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JAMES P WESTON: Hi.

Â Welcome back to Finance for Non-finance Professionals.

Â This is the second lecture of week one.

Â In our first lecture, we talked about interest rates in general.

Â And in this lecture about compounding and earning returns over time,

Â we're going to take those interest rates and put them to use.

Â So let's think about a very basic example.

Â Let's say you had $100 today.

Â And you put it in the bank.

Â You put it in the bank for one year earning 11% interest.

Â How much money would you have after one year?

Â Well that's a simple one.

Â You would have $100 that you put in originally

Â plus 11% of $100, which is $11.

Â So put those two together, the principal plus the interest.

Â And after one year, you would have $111.

Â That $111 represents the principal plus the 11% interest that you earned.

Â Now if you left that money in the bank for a second year--

Â you took that whole $111 and put it in for a second year,

Â how much money would you have after two years?

Â Well again, you're going to have the $111.

Â And then you're going to have the interest

Â that you earned during the second year.

Â Now you might think naively, well, if I'm earning 11% interest on the $100,

Â I'll get another $11.

Â So that would be $122.

Â But that's not quite right.

Â And the reason it's not quite right is compound interest.

Â You're going to earn interest on the $11 that you earned last year.

Â So you're going to have a little bit more than $122.

Â In fact, you're going to have on $123.21.

Â That extra little bit of money is that 11% on the $11 of interest

Â that you earned last year.

Â One of the nice things about compound interest

Â as that it grows exponentially.

Â So the longer you leave it in and the higher the interest rate,

Â the more interest you earn on the interest that you had before.

Â And that amount of money that you have explodes.

Â It explodes exponentially over time.

Â That's one of the beauties of compound interest.

Â The growth expands over time.

Â So that interest that you're earning on the interest

Â is what we call compound interest.

Â Because it's compounding exponentially over time.

Â Now let's say interest rates were 11% and that I put $1,000 in the bank

Â for five years instead of just the two.

Â Let's think about how much money you would have at the end of five years.

Â All right.

Â I've worked out a simple table for you here.

Â At year zero, which means right now, you have $1,000.

Â At the end of one year, let's think about what you've got.

Â If you have that $1,000 times the interest rate-- $1000 times 11%--

Â that's the interest that you would earn in the first year.

Â Plus your original principal gets you to $1,110.

Â Or, if I collect parentheses, 1,000 times 1 plus the interest rate.

Â OK.

Â Good.

Â How much would you have after two years?

Â So you're going to take that whole $1,110 and invest that at 1

Â plus 11%, principal plus interest.

Â After two years, you would have $1,232.

Â Now you might notice that the $1,110 was, as we said before,

Â 1,000 times 1 plus 11%.

Â So if I substitute this value in for that value,

Â you'll see that you have-- after two years-- 1,000 times

Â 1 plus 11% squared after two years.

Â There's that exponent in exponential growth.

Â How much would you have after three years?

Â Again, the same thing applies.

Â Take the $1,232.

Â Grow that at 1 plus 11%.

Â And how much would you have? $1,368.

Â And you'll remember just like in the last example, that $1,232?

Â Well that was 1,000 times 1 plus 11% squared.

Â So that $132 Is 1,000 times 1 plus 11 squared times 1 plus 11 gets me

Â to 1,000 times 1 plus 11 percent cubed.

Â To the third power.

Â Second time period, squared.

Â Third time period, cubed.

Â You can see where this is going maybe.

Â If I put it in for a fourth year, that's $1,368 times 1 plus 11%.

Â $1518.

Â Or 1,000 times 1 plus 11 percent to the fourth power.

Â After five years, $1685.

Â 1,000 times 1 plus 11 percent to the fifth power.

Â OK.

Â If you can see where this is going now, that formula

Â has got some regularity to it.

Â Every time we go out an additional period, what we're doing

Â is basically raising that exponent to one more power.

Â 1000 times 1 plus 11% times 1 plus 11% times 1 plus 11%.

Â That growth in the exponent of how we're earning that interest rate

Â is what we call exponential growth, or compound interest over time.

Â So the answer is $1,685.

Â 1000 grown at 11% over five years.

Â We can generalize that formula by induction from the example

Â that we just did.

Â The future value of any amount, how much I have in the future

Â is the present value of PV.

Â What I put in today, that's the $1000.

Â Times 1 plus r, the interest rate-- that was 11% in our previous example--

Â raised to the power of t, time.

Â How many times periods is that growing for?

Â So in our previous example, the present value is $1,000.

Â The interest rate was 11%.

Â And the t was 5.

Â And that's how we solve the problem.

Â The best way to think about this compound interest rate and to learn it

Â is to work a couple of examples.

Â So what I'd like to do now is move to the light board

Â and work through a couple practical applications with you.

Â All right.

Â I'd like to work a simple example with you of just taking money and putting it

Â in the bank over time.

Â And let's work through sort of the mathematics of compounding.

Â Let's say I take $1,000.

Â And I take that $1,000 and I put it in the bank at 11%.

Â Let's do that for five years.

Â And let's see how much money we have at the end of five years.

Â OK.

Â So if I take that $1,000 and put it in the bank,

Â how much do I have after one year?

Â After one year I'm going to have $1,000 plus

Â I'm going to have $1,000 times the interest that I earned on the $1,000.

Â That's at 11%.

Â So 1000 times 11%.

Â That's the interest.

Â And the $1,000 is my principle.

Â So at the end of the year, if I move the $1,000 out and just collect terms,

Â I'll have $1,000 times 1 plus 11%.

Â OK.

Â Now if I take that money and I put it in a bank for a second year,

Â how much am I going to have after two years?

Â After two years, I'm going to take that whole amount 1,000 times 1 plus 11%

Â and put that in the bank for the second year.

Â Because the second year, I've got my principal plus the interest

Â that I earned.

Â Compound interest.

Â 1,000 times 1 plus 11%.

Â That whole quantity times again 1 plus 11%, which I can just

Â rewrite as 1000 times 1 plus 11%.

Â 1 plus 11% times 1 plus 11% is 1 plus 11% squared.

Â So two years, 1 plus 11% squared.

Â How much am I going to have after three years?

Â Now I've put the money in the bank for the third year.

Â I'm going to have that whole quantity-- 1,000 times 1 plus 11%

Â squared-- I'm going to put that in the bank for the third year.

Â I'm going to have 1,000 times 1 plus 11% squared.

Â That's how much I have after two years.

Â And I'm putting that in the bank for the third year.

Â So times 1 plus 11% again.

Â And you can see how this is going to hang out here.

Â This is going to be 1000 times 1 plus 11%.

Â 1 plus 11% squared times 1 plus 11% is 1 plus 11% cubed.

Â OK.

Â So after one year, it was 1 plus 11%.

Â After two years, it was 1 plus 11% squared.

Â After three years, 1 plus 11% cubed.

Â You can kind of see where this is going.

Â After five years, how much am I going to have in the bank?

Â It's going to be 1000 times 1 plus 11% raised

Â to the five periods, fifth power.

Â And that's going to be equal to $1,685.

Â If I just put that in a calculator, and that's the answer to the problem.

Â So for five years at 11%, I'm going to take that 11%,

Â compound it up five years, multiply it by the amount that I'm putting in,

Â and that comes out to our answer of 1,685.

Â Now I'd like to work another example with you

Â where we're going to kind of flip the formula around and figure out

Â a rate of return.

Â All right.

Â In this example, I want to flip things around a little bit

Â and solve the formula similar to what we did the last example

Â but do it a little bit differently.

Â Let's say that I bought a piece of art, a painting for $700.

Â And then after three years, I solved it for $825.

Â OK.

Â So what I'd like to know is what kind of a rate of return

Â did I earn over those three years on an annual basis?

Â In other words, what is r?

Â So here, what did I buy it for today?

Â I bought it for $700.

Â That's really a present value.

Â Because that's money that I have today.

Â I'm going to sell it three years from now at $825.

Â So that's really a future value.

Â OK.

Â And then t was three years.

Â So what don't I know when I think about the formula of future value

Â present value?

Â Let's write it out and see.

Â We have that future value is equal to the present value times 1

Â plus r-- there it is-- to the t.

Â So I've got t.

Â That's three years.

Â I've got the present value.

Â That's $700.

Â And I've got the future value of $825.

Â But what I don't know is r.

Â So we're going to take this equation.

Â And we're going to solve for r.

Â So let's plug and chug.

Â My future value is 825.

Â My present value is 700.

Â 1 Plus r to the 3.

Â Now I want to solve this equation for r.

Â So let me divide through by 700.

Â That will isolate the 1 plus r cubed.

Â 1 plus r cubed then is equal to 825.

Â Over 700.

Â OK.

Â What do I need to do now?

Â I need to get rid of that 3.

Â So let me take the cubed root, which we could do easily

Â on a scientific calculator.

Â That'll get me to 1 plus r.

Â 1 plus r equals 825 over 700.

Â Whole thing cubed root.

Â And that's going to give me 1 plus r.

Â So all I need to do now is subtract one from each side.

Â And that's going to give me r.

Â I'll kind of move up here and do that.

Â So that's going to be r is equal to 825 over 700 cubed root minus 1.

Â And all I need to do now is take the cubed root of that ratio, subtract 1,

Â and that's going to get me to an answer of r is equal to 5.63%.

Â So if I bought something for 700, sold it three years later for 825,

Â how much did I earn per year on an annual basis?

Â That's an r of 5.63%.

Â So we can use this formula of connecting present value and future value

Â to move cash into the future, to pull it back into the present,

Â or to take what we bought and sold and compute compound annual growth rates.

Â