So let's generalize a bit

what we learned in the last presentation. Let's say I'm

borrowing P dollars. P dollars, that's what I

borrowed so that's my initial principal. So that's principal. r is equal to the rate,

the interest rate that I'm borrowing at. We can also write that

as 100r%, right? And I'm going to borrow

it for– well, I don't know– t years.

Let's see if we can come up

with equations to figure out how much I'm going to owe at

the end of t years using either simple or compound interest. So let's do simple first

because that's simple. So at time 0– so let's make

this the time axis– how much am I going to owe? Well, that's right when I

borrow it, so if I paid it back immediately, I

would just owe P, right? At time 1, I owe P plus the

interest, plus you can kind of view it as the rent on that

money, and that's r times P. And that previously, in the

previous example, in the previous video, was 10%. P was 100, so I had to pay $10

to borrow that money for a year, and I had to

pay back $110. And this is the same thing

as P times 1 plus r, right? Because you could

just use 1P plus rP. And then after two years,

how much do we owe? Well, every year, we just

pay another rP, right? In the previous example,

it was another $10.

So if this is 10%, every

year we just pay 10% of our original principal. So in year 2, we owe P plus

rP– that's what we owed in year 1– and then another

rP, so that equals P plus 1 plus 2r. And just take the P out,

and you get a 1 plus r plus r, so 1 plus 2r. And then in year 3, we'd owe

what we owed in year 2. So P plus rP plus rP, and then

we just pay another rP, another say, you know, if r is 10%, or

50% of our original principal, plus rP, and so that

equals P times 1 plus 3r.

So after t years,

how much do we owe? Well, it's our original

principal times 1 plus, and it'll be tr. So you can distribute this out

because every year we pay Pr, and there's going

to be t years. And so that's why

it makes sense. So if I were to say

I'm borrowing– let's do some numbers. You could work it out this way,

and I recommend you do it. You shouldn't just

memorize formulas.

If I were to borrow $50 at 15%

simple interest for 15– or let's say for 20 years, at the

end of the 20 years, I would owe $50 times 1 plus the

time 20 times 0.15, right? And that's equal to $50 times 1

plus– what's 20 times 0.15? That's 3, right? Right. So it's 50 times 4, which

is equal to $200 to borrow it for 20 years. So $50 at 15% for 20

years results in a $200 payment at the end. So this was simple

interest, and this was the formula for it. Let's see if we can do the same

thing with compound interest. Let me erase all this. That's not how I

wanted to erase it. There we go. OK, so with compound interest,

in year 1, it's the same thing, really, as simple interest, and

we saw that in the previous video. I owe P plus, and now the rate

times P, and that equals P times 1 plus r. Fair enough. Now year 2 is where compound

and simple interest diverge.

In simple interest, we would

just pay another rP, and it becomes 1 plus 2r. In compound interest,

this becomes the new principal, right? So if this is the new

principal, we are going to pay 1 plus r times this, right? Our original principal was P. After one year, we paid 1 plus

r times the original principal times 1 plus r rate. So to go into year 2, we're

going to pay what we owed at the end of year 1, which is P

times 1 plus r, and then we're going to grow that

by r percent. So we're going to multiply

that again times 1 plus r. And so that equals P

times 1 plus r squared.

So the way you could think

about it, in simple interest, every year we added a Pr. In simple interest, we

added plus Pr every year. So if this was $50 and this is

15%, every year we're adding $3– we're adding–

what was that? 50%. We're adding $7.50 in interest,

where P is the principal, r is the rate. In compound interest, every

year we're multiplying the principal times 1 plus

the rate, right? So if we go to year 3,

we're going to multiply this times 1 plus r. So year 3 is P times 1

plus r to the third. So year t is going to be

principal times 1 plus r to the t-th power. And so let's see

that same example. We owe $200 in this example

with simple interest. Let's see what we owe

in compound interest. The principal is $50. 1 plus– and what's the rate? 0.15. And we're borrowing

it for 20 years. So this is equal to 50 times

1.15 to the 20th power.

I know you can't read that,

but let me see what I can do about the 20th power. Let me use my Excel and

clear all of this. Actually, I should just use my

mouse instead of the pen tool to the clear everything. OK, so let me just

pick a random point. So I just want to– plus 1.15

to the 20th power, and you could use any calculator:

16.37, let's say. So this equals 50 times 16.37. And what's 50 times that? Plus 50 times that: $818. So you've now realized that if

someone's giving you a loan and they say, oh, yeah, I'll lend

you– you need a 20-year loan? I'm going to lend

it to you at 15%. It's pretty important to

clarify whether they're going to charge you 15% interest at

simple interest or compound interest. Because with compound interest,

you're going to end up paying– I mean, look at this: just to

borrow $50, you're going to be paying $618 more than if

this was simple interest. Unfortunately, in the real

world, most of it is compound interest. And not only is it compounding,

but they don't even just compound it every year and they

don't even just compound it every six months, they actually

compound it continuously.

And so you should watch the

next several videos on continuously compounding

interest, and then you'll actually start to learn

about the magic of e. Anyway, I'll see you

all in the next video..