EXAMPLE 3 Buying a House

Suppose that you buy a house with a $100,000 loan to be paid off over 30 years in equal monthly installments. Suppose that the interest rate for the loan is 6.00%. How much is your monthly payment?

Imagine changing the setup slightly so that instead of making monthly payments, you are supposed to pay off the entire principal and interest at the end. Meanwhile, you make payments to a savings fund that you’re building up to pay off the loan, and the savings fund earns the same rate of interest that the loan costs. The interest rate of 6.00% on the loan is compounded monthly, so the monthly rate is 0.5%. At the end of 30 years, the principal and interest on the loan would (by the compound interest formula) amount to

On the other hand, saving each month for 30 years at 6.00% interest compounded monthly, we know from the savings formula that you will accumulate

To make just the right amount to pay off the loan exactly, we need to solve the equation

for , getting as the monthly payment.

Now, $600,000+ is not what you will be paying over the course of the loan of $100,000! It is only an intermediate amount that we come across in calculating your payment. The total of your loan payments will be (“only”) — on a loan of just $ 100,000. (Usually, the bank will round up the regular monthly payment to the next nearest cent, with the consequence that the very last payment will be slightly less than the usual monthly payment.)

It is useful to note that for a 30-year mortgage at 6%, the monthly payment is almost exactly 0.6% of the amount of the loan—here, the payment is $599.55 and 0.6% of $100,000 is $600. Hence, for lower interest rates, the monthly payment will be a lower percentage of the amount of the loan, though the relationship is not proportional.