Subject: interest rate

Hi, my name is anthony.

This question is base on my interest. I would like to know the formula for calculating this example: If you borrow $10,000 from a bank with an APR of 11.7% to be paid off in 5 years, what is your monthly payment? Could you please show me step by step how this formula is developed. Thanks!

Hi Anthony,

I think that what you are looking for is called amortizing the loan. I want to illustrate the method with a smaller example. Suppose that I borrow $1,000 for 6 months at an annual interest rate of 12%. I have calculated the monthly payments to be $172.55. (I know that the calculation of the monthly payment is your question and I will return to it later.) The payment schedule is in the table below.

MonthPaymentInterestApplied to PrincipleUnpaid balance

The reasoning behind the schedule above is the following. At the first of month 1 I have borrowed $1,000 at 12% per year. Hence in one month the interest owing will be one-twelth of 12% of the principle which is $1,000. Thus the interest owing is  0.12/12 x $1,000 = $10.00. Hence $10.00 of the $172.55 payment goes toward the interest and the difference, $162.55, is applied to the princple. Thus at the beginning of month 2 I owe $1,000 - $162.55 = $837.45.

Thus in the second month I have borrowed $837.45 and hence owe  0.12/12 x $837.45 = $8.37 in interest. Therefore $8.37 of my $172.55 payment goes to interest and the remainder, $172.55 - $8.37 = $164.18 goes toward he principle. Thus at the beginning of the third month I owe $837.45 - $164.18 = $673.27.

This procedure continues for 6 months when the unpaid balance is $0. (The difference between $170.82 and $170.85 is roundoff error.)

The procudure to calculate the monthly payments for your loan of $10,000 at 11.7% per year for 5 years is as follows. Assume that the payment is $M per month. Complete the 60 rows in the table above that correspond to 5 years. In each of the 60 rows the Unpaid balance column will be an algebraic expression involving $M. The expression in the Unpaid balance column and the sixtieth row must be zero so set the exprression equal to zero and solve for $M. The result is the following formula for $M.

If $P is borrowed at an annual interest rate of r% for t years then the monthly payment $M that satisfies the procedure above is

For your situation the monthly payment is

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