Question

Gerald has taken out a loan of $100,000 today to start a business. He has agreed to repay the loan on the following terms:

• Repayments will be made on a monthly basis. The first repayment will be made exactly one month from today.

• The repayments for the first 5 years will cover interest only to help reduce the financial burden for Gerald’s business at the start.

• After the 5-year interest-only period, Gerald will make level monthly payments that will fully repay the loan after an additional 15 years (i.e. 20 years from today, the loan will be fully repaid).

• The interest charged is 5% p.a. effective

0 years have passed, and Gerald’s business is doing well. Further, he has made all the repayments on his loan so far as described above, and has just made the repayment due today. However, it has just been announced that the interest rate on Gerald’s loan will go up to 5.5% p.a. compounding semi-annually.

d) Calculate the new equivalent effective monthly rate on the loan. (1 mark)

e) Calculate the current loan outstanding (again, it is 10 years after the loan was initially taken out). Note that the new interest rate only applies from today onwards.

space (2 marks)

f) Because Gerald’s business is doing well, he decides to repay a lump sum of $10,000 immediately. To further reduce the amount of interest he is paying to the bank, he will increase his monthly repayments to $1,000 per month.

How many full repayments of $1,000 does Gerald have to make in order to fully repay this loan? (Note: Gerald may need to make a further, smaller payment in the subsequent month) (2 marks)

g) Calculate the size of the smaller payment.

Answer 1

d) Earlier at 5% pa interest rate, monthly interest rate was 5%/12 = 0.4167%

Now with interest rate change, 5.5% is the new interest which is compounded semiannually.

Hence, Effective Annual Rate would be (1 + 5.5% / 2)² - 1 = 5.58%

Therefore, monthly effective rate of the loan = 5.58% / 12 = 0.465%

e) After first 5 years, monthly payments to be made will be the installment or EMI for the loan which includes both Principal and Interest

Using the PMT function in excel or annuity formula, we can get the monthly payment

Parameters for calculation are as follows:

Interest rate = 0.4167%

Present Value = 100,000

Number of periods = 15*12 = 180

The monthly payment = 790.79 $

First monthly payment includes interest payment of 100,000 * 0.4167% = 416.67

The principal payment included in the first payment = Monthly payment - Interest = 790.79 - 416.67 = 374.13 $

Outstanding Principal at end of Year 1 = 100,000 - 374.13 = 99,625.87

Similarly, for next year, outstanding principal will be used for calculting interest payment. The monthly payment will remain same throughout the 15 years.

Using the above method, following payment schedule has been calculated:

By the end of 10 years, Gerald would have made 60 payments and below is the table to show the loan outstanding at the end of 10 years.

After the 60th payment, the outstanding loan would be 74,557.09 $.

f) As Gerald makes payment of 10,000 and increases monthly payment to 1000, this payment and increased monthly payment will go into 61st payment.

With new interest rate applied, interest payment of 61st period would be 74,557.09 * 0.465% = 346.42

Hence, principal paid in 61st period is 1000 - 346.42 = 653.58

With payment of 10,000, outstanding principal at the end of 61st period=74557.09 - 653.58 - 10,000 = 63,903.51

Below table shows how payment of 10,000 and 1000 monthly payment changes the schedule:

Using the above explained scenario, we can calculate the updated schedule and find out number of full 1000 $ payments are required.

Below is the table showing the end period where a full 1000$ payment is required:

As the above table shows, till period 136, Gerald need to pay full 1000$ payments.

Hence, the number of full 1000 $ payments required are 76 (inclusive of both 136 and 61)

g) In the subsequent month, that is 137th period, 990.48 principal is remaining. And an interest of 0.465% on it would be 4.60 $. Hence, the smaller payment that is required to be made = 990.48 + 4.60 = 995.08 $