In some senses, as I said before if you know how to draw time lines, you have arrived. What I'm going to do now is I'm going to show you what a time line means and you'll just laugh because it's silly, but I think it's important. By the way, just so that you know, my handwriting is nowhere near as perfect as the title on top, so if you expect it to be, you need to grow up. I'm going to just draw a time line. You should be able to take a word problem and put it along a time line and I put dots here. Why is this important? I think it's important because if you can take a real-life problem and put it on a time line, you achieved probably the most difficult part of taking a real-world situation and then using financial tools. In some senses, finance requires you to know what life is all about, what the problem is all about before you can use finance. I say this many times, it was an accident that I got to learn finance, but there is love and then there's finance. The gap is huge. Love is somewhere special, but being number two ain't bad. You need to understand life, love, and so on and then put it on a time line. You can put pretty much anything on a time line. Here's the first thing. At this point, we'll typically call this PV, present value. Then the number of periods are pretty obvious, so this is 1, 2, 3, and this is M. M reflects these time periods. The important thing to remember is that r is the interest rate that applies to one period and the real world typically that one period is a year. By that I mean, when you see interest rates being quoted for various stuff like a bank loan and so on, it'll be annual. That's just so that it makes sense, you can compare things kind of. In the beginning what I'll do is, I'll just take PV and I'll try to relate it to FV. We'll try to understand these two concepts. How does FV translate to PV, PV translate to FV, go back and forth and become very familiar today. But I remind you of one thing, just listening to me means it looks easy and that's the challenge of this class. I'll make it sound really easy. But your challenge is to do the problems and that's when you'll internalize because the word problem is the problem. If you can't figure out the word problem, this isn't going to help. Drawing a time line, bringing the word problem to it is what it's going to be about. I start off with simple problems and then make them more complicated. But today we have what we'll stick with is a single payment. Meaning, I will transfer something from PV to FV either for one year, two years, or 10 years, and vice versa. We could have stuff coming in here which is also dollars. Remember this is dollars and this is dollars. Could have dollars coming in here that's what is happening actually in most projects. But we're going to ignore that for the time being. The reason is, as I said, I want you to understand time value of money. We'll go slow in the beginning and then we'll take off. When you're on a plane, the pilot warns you, "Okay, fasten your seat belt, I'm going to take off." I'm going to warn you, when you hit the assignments and the problems, there'll be a warning. You better have your seat belt on. Get onto the problems, that's how you learn. You don't learn by just listening to me or anybody. Please recognize the importance of time line and I'm going to go back to the notion of how to think about time value of money, and how to take time lines and work them forward. We've talked about importance of time lines. I'm now going to jump into what I promised I'd do. I'm not going to pick a formula and just throw it at you, no. To the extent, I can and that's my challenge, is to talk about a problem and then create the formula, because I don't like formulas without understanding what's going on. But the main insight we're going to worry about is a dollar today is worth more than a dollar tomorrow. Or in other words, that's the essence of time value of money. The passage of time by itself has value. There are some reasons for it. As I said, you can go back and read up on them and we're going to assume what captures time value of money is the interest rate, the relationship between today and tomorrow or today and the future, and that interest rate we'll assume is positive. Let me start with an example. Suppose a bank pays a 10 percent interest rate per year and you are given a choice between two plans. By the way, I'll be going back and forth writing and stuff like that, but that'll hopefully make it more engaging. As I do a problem, you should do it with me. Then if the problem gets complicated, I'll give you more time and then we'll do it together and so on. These are your two choices. It's very simple, I either give you $100 today or I give you $100 one year from now. For the time being, let's keep our period one year. So the question really is, which one would you prefer, and why? As I said, I just don't want to know what you prefer, I want to know why the heck do you prefer It. Turns out, if you have thought about it even for a second, or even without thinking about it, you'll choose one of the two, and it's probably going to be the first one, right? So the goal here is to use the simple example to motivate something that is fundamental, that will build on. This is a future value problem and an example. So what I'm going to do is I'm going to try to walk with you, so try to think through this. Remember A was $100 now, this is A, and B was $100 in the future. That's what I meant, the timeline is extremely important. I'm giving you two very simple choices to actually recognize. Now, this is where popular financial press screws up and you won't believe it, but it's true. What we do in our head is we intuitively recognize that just the passage of time has an effect, and will have an effect, on the value of the money we are talking about or whatever it is. But we directly compare these two, and that's not the right thing to do. In other words, if you were to do this and I say this in my class and I'm going to say this to you. If you start comparing money across time directly with each other, it would be better if you stab me because you're basically telling me whatever you're teaching in finance is useless. So remember the first principle is you cannot compare money across time. That would only be meaningful if time had no value. What captures the value of time in this one scenario, is what? The interest rate. So let's try to work it a little bit better at this point in time. Let's do the future value, right? So what is already in the future? We know that this is already in the future. So the question is, I cannot compare this to this at time 0, but what can I do? I can either bring this back to time 0, so take this or carry this forward to the future. The reason I'm going to do carrying forward the future value first is I think it is easier to understand finance if you do that, and it also makes you think about the future and that's very important. Every decision that you make, every value-creating decision that you make, should force yourself to look in the future. This is where I think accounting I can make fun of. Accounting standing at time 0 where we are today is looking backwards. That's it, it's done. The past is over. So far you can derive very interesting implications from the past, and I don't mean to demean anything. All decisions ultimately involve your capability to look into the future, and that's what's challenging about it and that's what's awesome. Every decision has an impact on the future, and typically the painful part happens today. The better the idea, the more the pain today, but benefits a lot in the future. Like Google, it took a lot of effort to create, and now a lot of value is being created. Sticking to the simple problem, I think you know the answer to this, the answer to this will be $110. The reason is very simple, r is 10 percent. Let me just walk you through, talk you through, and then we'll do the formula. I know right now many of you are saying, "Come on, this is just too easy." Well, it'll build on itself and so we got to understand this piece. The 100 bucks that you had, you could put in a bank, and that 100, because the interest rate is positive, will be part of this 110. Because the interest rate is positive, you can't lose that 100 bucks. Then you're earning 10 percent interest, so what is 10 percent of a 100 bucks? Ten bucks. It's very obvious what's going on, that you, in the end, will have $110. As I promised you, what I'm not going to do is I'm not going to throw a formula at you till at least you've had some sense of where I'm going, and hopefully this simple example has motivated you to try to understand future value a little bit better. Now what I'm going to do is, I'm going to throw the concept at you. In this concept, what it says is the following, that the future value of anything that's carried forward has to have two components: one is the initial payment and in our example, it's 100, and the other is accumulated interest, which in our example is 10 bucks. So the problem becomes very straightforward. You put in 100 bucks, you get 100 bucks, but then you get 10 percent on the 100s, which is 10 bucks, so you get 110. This is the formula. If I were to ask you, what is it related to the problem that we just did? What is this P? P is your initial payment of 100 bucks. What is the r? R is the 10 percent. But the 10 percent is on what? Is on the P of $100. I know that 10 percent of 100 bucks is a fraction, 1/10 and this will be 10 bucks. But the way we write it, which looks is very straightforward, is we take P out of the picture. The P is common to the first one, therefore the one and rP, r times P. What you have put in the brackets is many times called future value factor. It's a factor because what does this 1 plus r reflect? Let's do it in our case. The 1 plus r in our case is 1.10. What's cool about this number is, it tells you the future value of one dollar. If you know the future value of one dollar, in this case it's 1.1, which is very simple, 1 plus the interest, you know the future value of any number. Because if the number is 100, you multiply 1.1 by 100, you get 110. If it's a million, you get 1.1 million, and so on, so forth. Many people conceptually emphasize that future value factor, and I'm going to just do it this one time. But you can go back to the notes and think about it like that. It'll be very helpful to you. Right now, what I'm not going to do is I'm not going to use any tool to elaborate on this formula. By that I mean, you don't need Excel to do this. Actually, you need Excel to do only when the problem becomes difficult to compute, not think about. The initial payment is P and the accumulated interest is r times P. That's the way you want to think about this problem.