Ask for Formula for Mortgage Interests
in this web http://www.fonerbooks.com/interest.htm, it has example to calculate mortgage.
for example: the loan = 100.000 interest 5%/year payment : 12.开发者_StackOverflow000/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 year.
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 year.
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.
There are all sorts of formulae for mortgage calculations here.
I know it's not quite what you asked for, but I've used these spreadsheets in the past to do my mortgage calculations. You might be able to extract the formulae from them.
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