The debit card, one of banking's indisputable marketing successes of recent years, is under attack.

Horror stories about fraud-particularly on the off-line cards that carry the MasterCard and Visa logos-are surfacing in the media, and some critics are taking the industry to task.

"Off-line debit is completely unsafe," said Edmund Mierzwinski, consumer program director of the U.S. Public Interest Research Group in Washington.

Such complaints may only compound headaches that the card associations are suffering on other fronts: a lawsuit by retailers complaining about debit fees, and decisions by car rental agencies not to accept debit cards at all.

The fraud problem arises from the fact that the cards are linked to checking accounts that can quickly be drained of funds by an unauthorized or unscrupulous person. Unlike the on-line debit cards associated with regional automated teller machine networks, the off-line MasterCard and Visa products do not require entry of personal identification numbers, and there can be a delay before it is apparent that the money is gone.

The off-line version is "a sloppy bank product" that exposes the consumer to fraud, bounced checks, disruptions of service, credit-report errors, and more, Mr. Mierzwinski said.

But consumers apparently like the debit idea-and the added convenience of the MasterCard or Visa logo printed on an ATM card, which makes a deposit account accessible from any credit-card-accepting merchant.

Visa said its U.S. member banks have issued a cumulative 46 million Visa check cards, more than double the 20.9 million of 1994. MasterMoney cards, at 15 million, have nearly quadrupled in that time.

In 1996, Visa check card purchases totaled $37.3 billion, up from $19 billion in 1994. Active cards were used eight times a month in 1996; the average purchase was for $39.

Brittain & Associates, an Atlanta-based research firm, said off-line debit transactions displaced a quarter of a billion checks at supermarkets alone in 1996.

Off-line cards are highly profitable. Merchant banks and card issuers collect fees mimicking those on credit cards, and the electronic transactions are highly cost-efficient.

But fraud is on the rise.

"Lost-and-stolen is the largest category of fraud" for First Chicago NBD Corp., said Thomas Tremain, vice president of electronic banking. The bank, which has issued 1.3 million off-line debit cards, has experienced "a worrisome increase in counterfeiting" in the past 18 months.

Off-line debit cards can be stolen and used with just a signature. Counterfeiting can be perpetrated the same way as with credit cards, but a deposit account balance-not a preauthorized credit line-is at stake.

Consumers have clear legal protections to fight credit card fraud. Debit fraud can require negotiating with banks to get the money back.

Once perceived as only for cash withdrawals, the trusty ATM card is increasingly being augmented with the MasterCard or Visa service, often through a replacement card arriving unsolicited by mail. The off-line concept is unfamiliar to many consumers, as are the newfound risks.

"Our biggest effort is on consumer education," said Irene Katen, a MasterCard vice president.

Brochures on the pros and cons of debit are available from both MasterCard and Visa. Their crucial message is to report a lost or stolen card as soon as possible.

Michael Beindorff, Visa U.SA's executive vice president of marketing, contended that fraud losses are "a trickle"-which they are, as a percentage of total card sales. He said consumer groups and the popular press concentrate on "what can go wrong, not what can go right."

"To the consumer who lost his card and had $500 emptied out of his bank account when it was time to make a mortgage payment, it's a serious problem," said Mr. Beindorff. "But in the scheme of things, the level of fraud is insignificant."

MasterCard said it experiences only 8 cents of fraud per $100 of debit transactions, with an average of $345 per affected account, compared to $993 on credit cards.

Visa combines credit and debit fraud numbers, which total about 10 cents per $100 and have been declining.

Debit cards are covered by Federal Reserve Regulation E. Cardholders are liable for a maximum $50 if they report a missing card or suspicious transaction within two days of discovery. Liability could rise to $500 if reporting is not timely and transactions occur that could have been blocked.

Mr. Mierzwinski wants the protection to be like Regulation Z, which limits the liability of credit card holders to $50 in all cases. He plans to take the issue up with Congress.

Banks are required to provide provisional credit within 20 days, while an investigation takes place. Many banks respond sooner. If the consumer's claim is valid, First Chicago covers bounced-check fees and all costs associated with the problem.

If fraud occurs, "we do everything we can to reimburse customers," said Mr. Tremain at First Chicago.

To fight the problem and prevent losses, banks are using techniques created for their credit card divisions. Many card issuers employ neural networks, a form of artificial intelligence, to flag suspicious transaction patterns. Cards sent in the mail often must be activated by calling an "800" number; other banks require consumers to complete a transaction using their PINs before an off-line transaction will be authorized.

Mr. Mierzwinski said fraud victims have been calling his office to complain about debit cards. A spokeswoman for the consumer advocacy group, coincidentally, had her own fraud experience.

Elizabeth Hitchcock, who told her story Tuesday on "Good Morning America," requested a MasterMoney card from a bank she did not identify. Soon after, she noticed a strange $75 charge on her statement. She notified the bank of this payment to a mail-order company. The bank canceled her card and replaced it with a new one, but she was not reimbursed.

Subsequently, a hotel stay in Boston was charged to her canceled number. Though she was not held responsible, she was shocked that the charge had been authorized even though the card number was invalid.

Ms. Hitchcock said her card never left her wallet but the number may have been pilfered when she ordered pizza over the phone. Numbers and expiration dates can also be retrieved from receipts. (On-line debit receipts typically contain only four digits from a customer's card.)

Ms. Hitchcock said she was "disturbed" by television commercials featuring Daffy Duck and Bob Dole that describe Visa check cards as more convenient than checks, without the hassle of providing the identification that would offer a measure of protection.

Marilyn Tredinnick, vice president of risk management at BankAmerica Corp., the country's fourth-largest Visa check card issuer, defended the commercial message. "It does say the money comes right out of your checking account," she said.

Though she stressed that education about risks is of the essence, "Do we want to alarm the public when we market a new product?" she asked. The disclosure materials that come with Bank of America cards are "really direct," she said.

Even so, said Ms. Hitchcock, if banks market the cards "as something to use without ID, my money is safer in my mattress."

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