Rs. 2,50,000 cash is equivalent to a perpetuity of Rs. 7500 payable at the end of each quarter. Then the rate of interest is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 12% p.a
The present value of a perpetuity of R per period at rate i per period is \(PV = \frac{R}{i}\). Here a period is a quarter:
\( \displaystyle 250000 = \frac{7500}{i} \)
\( \displaystyle i = \frac{7500}{250000} = 0.03 = 3\% \text{ per quarter} \)
A year has 4 quarters, so the annual rate (compounded quarterly) is
\(4 \times 3\% = 12\%\) p.a.
Why the other options are incorrect
Option (1) 4% p.a — Incorrect. At 4% p.a. the quarterly rate is 1%, and the perpetuity would be worth \(\frac{7500}{0.01}\) = Rs. 7,50,000.
Option (2) 6% p.a — Incorrect. At 6% p.a. the quarterly rate is 1.5%, and the perpetuity would be worth \(\frac{7500}{0.015}\) = Rs. 5,00,000.
Option (3) 3% p.a — Incorrect. 3% is the rate per quarter, since the payments are quarterly; the rate per annum is 12%.
CBSE Applied Mathematics (Class XII):
Perpetuity
