AALBORG, Denmark—Hans Peter Christensen got some unusual news when he opened his most recent mortgage statement. His quarterly interest payment was negative 249 Danish kroner.

Instead of paying interest on the loan he got a decade ago to buy a house in this northern Denmark city, his bank paid him the equivalent of $38 in interest for the quarter. As of Dec. 31, his mortgage rate, excluding fees, stood at negative 0.0562%.

It...