A bank's basic function is to "borrow short and lend long". In other words, it borrows money from depositors over the short term, promising to repay it on demand, while it lends most of that money out over the long term to borrowers, for instance in the form of 30-year mortgages. This difference between these time frames, known as maturity mismatch, leads to systematic problems for banking. It makes banks vulnerable to crises, because if all the depositors show up one day asking for their money, the bank can't give it to them, because it's been lent out to borrowers, so the bank becomes instantly insolvent, even if it had no financial troubles before the depositors were worried about the bank's solvency.
Indeed, this used to happen frequently in the 1800's and early 1900's, most prominently in the Great Depression, until the FDIC came about. The FDIC makes all the banks pay a premium, and in exchange, whenever there's a run on a bank, the FDIC gives the bank money so that it can meet all its depositors' demands (at least up to a cap, like a hundred thousand dollars per account).
My question is, in the absence of the FDIC, why wouldn't banks just obtain private deposit insurance? Whenever people have significant risks, even if they're small, they tend to buy insurance. You don't have a very great risk of dying tomorrow, or having a car accident, or having a flood in your house, but still you buy insurance just in case. Companies of all kinds do the same: stores buy liability insurance, fire insurance, etc. So why wouldn't banks insure their risks similarly?
And it's not like banks don't buy private insurance already. For instance, when they lend out money, they buy insurance in case the borrower defaults on a loan - it's called a credit default swap. (Those were partially responsible for the financial crisis of 2008.) So what reason would they have for not buying insurance in case their depositors' demands exceed their reserves?
Is the problem that the premiums they would have to pay on the free market would be too high to make banking profitable anymore? If that's the case, then does that mean that the FDIC is not charging actuarially fair premiums to banks right now?
Any help would be greatly appreciated.
Thank You in Advance.