This perpetuity calculator finds the present value of a payment stream that never ends — the level perpetuity PV = C ÷ r and the growing perpetuity PV = C ÷ (r − g) — the same formula hiding inside stock valuation and endowment math.

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Why "Forever" Costs a Finite Amount

A perpetuity pays C every year, forever. Its value is not infinite, because money arriving later is worth less today: the year-t payment contributes C/(1+r)^t, and that infinite sum is a geometric series that converges to PV = C/r. At a 5% discount rate, $1,000 a year forever is worth exactly $20,000 — and the intuition is a bank account: $20,000 at 5% throws off $1,000 of interest annually, forever, principal untouched. Add growth — payments rising at g per year — and the Gordon formula gives PV = C/(r − g), where C is the FIRST payment (arriving one period from now): $1,000 growing at 3% discounted at 8% is also $20,000, because growth and a higher rate cancel in the denominator. This is not a curiosity formula: it is the terminal value inside nearly every DCF, the Gordon growth model behind dividend discount valuation, the endowment spending rule at every university, and the consol bonds Britain issued in 1751 and only redeemed in 2015 — the market priced "forever" for 264 years using exactly this arithmetic.

The denominator (r − g) is where all the drama lives. When growth approaches the discount rate, value does not rise politely — it detonates: C = $1,000 at r = 8% is worth $25,000 at g = 4%, $50,000 at g = 6%, $100,000 at g = 7%. This is why valuation fights are fights about g: two honest analysts differing by one point of perpetual growth can differ by half the company's value. The discipline is that perpetual g cannot exceed the long-run growth of the economy itself (a company growing 6% forever eventually becomes larger than the world), and cannot exceed r at all — the formula's infinity is mathematics refusing an impossible assumption. Used within its limits, though, the perpetuity is the cleanest lens in finance: value = cash ÷ (return − growth), and every fancier model is this sentence with better manners.

Worked Examples (Verified)

  • Level perpetuity: $1,000/yr at 5% → PV = 1000/0.05 = $20,000.
  • Growing perpetuity: $1,000 first payment, r 8%, g 3% → 1000/0.05 = $20,000.
  • Sensitivity: same $1,000 at r 8%: g 4% → $25,000; g 6% → $50,000; g 7% → $100,000.
  • Endowment rule: to spend $50,000/yr forever at a 4% real return, the fund needs 50000/0.04 = $1,250,000.

Perpetuity Discipline

  • C in the growing formula is the NEXT payment (one period ahead) — using last year's dividend un-grown understates value by the factor (1+g); the DDM convention D1 = D0(1+g) exists precisely for this.
  • Perpetual g above long-run GDP growth (~4–5% nominal) is a red flag in any valuation — the company would outgrow its own economy.
  • Match the rate to the payments: nominal payments need a nominal rate; inflation-indexed payments need a real rate. Mixing them is the silent killer of endowment math.
  • Payments starting TODAY (perpetuity due) are worth PV × (1+r) — one payment arrives undiscounted. Consols and most textbook perpetuities are ordinary (first payment in one period).
  • Finite-but-long streams are near-perpetuities: a 100-year annuity at 5% captures 99.2% of the perpetuity's value — "forever" arrives sooner than intuition says.

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Frequently Asked Questions

How do you calculate the present value of a perpetuity?

For level payments: PV = C ÷ r — the annual payment divided by the discount rate. For payments growing at g per year: PV = C ÷ (r − g), with C the first payment, arriving one period from now. $1,000 a year at 5% is worth $20,000 either as a level stream at 5%, or as a stream growing 3% discounted at 8%. This calculator handles both — set growth to zero for the level case — and refuses the impossible case g ≥ r, where the honest answer really is infinity.

Why isn't a perpetuity worth infinite money?

Because distant payments are worth almost nothing today. At a 5% discount rate, the payment in year 100 contributes $1,000/1.05¹⁰⁰ ≈ $7.60; year 200 contributes about 6 cents; the infinite tail sums to a finite $20,000 — a geometric series converging because each term shrinks by the factor 1/(1+r). The bank-account view says the same thing: a finite principal at rate r generates the payment as interest forever. Infinity of payments, finitude of value — no paradox once discounting does its work.

What is the difference between a perpetuity and an annuity?

Duration. An annuity pays for a FIXED number of periods (a 30-year mortgage, a 20-year pension); a perpetuity never stops. Their formulas are kin — the annuity is the perpetuity minus a perpetuity starting after year n, which is why annuity values approach the perpetuity's as n grows: at 5%, the 50-year annuity is worth 91% of forever, the 100-year 99.2%. In practice "perpetuity" usually models the tail of something long — a mature company's cash flows, an endowment, a ground lease — where the finite/infinite distinction has stopped mattering numerically.

What is a growing perpetuity used for?

Primarily the terminal value in discounted-cash-flow valuation and the Gordon growth dividend model: after an explicit forecast period, a business is assumed to grow at a modest constant rate forever, and its value at that horizon is FCF ÷ (r − g). It also prices endowment spending (how much principal funds a growing annual grant), perpetual royalties and licenses, and some preferred stocks whose dividends step up. In every use, the fight is over g — the formula's output is hypersensitive to it, as the worked examples show.

What discount rate should I use for a perpetuity?

The rate that matches the risk and the currency of the payments: your required rate of return for an investment stream, the WACC for company cash flows (that's the DCF terminal value), a real (inflation-adjusted) rate for inflation-indexed payments. What you may not do is pair a high growth assumption with a low rate because the answer pleases you — r − g is the entire valuation, and honest inputs are the whole game. Sensitivity-test the pair: report value at g ± 1% and see whether your conclusion survives.

What is a perpetuity due vs an ordinary perpetuity?

Timing of the first payment. The ordinary perpetuity (the standard formula, and this calculator) pays first in ONE period. A perpetuity due pays immediately — so it equals the ordinary value plus one full undiscounted payment, equivalently PV × (1 + r). The distinction matters for leases and insurance products structured with payment-up-front terms; for stock dividends and endowments the ordinary convention is standard, since the next dividend is a period away.

Did real perpetuity bonds actually exist?

Yes — British consols, issued from 1751, paid a fixed coupon forever with no maturity; the last were redeemed only in 2015. The Netherlands issued a perpetual bond in 1648 for a water authority that still pays interest today. Their market prices across centuries tracked C/r through wars, inflations and rate regimes — the longest-running empirical test of this formula in finance, and a reminder that "forever" instruments are priced by exactly the division on this page.

Is this perpetuity calculator free?

Yes — free, instant, no sign-up. Enter the payment, discount rate and growth rate for the present value of a level or growing perpetuity.