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.

10 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.

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

b) 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.

c) 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)

d) Calculate the size of the smaller payment. (1 mark)

Answer 1

Loan principal after the 60 the monthly payment(ie. 5*12=60 mths.)= 100000

Monthly payment from end of 61st month can be calculated as follows:

Loan principal= 100000

Pmt.--Monthly pmt. Needs to be found out--

r= Rate of interest= 5%/12=0.4167% or 0.004167 p.m.

n= 15 yrs. *12=180 mths.

so, using the PV of ordinary annuity formula,

100000=Pmt.*(1-1.004167^-180)/0.004167

so, the monthly pmt. Will be

100000/((1-1.004167^-180)/0.004167)=

790.81

a) New equivalent effective monthly rate on the loan

As it is compounded semi-annually, we will find the effective semi-annual rate & then divide by 6 , to get the mthly. Rate--

(1+r)^2-1=5.5%

so, effective s/a r=2.71319% per s/a period

ie. 2.71319%/6=

0.004522

p.m.

0.4522%

b.Loan balance o/s after another 5 yrs. At original interest rates----ie. Another 60 mths. (10 yrs. From original time 0)

Original loan principal= 100000; Mthly. Pmt= 790.81 ,mthly r= 0.4522%, n=60 mths.

Loan balance=FV of the original loan amt.-FV of the 60 mthly. annuities

ie. (100000*(1+0.4167%)^60)-(790.81*((1+0.4167%)^60-1)/0.4167%)=

74557.98

c. So, from b. above, Loan balance pending (at end of 10 yrs. From t=0) is $ 74558 (rounded-off)

After the repayment of a lump sum of $10,000 immediately,the loan balance becomes

74558-10000=64558

so, equating the above PV of the loan , to the PV of the annuities of $ 1000

at the new monthly rate of 0.4522%---- for n no.of mths.

64558=1000*(1-(1+0.4522%)^-n)/0.4522%

solving for n,

no.of months = 76.5135 mths.

ie. No.of months of full repayment of $ 1000 =76

so, in 76 complete mths. He would have paid

1000*(1-(1+0.4522%)^-76)/0.4522%=

64195

& the smaller payment in the subsequent month will be

64558-64195=

363