Question 1 (a)

Deferred loan repaid in equal instalments

5 marks

Time to write: About 10 minutes

From an older CBCS paper

Question

Question 1 (a) · 5 marks

Mr. Z borrowed Rs. 2,40,000 as a loan, with interest payable at the rate of 10% per annum. He wishes, however, to begin repayment only at the end of third year from now. If the total loan and interest are to be repaid in 6 instalments, what should the annual payment be?

Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.

Before you write

Read off the question and its marks

10 minfor 5 marks

Read from the question

should theEnd with a one-line decision that follows from your figures.

What the paper leaves open

"Start repayment only at the end of third year" is read as: the first of the 6 instalments falls at the end of year 3, interest accruing on the loan for the first two years.

Write it in this order

Read the data; note exactly what is asked 1 min
1 Timing of the instalments and the two-year deferral 1 mark 1 min
2 Loan compounded to the end of year 2 1½ marks 2 min
3 Annuity factor and the annual instalment 2 marks 3 min
4 Check ½ mark 2 min
Check: arithmetic, units (₹, %), final answer underlined 1 min

1234

Try it first. About 10 minutes for 5 marks. Write it out, then open the solution.

Solution

Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.

Approach. Interest is compounded: interest earned becomes part of the principal at the end of each period. The first of the six instalments falls at the end of year 3 (paper note), so the loan grows with interest for two years; at the end of year 2 the six instalments (years 3 to 8) form an ordinary annuity. The present value of an annuity is the instalment multiplied by the sum of the present value factors, that is, the annuity factor.

Step 1: the loan with interest at the end of year 2. 2,40,000 × (1.10)² = 2,40,000 × 1.210 = ₹2,90,400.

Step 2: the annual instalment. The six instalments must have a value of ₹2,90,400 at the end of year 2. The annuity factor for 6 years at 10% is 4.355:

Instalment = 2,90,400 ÷ 4.355 = ₹66,681.97 a year, paid at the end of years 3 to 8.

Check. 66,681.97 × 4.355 = 2,90,400, and 2,90,400 ÷ 1.210 = 2,40,000, the amount borrowed. (With unrounded factors the instalment is ₹66,677.98.)

Where the 5 marks sit

#	Component	Marks
1	Timing of the instalments and the two-year deferral	1
2	Loan compounded to the end of year 2	1.5
3	Annuity factor and the annual instalment	2
4	Check	0.5
	Total	5

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