The additional interest that depositors will want from putting their money in a commercial bank rather than in something ultra-safe (like government debt or simply holding monetary base) will simply reflect the additional risk involved. Plus to insure against that risk, the insurance premium will have to reflect and be equal to the risk involved. Ergo the premium wipes out the extra interest.
However, that argument could be attacked on the grounds that insurance spreads risks, thus deposit insurance will in fact cut costs as compared to where depositors insure themselves.
My answer to the latter point is that it ignores the difference between on the one hand potentially catastrophic / ruinous risks and on the other hand, more manageable risks. To be exact, anyone undertaking a potentially ruinous risk will want a relatively large reward for doing so. Thus if that risk is spread in such a way that no individual faces a ruinous risk, the total cost of insurance will decline.
However, commercial banks are one huge “risk spreading” operation ANYWAY: i.e. a typical commercial bank might have a million or more depositors and an approximately equal number of borrowers. So risks are shared between those million or so depositors.
Ergo, commercial banks already and very largely dispose of the risk of ruinous loss (by that I mean depositors losing more than 75% or so of their money). And the latter idea is actually backed up by figures from the FDIC relating to the relationship between assets and liabilities of small FDIC insured banks which go bust. That is, it is unheard of for insolvent banks to have zero assets. Rather, assets normally amount to about 75% of liabilities, with an asset to liability ratio of less than 50% being a rarity.
As to large banks, they by definition involve even more risk sharing, so the chance of any depositor (absent deposit insurance) facing ruinous loss is even smaller.
As distinct from potentially ruinous risk, there are manageable or minor risks. In the case of the latter, working out the required insurance premium is a very mundane matter. To illustrate, if the chance of depositors losing 50% of their money in any one year is 1%, then a premium needs to be 1% of 50%, i.e. 0.05% of the capital sum insured. And that 0.05% won’t change if the risk is spread (e.g. if deposit insurance is implemented).
So I’m sticking to my story that deposit insurance achieves nothing.
That of course leaves the question as to whether depositors are not entitled to some form of near 100% safe form of saving or lodging their money. Well of course they are! And the way to do it has been set out by Laurence Kotlikoff, Matthew Kline and by this lot.