Question 1 (b)
Growing annuity: sum to fund a rising scholarship
7 marks
- Time to write
- About 14 minutes
From an older CBCS paper
Question
Question 1 (b) · 7 marksA person wishes to award a scholarship of Rs. 1200/- per year for the next 10 years. The first of the scholarships will be given after the end of one year, and after that the scholarship amount will grow at the rate of 5% a year to offset inflation. If the rate of interest is 10%, find the sum to be invested now for this scholarship.
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 scholarships are paid at the end of each year, the first at the end of year 1 (a growing annuity, C8).
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Reading of the growing annuity 2 marks 3 min
- 2 Formula and present value 3 marks 5 min
- 3 Check by discounting each payment 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): ten scholarships at the ends of years 1 to 10, the first ₹1,200, each later one 5% more than the one before; interest 10%. The sum to invest now is the present value of this growing annuity: each payment discounted at 10%.
Formula. PV = C ÷ (r − g) × [1 − ((1 + g) ÷ (1 + r))ⁿ] = 1,200 ÷ (0.10 − 0.05) × [1 − (1.05 ÷ 1.10)¹⁰] = 24,000 × [1 − 0.6280094] = 24,000 × 0.3719906 = ₹8,927.77.
Check by discounting each payment, as for any mixed stream of cash flows
| Year | Scholarship (₹) | PVF @ 10% | PV (₹) |
|---|---|---|---|
| 1 | 1,200.00 | 0.9091 | 1,090.91 |
| 2 | 1,260.00 | 0.8264 | 1,041.32 |
| 3 | 1,323.00 | 0.7513 | 993.99 |
| 4 | 1,389.15 | 0.6830 | 948.81 |
| 5 | 1,458.61 | 0.6209 | 905.68 |
| 6 | 1,531.54 | 0.5645 | 864.51 |
| 7 | 1,608.11 | 0.5132 | 825.22 |
| 8 | 1,688.52 | 0.4665 | 787.71 |
| 9 | 1,772.95 | 0.4241 | 751.90 |
| 10 | 1,861.59 | 0.3855 | 717.73 |
The PVs (computed on unrounded factors) add to ₹8,927.77. The sum to be invested now is ₹8,927.77.
Where the 7 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Reading of the growing annuity | 2 |
| 2 | Formula and present value | 3 |
| 3 | Check by discounting each payment | 2 |
| Total | 7 |
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