Question 141 markMCQSection A

The present value of a perpetuity of Rs R payable at the end of each period, when the money is worth i per period is :

Financial Mathematics
Present Value of a Perpetuity

Options

(A)RiRi
(B)R+RiR + \frac{R}{i}
(C)Ri\frac{R}{i}
(D)R−RiR - Ri

Official answer

The correct option is (C) R/iR/i.


Reason

  • A perpetuity is an infinite series of equal payments R at rate i per period.
  • Summing the geometric series of discounted payments gives present value = R/iR/i.
Answer keywords:perpetuitypresent valueR divided by iinfinite geometric seriesperiodic paymentinterest rate per periodfinancial mathematicsdiscounting

Marking scheme

  1. 1.1 mark: correct option (C) Ri\frac{R}{i}.
  2. 2.No marks for (A), (B) or (D).

Hint

Present value of an ordinary perpetuity is periodic payment divided by rate: Ri\frac{R}{i}.

Quick oral answer

The present value of an ordinary perpetuity is the periodic payment divided by the interest rate per period, that is R over i.

Common mistakes

  1. 1.Writing RiRi (multiplying), instead of dividing the payment by the rate.
  2. 2.Adding an extra R (giving R+RiR + \frac{R}{i}), which is the formula for a perpetuity due, not an ordinary perpetuity.
  3. 3.Forgetting that the rate i must be in decimal form matching the payment period.

Analysis and explanation

A perpetuity pays a fixed amount forever, and its present value is found by summing an infinite discounted series.


Concept

  • PV=R1+i+R(1+i)2+R(1+i)3+⋯PV = \frac{R}{1+i} + \frac{R}{(1+i)^2} + \frac{R}{(1+i)^3} + \cdots (infinite geometric series).
  • First term a=R1+ia = \frac{R}{1+i}, common ratio r=11+ir = \frac{1}{1+i}.
  • Sum=a1−r=R/(1+i)1−1/(1+i)=Ri\text{Sum} = \frac{a}{1 - r} = \frac{R/(1+i)}{1 - 1/(1+i)} = \frac{R}{i}.

Why the key is right

  • (C) Ri\frac{R}{i} is exactly this infinite-series sum.

Why the distractors are wrong

  • (A) RiRi multiplies instead of divides, giving a tiny wrong value.
  • (B) R+RiR + \frac{R}{i} adds an extra immediate payment, which applies only to a perpetuity due (payment at the beginning), not one payable at the end.
  • (D) R−RiR - Ri has no financial meaning here.

Interesting facts

The perpetuity formula underlies the valuation of irredeemable bonds and preference shares that pay a fixed dividend forever.

British government 'Consols', first issued in 1751, were real-world perpetuities paying interest indefinitely, and their price closely followed the R/iR/i rule.

Question details

Q14 of 44 questions

Board
CBSE
Class
Class 12
Subject
Applied Mathematics
Year
2026
Section
Section A
Marks
1
Type
MCQ
Chapter
Financial Mathematics
Topic
Present Value of a Perpetuity
Bloom's level
understand
Syllabus topic
Unit: Financial Mathematics — Perpetuity: Present Value
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

What is the difference between an ordinary perpetuity and a perpetuity due?

An ordinary perpetuity pays at the end of each period and has present value Ri\frac{R}{i}. A perpetuity due pays at the beginning of each period and has present value R+RiR + \frac{R}{i}, one extra immediate payment.

How can an infinite stream of payments have a finite present value?

Because each future payment is discounted, distant payments contribute vanishingly small present amounts. The infinite geometric series converges, summing to the finite value Ri\frac{R}{i}.