FOR a parent of a new, young driver, perhaps the only thing more agonizing than handing the car keys to a teen-ager is handing a sizable check to an auto insurance company.
In fact, adding a minor to an auto policy can double the cost, and some parents will do just about anything to curb those increases, from fibbing about the miles driven in a year to lying about their address to simply not reporting that they have a teen-age driver in the household.
''We know that unreported new drivers are a problem, but it is difficult to estimate how many,'' said Steve Witmer, a spokesman for State Farm Insurance.
It is also often unclear what will happen if you don't report your teen-ager to your insurer. The big fear for parents, of course, is that Junior will run the Cherokee through the front window of Tiffany's and the insurance company will ask, ''Junior who?'' That is a legitimate worry. Insurers insist that all drivers in a household be listed on a policy, and there are court rulings backing their decisions not to pay when an undeclared driver has an accident.
But insurance companies are also settling lawsuits from customers who contend that their policies as written cover teen-agers, so there is no need to report them to insurers.
So what's the truth? Your insurance company probably won't tell you this, but under most circumstances, Junior would be covered. The insurer may send you a nasty letter or even cancel your coverage, but it would still cover your child's accident.
''We would pay the claim,'' said John Walmsley, a spokesman for USAA, an insurance company in San Antonio. ''Our member, for sure, would get a letter saying don't let it happen again.'' Allstate also said it would pay up, as did Nationwide Insurance and State Farm.
Why do insurance companies pay if they don't have to? ''These companies spend a lot on advertising that portrays them as a member of the family,'' said Steven Goldstein of the Insurance Information Institute, a trade group in New York. ''The last thing they want to be seen as is a corporation trying to hurt parents who are struggling to raise teen-agers.''
Image aside, companies generally have a right to penalize fibbers. They can go back and assess the premiums customers should have paid. The insurers can also raise the premium on a new policy, or decline to renew. They can also accuse a policyholder of fraud and refuse to pay, which is more likely after bigger accidents.
''Typically, firms don't blanch when it comes to small claims, but when zeros are added to the dollar figure, they rethink their position,'' said Lorelie Masters, managing partner in the Washington office of Anderson, Kill & Olick, a law firm that often represents policyholders. Ms. Masters noted a 1979 case in which a 20-year-old who was not listed as a driver on his parents' policy drove his car into a light pole, two gates and a guardrail, causing $17,123 in damage. The insurance company denied benefits because the family had said their son would not drive the car. A Chicago court agreed that the insurer did not have to pay.
The issue of when a parent is legally required to tell an insurance company about a teen-age driver has long been murky. In December, the CNA Insurance Companies settled a class-action suit by agreeing to repay $3.1 million to about 15,000 policyholders whose rates increased when they added coverage for a teen-age driver in the middle of a policy period.
Lawyers for the policyholders said that the way the policy was written, all parents and children under 21 were already covered, whether or not the children were licensed. The lawyers argued that rates could not increase until the one-year or six-month policy was renewed.
''They were charged for insurance they already had,'' said Raymond Simon of Simon & Spitalli, the Chicago law firm that represented the policyholders, who added that other law firms had won similar settlements. So insurance companies have been rewriting contracts to close that loophole.
Like other companies faced with these suits, CNA says it already has that right. ''To my knowledge no company has ever admitted there was a problem with the language of the policy, and to my knowledge, none of these suits has ever gone to court,'' said Damian Sepanik, general counsel for CNA, which will pay an average of $147 to policyholders who seek reimbursement. The company said it settled the case because it was cheaper than going to court.
A study by the Insurance Research Council, a trade group in Wheaton, Ill., indicated that many Americans are willing to lie to lower their rates. About 20 percent of the 1,987 adults polled found it acceptable, to get a lower insurance rate, to list an older adult as the driver of a car actually driven by a minor. Another 23 percent said it was acceptable to lie about where the car was kept, and 32 percent said it was acceptable to underreport the miles driven annually.
''Most people negotiate in their minds the degree of acceptable cheating,'' Mr. Goldstein of the Insurance Information Institute said. ''There are always people who will get away with it, but why take the chance?''
William D. Smith of Kenmore, N.Y., near Buffalo, faced the reporting issue when it came time to insure his 17-year-old son.
''I heard from some acquaintances that you didn't have to tell the insurance company,'' Mr. Smith said. ''Some said, 'I never told them, and as far as I am concerned my kid is someone whom I lend my car to.' ''
Mr. Smith decided to insure his son, William, anyway, which added about $535 to his previous $1,545-a-year payment to the Auto Mutual Insurance Company of America.
''I decided to bite the bullet and report the new driver,'' he said. ''I just thought if he got in an accident there could be trouble.''
There are legitimate ways to lower the cost of insuring a teen-ager. Many insurance companies give discounts to good students and to graduates of certified driver-education schools. Delaying when a child starts to drive, say, to age 17 from 16, can also save a lot of money.
Statistics from the Insurance Institute of Highway Safety indicate that waiting a year also decreases the probability of an accident. The group's figures show that a 16-year-old is 42 percent more likely to have an accident than a 17-year-old.
Insurance companies say they have an easy solution to the problem of undeclared young drivers: getting the names and birth dates of all children in a family when first issuing insurance to the parents.
''This being the computer age, we store this information,'' said Bob Sohovich, a spokesman for Nationwide Insurance. ''And when the time comes, we ask if there is a new driver in the household.''
Photo: Although their annual insurance premium jumped $535, William D. and Patricia Smith of Kenmore, N.Y., did not hesitate to add coverage for their 17-year-old son, William. (Mike Groll for The New York Times) Graph: ''Dangerous Years'' shows the percentage of licensed drivers by age group who were involved in a traffic accident in 1990. (Source: Insurance Institute for Highway Safety)