Mortgage refinancing is paying off your existing loan with the money from a new loan against the same property.
You’re right that the bank that holds your mortgage would not be very motivated to do it if it were the only bank around if refinancing ends up saving you money, but…
…other banks exist. Nothing is stopping you from going to another bank, and refinancing with them at whatever the current market interest rate is. The other bank would be happy to get some business even if it is less profit than your original loan; they weren’t going to get the money you’re paying your current bank anyway.
This motivates your original lender to also offer refinancing so that they can keep your business, even if it means they get a little less return.
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