Chime
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Chime isn’t a tool that connects to your existing bank, it is the bank account. Open a Chime Checking account and a linked Savings account comes with it, and two features run quietly in the background once both are funded. Round Ups takes every debit card purchase (and qualifying credit card purchase) up to the next whole dollar and moves the difference into savings automatically. Save When I Get Paid does the same thing on a bigger scale, transferring a percentage you set, of each direct deposit, before the rest of your paycheck even hits checking.
There’s no monthly fee, no minimum balance, and no overdraft fee on the core accounts, which is the bigger draw for a lot of users over the saving mechanics themselves. The savings account pays an annual percentage yield well above the national average for a standard savings account, so the money set aside through round-ups actually earns something while it sits. It fits people who want a primary checking and savings setup that saves on autopilot, rather than a feature bolted onto a bank they already use.