Annual Percentage Rate Simplified

 June 4, 2001, Revised January 3, 2008

The APR is calculated as if the fees paid up front are allocated to each month over the life of the mortgage, and that the sum of the fees and the interest payment each month, when divided by the balance in that month, equals the APR.

“I read your columns on APR but I’m not too swift about numbers. Can you explain it in a way I can understand?”

That’s a challenge because the APR is the solution of a complicated mathematical equation. But I’ll give it a try.

Mortgage shoppers confront the APR as soon as they search for rate quotes, because under Federal regulations an interest rate quote must also show an APR. The rationale of this rule is that the APR reflects both upfront fees and the interest rate, and is therefore a more comprehensive measure of cost to the borrower than the interest rate alone.

However, borrowers have difficulty with the concept. How can you combine into one number interest that is paid every month over the life of the mortgage, and fees that are paid upfront?

While the fees are in reality paid upfront, the APR calculation assumes that the fees are paid over the life of the mortgage in the same manner as the interest. In the calculation, the sum of the interest payment in every period and the fees allocated to that period, as a percent of the balance, equals the APR.

To illustrate this, I’m going to assume a very simple and unrealistic mortgage. It is for $100,000 at 8%, with only 3 annual payments. Each payment is $38,803.36. Fees included in the APR are $1,000. The APR is 8.559%. I solved for the APR in the conventional way, using a computer.

At the end of year 1, the interest payment is 8% of $100,000, or $8,000 (see table). In addition, $559 of the original $1,000 in fees is allocated to year one. The total of $8,559 is 8.559% of the balance one year earlier.

Similarly, in each of the next 2 years, the sum of the interest payment and the upfront fee allocated to that year equals 8.559% of the balance.

It is assumed that until the fees are allocated, they earn a return equal to the APR. The original $1,000 plus the$147 interest earned on it over the 3 years just equals the sum of the fees allocated to each year.