Ask for Formula for Mortgage Interests


in this web, it has example to calculate mortgage.

for example:
the loan = 100.000
interest 5%/year
payment : 12.000/month

we got this table (on the web, 3rd table)

Year     Principal       Interest    Payment
One     100,000     5,000           12,000
Two     93,000          4,650           12,000
Three   85,650          4,282.5     12,000
Four    77,932.5    3,896.63    12,000
Five    69,829.13   3,491.46    12,000
Six     61,320.58   3,066.03    12,000
Seven   52,386.61   2,619.33    12,000
Eight   43,005.94   2,150.3     12,000
Nine    33,156.24   1657.81     12,000
Ten     22,814.05   1,140.7     12,000 

My question:

  • What is the formula for calculating
    interest in the X th year. For
    example: I want to know how much
    interest I have to pay in the seventh

  • What is the formula for calculating from year x to y for
    example: I want to calculate sum of
    interest from first year to seventh

Best Solution

This is pretty easy to solve by considering how the principal changes from year to year, so let pn be the principal after n years (so p0=100000 in this case). The amount that the principal decreases in a year is the payment minus the interest. The interest to be paid on year n is 0.05 * pn-1, so the principal decreases by 12000 - 0.05 * pn-1. So we finally get the formula

pn = pn-1 - (12000 - 0.05 * pn-1) = 1.05 * pn-1 - 12000

as a recursive equation for the amount of principal after n years.

Now, plugging into this equation the value for pn-1 we can unfold it one step to express pn in terms of pn-2. Continuing on, it's pretty easy to see that

pn = 1.05n * p0 - 12000 * (sum of 1.05i as i goes from 0 to n-1) = 1.05n * p0 - 12000 * (1.05n - 1) / 0.05 = 1.05n * (p0 - 12000/0.05) + 12000/0.05.

The last comes from the well-known formula for a geometric sum and then rearranging the terms a bit. Now you have the amount of principal after n years expressed in terms of the initial loan, interest rate, and yearly payment (I didn't compute the 12000/0.05 to show how the individual number figure in the formula). Then, the interest to be paid on year n is simply 0.05 * pn-1.

Computing the sum from year x to year y is now very simple, since there is only one term that varies as the year varies and that becomes a geometric sum.

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