Question

A loan of $100,000 is made today. This loan will be repaid by 10 level repayments, followed by a final smaller repayment, i.e., there are 11 repayments in total. The first of the level repayments will occur exactly 2 years from today, and each subsequent repayment (including the final smaller repayment) will occur exactly 1 year after the previous repayment. Explicitly, the final repayment will occur exactly 12 years from today. If the interest being charged on this loan is 8.5% per annum compounded half-yearly, and the final smaller repayment is $300, Calculate the loan outstanding exactly 11 years from today.

Answer 1

We have given the amount of final repayment that is $300. So if we find out what is the interest and principal amount included in that amount we can find out what is the loan outstanding at 11 years from today.

Suppose P is the loan outstanding at 11 years from today.

Then if we add interest to P we will get $300.

Now the formula for an amount which is half yearly compounded is this:

Here

A = Amount , in our case it is $300

i = Rate of interest, i.e. 8.5% or 0.085

When we solve, we will get P = 276.0382 (solution given below)

Then Interest = 300 - 276.0382 = 23.9618

But as per our requirement the amount we needed is $276.0382, that is the loan outstanding at exactly 11 years from today.

Now you might be thinking why we didn't use other information given in the question. We have to understand one thing that the interest is calculated on the loan outstanding(i.e the principal amount and doesn't include interest) whether compound interest or simple interest. So we only had to find the principal included in the last repayment, which is also the loan outstanding after payment of previous installment.