Q: In shopping online, I have run into something puzzling. All of the mortgage shopping sites you recommend show different combinations of interest rate and points, and in every case, the Annual Percentage Rate (APR) is lower on loans with lower rates. The APR makes low-rate/high-point mortgages look like bargains. Are they?

A: No, there are no bargains in this market. The APR is consistently lower on low-rate loans than on high-rate loans because it isn’t calculated properly. That isn’t the lenders’ fault; they must calculate the APR using government rules. But the rules don’t correspond to lender practice in pricing loans, or to borrower needs.

Lenders price loans with different rates so that their net return on investment will be about the same. Suppose they offer a 6.375 percent loan at a price of zero, meaning there are no upfront loan charges. This is called the “par mortgage.” Then on a 5.875 percent loan they are going to require an upfront payment that, combined with the 5.875 percent rate, will yield 6.375 percent. Similarly, on a 7 percent loan, they will offer a rebate that, combined with the 7 percent rate, will yield 6.375 percent.

In pricing loans that have different rates, lenders must make assumptions about when the loan will be repaid. The shorter the life of a loan with a rate below or above the par rate, the smaller the upfront payment or rebate required to generate the same yield as the par mortgage.

For example, to yield 6.375 percent, a 5.875 percent 30-year loan requires an upfront payment of 5.2 percent of the loan if the loan runs to term. But if the loan is paid off in six years, the required upfront payment is only 2.4 percent. Similarly, to yield 6.375 percent, a 7 percent 30-year mortgage requires a rebate of 6.9 percent if the loan runs to term, but only 3.1 percent if it is paid off in six years.

From lender rate/price quotes, it is possible to derive the implied assumptions about loan longevity. I did this for a 30-year fixed-rate mortgage on April 28, 2006, using data on www.Amerisave.com. The par mortgage had a rate of 6.375 percent, and I assumed that other rates were priced to yield 6.375 percent.

The 5.875 percent mortgage carried an upfront payment of 2.3 percent, which (to yield 6.375 percent) implied a life of 70 months. The 5.25 percent mortgage carried an upfront payment of 5.6 percent, which implied a life of 76 months. The 6.875 percent mortgage carried an upfront rebate of 1.8 percent, which implied a life of 49 months. The 7.5 percent mortgage carried an upfront payment of 3.3 percent, which implied a life of 39 months (note: The assumed length of life declines as the rate goes up because higher-rate mortgages are more likely to be refinanced).

The APR is a composite measure of the cost of credit to the borrower that takes account of all upfront lender charges or rebates, in addition to the rate. On the par mortgage, the APR is equal to the rate. If it used the same assumption about mortgage life as lenders, the APR for mortgages having different rates would be close to the par rate. But that is not the rule.

The rule is that the APR is calculated on the assumption that all mortgages run to term. This makes the APR lower than the par rate on all mortgages with rates below the par rate, and it makes the APR higher than the par rate on all mortgages with rates above the par rate. This pattern is wholly artificial and should be disregarded by borrowers. It is unfortunate that online lenders have to waste scarce screen space on it.

For the last 20 years I have been asking the Federal Reserve to drop the assumption used in calculating the APR, that all loans run to term. More than 90 percent of them don’t. The APR would become a useful measure if it was calculated using length-of-life assumptions that vary with the rate, as lenders do. It would be even more useful if it were calculated over the period each individual borrower expects to have the mortgage. With today’s technology, that is not difficult.

Meanwhile, if you have the money to pay points (referred to as “buying down the interest rate”), it is a good investment if you expect to have the mortgage at least four years. If you are cash-short, the rebate paid by a lender on high-rate loans can be used to defray settlement costs. However, it becomes extremely expensive if you don’t pay off the loan within three years.

The writer is Professor of Finance Emeritus at the Wharton School of the University of Pennsylvania. Comments and questions can be left at http://www.mtgprofessor.com.

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