Question

Problem 4 (Required, 25 marks) Recently, a company has borrowed an amount of $300,000 from the bank. The company will pay the interest payment interest due) to the bank at the end of every month until the loan principal is repaid. The bank charges interest rate at an annual nominal interest rate 10% compounded continuously. At the same time, the company makes a monthly deposit $600 into a sinking fund at the end of every month. The sinking fund earns interest at an annual nominal interest rate 8% compounded continuously. The company will repay the loan principal once the balance of the sinking fund at the end of some month (including the deposit made at that time) exceeds $300,000 (loan principal). Calculate the total amount paid by the borrower in the entire loan.

Answer 1

Let us first convert the 10% continuous rate into annual rate

e^0.1 = 1+r which gives r = 0.10517 or 10.517% compounded annually

The monthly interest paid is 300000 (e^(0.1/12)-1) = 2510.44

Now, the sinking fund is an annuity of $600 per month earning an interest of 8% annually compounded continuously

8% compounded continuousy = 8.3287% compounded annually

So, the no. of months (n) required for this fund to equal or exceed $300,000 is

600*((1+0.083287)^n/12-1) / ((1.083287)^(1/12)-1) > 300000

(1+0.083287)^n/12-1) > 3.3444

  (1+0.083287)^n/12> 4.3444

n > 220 or n=221 i.e. after 221 months , the company can pay back principal

Total amount paid by the borrower is 221 months of interest every month + Principal of Rs.300000

i.e. 221*2510.44+300000 = 554807.24 + 300000 = $ 854,807.24