> banks face a classic problem: their loans have long terms, but their deposits can be withdrawn at any time.
Borrow short and lend long - great work if you can get it. Of course, this practice is fundamentally unsound (it provides nasty game-theoretic incentives to participants), but it tends to work "well enough" in practice that no one really cares, especially when there is a lender of last resort who is able to print money at will.
To be fair, the lender of last resort isn't necessarily printing/creating money in the long term.
If it is simply a classic bank run, and everyone wants their money back now, the lender of last resort could take over all of the distressed bank's illiquid assets and, over time, recover some or all of the lent moneys.
In this case, what the lender of last resort is really doing is making all illiquid assets liquid.
If you think a bank run is going to happen, you want to withdraw your money ASAP, before it's all gone. Everyone else does the same - so you can get a self-fulfilling prophecy, even if everything would have been fine if everyone had stayed calm. For personal banking, this is largely mitigated by FDIC insurance, because the Fed can print enough money to cover small bank failures without anyone noticing or caring. However, if you're fucking around with collateralized debt obligations (CDOs)...2008 happens.
Banks actually pay insurance premiums to FDIC, which are held in a fund which is used to pay claims. If the fund is wiped out, practically speaking the Fed would cover it but that's not the normal method of operation.
The "Calculated Risk" blog reports on bank failure, and even in 2010 (11?) when literally hundreds of banks failed there were only a couple cases in which the FDIC had to pay anything out.
which states that "The fund had a balance of negative $7.4 billion as of Dec. 31, though that was an improvement from the $20 billion hole it was in at the end of 2009."
Borrow short and lend long - great work if you can get it. Of course, this practice is fundamentally unsound (it provides nasty game-theoretic incentives to participants), but it tends to work "well enough" in practice that no one really cares, especially when there is a lender of last resort who is able to print money at will.