Question

A loan of $100,000 is made today. The borrower will make equal repayments of $1357 per month with the first payment being exactly one month from today. The interest being charged on this loan is constant (but unknown).

For the following two scenarios, calculate the interest rate being charged on this loan, expressed as a nominal annual rate in percentage. Give your answer as a percentage to 2 decimal places.

(b) The term of the loan is unknown but it is known that the loan outstanding 2 years later equals to $76410.
(1.5 marks)

Today, Malorie takes out a 20-year loan of $200,000, with a fixed interest rate of 5.8% per annum compounding monthly for the first 3 years. Afterwards, the loan will revert to the market interest rate.

Malorie will make monthly repayments over the next 20 years, the first of which is exactly one month from today. The bank calculates her current monthly repayments assuming the fixed interest rate of 5.8% will stay the same over the coming 20 years.

(c) Calculate the total interest Malorie pays over this fixed interest period. (3 years)
(1.5 marks)

Today, Malorie takes out a 20-year loan of $200,000, with a fixed interest rate of 5.8% per annum compounding monthly for the first 3 years. Afterwards, the loan will revert to the market interest rate.

Malorie will make monthly repayments over the next 20 years, the first of which is exactly one month from today. The bank calculates her current monthly repayments assuming the fixed interest rate of 5.8% will stay the same over the coming 20 years.

(d) After the fixed interest period, the market interest rate becomes 6.8% per annum effective. Assuming the interest rate stays at this new level for the remainder of the term of the loan, calculate the new monthly installment.
(1.5 marks)

Answer 1

b). PV (current loan amount) = 100,000; N (number of payments made) = 24; FV (outstanding loan amount after 24 payments) = -76,410; PMT (monthly payment) = -1,357, solve for RATE.

Monthly rate = 0.4208% so annual nominal interest rate = 0.4208%*12 = 5.05%

c). PV (Loan amount) = 200,000; N (number of payments) = 20*12 = 240; rate (monthly rate) = APR/12 = 5.8%/12 = 0.4833%, solve for PMT.

Assuming fixed interest rate of 5.8% over term of loan, monthly payment = 1,409.88

Loan outstanding after 3 years: PV = 200,000; N = 3*12 = 36; rate = 0.4833%; PMT = -1,409.88, solve for FV.

Loan outstanding after 3 years = 182,617.69

Principal paid off in 3 years = total loan amount - loan outstanding = 200,000 - 182,617.69 = 17,382.31

Total amount paid in 3 years = PMT*number of payments = 1,409.88*36 = 50,755.75

Total interest paid in 3 years = Total amount paid in 3 years - principal paid off in 3 years = 50,755.75 - 17,382.31 = 33,373.44

d). PV (loan outstanding after 3 years) = 182,617.69; N (number of payments to be made) = 12*17 = 204; rate (monthly rate) = 6.8%/12 = 0.5667%, solve for PMT.

Monthly payment = 1,512.41