Financial Management and Business Data Analytics · Time Value of Money

Perpetuity and Growing Perpetuity: Formula and Examples

Updated 10 October 2026 · Fact-checked

A perpetuity is a cash flow that continues forever. Its present value is PV = C ÷ r, where C is the yearly cash flow and r is the discount rate. A growing perpetuity grows at a constant rate g, so PV = C1 ÷ (r − g), valid only when r is greater than g.

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 irredeemable preference shares with a fixed dividend, or a scholarship fund that pays the same amount every year.

It seems odd that infinite cash flows can have a finite value. It works because of discounting. Money received far in the future is worth very little today. Each year's cash flow is discounted more heavily, so the later flows add almost nothing and the total settles at a finite number.

The logic of the formula is simple. If you invest a sum at rate r and withdraw only the interest each year, the sum stays intact forever. So the sum needed to pay C every year is C ÷ r. That sum is the present value of the perpetuity.

A growing perpetuity has a first cash flow C1 at the end of year 1, and each later flow grows by a constant rate g every year. It is the base of the constant-growth share valuation model. Here PV = C1 ÷ (r − g). The formula only works if r is greater than g. If g is equal to or more than r, the value is not defined by this formula.

The formulas assume the first cash flow comes one period from now (an ordinary perpetuity). If the first flow comes today, treat it separately and add it to the value of the rest.

Key rules to remember

Present value of a perpetuity: PV = C ÷ r; C = fixed cash flow per period, r = discount rate per period. First cash flow comes at the end of period 1.
Present value of a growing perpetuity: PV = C1 ÷ (r − g); C1 = cash flow at end of year 1, g = constant growth rate. Valid only when r > g.
Cash flow in year n of a growing perpetuity: Cn = C1 × (1 + g)^(n − 1); Use it to find a later cash flow, or to get C1 from a given C0 using C1 = C0 × (1 + g).
Perpetuity due: PV = C + C ÷ r; Used when the first cash flow occurs immediately (at time 0).
Perpetuity starting later: PV today = [C ÷ r] ÷ (1 + r)^k; If the first flow comes at the end of year k + 1, the formula gives value at end of year k. Discount that value back k years.

How to solve Perpetuity and Growing Perpetuity questions

Use this method for any question on perpetuities, whether the cash flow is fixed or growing.

1Read the question and decide whether the cash flow is fixed (perpetuity) or grows at a constant rate (growing perpetuity).
2Note the discount rate r and growth rate g, and convert them to decimals.
3Find the exact timing of the first cash flow. Check whether it is at the end of year 1, today, or later.
4If growth is given and the question gives the latest cash flow C0, compute C1 = C0 × (1 + g). Do not put C0 in the formula.
5Check that r is greater than g for a growing perpetuity.
6Apply PV = C ÷ r or PV = C1 ÷ (r − g). This gives value one period before the first cash flow.
7If the first cash flow is not at the end of year 1, adjust: discount the result back to today, or add the immediate cash flow for a perpetuity due.
8Write the answer in rupees with a one-line conclusion, such as whether the price is worth paying.

Quickest way: One-period-before rule

When to use it: Use it in MCQs and short problems where timing of the first cash flow decides the answer.

Remember that the formula always values the stream one period before its first cash flow.
If the first flow is at year 1, the answer is today's value. Done.
If the first flow is at year 4, the formula gives the value at end of year 3. Discount it 3 years.
For growing perpetuity, plug in the first flow actually received, not the last one paid.
Sanity check: for positive growth (g > 0), the growing perpetuity value is higher than C1 ÷ r, because the denominator (r − g) is smaller than r.

Common mistakes in Perpetuity and Growing Perpetuity

Using the wrong cash flow in the growing perpetuity formula (using C0 instead of C1).
Questions often give the dividend just paid, and students plug it straight in.
Fix: Whenever the question says 'just paid' or 'current', multiply by (1 + g) first to get C1.
Using the formula when growth rate is equal to or higher than the discount rate.
Students apply the formula mechanically without checking r and g.
Fix: Always check r > g before applying PV = C1 ÷ (r − g). Otherwise the formula does not give a valid value.
Ignoring the timing of the first cash flow.
Students memorise PV = C ÷ r and forget it values the flow one period before the first payment.
Fix: Mark the first cash flow on a timeline. Discount or add the extra flow as needed.
Using the annual rate with monthly or half-yearly cash flows.
Rate and cash flow periods do not match.
Fix: Convert the rate to the same period as the cash flow, for example r = 12% a year is 1% a month for simple matching, then use C ÷ r with both on the same basis.
Writing the rate as 8 instead of 0.08.
Haste in calculation.
Fix: Convert percentages to decimals before dividing, or divide by 8% as 0.08 consistently.
Confusing annuity and perpetuity formulas.
Both involve a series of equal payments.
Fix: If the question gives a number of years, use the annuity formula. If it says 'forever', 'irredeemable' or 'indefinitely', use the perpetuity formula.

Worked examples

Example 1

An irredeemable preference share of Shree Textiles Ltd. pays a fixed dividend of ₹12,000 every year, first dividend due one year from now. If your required rate of return is 8%, what is the maximum price you should pay for it today?

Show the solution

The dividend is fixed and goes on forever, so it is a perpetuity.
C = ₹12,000 and r = 8% = 0.08.
First cash flow is at the end of year 1, so no timing adjustment is needed.
PV = C ÷ r = 12,000 ÷ 0.08.
PV = ₹1,50,000.

Answer: The maximum price is ₹1,50,000. Pay less than this to earn more than 8%.

Example 2

Mehta Industries has just paid a dividend of ₹10 per share. Dividends are expected to grow at 5% every year forever. If the required return is 15%, find the value of the share today.

Show the solution

Dividends grow at a constant rate forever, so it is a growing perpetuity.
The dividend just paid is C0 = ₹10, so C1 = 10 × 1.05 = ₹10.50.
Check r > g: 15% is greater than 5%, so the formula is valid.
PV = C1 ÷ (r − g) = 10.50 ÷ (0.15 − 0.05).
PV = 10.50 ÷ 0.10 = ₹105.

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

Exam tips

Look for words like 'irredeemable', 'forever', 'indefinitely' or 'in perpetuity'. They signal this topic.
In growing perpetuity problems, check whether the given dividend is D0 or D1. This decides the answer in most MCQs.
In written answers, state the formula, then the values, then the result. Step marks are given for each.
If the first payment is delayed, draw a quick timeline. It prevents the most common timing error.
In MCQs, the wrong options are often values from using C0 instead of C1 or from skipping the discounting. Compute the right value before looking at options.

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 difference between an annuity and a perpetuity?

An annuity pays equal amounts for a fixed number of periods. A perpetuity pays equal amounts forever. So the perpetuity formula C ÷ r needs no number of years, while the annuity formula does.

Why does a perpetuity have a finite present value?

Each later cash flow is discounted over more periods, so its present value gets smaller and smaller. The total of these shrinking values reaches a finite sum, which is C ÷ r.

When can I not use the growing perpetuity formula?

You cannot use PV = C1 ÷ (r − g) when g is equal to or greater than r. The formula needs r to be greater than g to give a valid value.

How do I value a perpetuity whose first payment is today?

Treat the first payment as a separate amount received now. Add it to C ÷ r, which values the remaining payments starting one year later. So PV = C + C ÷ r.