One warning about today, you've got to be an active listener, because we have done some finance, you know a little bit about what's going on. We will complicate it but the complication comes not just because of the formula, but because I'm going to make it more real world. You got to be engaged and you 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 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. Just a little bit, it's likely to be much more intense than last time. Let's get started. Today, the main issue we'll deal with put very simply is the notion of multiple payments. 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 forward or back, one payment, one amount of money, one piece of happiness, whatever just happening and you're trying to get a 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 is 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. For example, when Google was started, I doubt if they said we'll put in so much effort into creating something fantastic, which will last only for one period or two periods at best and things will happen only once. No, they were hoping, and reality proved them right, is that things will keep happening, it will expand and time will play a major role for hopefully forever. The first element of multiple payments, which I saw in textbooks is called an annuity. This is a special case. 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. It derives from the fact that payments are made back for an obligation. You'll understand in a second. Let me just first show you some terminology. 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 are personal examples. They apply to you personally. We will do a lot of corporate applications, applications of when you start a project in a firm and so on. But this class we'll start off doing a lot of applications that mean something to you. If you look over here, this is a timeline, another way of drawing a timeline, and I said, timelines are everything. If you would take a word problem and put it on a timeline, that's what we'll do in a second actually, you have arrived. Finance will just make your life so easy after that. Here's what an annuity is. An annuity pays C dollars three times in this chart. Something is very important for you to recognize. If you stare at this chart, you'll recognize that nothing is happening at Time 0. When I say time 0, I mean a specific point in time. What is1? End of Period 1, going from 0-1. What is 2? End of Period 2, and the period lasts from 1-2. What is 3? End of Period 3. One aspect of finance which you have to remember is there are some assumptions built into the formulas that you use, and here the assumption is the first payment of the annuity occurs one period from now. The reason for this is very simple. You will see in examples that the classic annuity if you think of an example, what is it? A classic annuity is alone. Why? Because you take out some money, if your bank gives you some money, and then you pay back. 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 and so forth. 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 C changing over three periods, that's becomes easier to do. 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. Let's fill in the cash flow here, I left it open. This is 0, nothing is happening at time 0, which is today, right now. So 0,1, 2, 3 are points in time, periods of time is 0-1, 1-2, 2-3. Time value of money is simply the existent of an interest rate per period. We'll come to that in a second. Suppose I ask you, "How many years to the end?" You should be able to say that there are three years left to the end. How did I figure that out? Very simple. Zero to one is one, 1-2 is another one, and 2-3 is the third one. Why am I doing this? I'm just giving you a sense of how many periods are left. Usually, this is something that should be second nature to you, but many times people start counting. How many periods are there? How many time periods are involved? They get confused. Don't worry. How many periods left here? Two. How many periods left here? One. How many periods left here? Zero. The thing to recognize is that there is no cash flow occurring at time Point 0, and there is a cash flow occurring at Point 3, which is the end of the period. Those are the two things important to recognize. Now, let's do the future value of this guy. Before, as I said, I jump into giving you the formula which textbooks tend to do, and 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, we did last time. What is the future value of this? I'll do it with you. It has to be 0, not because time is three years are not left, is because by convention, you do not get a cash flow at time 0. The notion is you take a loan and you start paying it one period later. If you do pay back some today, then the loan is lower. What's happening in the Period 2? Clearly, you're taking C, but for how many periods are you taking it forward? Well, it helps to have column number 3 which says, ''Years to the end," so you'll know it has to be 1 plus r square. Remember, you're carrying it forward. The reason why it's only square is because the first payment is at the end of the first year, and how many years are left from 1-3? Two. It's pretty straight forward. Second payment is C plus r. What am I doing? I'm actually just breaking up the problem into bite-size pieces to explain what's going on. Annuity gives you a C three times, not once, three times. Remember, we are 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. 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. That's what people will tend to do in their heads. In fact, Wall Street Journal, other articles I've read in famous newspapers tend to just approximate and say, "You're doing the three Cs." Well, they're not three Cs. Three Cs at different points in time are a totally different animal than paying three Cs at one point in time. That is simply because of time value of money. I'm going to move on and show you how to create the formula now. Again, I'm going to be a little slow in the beginning so that you understand what you're talking about. What's the formula? Remember, I'm doing future value of how many payments? Three payments. As I've told you before, the one thing not very good about this is my handwriting is pretty awesome. Yeah, so this is FV, future value. Okay, so what's the future value of the first guy? C,1 plus r squared. The first guy occurred in which period? One. Second, C, 1 plus r, and the third, C. The one cool thing about an annuity is that I can take C out in common and I can do one plus r squared, one plus r plus one. Let me ask you, what would this be? Remember last time we spoke about how if I got a factor, I can just multiply anything by it and I will 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, and three. At the end of those years, nothing happening at the time zero. The general formula, which I will write for you, turns out to be this, C one plus r raised to power n minus one. Why n minus one? I'll let you figure that out. N minus one is very obvious from here. If you stare at this, the annuity was three years, and this annuity is n years. So three is different from the two. Similarly, n is different from n minus one 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, depends who you are, you're the bank or you're the person. Nothing is happening at time zero. I want you to please understand that convention because it can confuse people and as I said, the nice thing about finance is there are not too many conventions. You don't have to memorize things, but this is a simple one you got to remember and keep in mind. What I'll do is I won't try to simplify these formulas. They 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.