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Geometric series sum to figure out mortgage payments

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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.
  • 16:02 - 16:04
    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.
  • 17:09 - 17:11
    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
  • 17:26 - 17:29
    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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Title:
Geometric series sum to figure out mortgage payments
Description:

Figuring out the formula for fixed mortgage payments using the sum of a geometric series

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Video Language:
English
Duration:
17:36

English subtitles

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