Present value of leftover payments
On a fixed-rate amortizing loan, each scheduled payment covers that month’s interest and reduces principal. After k payments, the remaining balance is the present value of the leftover payments at the same monthly rate. The mortgage page starts from a home price. Extra-payment payoff belongs on the loan-prepayment page. Loan-to-value then divides this balance by a property value.
Monthly payment = P × r ÷ [1 − (1 + r)^−n]
Remaining months = n − payments already made
Remaining balance = payment × [1 − (1 + r)^−remaining months] ÷ r
At 0%, remaining balance = original principal × remaining months ÷ n
Example
A $200,000 loan at 6% for 30 years has a monthly principal-and-interest payment of about $1,199.10. After 12 scheduled payments the remaining balance is about $197,543.98, so about $2,456.02 of principal and $11,933.19 of interest have been paid.
What this schedule leaves out
The original term must resolve to a whole number of months, and payments already made must be a whole number from 0 through that term. Extra principal, skipped payments, recasts, and biweekly posting change the true balance. Taxes, insurance, and HOA are not in this principal-only balance.