As for the reason: it currently costs the bank money to have your money (ECB interest rate on overnight loans is negative), in addition to the costs of operating the bank itself.
Last year, oil futures prices turned negative. Which means "wait, I can take delivery of oil AND get cash for it too." So why didn't people do this? Because there's a cost of cary to oil - I need to pick it up in a train car at a specific place in the US and I need to store it somewhere, which is expensive. So while they were "paying you to take the oil" you couldn't really just jump on it and do it.
This is an analogous situation to what's going on with a negative interest rate. You're right, you're literally better off taking the money out into a pile of cash. And if you only have a few thousand bucks you could just do that (but then you need to drive to the bank, figure out how to store it, drive back when the rate goes back up, etc.) But if you have a few hundred thousand bucks - what are you gonna do? Cash it out in hundreds and keep it in your garage? At some point the logistics and theft risk of that adds up too.
Meaning, similarly to the oil example, while the raw numbers indicate one strategy, the practicality of carrying the asset on your own tends to outweigh the benefit. So yeah, someone may chose to incur negative interest if their only other option is to pile up cash in the house where they have to worry about it being stolen, burning in a fire, etc.
Gresham's Law is that "bad money drives out good" - when you have two forms of currency circulating within the economy and one is inflating at a faster rate than the other, people will tend to spend the inflating currency and hoard the appreciating one. The inverse of that is that they will tend to hoard the appreciating currency and spend the inflating one, i.e. as a store of value, nobody with long time preferences is going to hold cash. Instead they pile into any asset with zero cost of carry and a liquid market, which was stocks from about 1980-2010 and now also includes art, cryptocurrency, real estate, NFTs, other securities, etc.
It occurs to me that this could play tricks on how we measure interest rates. Assuming no central bank interference, interest rates are set at the equilibrium between people who want to borrow cash and those who want to lend cash. But nobody serious about future returns wants to lend cash! They've exited the cash market entirely, and only touch it when they need to convert long-term holdings to short-term holdings. Instead, they trade amongst themselves in asset markets, which have appreciated far quicker than 3% CPIs and 2% nominal interest rates would suggest. It could be that the divorce between stock market returns (10-15% in recent years) vs. interest rates (2-3% in recent years) might indicate that all of the firms with high time preferences have exited the cash market, leaving behind only those firms who need cash now.
Nitpick: It's the equilibrium between people who want to have savings and those who want to have debt. When you have deflation no such mechanism exists which is why the central bank has to set the interest rate manually.
If a bank is forbidden from investing your money, and thus putting your savings at risk, then it costs them money to safely manage it.
Banks have found ways to reinvest your deposits. If the earnings there are higher than the storage costs they can share some of the earnings with you. But if they can’t, they have to charge for their service.
And they are also loaning out your money to others at a non trivial rate of interest.
A couple years ago I put a lot of my HSA into investments and the market tanked.
New and used car prices are like a jump rope this year, regardless...can't hide from the market in cash.
I can’t have a down payment plunge because Yellen made an announcement or (as I saw last year) a roller coaster of an election year.
If anyone charged me negative interests rates on the whole balance I would cancel my account.
This is very dependent on the country you are in. Some countries have very conservative risk tolerances for their banks which are imposed on them and watched closely.
With central banks the risks of loss is low enough that nobody needs the just keep my money safe service anymore - except now they do because rates are so low
Long-gone are the times when high street banks were based on loaning out the deposits from savings accounts. Building Societies do/did this, but of course the banks bought up most of those too in their spending sprees of M&A to become 'bigger and better'. Well, bigger anyway
Also, an interest rate of -0.5% is not that much worse than 0% -- they're both a few percentage points below inflation.
I would be willing to pay a lot of money for such a service. Of course, given that I already need it, I might as well choose a cheap option. If that cheap option happens to have a negative cost, I'm not going to complain.
Lafarge met with other banksters in a castle, and they had a big fight.