Investment Science - Understanding the mathematics of Mortgage payments - 1

Sunday, August 29, 2010

In the last post, we have seen the annuity formula and an example on how you pay a car loan for a given interest rate for a fixed period of time, in this post we will look into an example to illustrate how the mortgage period affects your total interest payment, for this we will look into an example problem from Investment Science by Luenberger (Problem 3.6).

Problem 3.6

The biweekly mortgage: Here is a proposal that has been advanced as a way for homeowners to save thousands of dollars on mortgage payments: pay biweekly instead of monthly. Specifically if monthly payments are x, it is suggested that one instead pay x/2 every two weeks (for a total of 26 payments per year). This will pay down the mortgage faster, saving interest. The savings are surprisingly dramatic for this seemingly minor modification - often cutting the total interest payment by over one-third. Assume a loan amount of $100,000 for 30 years at 10% interest, compounded monthly.

(a) Under a monthly payment program, what are the monthly payments and the total interest paid over the course of the 30 years?
(b) Under the biweekly program, when will the loan be completely repaid and what are the savings in total interest paid over the monthly program (You may assume biweekly compounding for this part.)

We know that the present value (P) of an annuity that pays an amount A every individual period for 'm' periods for 'n' years at an yearly interest rate 'r' is given by


(a) In monthly payment program

P = 100000, r = 10%, n = 30, m = 12, r/m = 10%/12 = 0.1 / 12 = 0.0083333, therefore


Solving, we get A = $877.568

Total interest paid over the course of 30 years = A * (n * m) - P = 877.568 * 30 * 12 - 100000  =  $215924.48

(b) In biweekly payment program

A = $877.586 / 2 = $438.784
m = 26, r/m = 10% / 26 = 0.1 / 26 = 0.0038461, therefore


Solving, we get n = 20.95

In the biweekly program, the loan will be repaid in 20.95 years (compared to 30 years in the monthly payment program)

Total interest paid in biweekly payment program = A * (n * m) - P = 438.784 * 20.95 * 26 - 100000 = $139005.645

Therefore total interest savings in biweekly payment program compared to monthly payment program = 215924.48 - 139005.645 = $76,918.835 or 76918.835 / 215924.48 = 0.3562 = 35.62% (Wow! when we dig deep into financial mathematics, we can always find a treasure!)

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