calculating mortgage the easy way!

Posted: September 29, 2012 in Mathematics

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Have you ever wondered how to calculate mortgage by hand?  I am taking about figuring it out on a piece of paper with just pencil and calculator.  Without the mortgage table, fancy formulas, mortgage solver programs you can easily find on the internet, or any other helps.  Alright let’s begin.

First the definitions:

  • A = The mortgage amount
  • n = number of payments
  • r = interest rate of each payment period, usually monthly interest or \frac{\mbox{annual interest}}{12}
  • p = the payment amount

Assume the payment is due at the end of each month, then the amount own in the first month will be charged an interest A(1+r).  After the mount p is pay, the bill of the second month will be (A(1+r)-p)(1+r).  And by the nature of mortgage, everything has to pay off at the end.  Hence we can come up with this equation,

((A(1+r)-p)(1+r)-p)(1+r)-...-p=0

Now expand the left side,

A(1+r)^n-p(1+r)^{n-1}-...-p(1+r)-p=0

A(1+r)^n=p+p(1+r)+\ldots+p(1+r)^{n-1}

Then apply the sum of geometric series,

A(1+r)^n=\displaystyle{p}\left(\frac{1-(1+r)^n}{1-(1+r)}\right)=p\left(\frac{(1+r)^n-1}{r}\right)

p=\displaystyle\frac{Ar(1+r)^n}{(1+r)^n-1}

p=\displaystyle\frac{Ar}{1-(1+r)^{-n}}

For example, a mortgage of $1,000,000 with monthly payment over 25 years at an annual interest rate of 3%.  By the above formula,

p=\displaystyle\frac{1000000(0.03/12)}{1-(1+0.03/12)^{-300}}=4742.11

The monthly payment is $4,742.11

Remark:

The other way of formulating the equation is to partition the mortgage amount A into n pieces,

A=A_1+A_2+\ldots+A_n

each A_i is different in size, then for each A_i,

A_i(1+r)^i=p

A_i=\displaystyle\frac{p}{(1+r)^i}

then,

A=\displaystyle\frac{p}{1+r}+\frac{p}{(1+r)^2}+\cdots+\frac{p}{(1+r)^n}

A=\displaystyle\frac{p}{1+r}\left(\frac{1-(1+r)^{-n}}{1-(1+r)^{-1}}\right)=\frac{p(1-(1+r)^{-n})}{r}

p=\displaystyle\frac{Ar}{1-(1+r)^{-n}}

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Comments
  1. Kenny says:

    How are you displaying latex? I say you post on the wordpress support page and I am having similar troubles. I tried placing

    i\int\frac{x^2-1}{(x^2+1)\sqrt{x^4+1}}dx

    into a post, and the latex does not compile. Are you using a particular plugin? I am using wordpress 3.4.2

  2. mathgarage says:

    I donno, I am using wordpress.com not wordpress.org

  3. Jonathan says:

    so how I calculate the remaining amount in 120th month?

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