The bank depends on the laws of large numbers. They don't need to make money on every customer -- just on average. There are several ways that zero interest makes sense to them:
- customer acquisition is expensive. Their marketing department knows exactly how much it costs in advertising and marketing costs, on average, to pick up a new customer. They even budget for this, so they have a "kitty" of funds to spend. Spending some of that money subsidizing a new customer's interest is a perfectly reasonable method.
- The zero interest period lapses - at which point a higher interest rate takes over. This is all but certain. A person could carefully manage this relationship so they pay it off before then, but the bank is betting the customer will not, or will not be able to, and will acquiesce to the interest. Many people don't really read their statements and won't even notice the interest.
- This only applies to balance transfers and/or cash advances, and these have high one-time fees, which more than make up for the zero interest.
- Membership has its privileges. You are a preferred customer with a lot of money on deposit or in investment accounts, on which they make a lot of money -- and the zero interest is a perk (perquisite).
- They are charging you a punishing interest rate, but are agreeing not to make you pay it unless you default. Then all the back interest is due, plus interest on the interest.
You asked about banks, and I don't think you see this last scheme in use very much by a bank. Here's why. First, customers absolutely hate it - and when you drop the interest bomb, they will warn their friends away, blow you up on social media, call the TV news consumer protectors, and never, ever, ever do business with you again. Which defeats your efforts in customer acquisition. Second, it only works on that narrow range of people who default just a little bit, i.e. who have an auto-pay malfunction. If someone really defaults, not only will they not pay the punishment interest, they won't pay the principal either! This only makes sense for secured loans like furniture or cars, where you can repo that stuff - with unsecured loans, you don't really have any power to force them to pay, short of burning their credit. You can sue them, but you can't get blood from a stone.