Interest (part 2) | Interest and debt | Finance & Capital Markets | Khan Academy

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..

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