Question 4 (d)
Present value of a deferred monthly annuity at an annual compounding rate
6 marks
- Time to write
- About 12 minutes
Question
Question 4 (d) · 6 marksFind the present value of an annuity consisting of 35 monthly payments of ₹150 each, the first being made at the end of 3 years and money is worth 8% 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
Part of "Solve any three"; 6 marks.
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Equivalent monthly rate 2 marks 3 min
- 2 Value of the annuity at month 35 2 marks 3 min
- 3 Present value 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.
Rate per payment period. Payments are monthly but money is worth 8% compounded annually, so the equivalent monthly rate j satisfies (1 + j)¹² = 1.08 (the same effective annual rate):
j = (1.08)^(1/12) − 1 = 0.0064340 a month.
Timing. The first payment is at the end of 3 years = month 36, the last at month 70 (35 payments). This is an annuity of 35 monthly payments deferred 35 months (its value one period before the first payment is at month 35).
Present value of the 35 payments at month 35 (ordinary annuity, P(n, j) = [1 − (1 + j)⁻ⁿ]/j):
P(35, j) = [1 − (1.0064340)⁻³⁵]/0.0064340 = 31.249499.
Value at month 35 = 150 × 31.249499 = ₹4,687.42.
Discount to the present (35 months): (1.0064340)⁻³⁵ = (1.08)^(−35/12) = 0.798940.
Present value = 4,687.42 × 0.798940 = ₹3,744.97.
| Step | Value |
|---|---|
| Monthly rate j | 0.0064340 |
| Annuity factor P(35, j) | 31.249499 |
| Value at month 35 (₹) | 4,687.42 |
| Discount factor for 35 months | 0.798940 |
| Present value (₹) | 3,744.97 |
The present value of the annuity is ₹3,744.97.
Where the 6 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Equivalent monthly rate | 2 |
| 2 | Value of the annuity at month 35 | 2 |
| 3 | Present value | 2 |
| Total | 6 |
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