New Zealand introduced deposit insurance in 2008, during the recent financial crisis. In the middle of an election campaign, and terrified that the failure of some finance companies could lead to worse things, the government made what I thought at the time and still think were some pretty bad bailout decisions. The potted history is here; more detail here.
The financial crisis now being over, the deposit guarantees scheme has been wound down in favour of Open Bank Resolution. The Reserve Bank of New Zealand has a primer on it here; here is their FAQ; here is more detail. But here's my version: when a bank goes into statutory management for insolvency or other serious problem, OBR first freezes a portion of depositors' accounts while keeping the bank open. They then burn the shareholder equity and the unsecured bondholder funds. If that covers it, depositors are fine and everything is unfrozen. If the bank still doesn't have enough to cover the depositors' claims, the depositors take a haircut.
It's a pretty nice setup. There is no banking holiday; all the banks keep running while OBR is underway. The incentives are set up correctly so that those with most control over how banks run their book are the ones burned first if things go wrong.
Matt Nolan provides the best critique of the whole thing. In short, he doesn't think the government can credibly commit to letting depositors take the haircut. Taxpayers will be on the hook if something bad happens, there's nothing we can do about it, and we might as well force banks to pay for insurance now because they'll get bailed out later regardless.
I think he's got the best argument against OBR, but I put a bit better than even odds that he's wrong. Why? Bailout pressure is endogenous to the risk imposed on depositors. And a bank would have to be in pretty rough shape to end up imposing any substantial haircut on depositors after having burned the unsecured creditors and the equity. Now the unsecured creditors and equity holders could work pretty hard to convince the depositors that the haircut would end up being really big so that they could pressure the government for a bailout that would also protect the big guys. But the first-stage deposit freeze would be an upper bound on depositor losses and anchor expectations.
Further, consider the government's approach to bailouts consequent to the Canterbury earthquake. The government refrained from bailing out anybody who hadn't purchased insurance and even, arguably, imposed a massive taking on red zone owners of bare uninsured land - despite plenty of very sad stories in the press about the consequences.
The Reserve Bank of New Zealand cannot bind future governments. But setting up the regime well in advance of a bank failure specifying that, no matter what else happens, the equity and (subordinated) bond holders get burned first gives those agents reasonable expectation that they should try to make sure that does not happen. If some future government defects by bailing out depositors, I'd expect it to happen only after burning through the equity and bondholders.
It is worth re-reading Rod Carr's speech on the topic from 2001. Open-Banking Resolution seems consistent with the goals he outlined.