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What I want to do in this
video is go over the math
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behind a mortgage loan.
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And this isn't really going
to be a finance video.
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It's actually a lot
more mathematical.
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But it addresses, at least in
my mind, one of the most basic
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questions that's at least been
circling in my head
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for a long time.
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You know, we take out these
loans to buy houses.
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Let's say you take out a
$200,000 mortgage loan.
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It's secured by your house.
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You're going to pay it over--
30 years, or you could
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say that's 360-- months.
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Because if you normally pay the
payments every month, the
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interest normally compounds
on a monthly basis.
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And let's say you're
paying 6%-- interest.
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This is annual interest, and
they're usually compounding
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on a monthly basis,
so 6% divided by 12.
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You're talking about
0.5% per month.
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Now normally when you get a
loan like this, your mortgage
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broker or your banker will look
into some type of chart or
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type in the numbers into some
type of computer program.
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And they'll say oh OK,
your payment is going
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to be $1,200 per month.
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And if you pay that $1,200 per
month over 360 months, at the
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end of those 360 months you
will have paid off the $200,000
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plus any interest that
might have accrued.
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But this number it's not
that easy to come along.
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Let's just show an example of
how the actual mortgage works.
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So on day zero, you
have a $200,000 loan.
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You don't pay any
mortgage payments.
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You're going to pay your
first mortgage payment
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a month from today.
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So this amount is going to be
compounded by the 0.5%, and
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as a decimal that's a 0.005.
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So in a month, with interest,
this will have grown to
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200,000 times 1 plus 0.005.
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Then you're going
to pay the $1,200.
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Just going to be minus 1,200
or maybe I should write 1.2K.
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But I'm just really just
showing you the idea.
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And then for the next month,
whatever is left over is
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going to be compounded
again by the 0.5%, 0.005.
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And then the next month you're
going to come back and you're
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going to pay this $1,200 again.
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Minus $1,200.
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And this is going to
happen 360 times.
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So you're going to
keep doing this.
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And you can imagine if you're
actually trying to solve for
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this number-- at the end of it
you're going to have this huge
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expression that's going to have
you know 360 parentheses over
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here-- and at the end, it's
all going to be equal to 0.
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Because after you've paid your
final payment, you're done
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paying off the house.
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But in general how did they
figure out this payment?
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Let's call that p.
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Is there any mathematical
way to figure it out?
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And to do that, let's get a
little bit more abstract.
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Let's say that l is equal
to the loan amount.
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Let's say that i is equal
to the monthly interest.
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Let's say n is equal to
the number of months
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that we're dealing with.
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And then we're going to set
p is equal to your monthly
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payment, your monthly
mortgage payment.
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Some of which is interest, some
of which is principle, but it's
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the same amount you're going to
pay every month to pay down
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that loan plus interest.
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So this is your
monthly payment.
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So this same expression I just
wrote up there, if I wrote it
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in abstract terms, you start
off with a loan amount l.
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After 1 month it
compounds as 1 plus i.
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So you multiply it times
1 plus i. i in this
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situation was 0.005.
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Then you pay a monthly
payment of p, so minus p.
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So that's at the
end of one month.
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Now you have some amount still
left over of your loan.
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That will now compound
over the next month.
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Then you're going to
pay another payment p.
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And then this process is going
to repeat 300 or n times,
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because I'm staying abstract.
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You're going to have
n parentheses.
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And after you've done this
n times, that is all
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going to be equal to 0.
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So my question, the one that
I'm essentially setting up in
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this video, is how
do we solve for p?
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You know if we know the loan
amount, if we know the monthly
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interest rate, if we know the
number of months, how
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do you solve for p?
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It doesn't look like this is
really an easy algebraic
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equation to solve.
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Let's see if we can
make a little headway.
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Let's see if we can rearrange
this in a general way.
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So let's start with an example
of n being equal to 1.
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If n is equal to 1, then our
situation looks like this:
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you take out your loan, you
compound it for one month, 1
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plus i, and then you pay
your monthly payment.
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Now this was a mortgage that
gets paid off in 1 month, so
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after that 1 payment you are
now done with their loan,
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you have nothing left over.
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Now if we solve for p, you
can now swap the sides.
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You get p is equal to
l times 1 plus i.
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Or if you divide both sides
by 1 plus i, you get p over
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1 plus i is equal to l.
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And you might say hey you
already solved for p
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why are you doing this?
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And I'm doing this, because
I want to show you a
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pattern that'll emerge.
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Let's see what happens
when n is equal to 2.
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Well then you start
with your loan amount.
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It compounds for one month.
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You make your payment.
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Then there's some
amount left over.
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That will compound
for one month.
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Then you make your
second payment.
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Now this mortgage only
needs two payments,
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so now you are done.
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You have no loan left over.
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You've paid all the
principal and interest.
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Now let's solve for p.
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So I'm going to color the p's.
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I'm going to make this p pink.
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So let's add p to both
sides and swap sides.
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So this green p will be
equal to all of this
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business over here.
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Is equal to l times 1 plus
i minus that pink p.
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They're the same p, I just
want to show you what's
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happening algebraically.
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Minus that pink p
times 1 plus i.
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Now if you divide both sides
by 1 plus i, you get p over
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1 plus i is equal to l times
1 plus i minus that pink p.
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Now let's add that pink p to
both sides of this equation.
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You get the pink p plus this
p plus p over 1 plus i is
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equal to l times 1 plus i.
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Now divide both
sides by 1 plus i.
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You get the pink p over 1 plus
i plus the green p, the same p,
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times-- it already is being
divided by 1 plus i, you're
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going to divide it again by 1
plus i, so it's going to be
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divided by 1 plus i squared
is equal to the loan.
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Something interesting
is emerging.
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You might want to watch the
videos on present value.
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In this situation, you take
your payment, you discount it
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by your monthly interest rate,
you get the loan amount.
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Here you take each of your
payments, you discount it, you
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divide it by 1 plus your
monthly interest rate to the
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power of the number of months.
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So you're essentially taking
the present value of your
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payments and once again,
you get your loan amount.
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You might want to verify this
for yourself if you want a
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little bit of algebra practice.
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If you do this with
n is equal to 3.
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I'm not going to do it just
for the sake of time.
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If you do n is equal to 3,
you're going to get that the
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loan is equal to p over 1 plus
i plus p over 1 plus i squared
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plus p over 1 plus
i to the third.
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If you have some time, I
encourage you to prove this for
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yourself just using the exact
same process that we did here.
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You're going to see it's going
to get little bit harry.
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There's going to be a lot of a
manipulating things, but it
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won't take you too long.
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But in general, hopefully, I've
shown to you that we can write
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the loan amount as the present
value of all of the payments.
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So we could say in general the
loan amount, if we now
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generalize it to n instead of
and n equals a number, we could
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say that it's equal to-- I'll
actually take the p out of the
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equation, so it's equal to p,
times 1 plus 1 over 1 plus i
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plus 1 over 1 plus i squared
plus, and you just keep
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doing this n times, plus 1
over 1 plus i to the n.
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Now you might recognize this.
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This right here is a
geometric series.
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And there are ways to figure
out the sums of geometric
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series for arbitrary ends.
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As I promised at the beginning
of the video this would be
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an application of a
geometric series.
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It's equal to the sum of 1 over
1 plus i to the, well I'll use
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some other letter here, to
the j from j is equal to 1.
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This is to the one power you
could view this is to the first
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power to j is equal to n.
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That's exactly
what that sum is.
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Let's see if there's any simple
way to solve for that sum.
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You don't want to
do this 360 times.
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You could, you'll get a number,
and then you could divide
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l by that number, and you
would have solved for p.
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But there's got to be simpler
way to do that, so let's see
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if we can simplify this.
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Just to make the math easier,
let me make a definition.
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Let's say that r is equal
to 1 over 1 plus i.
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And let me call
this whole sum s.
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This sum right here
is equal to s.
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Then if we say r is equal to
each of these terms then s is
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going to be equal to this is
going to be r to
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the first power.
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I'll write r to first this is
going to be r squared, because
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if you square the numerator
you just get a 1 again.
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So this is plus r squared plus
r to the third, plus all
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the way this is r to the n.
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And I'll show you
a little trick.
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I always forget the formula,
so this is a good way to
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figure out the sum of
a geometric series.
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Actually this could be used to
find a sum of an infinite
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geometric series if you
like, but we're dealing
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with a finite one.
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Let's multiply s times r.
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So r times s is going
to be equal to what?
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If you multiply each of these
terms by r, you multiply r
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to the first times r
you get r squared.
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You multiply r squared times
r you get r to the third.
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And then you keep doing that
all the way, you multiply r--
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see there's an r to the n minus
one here-- you multiply that
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times r, you get r to the n.
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And then you multiply r to
the n times r, you get
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plus r to the n plus 1.
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All this is right here is all
of these terms multiplied
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by r, and I just put them
under the same exponent.
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Now what you can do is you
could subtract this green
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line from this purple line.
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So if we were to say s
minus rs, what do we get?
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I'm just subtracting this
line from that line.
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Well, you get r1 minus 0,
so you get r to the first
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power minus nothing there.
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But then you have r squared
minus r squared cancel out r
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to the third minus r to
the third cancel out.
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They all cancel out, all the
way up to r to the n minus r to
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the n cancel out, but then
you're left with this
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last term here.
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And this is why
it's a neat trick.
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So you're left with minus
r to the n plus 1.
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Now factor out an s.
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You get s times 1 minus r-- all
I did is I factored out the s--
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is equal to r to the first
power minus r to the n plus 1.
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And now if you divide
both sides by 1 minus
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r, you get your sum.
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Your sum is equal to r minus r
to the n plus 1 over 1 minus r.
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That's what our sum is
equal to, where we defined
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our r in this way.
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So now we can rewrite this
whole crazy formula.
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We can say that our loan amount
is equal to our monthly
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payment times this thing.
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I'll write it in green.
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Times r minus r
to the n plus 1.
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All of that over 1 minus r.
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Now if we're trying to solve
for p you multiply both sides
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by the inverse of this, and you
get p is equal to your loan
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amount times the
inverse of that.
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I'm doing it in pink,
because it's the inverse.
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1 minus r over r minus
r to the n plus 1.
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Where r is this
thing right there.
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And we are done.
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This is how you can actually
solve for your actual
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mortgage payment.
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Let's actually apply it.
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So let's say that your loan
is equal to $200,000.
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Let's say that your interest
rate is equal to 6% annually,
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which is 0.5% monthly which
is the same thing as 0.005.
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This is monthly interest rate.
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And let's say it's a 30 year
loan, so n is going to
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be equal to 360 months.
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Let's figure out what we get.
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So the first thing we want
to do is we want to figure
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out what our r value is.
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So r is 1 over 1 plus i.
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So let's take 1 divided by
1 plus i so plus 0.005.
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That's what our monthly
interest is, half a percent.
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So 0.995 that's what
our r is equal to.
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Let me write that down, 0.995.
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Now this calculator doesn't
store variables, so I'll
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just write that down here.
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So r is equal to 0.995.
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We just used that right there.
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I'm losing a little bit
of precision, but I
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think it will be OK.
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The main thing is I want to
give you the idea here.
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So what is our payment amount?
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Let's multiply our loan amount
that's $200,000 times 1 minus
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r, so 1 minus 0.995 divided by
r which is 0.995 minus 0.995 to
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the of the-- now n is 360
months, so it's going to be
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360 plus 1 to the 361 power,
something I could definitely
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not do in my head, and then I
close the parentheses, and my
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final answer is roughly $1,200.
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Actually if you do it with the
full precision you get a little
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bit lower than that, but this
is going to be roughly $1,200.
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So just like that, we were
able to figure out our
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actual mortgage payment.
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So p is equal to $1,200.
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So that was some reasonably
fancy math to figure out
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something that most people deal
with everyday, but now you know
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the actual math behind it.
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You don't have to play with
some table or spreadsheet to
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kind of experimentally
get the number.
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