## Thursday, June 18, 2009

### Overpayments on Loans

Someone came up to me today and asked about what are known as Bi-weekly loans. The statement he made at first was slightly counter intuitive, "You can make half your monthly payment and pay off your loan in six to eight years earlier." I was doubtful, and the math did not yield the result, if anything more compounding cycles means you would pay more over 30 years not less.

The "trick" or lie (if your the bank) is that with the Bi-weekly schedule you actually make what is equivalent to an extra month payment every year. My contention was that if you overpaid the mortgage by the same amount each month that would sum up to what you where effectively doing with a Bi-weekly schedule you would wind up in the same position or near the same position, so lets take a look at that.

Resolving our relationship X from the previous article for nt yields:

$nt=-\frac{\ln(1-(r/t)(1/(X+OP))(P-D))}{\ln(1+r/t)}$

So if you overpay OP by X/12 each schedule you will find that you will pay the loan off by about the same amount of time. Example r = 4%, t = 12, X = 1432.25, P = 300,000. Note that a payment of 1432.25 is what it takes pay of the loan in 30 years. Applying this the equation above we find that overpaying by the specified amount would yield 310.78 terms or you would be able to pay it off in 25 years and 11 months, the last month you would make a reduced payment of \$1211.

So if you don't want to refinance your loan, just make the overpayment and make sure to write on the stub that you want the additional amount to be applied to the principal.