Question 1 OR (b)
Future value of a deferred annuity
7 marks
- Time to write
- About 14 minutes
From an older CBCS paper
Question
Question 1 OR (b) · 7 marksA 4-year annuity of Rs. 3600/- per year is deposited in a bank account paying 9% interest, compounded yearly, with the annuity payments beginning in year 12 from now. What is the annuity's future value? (7)
At 9%, the Compounded Value Annuity factors are:
| Year | Yr. 1 | Yr. 2 | Yr. 3 | Yr. 4 | Yr. 5 | Yr. 6 | Yr. 7 | Yr. 8 | Yr. 9 | Yr. 10 | Yr. 11 | Yr. 12 | Yr. 13 | Yr. 14 | Yr. 15 | Yr. 16 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9% (CVA factor) | 1.0000 | 2.0900 | 3.2781 | 4.5731 | 5.9847 | 7.5233 | 9.2004 | 11.0285 | 13.0210 | 15.1929 | 17.5603 | 20.1407 | 22.9534 | 26.0192 | 29.3609 | 33.0034 |
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
What the paper leaves open
The four deposits are made at the ends of years 12, 13, 14 and 15; the future value is taken at the date of the last deposit, the end of year 15, using the 4-year factor. The annuity factors are given to four decimals (the paper prints three).
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Reading of the deferred annuity 2 marks 3 min
- 2 Future value with the 4-year factor 3 marks 5 min
- 3 Check 2 marks 3 min
- Check: arithmetic, units (₹, %), final answer underlined 2 min
Try it first. About 14 minutes for 7 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.
Reading (paper note). The four deposits of ₹3,600 are made at the ends of years 12, 13, 14 and 15. The future value of an annuity is the payment multiplied by the compound value annuity factor for the number of payments; the delay before the first deposit does not change the factor, only the date at which the value is measured, the date of the last deposit.
Future value at the end of year 15 = 3,600 × CVAF (9%, 4 years) = 3,600 × 4.5731 = ₹16,463.16.
Check by compounding each deposit to the end of year 15 (compound value factors at 9%)
| Deposit at the end of year | Years compounded | Factor (1.09)ⁿ | Value at the end of year 15 (₹) |
|---|---|---|---|
| 12 | 3 | 1.2950 | 4,662.00 |
| 13 | 2 | 1.1881 | 4,277.16 |
| 14 | 1 | 1.0900 | 3,924.00 |
| 15 | 0 | 1.0000 | 3,600.00 |
The four values add to ₹16,463.16, agreeing with the annuity-factor result. The year-12 to year-16 factors printed in the table (20.1407 and beyond) are not needed: they would apply to deposits from year 1.
Where the 7 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Reading of the deferred annuity | 2 |
| 2 | Future value with the 4-year factor | 3 |
| 3 | Check | 2 |
| Total | 7 |
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