-
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.
