We've done this problem and this problem is a PMT problem. However, what's given to you here? The PMT is given to you, you'll figured it out. Now I'm going to do a future value problem with PMT figuring in but exactly the opposite. Let's go there. Then, I will take a natural break. The next problem, let's just read it. By the way, if at this point you're feeling a little tired, you've had too much of future value of an annuity, take a break. That's okay because I think it's much more important you understand bite-size pieces, and you can always join me in a minute. I'm not going anywhere, I'm here. Let's get started with example number 2, of an annuity. I'm going to again draw a timeline and hopefully, we'll get so familiar with doing this. I promised myself today that I'm going to go slow and I'm going to squeeze every ounce of energy from a problem. That's what the beauty of finance is. Zero here, and how much here? Twenty five. Now, what do I know in this problem? I know that per year, the interest rate again is 8 percent. Just for simplicity, the same number. The numbers will change depending on who you are. Now I say, I know the future value. I'm using dollars again just for simplicity. Let's pause. Last time, I jumped straight into PMT knowing that it's a PMT problem. But it's not the kind of problem that dictates what you're looking for, it is what information you need to be answered. Here, I know future value. The question which is being asked is, suppose you want $500,000 when you retire 25 years from now, how much must you invest each year starting at the end of this year if the interest rate is 8 percent? Now I repeat again, I'm choosing 8 percent, but actually you are choosing 8 percent, not the exact number, but the strategy. For 8 percent, you better be invested in something risky. You're not going to get 8 percent from a bank. Five hundred thousand dollars, you need at retirement, and you're using 8 percent. Let me ask you this. When we go to Excel in a second, you will use the PMT function. Why? Because that's the guy I don't know, and that's the guy I'm trying to solve for. Let's do it. The good news is, I have the same problem setup over there, but now what do I do? This actually helps. I have the last problem. What do I do? I change FV to PMT. Why did I do that? Because as I said earlier in this particular example, I do not know one of them, and that's PMT. What's the interest rate? Eight percent. How many years? In the previous problem it was 40 years. In this problem, we have, I believe, 25. If I make a mistake, that's one of the times you can catch me and fix it faster than me. The number of periods, PMT, and then the next information. This is a little bit important because Excel has a system which you got to follow, otherwise you're on your own. If after the number of periods there is a symbol called PV, which we know what it is. Do we know the PV of this problem? Answer is no. So we've got to put 0 because we don't have a number there, and then we type FV, 500. Hopefully, when I say if I have all the numbers right, and I'm doing it in real-time with your simply to make you recognize that you can do it. You can do it just like I did it. The reason again, I'm using a calculator is simply because the number that I need to calculate has got 25 operations involved. The only operation that's simple is the last one. But in this case, I don't know the last one either. I don't know 25 of them. The PMT, so how do I figure that out? I have to use a calculator or do stepwise, very slowly the problem, and we'd be here forever. Okay. Let me call it 6,840. So I'm going to now go back to the problem. The answer to this is in dollars, $6,840. Why am I making it 6,840? Why not 6,839.1? Because we're a family now. I'm not going to worry about decimals and you don't need to worry about them, at least in the classroom, in real life, probably, yes. Okay, 6,840. Let's first convenience assume that it's about 7,000, approximately. What's going on here? I need to put away $6,840, and I'll approximate it at approximately $7,000. How many times? Twenty-five times to end up having 500 bucks. All right. Why did I approximate even 6,840 by 7,000? Let me ask you the following question. Suppose the interest rate was zero, in other words, there was no value to time, what would you have if you invested $7,000 25 times? You just multiply 7,000 by 25, right? What do you do? You take $7,000 multiply it by 25, you have 175. Why did I do this? Again, as in finance, pause and say, compounding, right? If I didn't have eight percent rate of return, I would have only $175,000. That's not little, and by no means am I saying it's throwaway money. But compare the 175 to the 500. What's going on? The eight percent is helping me, and here's my little take before we take an actual break on this. I hope you understand this problem. Secondly, I hope you recognize that the eight percent is coming from where? The market. I hope you realize now why the market is so awesome. Because you're not doing anything. I'm not doing anything when I put away my money in my retirement. What am I giving in this example? I'm putting away 6,840, right? I understand that could be my hard-earned money, but the fact is the externality, the positive benefit a market provides to people for their ability to benefit from the economy at the rate of eight percent is phenomenal. You see what I'm saying? So what's going on here is that my money goes to somebody with great ideas, who's able to earn some money and I still. Can earn eight percent. I don't want you to ever forget the beauty of markets. Beauty of markets is an ability for all of us to share. Not one person, all of us. That's the beauty of it. The unfortunate thing about life, as I said once in a while I'll go into life is that not everybody has this opportunity. Yes, we can all say that everybody's not working hard enough, but sometimes it's difficult to make money. It's difficult to have jobs. A lot of people these days don't have the ability to even invest. Let's do this, let's take a break right now. We have spent a lot of time on two problems. I do not want you to exhaust yourself, but I want you to think about these issues. One last thought, while we're offline, redo these problems and double-check them. Let me explain what I mean. Make 6,840 your payment. Make number of years, 25, and make the interest rate eight percent. Solve the problem for what the future value will be. What should your answer be? Five hundred thousand. Tell me what's cooler than that. It's internally consistent. It's got to be, if it's not, it's not finance. Okay? Take a break and I'll also take a pause, and we'll come back and start off with present value of annuities. Keep smiling. Thank you.