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:

An endowment promises $10,000 a year forever, and the trustee's required return is 5%.

Step 1: The level perpetuity. $10,000 / 0.05 = $200,000.00

Step 2: As a multiple of the payment. $200,000.00 / $10,000 = 20.0x, which is simply 1/0.05

Step 3: The same stream growing 2% a year. $10,000 / (0.05 - 0.02) = $10,000 / 0.03 = $333,333.33

Two points of growth adds 67% to the value, because the denominator fell by 40%. Everything on this page is a division by a small number, and small numbers are where valuations go wrong.

Now the question that surprises people: how much of that $200,000 is the distant future?

Step 4: A thirty-year annuity of the same payment. $153,724.51

Step 5: The value of everything after year thirty. $200,000.00 - $153,724.51 = $46,275.49

Step 6: That tail as a share of the whole. $46,275.49 / $200,000.00 = 23.1%

Three quarters of an infinite stream's value arrives in the first thirty years. Payments from year 100 onward are worth almost nothing at 5%, which is why a perpetuity is a reasonable approximation for a long-lived asset rather than a mathematical curiosity.

The discount rate is the dominant variable:

Step 7: Halve the rate to 2.5%. $10,000 / 0.025 = $400,000.00, exactly double, at 40.0x the payment

Step 8: What that does to the tail. The thirty-year annuity rises only to $209,302.93, so the post-year-thirty tail is now $190,697.07, or 47.7% of the value

Halving the rate does not just double the answer; it moves half the value beyond the point most forecasts even attempt.

The growing case is more fragile still:

Step 9: Raise growth to 4.5% against the 5% discount rate. $10,000 / (0.05 - 0.045) = $10,000 / 0.005 = $2,000,000.00

Step 10: Compare that to the level case. $2,000,000.00 / $200,000.00 = 10 times the value, from a growth assumption half a percentage point below the discount rate

Step 11: The boundary. At g equal to or above i the denominator reaches zero or turns negative, and the calculator reports divergence rather than a number, because the value is genuinely infinite

Step 10 is the terminal-value problem in every DCF, stated plainly. Analysts argue over the growth rate to a tenth of a point precisely because the last tenth is worth more than everything before it.

Finally, deferral, which works in the opposite direction:

Step 12: The same perpetuity starting in twenty years. $200,000.00 / 1.05^20 = $75,377.90

Waiting two decades costs 62% of the value of an otherwise identical infinite stream.

• 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%.

• U.S. Securities and Exchange Commission, Investor.gov, the official authority for the investing this calculator relates to. investor.gov

Also consulted: 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.