I'm not sure you understand how an equity cushion works.
Suppose you start a company with 1,000 kg of gold. No other assets, no liabilities. Currently, that gold is worth about 60 million USD.
Now you issue 1,000,000 'stable' tokens. Each can be redeemed for one USD. For simplicity, assume that you just give them away.
At current gold prices, your tokens are 60x over-capitalised. If the gold price collapses 90%, your tokens are still 6x over-capitalised. That's what I mean by an equity cushion.
A 'run' is everyone trying to redeem their tokens. If that happens, you just sell one million dollars worth of gold, and pay that obligation. You are not running out of anything here.
In historical practice in eg Scotland, the banks that issued private bank notes there kept an equity cushion of about 30% around. That means for every 70 pounds in liabilities, they had 100 pounds of assets on their balance sheet. (For comparison, in our example of 60x over-capitalisation, that's an equity cushion of 59/60 = ~98%. Or for the 6x over-capitalisation, it's 5/6 = ~83%.)
It's called an equity cushion, because on a balance sheet the excess of assets shows up at equity.
That equity on the books is very different from the banks shares. It's the difference between 'book value' and 'market value'. See https://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile... for a table with price-to-book-values (PBV). You can also look up that value for any specific firm online.
> I don't think it really matters what the underlying assets are. Once the bank runs, it'll probably run dry.
That might be true in the most trivial sense: if a bank has plenty of underlying assets, like our example that was 6-60x over-capitalised, there will never be a run. So the 'once the bank runs' condition would never happen.
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Technical detail: I completely avoided the question of reserves here. Mostly, because it's not actually very interesting. The bank will keep some reserves on hand, so that they can pay out routine redemptions immediately. They would be well advanced to follow the role model of the Scottish banks: their notes came with an 'option clause'.
The 'option clause' says that the bank can either redeem notes right away at par, or they can opt to 'pay' with a super-senior IOU that accumulates a punitive interest rate but that the bank can choose to pay back any time.
That option gives the bank time to liquidate assets, if they ever ran out of reserves. The punitive interest rate re-assures customers that the bank would not abuse that privilege it willy-nilly.
(Of course, that mechanism only helps with illiquidity. It does not protect against insolvency. But that's fine: it's better for the economy when insolvent companies cease operations.)