) squared. Remember, you're carrying it forward. And the reason why it's only squared, is because the first payment is at the end of the first year and how many years are left from one to three? Two. It's pretty straightforward. Second payment is C (1+r). So, what am I doing? I'm actually just breaking up the problem into bite size pieces to explain what's going on. So annuity gives you C three times, not once, three times, remember we're talking about multiple payments. And the last one is C. So you see what's going on, it's pretty straight forward and I'll let you look at this for a second. What's going on is, I have three Cs happening at different points in time. And let me ask you this, if there was no time value of money, how many Cs do you have? Answer's very simple. You have three Cs. And that's what people attempt to do in their heads. In fact, Wall Street Journal, other articles I've read, and famous newspapers, tend to just approximate and say, you're doing the three Cs. Well, they're not three Cs. Three Cs have different points in time, are a totally different animal then paying three Cs at one point in time. And that is simply because of time value money, okay? So I'm going to move on and show you how to create the formula now, okay? So again, I'm going to be a little slow in the beginning so that you understand what you are talking about. And so, what's the formula? Remember, I'm doing future value of how many payments? Three payments. And as I've told you before, the one thing not very good about this as my handwriting is pretty also. So this is FV, future value. Okay, so what's the future value of the first guy? C(1+r) squared. And the first guy occurred in which period? One. Second C(1+ r) and the third, C. The one cool thing about an annuity is that I can take C out in common, right? And I can do C[(1+r) squared + (1+r) + 1]. Right? So, let me ask you what would this be? Remember last time, how we spoke about how if I got a factor I can just multiple anything by it and I'll get what I want. Well, this is what? This is the future value factor of what? Not one payment but three Cs happening three times. In year one, two, three. At the end of those years nothing happening at times zero, right? So the general formula, which I will write for you turns out to be this, C [(1+r) raised to power n-1. Why n-1? I'll let you figure that out N-1 is very obvious from here. If you stare at this, the annuity was three years and this annuity is n years. Right? So, three is different from the two. Similarly, n is different from n-1 by just one period and that goes back to the issue I earlier emphasized. This is all happening because of the fact that you don't get any payments or you don't make depends who you are, your bank or your person. Nothing is happening at times zero and I want you to please understand that convention because it can confuse people. And as I said, the nice thing with finance is that there are not too many conventions, right? And you don't have to memorize things, but this is a simple one. You got to kind of remember and keep in mind, okay? What I'll do is, I won't try to simplify these formulas, there are simplifications available and we can do all of that stuff. What I'll do on the website is, I have provided you the short form formulas of all of these, if you need to use them. However, we'll jump in class straight to, not the final formula not simplifying things it's a waste of your time and my time. I think what's much more useful to do is to jump directly into doing examples.