Question 4 (a)
Cash price of a land lease: a deferred annuity
6 marks
- Time to write
- About 12 minutes
Question
Question 4 (a) · 6 marksAttempt any three of the following:
- A piece of land leased with eucalyptus plantations will begin to yield timber and oil from the end of the sixth year, continuing for 25 years in all, and is expected to give an annual income of ₹1,50,000. Find the cash price of the land lease if money is worth 8% per annum.
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
- any threeThe paper asks for any 3, and this is one of them. Only 3 are marked: a fourth earns nothing.
- FindFormula first, then the working, then the answer with its unit (₹, %, units).
What the paper leaves open
Q4 is "Attempt any three" of four parts (a) to (d); each part is read as 6 marks, from the marker (6×3) printed after part (d).
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Identifying the deferred annuity 25 payments, deferred 5 years 2 marks 3 min
- 2 Present value of the annuity 2 marks 3 min
- 3 Discounting to the present; cash price 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 lease yields ₹1,50,000 a year at the end of years 6, 7, …, 30 (25 payments in all, the first at the end of the sixth year). Its cash price is the present value of this deferred annuity at 8% a year.
Method. The present value at time 0 of an ordinary annuity of A for n years at rate i is A·P(n, i), with P(n, i) = [1 − (1 + i)⁻ⁿ]/i. Here the annuity of 25 payments is valued first at the end of year 5 (one period before the first payment) and then discounted back 5 years, as a single sum P = A/(1 + i)ⁿ.
P(25, 0.08) = [1 − (1.08)⁻²⁵]/0.08 = 10.674776.
Value at the end of year 5 = 1,50,000 × 10.674776 = ₹16,01,216.43.
(1.08)⁵ = 1.469328.
Cash price (value now) = 16,01,216.43/1.469328 = ₹10,89,761.00 (about ₹10.90 lakh).
Equivalently, PV = 1,50,000 × [P(30, 0.08) − P(5, 0.08)] = 1,50,000 × (11.257783 − 3.992710) = 1,50,000 × 7.265073 = ₹10,89,761.00.
Answer: the cash price of the land lease is ₹10,89,761 (to the nearest rupee).
Where the 6 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Identifying the deferred annuity (25 payments, deferred 5 years) | 2 |
| 2 | Present value of the annuity | 2 |
| 3 | Discounting to the present; cash price | 2 |
| Total | 6 |
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