Question 1 OR (b)
Deposit needed today for two future withdrawals
8 marks
- Time to write
- About 16 minutes
From an older CBCS paper
Question
Question 1 OR (b) · 8 marksSuppose a deposit is to be made at year zero in an account which earns 8% compounded annually. It is desired to withdraw ₹8000 two years from now and ₹6000 five years from now. How large must the year zero deposit be if it is to produce these future payments?
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
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Approach: deposit = present value of the future withdrawals 2 marks 4 min
- 2 Discount factors at 8% for years 2 and 5 2 marks 3 min
- 3 Present values and the deposit required 3 marks 5 min
- 4 Conclusion and check 1 mark 2 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Short on time? End with step 4's one-line conclusion, whatever happens: never leave the figures without it.
Try it first. About 16 minutes for 8 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. The deposit needed today is the present value of the two withdrawals at the 8% the account earns: a sum due in the future is converted into today's value by discounting it, P = A × PVIF, and for a series of uneven sums the present value is the total of the separate present values of each sum. The question prints no factors, so present-value table factors to 3 decimals are used: at 8%, 0.857 for year 2 and 0.681 for year 5.
Present value of the withdrawals at 8%
| Withdrawal | End of year | Amount (₹) | PVIF @ 8% | Present value (₹) |
|---|---|---|---|---|
| First | 2 | 8,000 | 0.857 | 6,856 |
| Second | 5 | 6,000 | 0.681 | 4,086 |
| Deposit required at year zero | 10,942 |
Conclusion. A deposit of ₹10,942 at year zero, earning 8% compounded annually, will fund a withdrawal of ₹8,000 at the end of year 2 and ₹6,000 at the end of year 5.
Check. With exact factors (1 ÷ 1.08 raised to the power of the year) the deposit is ₹10,942.21; the 21 paise difference is only the rounding of the table factors. Compounded at 8%, ₹10,942.21 grows to ₹12,763 by the end of year 2; after the ₹8,000 withdrawal, the ₹4,763 left grows to ₹6,000 by the end of year 5, which is exactly the second withdrawal, so the account is left at nil.
Where the 8 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Approach: deposit = present value of the future withdrawals | 2 |
| 2 | Discount factors at 8% for years 2 and 5 | 2 |
| 3 | Present values and the deposit required | 3 |
| 4 | Conclusion and check | 1 |
| Total | 8 |
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