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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. 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. Over the 3 years, the sum of the fees allocated to each year equals the total fees paid upfront plus assumed interest on those fees.

"In checking advertisements by mortgage lenders, several showed APRs on fixed-rate mortgages that were lower than the interest rate. Is that possible...?"

No, it's a mistake. On fixed-rate mortgages, the APR will never be lower than the interest rate. Because of the fees included in the APR, in most cases the APR is higher than the interest rate. This was the case in the example above.

If the lender pays all the fees included in the APR, then the borrower doesn't have to pay them and the APR will be the same as the interest rate. Lenders generally offer different combinations of interest rate and points, including negative points, referred to as "rebates". Rebates are used to pay fees.

Some lenders make a practice of offering rebates that will just cover all the fees included in the APR. One of the reasons lenders do this is that when the APR and the interest rate are the same, borrowers are much less likely to ask them to explain what the APR is.

One of the major problems with the APR is that some upfront fees are not included in it. Examples are title insurance, appraisal and credit report fees, and transaction taxes. If the lender's rebate is large enough to cover these fees, as well as all the fees included in the APR, the APR should be below the interest rate, but it isn't.

Under the rules, any fees not included in the APR are ignored in the calculation. This means that the APR on a "no-cost" mortgage where the lender pays all the settlement costs is misleadingly high.