Perpetuity Calculator (Level, Growing and Deferred)

A perpetuity is a cash flow that continues forever. Despite the infinite stream, its present value is finite and remarkably simple:

The value is always the payment divided by the discount rate, so at 5% a perpetuity is worth exactly twenty times its annual payment, and at 4% exactly twenty-five times.

Two consequences follow, and both matter more than the formula.

Perpetuities are brutally rate-sensitive. Halving the discount rate exactly doubles the value. This is why long-duration assets valued as perpetuities -- infrastructure, ground rents, terminal values in a DCF -- swing so violently when rates move.

Almost all the value is near-term. At 5%, less than a quarter of a perpetuity's value comes from beyond year thirty. The infinite tail contributes far less than intuition suggests.

Level perpetuity:

Growing perpetuity, where each payment is g larger than the last:

This requires i > g. If growth equals or exceeds the discount rate the series diverges and the value is genuinely infinite; the calculator reports that rather than returning a misleading number.

Deferred perpetuity, starting n years from now, is the ordinary value discounted back:

$10,000 a year at 5%:

• Level perpetuity: $10,000 ÷ 0.05 = $200,000, exactly 20× the payment
• Growing at 2%: $10,000 ÷ (0.05 − 0.02) = $333,333.33
• A thirty-year annuity of the same payment: $153,724.51
• So everything beyond year thirty is worth $46,275.49, just 23.1% of the total

Halving the discount rate to 2.5%: value doubles to $400,000. Exactly, not approximately.

Growth raised to 4.5%: with only half a point between growth and discount rate, the value explodes to $2,000,000 -- ten times the level perpetuity. This is the fragility of terminal value assumptions in a DCF.

Deferred twenty years: the same infinite stream is worth $75,377.90, because $200,000 discounted at 5% for twenty years is a little over a third of its undeferred value.

• Whether anything truly lasts forever. Perpetuity is a modelling convenience. Real ground rents get enfranchised, real companies fail, and real infrastructure concessions expire.
• Credit risk. The formula assumes payments arrive with certainty. A risky perpetuity needs a higher discount rate, and the value falls proportionately.
• Inflation. Use either a nominal payment with a nominal rate, or a real payment with a real rate. Mixing them is a common and serious error.
• Tax on the income stream.
• Variable growth. Real cash flows rarely grow at a single constant rate forever. Multi-stage models exist for that.
• Payment frequency. This assumes annual payments in arrears. Quarterly or monthly streams need the rate converting to match.
• Callability. Many real perpetual instruments, such as perpetual bonds, can be redeemed by the issuer, which caps the upside.

• Setting growth close to the discount rate. As g approaches i the denominator approaches zero and the value approaches infinity. A terminal growth rate of 4.5% against a 5% discount rate produces a $2 million valuation from a $10,000 payment, which is arithmetic rather than analysis.
• Assuming the far future matters most. It does not. At 5%, more than three quarters of the value arrives within thirty years.
• Mixing real and nominal. A payment stated in today's money discounted at a nominal rate will understate the value substantially.
• Using a perpetuity where a finite annuity is correct. A thirty-year lease is not a perpetuity, and treating it as one overstates its value by about 30% at a 5% discount rate.
• Forgetting the first payment timing. The standard formula assumes the first payment arrives one period from now. A payment today makes it a perpetuity due, worth one payment more.
• Ignoring the rate sensitivity when rates move. A move from 5% to 4% raises the value by 25%, not by 1%.

• Standard present value mathematics. The level perpetuity, growing perpetuity and deferred perpetuity formulas shown above are closed-form results with no jurisdictional content.
• The growing perpetuity formula is the Gordon growth model, which requires the discount rate to exceed the growth rate for convergence.