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 the PV. Let's read the problem very carefully and just so that I can write on this. You see I press 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. I love technology. There's a person here sitting next to me who's an awesome person and without him, I wouldn't be doing this honestly. 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? 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. Suppose the interest rate is five percent, and I think we are becoming very real with interest rates. We've gone from eight to five now, that's fair. The real world looks like right now. 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. Another use of Excel is it comes with a timeline with the ASL being zero and so on. 25,0,1. Remember more points in time by one then the number of periods. What do I know here? I'm standing here, what do I know? I know that I need $10,000, 25 times. This to me is C or PMT. I know PMT, but what do I want to figure out? 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'll 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. The story is not over. What do I have to do? 10,000 divided by 1 plus r squared. How many times? Twenty five times. Now if I had to do the same thing 25 times, life is easy. But again, who's messing with my fun? Compounding. Because every time I add another one, this two become 3,4,5,25 times. What do I have to do? When in desperation for calculation, go to Excel. I'm going to go to Excel and the good news is the previous problem is already there. I do not want to do a PMT first because I already know PMT, but I want to do now what? PV. Interest rate is now not eight percent, but it's five percent. How many years left? I believe in our problem that we were looking at, just let me just confirm what it is. We do have 25 years left, so we're 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 140,939.45, what does that tell me? I better have $140,939 in the bank to satisfy the need of spending 10,000 in the future. Let me go back to the problem and tell you what's going on. I need in the bank 140,939 and let me just for convenience call it $141,000. Let me ask you this, if I spent $10,000 every year and the interest rate was zero, how many times will I spend 10,000 25 times? Ten, is 250,000. The number that I'm going to put in the bank is nowhere close to 250. It's off by at least a 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 five percent rate of return. 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. This problem gives you a sense of how to do PV and remember the PV is discounting. Every $10,000 in the future is becoming less so today and the last $10,000 is being discounted by 25 years. What happens is you are not multiplying $10,000 by 25 to get this answer, because of compounding and 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. We'll do this in a context and why am I encouraging you to think like that? It's simply to confirm that this number is right under the assumptions. One final comment before we move on to the next problem. The interest rate is 5 percent here so if the world is the same as our previous problem, it requires a strategy that is less risky than the 8 percent to follow. 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 10, you know the 25 and basically, you know the 5. Of course, the 5 won't be 5 the more risk you take, but that's true about anybody making an investment. 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 have become familiar with doing these 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 anytime, 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. Why am I doing this and spend a lot? You'll see this is a classic finance problem in a real-world sense. 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 percent and the 10 percent, you'll see is artificial because I want to make my life a little easy here. Hopefully, you don't need to pay 10 percent, 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. 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. Remember when you come to school, you're also giving up an opportunity of working. When you do your calculations to go to school, it's not a minor thing, 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 as, how do people survive if everything is for free and so on. But I personally feel this $100,000 number 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. 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 percent. 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 simply because it's just too high for many standards. Anyway. At 10 percent, simply because you'll see later, it will help us with the calculations. 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? If you tell me no, I think the issue is with you, not with me. This is a real-world problem. The only thing that will change is the numbers. The quick question to you is who decides $100,000? A hundred thousand 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. Let's assume you'll borrow $100,000 which is two years' worth 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 percent? This is a very interesting question and goes back to my emphasis on markets. If there's one person in the whole market deciding the interest rate, that person is called a monopolist. If in finance or borrowing and lending money, there is one person who'll get screwed, us, the customers, or the people borrowing money. That's why competitive markets are important. Competitive markets are important so that the consumer benefits not the producer necessarily. 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, that's the beauty of markets. Anyway the 10 percent hopefully it's coming out of competition. Why? Because if there are three banks, you'll always go to the lowest interest rate, so competition among banks on the internet hopefully is getting better to help you get a reasonable interest rates. R per period is 10 percent. How did you decide 5-years? Well, it's again an interaction between you and the bank and interest rates will vary depending on the periodicity or the length of the, sorry, the maturity of the loan and so on. Let's keep that issue right now in the risk category. 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? Let's do this problem. 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. Let's do it without screwing things up obviously. What was our problem now? I know my PV and I'm going to calculate what? PMT. This is a number I should know and the bank should tell me. The interest rate is how much? Interest rate is 10 percent, and how many years? Not 25, five, and the next number is PV fortunately, if I see it right. You've got to keep your eye on the ball. How much did I borrow? A 100,000, huge number. The answer to this question is 26,380 and I think what this is telling you is, and I'm rounded things off again without decimals. It's telling you that I, or you, whoever is borrowing the money, will get a $100,000 today, but we'll have to pay $26,000 plus 380, five times. Just pause there. This looks like a huge number. Now, don't get fixated on the number. If you were borrowing 10,000, it will be less. If you're borrowing 50,000, it will be less. If you're borrowing one million, the payment will be more. But there is a one-to-one relationship between what you're borrowing and what you have to pay. Let me ask you this question. Suppose the interest rate was zero, suppose you could go to the bank and just get a $100,000 and not pay any interest. How much would you pay every year? Pretty simple. Take a $100,000 and divide by five. In this case, you're paying 6,380 more every period, every year. For simplicity, we kept the year as a fixed quantity, not a month. We'll get to that in a second. 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 five, r of 10 percent. 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 a $100,000 because you're just going back and forth with the same problem. So $26,380 is the amount of money that you will have to pay every year on a 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 problem to show you how great and awesome finance is. 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 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, won't be the same. Let's take a break, we'll come back and deal with this issue in a second.