Question 4 (c)
Amount of a deferred annuity of ten payments
6 marks
- Time to write
- About 12 minutes
Question
Question 4 (c) · 6 marksMr. Devendra makes 10 annual payments of ₹1,20,000 each, the first being made at the end of 4 years. Find the amount of the annuity if money is worth 6% compounded annually.
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
Read from the question
- FindFormula first, then the working, then the answer with its unit (₹, %, units).
What the paper leaves open
The paper sets this part under the instruction to answer any three questions, and gives it 6 marks.
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Recognising the deferred annuity and its amount 2 marks 3 min
- 2 Future value factor 2 marks 3 min
- 3 Amount 2 marks 3 min
- Check: arithmetic, units (₹, %), final answer underlined 2 min
Try it first. About 12 minutes for 6 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.
The payments of ₹1,20,000 are made at the end of years 4, 5, …, 13: an annuity of 10 annual payments deferred 3 years. The amount of an annuity is its value at the time of the last payment (its future value). Nothing is paid or earned during the 3 years of deferment, so by our reasoning the amount is that of a 10-payment ordinary annuity. For an annuity regular, with the first payment at the end of the first period, the future value of periodic payments is:
A = R × [(1 + i)ⁿ − 1]/i, with R = 1,20,000, i = 0.06, n = 10.
(1.06)¹⁰ = 1.790848; [(1.06)¹⁰ − 1]/0.06 = 0.790848/0.06 = 13.180795.
A = 1,20,000 × 13.180795 = ₹15,81,695.39, the amount at the end of year 13.
| Payment at end of year | 4 | 5 | … | 13 |
|---|---|---|---|---|
| Years of interest to year 13 | 9 | 8 | … | 0 |
(Payment k earns interest for 13 − (k + 3) years: the sum 1,20,000[(1.06)⁹ + (1.06)⁸ + … + 1] gives the same amount.)
For reference: the present value of the deferred annuity at time 0 is 15,81,695.39/(1.06)¹³ = ₹7,41,560.52.
Where the 6 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Recognising the deferred annuity and its amount | 2 |
| 2 | Future value factor | 2 |
| 3 | Amount | 2 |
| Total | 6 |
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