Question

When you purchased your​ house, you took out a​ 30-year annual-payment mortgage with an interest rate of 5 % per year. The annual payment on the mortgage is $ 17 comma 573. You have just made a payment and have now decided to pay the mortgage off by repaying the outstanding balance. a. What is the payoff amount if you have lived in the house for 18 years​ (so there are 12 years left on the​ mortgage)? b. What is the payoff amount if you have lived in the house for 21 years​ (so there are 9 years left on the​ mortgage)? c. What is the payoff amount if you have lived in the house for 18 years​ (so there are 12 years left on the​ mortgage) and you decide to pay off the mortgage immediately before the 18 th payment is​ due?

Answer 1

The mortgage amount outstanding (balance remaining to be paid) at any point in time is equal to the present value (at that time) of the remaining mortgage payments discounted at the designated mortgage interest rate.

Mortgage Rate = 5 %, Mortgage Tenure = 30 years, Annual Mortgage Payments = $ 17573

Mortgage Amount = 17573 x (1/0.05) x [1-{1/(1.05)^(30)}] = $ 270140.1

(a) Remaining Mortgage Tenure of 12 years implies that 12 annual payments of $ 17573 each are left and the total present value of these payments should equal the balance outstanding.

Outstanding Balance = Payoff = 17573 x (1/0.05) x [1-{1/(1.05)^(12)}] = $ 155753.9

(b) Remaining Mortgage Tenure = 9 years

Outstanding Balance = Pay Off = 17573 x (1/0.05) x [1-{1/(1.05)^(9)}] = $ 124905.8

(c) In this case the outstanding balance would also include the interest accumulated on the mortgage balance since the 17th payment, as the 18th payment is yet to be paid.

Outstanding Balance after 17th payment = 17573 x (1/0.05) x [1-{1/(1.05)^(13)}] = $ 165073.3

Interest Accumulated = 165073.3 x 0.05 = $ 8253.663

Total Outstanding Balance = 165073.3 + 8253.663 = $ 173326.9