Yes, a new mortgage lender pays off the balance of the loan with the previous mortgage lender. And you now owe the balance to the new lender.
As far as the old lender is concerned, having a mortgage paid off because of refinancing is no different from a mortgage being paid off due to a house being sold. Mortgages very commonly get paid off before the loan matures (typically 15 or 30 years) for a variety of factors like a house being sold, owner death, refinancing.
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