Question

A young professional couple buy a house for $600,000. They make a down payment of $60,000 and agree to amortise the rest of the debt with quarterly payments made at the end of each quarter over the next 20 years. The interest on the debt is 12% per annum compounded quarterly.

5(a). (i) What type of annuity is this?

(ii) How many payments are made in total?

(iii) What is the interest rate per period?

(iv) Write down an explicit expression for the quarterly payments.

(v) Determine the amount of each payment (to the nearest dollar).

5(b). Determine the sum of all the payments including the down payment. (Give your answer to the nearest $100.)

5(c). Explain why the total amount of interest the couple will pay is $890,400 (to the nearest $100).

5(d). Give an expression for the unpaid balance of this loan after 12 years, and give the value of the unpaid balance (to the nearest $100)

Answer 1

(5)

(a) Purchase Price = $ 600000, Down Payment = $ 60000, Borrowing = 600000- 60000 = $ 540000, Repayment Frequency: Quarterly, Tenure = 20 years or (4 x 20) = 80 quarters, Interest Rate = 12% per annum or (12/4) = 3 % per quarter.

(i) As all payments are received at the end of the quarter, this is an ordinary annuity.

(ii) As payments are made 4 times per annum and there are 20 years in total, the number of payments is (4x20) = 80

(iii) Interest Rate per Period = applicable interest rate per quarter = 12/4 = 3%

(iv) Let the quarterly payments be $ N

Therefore, 540000 = N x (1/0.03) x [1-{1/(1.03)^(80)}]

(v) Value of N = 540000 /  (1/0.03) x [1-{1/(1.03)^(80)}] = 540000 / 30.20076 = $ 17880.34

(b) Sum of All Payments = Down payment + Number of Payments x Quarterly Payment = 60000 + 80 x 17880.34 = $ 1490427.415

(c) Total Interest Paid = 1490427.415 - 60000 - 540000 = $ 890427.415 ~ $ 890400

(d) Unpaid Loan Balance after 12 years = Sum of the Present Values of the remaining quarterly payments as at the end of Year 12 = 17880.34 x (1/0.03) x [1-{1/(1.03)^(32)}] = $ 364558.115 (as after 12 years the number of quarters remaining is 4x8 = 32)