The banks warn that students can be quickly dragged into financial difficulties if they have two overdraft facilities on which to draw. But the banks themselves - now offering larger free overdraft facilities than ever before - also have a lot to lose if more students start collecting current accounts.
Student packages are so competitive that they are only worthwhile to the banks if they retain the customers after graduation.
One of the commonest tales to be heard in student bars is the story of the freebie-loving fresher who opens several bank accounts to take advantage of their free offers.
So not only does he get the four free cinema tickets from the TSB but he also gets pounds 25 each from the Midland and NatWest, pounds 10 from the Royal Bank of Scotland, a clock radio from Northern Bank, Barclays' pounds 10 record voucher and Lloyds' railcard.
However, the really shrewd operators tend to keep their own counsel. What excites them is not so much the free offers but the possibility of borrowing perhaps pounds 2,000 or pounds 3,000 without being charged interest.
Most banks offer a free overdraft facility of pounds 400 and Halifax Building Society has a slightly lower limit of pounds 300. Even at today's low interest rates a student could earn pounds 600 during his undergraduate career by opening six bank and building society accounts, taking advantage of the free overdraft facility and putting the funds in an instant access building society account.
Although the banks are beginning to work hard at discouraging students from operating more than one account, a substantial minority still manage it. Even among the student customers of the traditional big four, about 6 per cent are thought to have opened accounts with at least two banks.
The banks are now working together to clamp down on the practice. On their bank account application forms students are asked if they have any other account. If they say they have they will usually be asked to explain the need for a second one.
If there is no good reason they may not be offered the benefits of a student package - the freebies, the free overdraft and, in many cases, the commission-free travellers' cheques.
The banks are less concerned about students who also operate building society accounts even though Halifax, for example, has set itself up as a rival to the banks in the student sector, offering a market-leading 4.25 per cent gross interest rate on its student current account.
Local bank managers are given considerable latitude, but head offices now tend to require students to pay either their grant or a cheque from their parents into the account before they will release the cinema tickets or draw down the overdraft facility.
Since many students now get money from their parents as well as from their local authority they have the financial means to open more than one account.
Many branches now require students to show them the letter from their local education authority proving that it is paying for their tuition fees. When the bank gets this letter it stamps it to prevent the student opening another account without the second bank being alerted.
Banks are often extremely anxious about people who operate two current accounts. Neither can be sure it has a clear picture of its customer's financial circumstances. Some customers will run up two overdrafts and some will go in for 'cross-firing' - alternately feeding one account from the other to create the impression of a series of temporary emergencies rather than a permanent crisis.
In some circumstances, however, banks are quite happy to open a second account. Alan Kitto, Barclays' student officer in Cardiff, said: 'There might be four or five students sharing a house who want to pay their bills through one account. We would be perfectly happy with that.'
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