Corporate Accounting and Financial Management · Time Value of Money

Perpetuity and Growing Perpetuity: Present Value Formulas

Updated 11 October 2026 · Fact-checked

A perpetuity is a cash flow that continues forever. Its present value is the cash flow divided by the discount rate: PV = C ÷ r. A growing perpetuity grows at a constant rate g each year, so PV = C1 ÷ (r − g), valid only when r > g. C1 is the next cash flow.

Understand Perpetuity and Growing Perpetuity

An annuity pays a fixed amount for a fixed number of years. A perpetuity pays a fixed amount every period with no end date. Examples are an irrecoverable preference share dividend or a scholarship fund that pays out forever.

You cannot add up infinite cash flows one by one. But each later cash flow is worth less today, so the total present value is a finite number. That is why a simple formula exists.

The logic is simple. If you deposit a sum PV at rate r, it earns PV × r each year. If you withdraw only that interest, the deposit stays intact forever. So the annual cash flow C equals PV × r, which gives PV = C ÷ r.

A growing perpetuity has cash flows that rise by a constant rate g every year. The first cash flow C1 comes at the end of year 1. Then C2 = C1 × (1 + g), and so on. The present value is C1 ÷ (r − g). This works only if r is greater than g. If g is equal to or more than r, the formula gives no valid value.

This idea is the base of equity valuation under the constant-growth dividend model, where the price of a share is next year's dividend divided by (ke − g). You will meet it again in Gordon's Model and share valuation.

Key rules to remember

Present value of a perpetuity: PV = C ÷ r; C is the fixed cash flow per period, received at the end of each period starting one period from now. r is the discount rate per period.
Present value of a growing perpetuity: PV = C1 ÷ (r − g); C1 is the cash flow at the end of the first period. Valid only when r > g and growth is constant forever.
Next cash flow from current cash flow: C1 = C0 × (1 + g); Use this when the question gives the cash flow just paid (C0), not the next one.
Perpetuity due: PV = C + C ÷ r; Use when the first payment is made today. It is the first payment plus an ordinary perpetuity.
Implied rate or cash flow: r = C ÷ PV ; C = PV × r; Rearrange the formula when the price is known and the rate or cash flow is asked.

How to solve Perpetuity and Growing Perpetuity questions

Use this method for any perpetuity question. It keeps timing and growth correct, which are the two places marks are lost.

1Read the question and decide: is the cash flow fixed (perpetuity) or growing at a constant rate (growing perpetuity)?
2Note when the first cash flow arrives. End of year 1 is the standard case. If it is received today, it is a perpetuity due.
3Identify the cash flow. If growth is given and the question gives the latest dividend or cash flow already paid, compute C1 = C0 × (1 + g).
4Convert the rate to a decimal. Check that r and the cash flow period match, for example a monthly cash flow needs a monthly rate.
5For a growing perpetuity, check that r > g before you apply the formula.
6Apply PV = C ÷ r or PV = C1 ÷ (r − g) and show the substitution.
7If the first cash flow starts later than year 1, find the value one year before the first flow, then discount it back to today using the single-sum factor.
8State the answer in rupees with a one-line conclusion.

Quickest way: Divide, then check timing

When to use it: Use this when time is short and the question is a direct valuation with the first cash flow one period away.

Fixed cash flow: divide the cash flow by the rate. ₹5,000 at 10% gives ₹50,000.
Growing cash flow: subtract g from r, then divide C1 by the result.
If the question gives C0, multiply by (1 + g) first.
If the first cash flow is delayed, the formula gives the value one period before that first flow. Discount that value back for the remaining periods.

Common mistakes in Perpetuity and Growing Perpetuity

Using C0 instead of C1 in the growing perpetuity formula
Students see the latest dividend in the question and put it straight into the formula.
Fix: Read the wording. If the cash flow has already been paid, multiply by (1 + g) to get C1 before dividing by (r − g).
Applying the growing perpetuity formula when g is equal to or more than r
Students substitute numbers without checking the condition.
Fix: Always compare r and g first. The formula needs r > g. Otherwise state that the formula is not applicable.
Ignoring a delayed first cash flow
Students apply PV = C ÷ r even when payments start in, say, year 4.
Fix: The formula values the stream one period before the first payment. Then discount that value back to today for the remaining years.
Treating a perpetuity due like an ordinary perpetuity
Students forget that the first payment today is not discounted.
Fix: Add the first payment to C ÷ r. Or multiply the ordinary value by (1 + r).
Mixing percentage and decimal forms
Writing C ÷ 10 instead of C ÷ 0.10 changes the answer ten times over.
Fix: Convert the rate to a decimal before dividing, and check that the answer looks sensible.
Confusing annuity with perpetuity
Both involve equal payments, so the difference in the number of periods is missed.
Fix: Look for words like forever, irredeemable or indefinitely. If there is a fixed number of years, use the annuity formula instead.

Worked examples

Example 1

An irredeemable preference share pays a fixed annual dividend of ₹1,20,000 for ever, with the first dividend due one year from now. If the required rate of return is 8%, find the present value of the dividend stream.

Show the solution

The cash flow is fixed and never ends, so it is a perpetuity.
C = ₹1,20,000 and r = 8% = 0.08.
PV = C ÷ r = 1,20,000 ÷ 0.08.
PV = ₹15,00,000.

Answer: The present value is ₹15,00,000.

Example 2

A company has just paid a dividend of ₹10 per share. Dividends are expected to grow at 5% a year for ever. If the required rate of return is 15%, find the value of the share today.

Show the solution

Dividends grow at a constant rate for ever, so use the growing perpetuity formula.
The dividend just paid is C0 = ₹10, so C1 = 10 × 1.05 = ₹10.50.
r = 15% = 0.15 and g = 5% = 0.05. Check that r > g. It is.
PV = C1 ÷ (r − g) = 10.50 ÷ (0.15 − 0.05) = 10.50 ÷ 0.10.
PV = ₹105.

Answer: The value of the share today is ₹105.

Exam tips

Underline words like forever, irredeemable, indefinitely and constant growth. They signal a perpetuity.
Always write which cash flow you are using, C0 or C1. Examiners look for this step in growing perpetuity answers.
Write the formula first, then the substitution, then the answer. Step marks are given even if the final figure slips.
Questions often combine this topic with share valuation or cost of equity, so be ready to rearrange the formula to find r or g.
If the first payment is delayed, show the discounting step clearly as a separate line.

Practice questions from Time Value of Money

Perpetuity and Growing Perpetuity in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Perpetuity and Growing Perpetuity: frequently asked questions

What is the formula for the present value of a perpetuity?

PV = C ÷ r, where C is the fixed cash flow per period and r is the discount rate per period. The first cash flow is assumed to come at the end of the first period.

What is the difference between an annuity and a perpetuity?

An annuity pays a fixed amount for a limited number of periods. A perpetuity pays for ever. So an annuity needs the number of periods in its formula, while a perpetuity does not.

When can I not use the growing perpetuity formula?

You cannot use it when the growth rate is equal to or higher than the discount rate. It also assumes growth is constant for ever. If growth changes over time, value each stage separately.

Is the growing perpetuity formula used in share valuation?

Yes. The constant-growth dividend model values a share as next year's dividend divided by (required return − growth rate). It is the same growing perpetuity formula applied to dividends.