Thousands of Americans gave themselves a cheaper mortgage for Christmas. Applications for 12 percent fixed-rate mortgages guaranteed by the Federal Housing Administration or the Veterans Administration are running at record rates.

Nearly half of these applications are coming from homeowners refinancing their present mortgages. Some banks and S&Ls are also offering conventional mortgages in the 12 1/2 percent range, usually at variable interest rates--another option for people eager to refinance.

Anyone holding a mortgage at 16 or 17 percent should have a chat with his or her lender about lowering the interest rate. If you're paying off a second mortgage held by the person who sold you the house, this may be the moment to get out from under.

You might be able to combine your first and second mortgages into a single, cheaper loan. If nothing else, you should try to refinance that expensive second mortgage on better terms.

Anyone can qualify for an FHA mortgage, as long as he or she is credit worthy and the house is sound enough to pass minimal appraisal standards. But these loans are generally useful only for medium-priced homes. The most you can borrow on FHA in many cities is $67,500 (although larger amounts are available in especially expensive areas like Hawaii).

Larger mortgages may also be had through the Veterans Administration, for eligible service men and women or the surviving spouses of an eligible veteran who died as a result of military service. Both the FHA and VA mortgages are still being offered at fixed rates for 30 years, although you may also apply for flexible-payment plans.

Whether or not to refinance your present mortgage depends on arithmetic and on your personal expectations. Here's how to decide:

First, find out what interest rate you can get from your present lender or from competing lenders, and ask what your monthly payment would be. Compare this with your present payment, to see how much you would save every month.

Second, estimate how much longer you expect to live in your present house. If you expect to stay just two more years, multiply your potential monthly savings by 24 months. If you'll be staying 10 years, multiply the monthly savings by 120 months. And so on.

Third, find out what it will cost up front to take out a new mortgage at a lower interest rate. Among the likely charges: two to five "points" (one point equals 1 percent of the mortgage loan), prepayment penalties, title search, appraisals and legal fees. Your present lender will generally charge less than a competing lender.

The points paid at refinancing are generally deductible on your income-tax return. So to find your after-tax cost for this part of the transaction, reduce the value of the points by your tax bracket. For example, if you pay $1,000 in points in the 40 percent bracket, your write-off is $400, so your real cost is only $600.

Finally, add up all your costs and compare them with the savings you expect from refinancing. If savings exceed costs, refinance.

Some people will want to refinance even if savings do not exceed costs. For example, if you have a $10,000 second mortgage coming due in two years and don't know how you'll pay it, it makes good sense to refinance for a longer term even at a higher interest rate.

Savings and loan associations and banks write some FHA-VA mortgages. But a majority of these low-rate loans come through mortgage-lending companies. A real-estate broker should be able to direct you to such a company, or look for one in the Yellow Pages.

Some homeowners with low-interest, fixed-rate mortgages might be forced to refinance their homes, in order to pay for college or invest in a business.

If you're in this situation, explore a special program launched by the Federal National Mortgage Association (Fannie Mae), which might provide you with a loan at a below-market rate.

Mortgage lenders have sold 2 million conventional, fixed-rate FHA and VA mortgages to Fannie Mae, in order to get cash to make new mortgages. If your loan was among those sold, Fannie Mae will swap it for a new and larger mortgage at a "blended" rate--something above your old rate but less than the rates now being charged on new mortgages, and with no prepayment penalty.

Your lender should be able to tell you if your loan has been sold to Fannie Mae, or ask a real-estate broker. The National Association of Realtors in Washington, D.C., is running a "Realtors' national mortgage-access program"--a hot-line service that Realtors can call to find out whether your mortgage can be swapped at a blended rate.

Why would brokers help with your refinancing? Because they hope to get your business when you finally sell the house.