In continental capitals and bank boardrooms there is a common fear. It is that the slow jog of deposits leaving banks in Greece and, more recently, Spain, may turn into a full-blown run that quickly spreads from bank to bank, and then from country to country. There have already been some warning signs, such as a sudden acceleration of deposit outflows from Greek banks in May.

A fierce debate is now taking place as to the best way to avert a run that, if it started, might be difficult to contain and could lead to massive capital flight from the euro zone's peripheral countries, which have 1.8 trillion Euro(2.2 trillion Dollar) in household deposits (see chart). Increasing numbers of people think the answer is greater financial integration. On May 30th the European Commission said there ought to be “full economic and monetary union, including a banking union; integrated financial supervision and a single deposit guarantee scheme”.

The first step is to shore up confidence in the region's banks by making sure they have enough capital to withstand a crisis. It is far cheaper to recapitalise banks, after all, than to stand behind all of their deposits. Yet such efforts have been bungled time and again. Europe has twice over the past two years tried to reassure depositors and investors that its banks are sound by subjecting them to “stress tests” that were supposed to mimic an economic downturn. In each case the tests were soon followed by revelations of deep capital holes in some banks (newly nationalised Bankia among them). Since some national regulators have lost the confidence of markets, they are having to bring in outsiders to assess how much capital their banks need.

Actually raising the capital is the next big problem for countries such as Spain or Italy, which are already struggling to convince markets that their public debt is sustainable. Ideally it should come from the European Stability Mechanism (ESM), Europe's new bail-out fund, as a direct capital injection into banks rather than as loans to governments, which then use the money to recapitalise their ailing lenders.

Injecting capital is politically difficult. Core countries such as Germany fret they will lose a lever of influence over government policies in peripheral countries by handing over equity. They also stand a greater chance of losing money if the ESM takes on the risk of bank investing, not least because they know even less about the balance-sheets of individual lenders than those of national governments. Peripheral countries are less than keen on handing ownership of important banks to bureaucrats in Brussels. And unless the capital is accompanied by supervisory reforms, local regulators may encourage banks to lend more freely at home since the risk of loss will have been exported.

Recapitalising banks would not put the catch on every trigger for a run, however. The worry among depositors is not just that their bank will go bust, it is also that their deposits in euros may overnight turn into a less valuable currency. So savers in the periphery would need some additional reassurance that their money is safe.

The job of providing this extra comfort usually falls to national deposit-insurance funds, ideally ones that are prefunded with assets worth about 1.5% of insured deposits, as is the case in America and Europe. If a medium-sized bank goes bust, prefunded schemes can usually pay depositors immediately; European countries insure qualifying deposits up to 100,000 Euro. For big banks, or for systemic crises, funds are usually backed by their governments.

Such schemes are fine in normal times, but do not do much to reassure people that their deposits won't be repaid in a different currency (or that their money will be safe if their own government cannot stand behind the scheme). That has prompted calls for a European insurance fund which would guarantee repayment in euros. The idea is appealing, but the politics and logistics of a credible guarantee are daunting.

Household and corporate deposits across the euro area total some 7.6 trillion Euro. Assuming you wanted to limit the insurance to household deposits only, the figure is still 5.9 trillion Euro. Not all of this would be covered by a guarantee. A European Commission study in 2010 reckoned that 72% of deposits (and 95% of deposit accounts) fall under the 100,000 Euro limit. What's more, a European fund would not have to be big enough to deal with simultaneous deposit runs across all of Europe but only with ones in the periphery, since money would presumably flow to banks in core countries such as Germany. So the fund would have to be big enough to cover only some 1.3 trillion Euro in insured deposits in the periphery.

The question then is how much of that amount a prefunded scheme would have to set aside. Economists at Citigroup reckon that a fund ought to start with a baseline of 2% of insured deposits, and then top up that amount with an additional premium to reflect the risk that peripheral countries may leave the euro and that their currencies would then depreciate. Assuming a less-than-10% chance of all the peripheral countries leaving, and that when they did their currencies would fall by 30%, a prefunded scheme would need 154 billion Euro-198 billion. Such a fund should ordinarily be financed by a small insurance premium on deposits, but since the money is needed to restore confidence now, it would have to be backed by the ESM.

Again, none of this is easy to achieve. Banks and voters in core countries (let alone Britain, whose position in a more integrated European banking system is very muddy) would be reluctant to insure peripheral deposits. Without beefed-up European supervision, it could lead to banks taking too many risks. Critics say that even a prefunded scheme would soon be depleted were a run to take hold. But a flawed scheme would be better than nothing.