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A couple wish to buy a house for $520,000
A couple wish to buy a house for $520,000. They have saved enough for a 20% down payment, and will finance the remainder with a 30-year fixed rate mortgage. Their budget allows for a monthly payment of $2,100. (For this problem, ignore agents fees, closing costs, taxes, and insurance.) Show all work. (a) (5 pts) What is the largest annual effective interest rate for their loan that will allow them to afford this purchase? Give your answer as a percentage correct to 3 decimal places. (b) (5 pts) Suppose they take out a loan at 4.125% annual effective interest, and make monthly pay- ments of $2,100. What is the outstanding balance on the loan after 10 years of payments?
Expert Solution
Loan amount, PV = House cost x (1 - downpayment) = 520,000 x (1 - 20%) = 416,000
Nper = Number of mon ths in 30 years = 12 x 30 = 360
PMT = 2,100
FV = 0
Part (a)
Hence, monthly interest rate = Rate (Nper, PMT, PV, FV) = Rate (360, 2100, -416000, 0) = 0.3724%
Hence, the anual effective interest rate = (1 + r)m - 1 = (1 + 0.3724%)12 - 1 = 4.561%
Part (b)
Annual effective interest rate, R = 4.125%
Hence, monthly rate = r = (1 + R)1/m - 1 = (1 + 4.125%)1/12 - 1 = 0.3374%
After 10 years payment, Nper, n = number of months in 10 years = 12 x 10 = 120
Loan outstanding after 10 years = PV x (1 + r)n - FV of all the payments = 416,000 x (1 + 0.3374%)120 - FV (0.3374%, 120, 2100, 0) = 313,198.06
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