What I'm going to do is I'm going to introduce the concept of compounding before I do another problem and then what I would like you to do is do a problem with compounding. Simple interest rate we've talked about. Simple interest is very simple, which is if I give $100,000 to the bank and it pays 10 percent every year, you get $10,000 every year, but that's not how the world is. Let's do compound interest. Suppose you take the same problem. You plan to attend a business school and you will be forced to take out a $100,000 loan at 10 percent, but now the loan characteristics changes. What are your monthly payments given that you'll have to pay back the loan in five years? Remember what have I changed? Have I changed the amount you're borrowing? No. Have I changed the interest rate you're borrowing it yearly? Actually, yes, but we will see that in a second. The interest rate is written as 10 percent. The thing that I've changed on you is another real world's twist, which I think is important. I've changed the annual payments to monthly payments and the number of years are the same. Quick question; what is your monthly payment and what is the real annual interest rate? These are the two things we will do, and I will show you how to do this, and I will expect you to understand and do it on your own but I'll go through the steps. What has changed? R is 10 percent annual but my payment is monthly. So five years implies how many months? 60 months. If the interest rate is 10 percent annual, and by the way, the good news is that's how interest rates are quoted, what is the interest rate per month? It's 0.1 divided by 12, and the reason is there are 12 months. Why is five years 60 months? Because there are 12 months in a year. What is PV? 100,000. What are we trying to figure out? PMT, but we are not doing five of them. We are doing how many of them? 60. A lot of people get very confused when the periodicity of the interest payment changes. You could have monthly interest, you could have annual interest, you could have daily interest, you could have quarterly interest. The way I think about this problem is to remember you are in control. The best way to deal with this problem is to change the timeline. Do this. Start at 0, put 100,000, then how many periods? One through 60. Why? Because I'm doing the problem monthly. What is the interest rate per period? Apples to apples, remember, take 0.1 and divide by 12. Does it make sense? I'm being internally consistent. Why is this a good way to solve the problem? Because the problem now fits what I know already. What will happen? Let's go on a spreadsheet and do this problem and you'll see. The good news is I have this problem solved for an annual basis. What do I do? I just divide 0.1 by 12. What have I done? I've converted the interest rate to monthly. Then what do I do? Whatever I have divided the interest rate by, I have to multiply the number of years by the same amount and it's 60. What is $100,000? It hasn't changed. My payment amount is 2124. I'm paying about $2,125 per month to repay the loan. You see what I've done? I've just simply taken the fact that I know that I can mess with the timeline. So I made n 60, I've made r 0.1 divided by 12, and my PV, the amount of loan was 100,000. I came up with, I believe 2,125. Let me just double-check that the number is right. Yes, it is. The question now is, this is question number 1, how many of this will I pay? Obviously, 60. I will encourage you to do one exercise. What is the present value of paying 2125 at that interest rate 60 times answer is has to be this amount of the loan. Let me ask you, how much will you owe after 30 months? Very simple, make PMT 2125 which you just calculate, r is what? 0.1 over 12. N is how much? Remember, where are you standing now? You're standing at point 30, looking forward to 60, m is 30. Do the PV of this. You have the amount of money you owe the bank. It's so simple, you don't even have to do that whole table. The reason I went through this problem in detail with the annual and monthly, is simply to emphasize to you that it is extremely important for you to recognize that finance is very logical and you take yourself to the problem and not let the problem scar you. Okay, what is the actual interest rate? How much is my annual r actually? This is a good question to ask. Here's for you to pause and think. The stated r is 10 percent, but the actual r can't be 10 percent, it has to be more and the reason is, again, pause, compounding. Let's just quickly do that and then I encourage you right after that to take another break. As I said, today is a little bit intense and I want to emphasize we have done the timeline; the formula, very simple, so what I'm going to do is I'm going to just use this formula and explain. If I put $1 at what interest rate? R is annual, 10 percent, this is always annual. What is K? K is the number of periods that are within that year, so K is 12 here. Why? Because it's monthly. How many periods, 12-month period, 12 raised to power 12. What is this? This number is the future value of $1 after 12 intervals, so what is the interest rate being charged? Is that minus 1, and I would encourage you to do this calculation. The answer is, you should know this number is greater than 10 percent. Why? Because you are paying 10 percent annually, but actually that's not true. You're not paying 10 percent annually, you're paying 10 percent divided by 12 monthly and with compounding raised to power 12, this works out to actually be about 10.47 percent. Now you may think that 0.5 percent is not a big deal, It is, especially if you're borrowing a lot of money, it's a big deal. The difference between this and this, these two, this is stated and this is actually the real interest that you're being charged. I hope you take a break now, what we have done is we have taken a loan problem, we have dissected it because it just reflects everything awesome about finance. Then what we have done is we have gone back and said to ourselves, what if we took the loan and we made it a monthly loan? No problem. If you know finance and you are thinking clearly, you're logical, you won't mess with anything except your timeline, so your month matches the period and if they are five years they're, 60 months, the interest rate changes accordingly and changes accordingly and then your payment is calculated based on the same kind of information you give. Okay, so I would at this point again take a break, we have a little bit left, one problem left, and a little bit of concept for today, and then we'll call it quits. It's a long day today, but I therefore encourage you to just take a little bit of a break.