Why do companies buy insurance to protect against all kinds of other existential risks but apparently don’t have any kind of insurance to protect against destruction of their financial infrastructure? Why is banking an exception?

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How could you insure a risk so large?

It's generally a low risk of bank insolvency though, so in theory it shouldn't be too expensive no?

The problem is that it’s too systemic

Sure, if all banks were actually isolated and used sound money, you could insure against it. But in our economy, all the banks are intertwined and the currency itself is a risk. Once the dominos start falling the cost to insure the banks is way higher than letting them fail, hence why the only people who insure them are governments

It’s not exactly like car insurance where at most a couple cars need to be covered at a time and risk is relatively low and well distributed.

They do, but,

Interest is almost directly related to cost of risk.

If you are earning 3% interest on whatever you're doing, assuming market is being efficient, you have a 3% risk of loss.

Paying for insurance would cost you the equivalent of about 3% of what you have in premiums.

So if this was an efficient market and the risk was properly calculated, to be completely risk free on your 3% investment, you would have to pay 3% for insurance plus whatever extra administrative costs the insurance company needs.

In other words, insuring completely against all risk would bring your investment return to 0% or less. So the only way to earn anything is to take risks. Otherwise you might as well just keep the cash in the bank, since risk free insurance would just eat your investment returns.

Incidentally, the reason 2008 happened was because insurance company miscalculated the risk and charged something like 3% premium on a 6% return, effectively creating a risk-free 3% return, which is impossible. Then when claims started coming in, the insurance realized they weren't charging enough and didn't have enough in their insurance pool to pay out the claims, causing them to go bankrupt, and everyone who thought they had insurance to suddenly become completely exposed.