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Math Help - Mortgage Formula

  1. #1
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    Mortgage Formula

    Hello. I am wondering about a specific mortgage formula.
    The formula I found:
    P = monthly payment
    L = principal
    c = monthly interest
    n = total months of amortization

    Thus:

    P = L[c(1+c)^n]/(1+c)^n-1

    Question: Is this the proper formula? Why is it the proper formula?


    I can use the formula, but I do not understand it. I want to get to the point where I can explain how the formula is derived in english.


    The numerator: L[c(1+c)^n]
    This is trying find the 'money' that is being paid. The term (1+c)^n gives you the total interest that you pay over the life of the loan.
    Since we are trying to find the monthly payment, we don't care about that. So, we take the total interest that is paid over the life of the loan, and scale it from the life of the loan to a monthly interest payment, hence c(1+c)^n .

    The denominator: (1+c)^n-1
    This is where I get confused. In the numerator, we found the real interest rate. In the denominator, what are we trying to find?
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  2. #2
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    Quote Originally Posted by thenextbesthang View Post
    The formula I found:
    P = monthly payment
    L = principal
    c = monthly interest
    n = total months of amortization
    Minor changes:
    P = monthly payment
    A = amount borrowed (principal)
    i = monthly interest
    n = total number of months (amortization)

    The formula to calculate the monthly payment P is:
    P = Ai / (1 - k) where k = 1 / (1 + i)^n

    The total interest that will be paid is:
    Pn - A
    In other words, sum of all payments - amount borrowed

    Example:
    P = ?
    A = 200000
    i = .01 (12% annual cpd monthly)
    n = 240 (20 years)

    P = 200000(.01) / [1 - 1 / (1.01)^240] = 2202.1722...

    Total interest = 240(2202.1722..) - 200000 = 328521.3441...

    Hope that helps...
    Last edited by Wilmer; April 27th 2010 at 10:23 AM. Reason: Add example
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  3. #3
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    Quote Originally Posted by Wilmer View Post
    Minor changes:
    P = monthly payment
    A = amount borrowed (principal)
    i = monthly interest
    n = total number of months (amortization)

    The formula to calculate the monthly payment P is:
    P = Ai / (1 - k) where k = 1 / (1 + i)^n

    The total interest that will be paid is:
    Pn - A
    In other words, sum of all payments - amount borrowed


    Example:
    P = ?
    A = 200000
    i = .01 (12% annual cpd monthly)
    n = 240 (20 years)

    P = 200000(.01) / [1 - 1 / (1.01)^240] = 2202.1722...

    Total interest = 240(2202.1722..) - 200000 = 328521.3441...

    Hope that helps...
    So firstly, I want to thank you because I can conceptualize it now.
    However, since I am pretty bad at thinking in terms of math...I still have questions.

    P = Ai / (1 - k)
    k = 1 / (1 + i)^n

    -> P = Ai / [1 - (1 / (1 + i)^n]
    How do you get from that last line to interest paid = P*n - A
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  4. #4
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    P = Ai / (1 - k)
    k = 1 / (1 + i)^n

    -> P = Ai / [1 - (1 / (1 + i)^n]
    How do you get from that last line to interest paid = P*n - A

    Not sure what (or why) you're asking

    If you lend me $100 and I pay you back with 3 payments of $40,
    I've paid you 20 bucks in interest, right?
    Which is 3*40 - 100, right?
    Or the A=100 and the P=40 and the n=3, right?
    Same thing for ANY loan/mortgage...
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  5. #5
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    Quote Originally Posted by Wilmer View Post
    P = Ai / (1 - k)
    k = 1 / (1 + i)^n

    -> P = Ai / [1 - (1 / (1 + i)^n]
    How do you get from that last line to interest paid = P*n - A

    Not sure what (or why) you're asking

    If you lend me $100 and I pay you back with 3 payments of $40,
    I've paid you 20 bucks in interest, right?
    Which is 3*40 - 100, right?
    Or the A=100 and the P=40 and the n=3, right?
    Same thing for ANY loan/mortgage...
    Yeah that I get. Its more a question of the 'language' of math.
    I am asking this, because if I am having a conversation with someone and they say 'I have a 300k/30year/6% fixed mortgage' I want to be able to approximate in my head around number their monthly payments should be.

    lend amount (300k),
    number of payments (360),
    yearly payments of interest(.06 of principal (18k)
    monthly payments of interest (.06/12 = .005 of principal (1.5k))

    The total interest paid to the bank over the course of the loan = interest paid to the bank per month ^ number of months = 1.005^360 = 6.022575.

    So then the total interest payments would be about 6*300k = 1.8 million , then 1.8 million - 300k = 1.5 million.......????
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