In May, 2003, I received this letter:

My reply at the time was that the lender had not made a mistake, that the APRs on many adjustable rate mortgages (ARMs) at that time were below their initial interest rates. The reason is that short term interest rates at that time were extremely low, which reduced the APRs on ARMs. The APR could fall below the interest rate, even when there were upfront loan fees that raised the APR. How that works will be explained further below.

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 lender fees and the interest rate, and is therefore a more comprehensive measure of cost to the borrower than the interest rate alone.

In calculating the APR, it is assumed that the lender fees are paid over the life of the mortgage, as an increment to the interest payment. 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. See Annual Percentage Rate Simplified.

On a fixed-rate mortgage, any upfront fees paid to the lender must result in an APR higher than the interest rate. Since the interest rate remains the same over the life of the loan, the addition of fees brings the APR above the rate. On an ARM, however, the tendency of fees to raise the APR above the rate can be more than offset by low rate indexes.

On an ARM, the quoted interest rate holds only for a specified period, which can range from a month to 10 years. In calculating an APR, therefore, some assumption must be made about what happens to the rate at the end of the initial rate period.

ARMs first burst on the scene in the early 80s, a period of very high interest rates. In calculating the APR on an ARM at that time, it was assumed that the initial rate lasted through the life of the loan. This led to absurdly low APRs on ARMs with low "teaser" rates that held for only a short period – in some cases, for only a month.

So the Federal Reserve, which administers Truth in Lending, changed the rule for calculating the APR on an ARM. It said that the APR calculation should use the initial rate only for as long as it lasted, and then should use the rate that would occur if the interest rate index used by the ARM stayed the same for the life of the loan. This is called a "no-change" or "stable- rate" scenario.

Under a stable-rate scenario, at the end of the initial rate period, the interest rate used in calculating the APR adjusts to equal the "fully-indexed rate", or FIR. The FIR is the value of the interest rate index at the time the ARM was written, plus a margin that is specified in the note.

Assuming zero loan fees on an ARM, the APR will be below the interest rate if the FIR is below the interest rate, and vice versa.

The indexes used by ARMs are short-term rates. A common one is the one-year Treasury rate, which I will use in my illustrations. In April, 1995, that rate was about 6.25%, in April 2003, it was down to about 1.25%, and in November, 2006 it had climbed back to about 5%.

An ARM that uses this index, say a 5 /1 on which the initial rate holds for 5 years, might have a margin of 2.75%. The initial rate would change over time but much less than the index it uses, as shown below.