I hope all of you get a sense of this very simple problem because I think what the future problems will be, will be complicated versions of this. I get very excited with simple stuff too because it's cool. How much will $100 become after two years? Now, what have I done? I have taken the one-year problem and made it into a two-year problem. Let's see how to do that. I really would appreciate it if you did what I'm doing, either now or later. By the way, I'll do this for relatively simple problems and let you work with the more difficult ones. How many periods do I have? I have 0, 1, and 2. As I said, the length of the period is a year, but that's artificially chosen. I'm just choosing it for simplicity. It can be anything, you'll see that in a second. So one-year, two-years. What does the question asking me? You put 100 bucks in the bank and you're asking yourself, how much will it become at this point? What is the future value of this? The question is pretty straightforward, but it is a little bit complicated. Here's what the answer will be, and I'll tell you the answer first. The answer will be $121. It's clearly more than $110, but it's $121. In all of this, that $1, this guy, if you understand where that's coming from, you'll see how it'll blow your mind when you increase the number of periods. Here's a simple way of understanding what's going on. I'm going to do the example without a formula, or whatever, the formula that we already know. Can you tell me how much will this be at this point? Do we know how to solve a one-period problem? Of course, we do. We know this is $110. Why? Because this was $100 times 1.1. So you don't need the formula to do this hopefully, but you could do it in our heads. But now notice what has happened. I can use the same one-period concept conceptually to move one period forward. What will this amount, which is $110 be after one year? What will you do again, you'll take $110, multiply again by 1.1. That should give you an answer of $121. What's going on here? Answer is very simple, you are doing a one-period problem twice. Think of a bank, it's dumb. Not people, the bank. The bank is looking in the first period and saying what's going on at time 0, you have $100. At the end of one period, what does it do? Given interest rate of 10 percent, it says now you have $110. But the bank doesn't know the difference between the $10 that you didn't have in the first year, but now have. So $110, bucks are the same. It keeps you now have rightly so $110 and takes it forward another period, it becomes $121. What's going on? Where is that $1 coming from? If you think about it, you're getting $10 here and $10 here. That's one way to think about it. Why? Because this $10 is 10 percent of this for the first period, and this $10 is again a 10 percent of this in the first period. If you add up those, you have 100 plus 10, plus 10. You 120 at the end. You have 100 plus 10 plus 10, so you have 120. You say, "How did I go from $120 to $121?" The answer is very simple. What we have ignored in all this is this $10, which was not here, is added here, will also earn interest over the second period. What is 10 percent of $10? $1, so plus $1 is $121. It's pretty straightforward. I'm writing all over the graph, but I want you to understand that this is not complicated. The complication is simply coming because if you're thinking, you're not thinking about the 10 bucks that comes as interest, we'll also start earning interest in the next period. I've given you a sense of what is the future value of a 100 bucks two years from now. The concept and formula, let me just repeat one more time so that you can understand. The formula says this, "If I have p at time 0, after one year, it will be p 1 plus r. After two years, what will it be? P 1 plus r times 1 plus r." Why? Because this p, in our case, was 100, but after one year, this whole thing has become a 110. Then when you carry it forward, again, it'll become a 121, and turns out, p times 1 plus r squared is exactly equal to a 121. Now, isn't this cool? The formula is telling you exactly what's going on, instead of me throwing the formula at you. Formula makes sense, but here's where Einstein got blown away too. Einstein said this: Einstein's most famous equation was E equals mc squared. Now, it's square and square are here; they're common to the two. But, turns out, if I have 100 years passing by, if 2 were to increase to 100, what would this formula become? It would become p times 1 plus r^100. Even Einstein saw compounding work, that is interest, on interest, on interest, in this case, it was only one buck initially over two periods. Interest, on interest, on interest, is so powerful that in fact, I would give this advice to you: anytime you're asked to finance a question, say the answer is compounding, and you are likely to be right 90 percent of the time. The only thing you want to do is, you want to look intelligent. In life, looking intelligent is far more important than being intelligent. What you want to do is, you want to pause and say, "Is it compounding?" Because what that'll do is it'll make people think you are really cool, something they don't. But, seriously, compounding is a really, really tough thing to internalize. What I'm going to do now is I'm going to take advantage of Excel, and I promised you that I won't teach Excel, but I'm going to do a problem where I'll be forced to use Excel. Let's stare at this problem. If you want to take a break right now, this may be a great time to take a break because we have done Future Value, though we actually could, by hand, do the calculation. Repeat again in words, I will. Hundred dollars after one year 110. Why? I got 10 percent,10 bucks over one year, I have a 110. After two years, what's happened? One way to think about it, which is very intuitive, is how much do I have after one year? If the bank is still there, of course, it's a 110. I told you I won't talk about risks, so I'm assuming the bank is still there. Hundred and ten you still have, and after two years it would have become a 121. The real thorn in your side is that one buck. If you understand that one buck comes simply from the fact that you now have 10 more dollars after one year, which is also earning 10 percent because it didn't do any harm to anybody, it's just like the 100. What did it do? A 10 percent of that; that's the one buck. Now, that is what is compounding's power: interest on interest, but it's only one buck. Otherwise, if you didn't have interest on interest, you would still have a 120; 10 bucks each year on the original 100, now you have 121. It says, "What's the big deal here?" Let me try another example, and then I'll give you some examples which are really awesome. Just the simple idea. I think if you understand compounding as how difficult it is for a human being to internalize, you'll understand why finance is viewed as so difficult, but if you understand the intuition, it's pretty straightforward. Let's do this problem.