Free perpetuity calculator
Value a payment that lasts forever in two seconds. Enter a payment and a discount rate, then switch to a growing perpetuity to add a constant growth rate. The calculator returns the present value — payment ÷ rate for a level perpetuity, or payment ÷ (rate − growth) for a growing one — updated live, as you type.
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Estimates only, based on the values you enter. Not financial advice.
Results are estimates. Consult a professional.
What is a perpetuity?
A perpetuity is a stream of equal cash payments that continues forever — an infinite series with no maturity date. It sounds like it should be worth an infinite amount, but it is not. Because of the time value of money, each payment further into the future is worth less today, and those shrinking present values add up to a finite, calculable number. This perpetuity calculator returns that number the moment you enter a payment and a discount rate.
The classic real-world example is a consol — a British government bond, first issued in 1751, that paid interest forever with no repayment of principal. The same math prices a share of preferred stock with a fixed dividend, and it sits at the heart of the terminal value in a discounted-cash-flow valuation. Whenever a payment is expected to last indefinitely, a perpetuity is the tool that puts a price on it today.
The perpetuity formula
For a level (zero-growth) perpetuity, the formula is the simplest in all of finance — you divide the payment by the discount rate. This is the same formula used by Corporate Finance Institute, Omni, and Wall Street Prep:
The reason an infinite stream collapses to one clean term is convergence. The full present value is an infinite geometric series — payment/(1+r) + payment/(1+r)² + payment/(1+r)³ + … — and because every term is smaller than the last, the sum converges to exactly payment / r. The calculator above evaluates this directly.
Growing perpetuity formula (Gordon growth)
Many real cash flows do not stay flat — they grow. A company's dividends rise, rents are bumped each year, and corporate cash flows drift up with inflation. A growing perpetuity models payments that increase at a constant rate g forever. The formula divides the next payment by the discount rate minus the growth rate:
This is the Gordon Growth Model — the workhorse of dividend valuation and DCF terminal values. Notice that when the growth rate is zero, the formula collapses straight back to the level perpetuity PV = payment / r, so the level case is simply the growing case with g = 0. Toggle the calculator from Ordinary to Growing and add a growth rate to apply it.
Ordinary vs. growing perpetuity
The only structural difference between the two is whether the payment stays flat or rises each period. That single change makes a large difference to the present value, because even a small steady growth rate compounds across an infinite horizon.
| Ordinary (zero-growth) perpetuity | Growing perpetuity | |
|---|---|---|
| Payment over time | Stays level forever | Grows at a constant rate g |
| Formula | PV = payment / r | PV = payment / (r − g) |
| Typical examples | Consols, preferred stock | Dividend valuation, DCF terminal value |
| Convergence condition | r > 0 | r > g |
Same payment, same discount rate — adding growth raises the present value. Source: Corporate Finance Institute; Wall Street Prep.
How to calculate the present value of a perpetuity
Calculating a perpetuity by hand is a short, three-step process. The calculator collapses it into one, but seeing the steps makes the number trustworthy.
- Identify the payment. Take the cash flow for the next period — the annual dividend, coupon, or rent. For a growing perpetuity, use the Year-1 amount (this period's payment grown once if needed).
- Pick a discount rate. Use the required return for that risk level — a market yield, a cost of equity, or your own hurdle rate. Enter it as a percent.
- Divide. For a level perpetuity, PV = payment / r. For a growing one, subtract the growth rate first: PV = payment / (r − g). The calculator does this live as you type.
A worked example using the perpetuity calculator
A preferred stock pays a fixed $5 annual dividend and your required return is 8%. You want to know what one share is worth — and then what happens if, instead, that payment were expected to grow 3% a year.
Step 1 — Enter the level perpetuity
Set the payment to $5 and the discount rate to 8%, leaving the toggle on Ordinary. The calculator divides the payment by the rate: $5 ÷ 0.08 = $62.50 per share.
Step 2 — Switch to a growing perpetuity
Now flip the toggle to Growing and add a 3% growth rate. The calculator subtracts growth from the discount rate first: $5 ÷ (0.08 − 0.03) = $5 ÷ 0.05 = $100.00 per share.
| Input | Ordinary | Growing (g = 3%) |
|---|---|---|
| Payment | $5.00 | $5.00 |
| Discount rate (r) | 8% | 8% |
| Growth rate (g) | — | 3% |
| Effective spread (r − g) | 8% | 5% |
| Present value | $62.50 | $100.00 |
$5 ÷ 0.08 = $62.50; $5 ÷ (0.08 − 0.03) = $100.00. A 3% growth assumption raises the value by 60%.
Why the discount rate must exceed the growth rate
The growing-perpetuity formula only works when r is greater than g. This is not a technicality — it is what makes the answer exist at all. As long as the discount rate beats the growth rate, each future payment's present value shrinks faster than the payment grows, the infinite series converges, and the present value is finite.
- If r = g, the denominator (r − g) is zero and the present value is mathematically infinite — division by zero. The calculator returns no result rather than a fake number.
- If r < g, the payments grow faster than they are discounted, so the series diverges. The formula would spit out a negative value, which is meaningless — a growing perpetuity can never be worth less than nothing.
- If r > g, the series converges and you get a sensible, finite present value. This is the only valid case.
Real-world examples of perpetuities
True forever-payments are rare, but the perpetuity formula is used widely because many cash flows behave like one or are deliberately modeled as one:
- Consols and perpetual bonds. British consols paid a fixed coupon with no maturity. A consol paying £2.50 a year at a 4% yield was priced at £2.50 ÷ 0.04 = £62.50 — a textbook level perpetuity.
- Preferred stock. Preferred shares pay a fixed dividend with no maturity, so they are valued exactly like a level perpetuity: dividend ÷ required return.
- Dividend valuation (Gordon growth). A stock whose dividend grows at a steady rate forever is valued as a growing perpetuity — the foundation of the dividend discount model.
- DCF terminal value. In a discounted-cash-flow model, the value of all cash flows beyond the explicit forecast is captured as a growing perpetuity: terminal value = final-year cash flow × (1 + g) ÷ (WACC − g).
- Endowments and scholarships. A fund designed to pay out a fixed amount every year in perpetuity is sized using the same formula in reverse — the principal needed = annual payout ÷ return.
For a stream of payments with a fixed end date rather than an infinite one, use the present value of an annuity calculator. To price a single future amount, use present value; to value a stock from its yield, see the dividend yield calculator.
Perpetuity vs. annuity: what is the difference?
A perpetuity and an annuity are close cousins — both are streams of equal periodic payments. The difference is the end date. An annuity makes a fixed number of payments and then stops; a perpetuity makes payments that continue forever, with no end date.
| Annuity | Perpetuity | |
|---|---|---|
| Number of payments | Fixed (n periods) | Infinite — never ends |
| End date | Has a defined maturity | None |
| Present value formula | PMT × [1 − (1 + r)^−n] / r | payment / r |
| Common examples | Mortgages, car loans, pensions | Consols, preferred stock, terminal value |
An annuity is a perpetuity with a finish line; a perpetuity is an annuity with n set to infinity. Source: Wall Street Prep; MyAccountingCourse.
Mathematically, the perpetuity formula is the annuity formula with the number of periods pushed to infinity — the (1 + r)^−n term shrinks to zero, leaving just payment / r. That is why the perpetuity is the simpler of the two.
Formula sources and methodology
This calculator uses the standard perpetuity formulas: PV = payment / r for a level perpetuity and PV = payment / (r − g) for a growing (Gordon) perpetuity, with rates entered as percentages and converted to decimals before dividing. It returns no result when the series does not converge — when r ≤ 0 for a level perpetuity, or when r ≤ g for a growing one — rather than display an infinite or negative value. The figures match the conventions documented by Corporate Finance Institute and Wall Street Prep.
Corporate Finance Institute — Perpetuity: Definition, Formula, Examples.Wall Street Prep — Perpetuity & Growing Perpetuity: Formula + Calculator.Frequently asked questions about the free perpetuity calculator
About this perpetuity calculator
This perpetuity calculator runs entirely in your browser. Every figure you enter stays on your device — nothing is sent to a server, logged, or shared. It applies PV = payment / r for a level perpetuity and PV = payment / (r − g) for a growing one, and returns no value when the series fails to converge (r ≤ 0, or r ≤ g for a growing perpetuity) rather than showing an infinite or negative number — updating instantly as you type.
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