I am trying to understand the fractional reserve banking system (with say 10 % reserve ratio). If a bank becomes insolvent because 100 % of the loans of a bank go bust, then who really becomes liable for all that money lent out in excess of the bank's value ? Let us say, the depositors are all FDIC-insured and the stockholders of the bank lose all their money, but isn't the loaned amount still larger than the (stock value + deposit amount) ? So perhaps all that happens is that this money simply gets written off (and adds to inflation) ?
Let's assume (as you do if I understand well) that the bank's only liabilities are deposits, so that the bank's equity valuation is simply the value of assets minus deposits. In order to provide loans the bank will have to hold 10% of those funds as reserves in its central bank account (the account it holds at the central bank, written in central bank money or base money), the the maximum amount the bank could lend out would be 90% of its liabilities + equity.
Then if 100% of the loans defaulted at 100% loss severity, 10% of assets would remain, to be shared between the depositors and the equityholders. By the traditional laws of bankruptcy and claims priority, the equityholders are the first to incur losses, so they would lose all their money (assuming the depositors represent at least 10% of the bank's balance sheet). Depositors would incur the remaining loss. At that point, if the FDIC had previously insured all depositors, it would have to repay them what they had lent to the bank, and the FDIC Deposit Insurance Fund would suffer from the loans' defaults. If you assume no external debt for the bank, then by accounting equality the total loan amount cannot be higher than the equity + deposits. (Assets = Liabilities + Owners' Equity)
The above answer is correct, but I'll put my own spin on this.
When a bank goes bankrupt, it actually contributes to deflation actually and not inflation.
In reality a genuine bankrupcy rarely happens... The FDIC will take over the bank before it gets that far and merge their remaining assets in with a larger bank.