JAMES P WESTON: Hi. Welcome back to Finance for Non-finance Professionals. This is the second lecture of week one. In our first lecture, we talked about interest rates in general. And in this lecture about compounding and earning returns over time, we're going to take those interest rates and put them to use. So let's think about a very basic example. Let's say you had $100 today. And you put it in the bank. You put it in the bank for one year earning 11% interest. How much money would you have after one year? Well that's a simple one. You would have $100 that you put in originally plus 11% of $100, which is $11. So put those two together, the principal plus the interest. And after one year, you would have $111. That $111 represents the principal plus the 11% interest that you earned. Now if you left that money in the bank for a second year-- you took that whole $111 and put it in for a second year, how much money would you have after two years? Well again, you're going to have the $111. And then you're going to have the interest that you earned during the second year. Now you might think naively, well, if I'm earning 11% interest on the $100, I'll get another $11. So that would be $122. But that's not quite right. And the reason it's not quite right is compound interest. You're going to earn interest on the $11 that you earned last year. So you're going to have a little bit more than $122. In fact, you're going to have on $123.21. That extra little bit of money is that 11% on the $11 of interest that you earned last year. One of the nice things about compound interest as that it grows exponentially. So the longer you leave it in and the higher the interest rate, the more interest you earn on the interest that you had before. And that amount of money that you have explodes. It explodes exponentially over time. That's one of the beauties of compound interest. The growth expands over time. So that interest that you're earning on the interest is what we call compound interest. Because it's compounding exponentially over time. Now let's say interest rates were 11% and that I put $1,000 in the bank for five years instead of just the two. Let's think about how much money you would have at the end of five years. All right. I've worked out a simple table for you here. At year zero, which means right now, you have $1,000. At the end of one year, let's think about what you've got. If you have that $1,000 times the interest rate-- $1000 times 11%-- that's the interest that you would earn in the first year. Plus your original principal gets you to $1,110. Or, if I collect parentheses, 1,000 times 1 plus the interest rate. OK. Good. How much would you have after two years? So you're going to take that whole $1,110 and invest that at 1 plus 11%, principal plus interest. After two years, you would have $1,232. Now you might notice that the $1,110 was, as we said before, 1,000 times 1 plus 11%. So if I substitute this value in for that value, you'll see that you have-- after two years-- 1,000 times 1 plus 11% squared after two years. There's that exponent in exponential growth. How much would you have after three years? Again, the same thing applies. Take the $1,232. Grow that at 1 plus 11%. And how much would you have? $1,368. And you'll remember just like in the last example, that $1,232? Well that was 1,000 times 1 plus 11% squared. So that $132 Is 1,000 times 1 plus 11 squared times 1 plus 11 gets me to 1,000 times 1 plus 11 percent cubed. To the third power. Second time period, squared. Third time period, cubed. You can see where this is going maybe. If I put it in for a fourth year, that's $1,368 times 1 plus 11%. $1518. Or 1,000 times 1 plus 11 percent to the fourth power. After five years, $1685. 1,000 times 1 plus 11 percent to the fifth power. OK. If you can see where this is going now, that formula has got some regularity to it. Every time we go out an additional period, what we're doing is basically raising that exponent to one more power. 1000 times 1 plus 11% times 1 plus 11% times 1 plus 11%. That growth in the exponent of how we're earning that interest rate is what we call exponential growth, or compound interest over time. So the answer is $1,685. 1000 grown at 11% over five years. We can generalize that formula by induction from the example that we just did. The future value of any amount, how much I have in the future is the present value of PV. What I put in today, that's the $1000. Times 1 plus r, the interest rate-- that was 11% in our previous example-- raised to the power of t, time. How many times periods is that growing for? So in our previous example, the present value is $1,000. The interest rate was 11%. And the t was 5. And that's how we solve the problem. The best way to think about this compound interest rate and to learn it is to work a couple of examples. So what I'd like to do now is move to the light board and work through a couple practical applications with you. All right. I'd like to work a simple example with you of just taking money and putting it in the bank over time. And let's work through sort of the mathematics of compounding. Let's say I take $1,000. And I take that $1,000 and I put it in the bank at 11%. Let's do that for five years. And let's see how much money we have at the end of five years. OK. So if I take that $1,000 and put it in the bank, how much do I have after one year? After one year I'm going to have $1,000 plus I'm going to have $1,000 times the interest that I earned on the $1,000. That's at 11%. So 1000 times 11%. That's the interest. And the $1,000 is my principle. So at the end of the year, if I move the $1,000 out and just collect terms, I'll have $1,000 times 1 plus 11%. OK. Now if I take that money and I put it in a bank for a second year, how much am I going to have after two years? After two years, I'm going to take that whole amount 1,000 times 1 plus 11% and put that in the bank for the second year. Because the second year, I've got my principal plus the interest that I earned. Compound interest. 1,000 times 1 plus 11%. That whole quantity times again 1 plus 11%, which I can just rewrite as 1000 times 1 plus 11%. 1 plus 11% times 1 plus 11% is 1 plus 11% squared. So two years, 1 plus 11% squared. How much am I going to have after three years? Now I've put the money in the bank for the third year. I'm going to have that whole quantity-- 1,000 times 1 plus 11% squared-- I'm going to put that in the bank for the third year. I'm going to have 1,000 times 1 plus 11% squared. That's how much I have after two years. And I'm putting that in the bank for the third year. So times 1 plus 11% again. And you can see how this is going to hang out here. This is going to be 1000 times 1 plus 11%. 1 plus 11% squared times 1 plus 11% is 1 plus 11% cubed. OK. So after one year, it was 1 plus 11%. After two years, it was 1 plus 11% squared. After three years, 1 plus 11% cubed. You can kind of see where this is going. After five years, how much am I going to have in the bank? It's going to be 1000 times 1 plus 11% raised to the five periods, fifth power. And that's going to be equal to $1,685. If I just put that in a calculator, and that's the answer to the problem. So for five years at 11%, I'm going to take that 11%, compound it up five years, multiply it by the amount that I'm putting in, and that comes out to our answer of 1,685. Now I'd like to work another example with you where we're going to kind of flip the formula around and figure out a rate of return. All right. In this example, I want to flip things around a little bit and solve the formula similar to what we did the last example but do it a little bit differently. Let's say that I bought a piece of art, a painting for $700. And then after three years, I solved it for $825. OK. So what I'd like to know is what kind of a rate of return did I earn over those three years on an annual basis? In other words, what is r? So here, what did I buy it for today? I bought it for $700. That's really a present value. Because that's money that I have today. I'm going to sell it three years from now at $825. So that's really a future value. OK. And then t was three years. So what don't I know when I think about the formula of future value present value? Let's write it out and see. We have that future value is equal to the present value times 1 plus r-- there it is-- to the t. So I've got t. That's three years. I've got the present value. That's $700. And I've got the future value of $825. But what I don't know is r. So we're going to take this equation. And we're going to solve for r. So let's plug and chug. My future value is 825. My present value is 700. 1 Plus r to the 3. Now I want to solve this equation for r. So let me divide through by 700. That will isolate the 1 plus r cubed. 1 plus r cubed then is equal to 825. Over 700. OK. What do I need to do now? I need to get rid of that 3. So let me take the cubed root, which we could do easily on a scientific calculator. That'll get me to 1 plus r. 1 plus r equals 825 over 700. Whole thing cubed root. And that's going to give me 1 plus r. So all I need to do now is subtract one from each side. And that's going to give me r. I'll kind of move up here and do that. So that's going to be r is equal to 825 over 700 cubed root minus 1. And all I need to do now is take the cubed root of that ratio, subtract 1, and that's going to get me to an answer of r is equal to 5.63%. So if I bought something for 700, sold it three years later for 825, how much did I earn per year on an annual basis? That's an r of 5.63%. So we can use this formula of connecting present value and future value to move cash into the future, to pull it back into the present, or to take what we bought and sold and compute compound annual growth rates.
