Question

QUESTION 2 continued. a) This question relates to loan repayments and loan terms. James and Mary Hall wish to borrow $750,000 to buy a home. The loan from the Federal Bank requires equal monthly repayments over 25 years, and carries an interest rate of 4.5% per annum, compounded monthly. The first repayment is due at the end of one month after the loan proceeds are received. You are required to calculate:

i) The effective annual interest rate on the above loan (show as a percentage, correct to 3 decimal places).

ii) The amount of the monthly repayment (consisting of interest and principal repayment components) if the same amount is to be paid every month over the 25 year period of the loan.

iii)The amount of $Y, if - instead of the above – the Federal Bank agrees that James and Mary will repay the loan by paying the bank $3,000 per month for the first 12 months, then $3,500 a month for the next 12 months, and after that $Y per month for the balance of the 25 year term.

iv) How long (in years and months) would it take to repay the loan if, alternatively, James and Mary decide to repay $4,400 per month, with the first repayment again being at the end of the first month after taking the loan, and continuing until the loan was repaid. [HINT: The final repayment is likely to be less than $4,400, and will be paid one month after the final full installment of $4,400 is paid.)

Answer 1

Solution i:

Rate of interest on loan = 4.50%

Monthly rate of interest = 4.50% / 12 = 0.375%

Effective annual interest rate = 1 *(1+r)^12 - 1 = 1*(1+0.00375)^12 - 1 = 4.594%

Solution ii:

Loan amount = $750,000

Period = 25 years = 300 monthly period

Monthly rate of interest = 0.375%

Monthly repayment amount = $750,000 / cumulative PV factor at 0.375% for 300 periods

= $750,000 / 179.9103 = $4,168.74

Solution iii:

Present value of repayment = Loan amount

$3,000* cumulative PV factor at 0.375% for 1 to 12 period + $3,500 * cumulative PV factor at 0.375% for 13th to 24th period + $Y * cumulative PV factor at 0.375% for 25th to 300th period = $750,000

$3,000*11.71255 + $3,500 * 11.19811 + $Y * 156.9997 = $750,000

Y = $4,303.63

Solution iv:

Loan amount = $750,000

Monthly repayment = $4,400

Let its takes n monthly period in making this payment

Now

$4,400 * Cumulative PV factor at 0.375% for n periods = $750,000

Cumulative PV factor at 0.375% for n periods = $750,000 / $4,400 = 170.454545

This PV factor belongs at n = 273 monthly periods

Therefore it will take 22 years and 9 months to repay the loan.