To open an account, you must accept a geas spell that you won't attempt to swindle the bank or rob it.

Leomund's tiny hut is actually super useful as a vault door.

Detect thoughts, possibly as a medallion of thoughts, could be pretty useful too. Maybe hire a mind flayer or a doppelganger for this role. Flumphs can't actually read minds, but they can eavesdrop on telepathy, so they could keep an inner eye on the mind flayers. Actually, even if you aren't hiring mind-reading monsters, it's probably a good idea to have some flumphs around just in case. You never know when you'll need one.

Wand of magic detection and wand of enemy detection are not too rare.

Glyph of warding can be loaded with all kinds of interesting non-lethal options, like "suggestion: Instead of risking further magical traps, why not go to the local constabulary and confess to breaking and entering?"

Many of your magical defenses could be hidden by Nystul's magic aura, which is easy to make permanent.

Sending, or sending stones, are great for transmitting simple messages long distances, for example, to update ledgers. But sending is a 3rd-level spell. A cheaper version is animal messenger.

Bank branches can also mail each other the old fashioned way, but the mail can be secured cheaply using illusory script. But that spell only lasts 10 days. A cheaper way to secure a message might be to record it on a small clay tablet, then smash that to pieces, and send the pieces with different couriers. Once all the couriers are at the destination, a few hours of mending cantrips should restore the message. That works for messages over 25 words, too. Or, you could grab a stone and ritually cast magic mouth on it, containing the message you want; the recipient would need to know how to trigger it. If the message is longer than 25 words, I don't see why you can't put multiple magic mouths on the same stone. Of course, each one costs 10 gp, so it's not the cheapest postal system.

Illusory script also has some interesting applications for creating cashier's checks, letters of credit, promissory notes, certificates of deposit, bearer bonds, etc., in ways that are difficult to forge. The hidden message could contain secret code words, numeric hashes, or your signature, and the "creatures you designate" could theoretically be open-ended, like "registered agents of the Royal Bank of Geldland." It's castable as a ritual, too.

Unfortunately, it only lasts 10 days. If we want to step slightly outside of RAW, I think it's reasonable to assume that a dedicated bank can research a version that lasts indefinitely, maybe for a modest material component that gets consumed; say, 1 platinum piece (which is on par with magic mouth). Or just make it a 2nd-level spell; but, that makes it harder to cast. An interesting variant would be a ritual that allows someone else to write the hidden message when you cast it: This would allow you to create traveler's cheques, with the signature hidden. The bearer could then counter-sign the cheque and present it to the bank agent, who would compare the two signatures. A forger wouldn't be able to copy the signature from the cheque because it's hidden.

3E actually had a great spell to prevent forgery and counterfeiting, arcane mark. The mark was unique to each caster. 5E prestidigitation has a similar function but it only lasts an hour. Also I think there's a sidebar in SCAG that recants the "uniqueness" aspect. Still, I'd imagine a dedicated bank would research some magical way to create a hard-to-forge seal for its documents, similar to how modern currency has holograms and hidden messages and so forth. Sympathetic magic might work: maybe each bank branch has a hair clipping from the manager of every other bank branch; important financial documents are signed in blood by a bank branch manager; and then there's a low-level spell to detect whether two biological samples came from the same creature, and is used to compare the blood to the hair. (A spell like this would have tremendous implications for forensics, too.) Of course such magic could be fooled; Nystul's magic aura can fool lots of things.

Finally, if you don't mind consorting with devils (and why would you? You're a bank! *ba-dum ching*), I'd expect imps to be relatively easy to summon and bind. With the right motivation, these creatures could become the bank's information network, delivering reliable messages for keeping ledgers up-to-date. Heck, the average imp has Int 11, Wis 12, and Cha 14 -- sounds like management material!

EDIT: I totally forgot about find familiar. Your telepathy with your familiar, and your ability to see and hear through it, extends to a range of 100 miles. That's reasonably far. As long as each bank branch was within 100 miles of another bank branch, the branch manager could leave their familiar at another branch and communicate telepathically through that, relaying messages back to the home office.

All this stuff about using magic to transmit messages sounds time-consuming and tedious, but in pseudo-medieval fantasy times, I'm guessing the bank caters mostly to a few rich customers, and so there would only be updates every few days or so.

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