Okay, the first example I'll do of a PV of an annuity is just like a future value. I'll give you the annuity, and I will ask you to calculate PV. So let's read the problem very carefully. How much money, and just so that we are, I can write on this. You see I've pressed a button here to be able to write all over and then go back. So I do a lot of cool things that you can't see. Okay, anyway, so I love technology. And there's a person here sitting next to me who's an awesome person and without him I wouldn't be doing this, to be honest. How much money do you need to put in the bank today so that you can spend $10,000 every year for the next 25 years, starting at the end of this year? So think about this. What kind of problem is this? This is a problem where you need money today, and you are thinking about using it and depleting it over time, all right? So, suppose the interest rate is 5%, and I think we have become very real with interest rates, we've gone from eight to five now, yeah that's fair, the real one looks like right now. So, what I'm going to do is, I'm going to draw a timeline. And I'm sorry, this is something I'll force you to do. And another use of Excel is, it comes with the timeline, with the ASL being 0 and so on. Okay, so 25, 0, 1. So remember, more points in time by 1 than the number of periods, right? So what do I know here? I'm standing here. What do I know? I know that I need $10,000 25 times, right? So this to me is C, or PMT. So I know PMT but what do I want to figure out? Okay I need to spend $10,000 but I need to have it in my bank because maybe I just want to retire. Or maybe over the next 25 years I'm working but I want $10,000 set aside for some needs that are important for me. Whatever the motivation you've decided to do this, I will have to figure out PV. Now if I did it the long way, what will I have to do? 10,000 divided by 1 plus R, right? Story is not over. What do I have to do? 10,000 divided by 1 plus R squared. And how many times? 25 times. Now if I had to do the same thing 25 times, life is easy. But again, who is messing with my fun? Compounding, because every time I add another one, this 2 becomes a 3, 4, 5, 25 times. Okay, so what do I have to do? When in desperation for calculation, go to Excel. Okay, so I'm going to do, I'm going to go to Excel and the good news is, the previous problem is already there. So I do not want to do a PMT first because I already know PMT. But I want to do now what? PV, right? Interest rate is now, not 8%, but it's 5%. How many years left? I believe in our problem that we were looking at, just let me confirm what it is. We do have 25 years left. And so we are okay on that. What is the PMT? Well, I know my PMT. And I'm going to remove that last element because it's not needed. So let's say. So $140,939.45. So what does that tell me? I better have $140,939 in the bank to satisfy the need of spending $10,000 in the future. So let me go back to the problem and tell you what's going on. So i need in the bank 140,939, and let me just for convenience call it $141,000, right? So let me ask you this. If I put, spent $10,000 every year, and the interest rate was zero, right? How many times will I spend $10,000? 25, times 10 is $250,000, right? So the number that I'm going to put in the bank is nowhere close to 250. It's off by at least 110, approximately. Why is that? Because when I put $141,000 in the bank, the world is helping me. The ingenuity of the world is helping me in the form of 5% rate of return. Right? So the good news here is, when you do PV in this case, you have to put much less than what you need in the future. Simply because as the money is being withdrawn, the remaining money is growing in value. Because again of the positive interest-rate. So this problem gives you a sense of how to do PV. And remember the PV is discounting. So every $10,000 in the future is becoming less so today. And the last $10,000 is being discounted by 25 years. So what happens is, you are not multiplying 10,000 by 25 to get this answer. Because of compounding on a positive interest rate, you're getting an answer of $141,000. I would encourage you to do this in your own time. Try to, after we are done with this class, see how much money are you left with at the end of 25 years if you carry this money forward. And we'll do this in a context. Why am I encouraging you to think like that? It's simply to confirm that this number is right and the assumptions. One final comment before we move onto the next problem. The interest rate is 5% here, so if the world is the same as our previous problem, it requires a strategy that is less risky than the 8% before it. So I just wanted to bring that risk thing that's at the back of your mind into the picture, just to show you. You know the ten. You know the 25. And basically, you know the five. Of course, the five won't be five, the more risk you take. But that's true about anybody making an investment, right? So please remember that. This problem helps you a ton. What I'm going to do now is, if you want to take a break, this is a natural time. But I'm going to, because we've become familiar with doing this kind of problems, I'm going to take the next problem example on right away. But as I said, I always take pauses for you to take ownership. In spite of the fact that you could pause me any time, I think it's good for me to tell you what I think would be a good time for you to pause. Okay, guys. Now I'm going to do a problem on which I will spend a lot of time. And why am I doing this if I'll spend a lot of time? You see, this is a classic finance problem in the real world sense. So, here goes the problem and please read with me. I'm going to try to highlight things as I go along. You plan to attend a business school and you will be forced to take out $100,000 in a loan at 10%. And the 10% you'll see is kind of artificial because I want to make my life a little easy here. Hopefully you don't need to pay 10%, you need to pay much less, but the $100,000 is a fact of life. It costs about that much for two years worth of tuition. And I teach at the business school, and clearly I benefit from the value provided by business school education and people's willingness to pay. But I want to emphasize this is not a small number. And remember, when you come to school, you're also giving up an opportunity of working. So when you do your calculations to go to school, it's not a minor thinking. And being in education, a part of me really believes that things like this class I am doing, this should be a large part of the future. Of course, it brings up the question of how did people survive if everything is for free and so on. But I think, I personally feel, this $100,000 is a bit too high even if I benefit personally from it. Anyway, you want to figure out your yearly payments given that you will have five years to pay back the loan, right? So what do I know? We call this guy n, we call this guy what? PV, FV, PMT? Well, we call it PV. Why? Because when I walk out from the bank with $100,000 of loan, I have the money today. But what do I have to do? In this case, pay a hefty 10%, but I emphasize again, the good news is the interest rate is not that high, and should not be that high. Let me throw in the word should there too because, simply because it's just too high, from any standard. Anyway, so at 10% simply because you will see later it will help us with the calculations. So the first question to ask ourselves is, let's draw the timeline. And in this case, there's 5, 0, 1, 2, 3, 4. Now I'm going to ask you, is this a real world problem? And if you tell me no, I think the issue's with you, not with me, [LAUGH] right? This is a real world problem, the only thing that will change is the numbers, right? So the quick question to you is, who decides 100,000? The 100,000 is here, who decides it? Well, all of us collectively. The fee is determined by the school or the university, wherever you're going. And the amount you need to borrow depends on your ability to finance the education. Right? So you'll borrow, let's assume you'll borrow 100,000, which is two years of tuition. And of course, you need to spend money on yourself, too. Let's keep that aside for a minute. Who decides the 10%? This is a very interesting question, and goes back to my emphasis on markets. If it is one person in the whole market deciding the interest rate, that person is called a monopolist. And if, in finance, or borrowing and lending money, there is one person you know who will get screwed. Us, the customers, or the people borrowing money. So that's why competitive markets are important. Competitive markets are important so that the consumer benefits, not the producer necessarily. And by the way, all of us are the same people. It's not us versus them. I'm just emphasizing that markets are for people. Markets are not for one person or a few of us. All right, that's the beauty of markets. Anyway, so the 10% hopefully is coming out of competition, why? Because if there are three banks, you'll always go to the lowest interest rate, right? So, competition among banks on the Internet hopefully is getting better to help you get a reasonable interest rate. Set R per period is 10%. And how did you decide five years? Well, it's again an interaction between you and the bank. And interest rates will vary depending on the periodicity or the maturity of the loan and so on. So let's keep that issue right now in the risk category, right? So you have five years to pay it back, and the question I'm asking is what? How much will you pay every year, per year, right? So, let's do this problem. And to do this problem, I have to go to Excel and I'm going to now try to do things a little quickly on Excel. So let's do it without screwing things up, obviously. So, what was our problem now? I know my PV and I'm going to calculate what? PMT, right? This is a number I should know. And the bank should tell me. So the interest rate is how much? Interest rate is 10%, and how many years? Not 25, 5. And the next number is PV fortunately, if I see it right. Yup. And you got to keep your eye on the ball. And how much did I borrow? 100,000, right? So, huge number. So the answer to this question is 26,380. And I think what this is telling you, and I kind of rounded things off again without decimals. It's telling you that I, or you, whoever's borrowing the money, will get $100,000 today, but will have to pay $26,000 plus 380 five times. So just pause there. This looks like a huge number. Now, the number, don't get fixated on the number. If you were borrowing 10,000, it would be less. If you were borrowing 50,000, it would be less. If you were borrowing 1 million, the payment would be more. But there is a one to one relationship between what you're borrowing and what you have to pay. So let me ask you this question, suppose the interest rate was zero, suppose you can go to the bank and just get $100,000 and not pay any interest. How much would you pay every year? Pretty simple, take $100,000 and divide it by 5. In this case, you're paying 6,380 more every period, every year. And for simplicity we kept the year as a fixed quantity, not a month. We'll get to that in a second. So what I'm going to do now is I'm going to take the 26,380 and do this. What should be the present value of this, with n=5, r of 10%, what should be the PV? If you can answer that question, you know how to mess with Excel, actually you know how to do something very profound. If you do this exercise which I encourage you to do, it has to be $100,000, right? Because you're just going back and forth with the same problem. Okay? So $26,380 is the amount of money that you have to pay every year on the loan. What I'm going to do next, and I'm going to take a break now, and I think you need to take a break, is you know how to do this problem, I'm going to now use this problems to show you how great and awesome finance is. And after that, I will do a couple of other problems and get you completely internalized with the class today. As I had promised, the class is in intense because I'm doing problems. I'm bringing in the real world. If I were just doing the formulas, you'd be much happier because time would just be passing by quickly. But the learning, I believe, wouldn't be the same. Okay? So let's take a break, we'll come back, and deal with this issue in a second.