Question

Problem 2 (Required, 25 marks) Tina borrows an amount $500000 from the bank and agrees to repay the loan by 4n level monthly payments (with amount X) made at the end of every month. The first repayment will be made 1 month after today. You are given that • The loan charges interest at an annual nominal interest rate 5.9% convertible continuously. • The outstanding balance at 25th repayment date is OLBs = 397021.93. (a) Calculate the interest due and principal of (n + 1) th repayment. ( Hint: Find X first) (b) Just after nth repayment, Tina decides to increase the monthly repayment amount to Y such that the remaining outstanding balance can be repaid completely by 2n repayments (instead of 3n repayments) By speeding up the repayment process, Tina can save some interest payment. Calculate the amount of interest saved by adopting this new repayment schedule. ( Hint: To do so, you need to calculate the total amounts of interest payments under the old repayment schedule and new repayment schedule respectively.)

Answer 1

The continuous interest rate can be converted into annual interest rate with the formula

e^0.059 = 1+r which gives the value of r = 6.077% p.a. compounded annually

a) Now, after 25th month the Outstanding balance can be measured as the difference of the loan amount and total repayment amount , each at the end of 25th period

Value of loan at 25th repayment date= 500000*(1+.06077)^(25/12) = 565389.27

Value of all level payments at 25th repayment date = X*(1.06077^(25/12)-1)/(1.06077^(1/12) -1) = 26.536 X (including 25th repayment)

Now, we have 565389.27-26.536X = 397021.93

Therefore, X = $6344.865

Now, total repayment period can be calculated as -

6344.865 *(1- (1/ 1.06077)^(4n/12)/ (1.06077^(1/12)-1)= 500000

( 1-(1/1.06077)^(4n/12) ) = 500000/1287419 =0.388374

1.06077^(4n/12) = 1.634986 which gives us 4n =100, n =25

Now, the principal of (n+1)th repayment i.e. 26th repayment is that remaining at the end of 25 years i.e. $397021.93

and the interest due is 397021.93 * ((1.06077)^(1/12)-1) = $1956.667

b) Now, after 25th payment, Tina wants to pay in 50 installments instead of 75 installments

So , the present value of those 50 installments should be equal to $397021.93 , i.e. Outstanding balance after 25th repayment

Y * (1-(1/1.06077)^(50/12)) / (1.06077^(1/12)-1) = 397021.93

Y *44.21985 = 397021.93 Therefore Y = $8978.363 per month

So the total payment in 50 installments would be = 8978.363*50 = $448918.17  

So, interest paid in this case = 448918.17- 397021.93 = $51896.24

If the earlier schedule was followed , total payment = X *75 = 6344.865* 75 =$475864.88

and interest paid = 475864.88-397021.93 =$78842.95

So, Interest saved = $78842.95-$51896.24 = $ 26946.7