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 :

Q: 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 :

The correct option is **(C) $R/i$**. **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 = $R/i$.

Financial Mathematics

Present Value of a Perpetuity

Options

(A)

Ri

(B)

R + \frac{R}{i}

(C)

\frac{R}{i}

(D)

R - Ri

Official Answer

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

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 = $R/i$ .

perpetuitypresent valueR divided by iinfinite geometric seriesperiodic paymentinterest rate per periodfinancial mathematicsdiscounting

Marking Scheme

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

Hint

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

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.

Analysis & Explanation

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

Concept

$PV = \frac{R}{1+i} + \frac{R}{(1+i)^2} + \frac{R}{(1+i)^3} + \cdots$ (infinite geometric series).
First term $a = \frac{R}{1+i}$ , common ratio $r = \frac{1}{1+i}$ .
$\text{Sum} = \frac{a}{1 - r} = \frac{R/(1+i)}{1 - 1/(1+i)} = \frac{R}{i}$ .

Why the key is right

Why the distractors are wrong

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

Common Mistakes

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

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 $R/i$ rule.

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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 $\frac{R}{i}$ . A perpetuity due pays at the beginning of each period and has present value $R + \frac{R}{i}$ , 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 $\frac{R}{i}$ .