Perpetuity Calculator

A payment that never stops

Suppose something pays you 90,000 a year and never stops. Not for thirty years, not for a hundred. Forever.

The instinct is that an infinite stream must be worth an infinite amount. It is not, and the reason is the whole idea behind discounting. A payment far enough in the future is worth almost nothing today, and payments keep getting further into the future, so the sum settles on a finite number.

A remarkably simple one, in fact.

PV = Payment / Discount rate

Ninety thousand at 7 percent is 90,000 divided by 0.07, which is 1,285,714.29. That is what an endless stream of ninety thousand a year is worth today.

This calculator adds one more term for a stream that also grows:

PV = Payment / (Discount rate - Growth rate)

Which is the same formula with growth subtracted from the denominator, and it is the reason the tool asks for three numbers rather than two. Leave growth at zero and you get the flat version.

Why forever is worth so little more than fifty years

This is the part worth sitting with, because it changes how you read every long dated valuation.

A perpetuity factor is 1 divided by the rate. At 7 percent that is 14.2857, meaning an endless stream is worth 14.29 times one payment.

Now watch a finite annuity climb toward it. These are annuity factors at the same 7 percent, from our PVIFA calculator:

Fifty years of payments captures 96.6 percent of what an infinite stream is worth. Every year from the fifty first onward, stretching out to the end of time, is worth the remaining 3.4 percent between them.

Two useful consequences.

A perpetuity is a fine approximation for anything long. If a stream runs fifty years or more, treating it as endless overstates the value by a few percent and saves a great deal of arithmetic.

Very long term promises are worth less than they sound. An arrangement promising payments for a century is worth almost exactly the same as one promising them forever, and only about 3.5 times what a twenty five year version is worth. The far future barely registers.

The rate drives how quickly this bites. At 12 percent, the perpetuity factor is only 8.33, and the convergence is faster still. At 3 percent it is 33.3 and the distant years matter considerably more. Low rates are what make long horizons valuable, which is the same mechanism that makes long bonds so sensitive to rate changes.

Adding growth, and the trapdoor underneath it

Growth goes into the denominator, and that placement is what makes this formula both powerful and dangerous.

On the same ninety thousand at a 7 percent discount rate:

Read the two ends. Moving the growth assumption from 0 to 6 percent multiplies the answer by seven. And the last step alone, from 5 to 6 percent, doubles it.

That is not the model misbehaving. It is arithmetic: as growth approaches the discount rate the denominator approaches zero, and dividing by a number approaching zero sends the answer toward infinity.

Which is why the calculator refuses to run when growth is greater than or equal to the discount rate. There is no finite answer there. A stream growing at least as fast as it is being discounted never shrinks in present value terms, so the infinite sum genuinely is infinite.

Note that this is different from a growing annuity with a fixed number of payments, where growth above the discount rate is perfectly fine and produces a sensible number. Our growing annuity calculator handles that case. The distinction is entirely about whether the stream ends.

The practical warning: this is the most assumption sensitive formula in ordinary finance. A valuation that leans on a growing perpetuity is really a statement about the growth rate, and small disagreements about that one number produce enormous disagreements about value. If you are using it, run it across a range and report the range rather than a point.

A sanity rule from the same reasoning: no company can grow faster than the economy forever, so a long run growth rate above the economy's nominal growth is not a bold assumption, it is an impossible one.

Which dividend goes in the box

The field is labelled dividend, and there is a convention here that catches people out, so it is worth being explicit.

When this formula is used to value a share, which is what the labels suggest, the payment should be next period's dividend, not the one just paid.

If a company has just paid 90,000 and dividends grow at 3 percent, next period's is 92,700, and that is the figure that belongs in the box. Entering 90,000 instead undervalues the stream by exactly 3 percent, since every payment in the series is shifted down by one period of growth.

The rule in general form: the payment entered is the one arriving at the end of the first period. If you have the current figure, multiply it by one plus the growth rate first.

Two more things the formula quietly assumes, and both are worth knowing before trusting a number.

The discount rate must exceed the growth rate permanently, not just this year. And the discount rate here is the return required by whoever receives the payments, which for a share is the cost of equity. Our WACC calculator helps establish it where the funding is a mix of debt and equity.

Where an infinite stream is a reasonable assumption

Nothing actually pays forever, so the question is always whether the assumption is close enough to be useful. Sometimes it clearly is.

Terminal value in a cash flow model. The commonest use by far. Project cash flows explicitly for five or ten years, then use a growing perpetuity to capture everything beyond the forecast horizon. That terminal value is often the majority of the total, which is why the growth assumption deserves the scrutiny it rarely gets.

Valuing a mature dividend payer. A stable company paying a steadily rising dividend is close enough to a growing perpetuity for the model to be informative.

Freehold property and ground rents. Income with no natural end date, which is why the capitalisation rate used in property valuation is a perpetuity formula wearing different clothes.

Preferred shares and undated bonds. Fixed payments with no maturity, which is the simple perpetuity in its purest form.

Endowments and trusts. Working out what a fund can pay out indefinitely without depleting is the same calculation run backwards.

Where it does not fit: anything with a genuine end date, anything whose payments are irregular, and any business whose growth is currently far above what it can sustain. For those, a finite model is honest and a perpetuity is wishful.

Questions people ask

How can an endless stream have a finite value?

Because each payment is discounted more heavily than the one before, so the contributions shrink toward nothing and the total settles. At 7 percent, everything beyond year fifty is worth about 3.4 percent of the whole.

Why does it refuse when growth is at or above the discount rate?

Because there is no finite answer. A stream growing at least as fast as it is discounted does not shrink in present value, so the infinite sum does not converge.

Should I enter this year's dividend or next year's?

Next year's, meaning the payment arriving at the end of the first period. If you have the current figure, multiply it by one plus the growth rate.

What growth rate is reasonable?

For a long run assumption, something at or below the nominal growth rate of the wider economy. Nothing can outgrow the economy indefinitely, so a higher figure is not aggressive, it is impossible.

How is this different from a growing annuity?

A growing annuity has a fixed number of payments and tolerates any growth rate. A growing perpetuity never ends and requires growth below the discount rate.

Is this the same as a capitalisation rate?

Closely related. Dividing annual income by a capitalisation rate to value a property is the perpetuity formula, with the cap rate playing the part of the discount rate less growth.

Why does my answer change so much when I adjust growth?

Because growth sits in the denominator. Moving it from zero to 6 percent against a 7 percent discount rate multiplies the value by seven. Always run a range rather than a single figure.

References

A note on the sources. Valuing an income interest that continues indefinitely is not merely a modelling convenience: United States Treasury regulations govern the valuation of life interests, term interests and remainder interests using prescribed actuarial factors and a prescribed rate, and they permit exact computation by software rather than reliance on rounded tables. The idea that a share's value rests on the return it earns relative to the return investors require, which is what the discount rate in this formula represents, is set out by CFA Institute. The compounding relationship that makes distant payments contribute so little is defined by the Securities and Exchange Commission's investor education office.

1. Cornell Law School, Legal Information Institute, 26 CFR § 1.7520-1, Valuation of annuities, unitrust interests, interests for life or terms of years, and remainder or reversionary interests, on actuarial factors and exact computational methods. https://www.law.cornell.edu/cfr/text/26/1.7520-1
2. CFA Institute, Residual Income Valuation, on intrinsic value as a function of the return a company earns relative to the required return on equity. https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/residual-income-valuation
3. Internal Revenue Service, Section 7520 interest rates, the monthly prescribed rate for valuing annuities, life estates and remainder interests. https://www.irs.gov/businesses/small-businesses-self-employed/section-7520-interest-rates
4. Brealey, R.A., Myers, S.C., and Allen, F., Principles of Corporate Finance, McGraw-Hill Education, chapters on perpetuities, growing perpetuities and terminal value.