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What Is a Perpetuity?
A perpetuity is a stream of cash flows that continues forever, an infinite annuity. The concept may feel abstract, but perpetuities appear constantly in modern finance: preferred-stock dividends, perpetual government bonds (the British consol is the classic example), ground rents, royalty streams, and the terminal piece of every discounted-cash-flow valuation. This perpetuity value calculator, sometimes shortened to a perpetuity calculator, lets you find the present value of any such never-ending stream using a single closed-form formula, then test how that value reacts to changes in the discount rate or growth rate.
The reason perpetuities matter in valuation is that most economically valuable assets generate cash flows over very long horizons, and forecasting each year individually quickly becomes pointless. By treating long-horizon cash flows as a perpetuity, analysts collapse decades or centuries of forecasting into one tidy expression. The tool above implements both versions: a constant perpetuity for fixed cash flows and a growing perpetuity for cash flows that escalate at a steady rate. For broader equity valuation that builds on perpetuity theory, see our dividend discount model calculator.
Investopedia's overview at Investopedia's perpetuity guide is a useful primer if you want additional background on the history and applications of the concept before running the calculator. Most professional valuation courses introduce the perpetuity formula in the first few lectures because so many later results (bond pricing, equity valuation, real-estate cap rates) are special cases of it.
The Perpetuity Formula Explained
The constant perpetuity formula is PV = C / r, where C is the periodic cash flow and r is the discount rate per period. The derivation comes from summing an infinite geometric series of discounted cash flows: PV = C/(1+r) + C/(1+r)² + C/(1+r)³ + … which converges to C / r whenever r is positive. That single ratio is one of the most-used shortcuts in finance because it bypasses the need to forecast every individual future payment. Some analysts simply call this a perpetuity PV calculator, since the whole point of the exercise is converting a cash flow stream into a single present value.
A worked example makes the result tangible: a preferred share paying a $5 annual dividend with a 6% required return has an intrinsic value of $5 / 0.06 = $83.33. If required returns drop to 4%, the same share becomes worth $5 / 0.04 = $125, a 50% jump for a 2-percentage-point move in r. This extreme sensitivity is the single most important property of any perpetuity-style valuation, and it is exactly what the sensitivity table in the calculator's constant-perpetuity tab is designed to make visible.
Notice that the formula does not depend on time. There is no n for the number of periods because the perpetuity has no end. The formula also assumes the first cash flow arrives one period from now, if you receive a payment today as well, you simply add C to the result to convert to a perpetuity-due. For finite cash flow streams, use an discounted cash flow calculator instead. The CFA Institute's curriculum on discounted dividend valuation walks through the perpetuity derivation in depth and is the standard reference for professional analysts.
Growing Perpetuity and the Gordon Growth Model
Few real cash flows stay flat forever. Inflation alone pushes nominal cash flows higher each year, and real businesses tend to grow their distributions over time. The growing perpetuity formula handles this by allowing cash flows to grow at a constant rate g: PV = C₁ / (r − g), where C₁ is the cash flow one period from now. Myron Gordon popularized this formula in 1959 for equity valuation, which is why it is widely known as the Gordon Growth Model. Run it through the growing perpetuity calculator tab whenever the cash flow you are valuing is expected to escalate rather than stay flat.
The growing perpetuity formula has one critical constraint: r must be strictly greater than g. If r equals g, the denominator is zero and the value is undefined; if r is less than g, the formula returns a negative number and the underlying geometric series fails to converge. Economically, no real asset can grow its cash flow faster than the cost of capital forever, such an asset would eventually consume the entire economy. The perpetuity value calculator blocks invalid inputs and shows an error message whenever this rule is violated on the growing perpetuity calculator tab.
Aswath Damodaran at NYU Stern has written extensively about the dangers of pushing g too close to r, see his materials at NYU Stern's valuation resources for a thorough discussion. A practical rule of thumb is that g should never exceed the long-run nominal GDP growth rate of the economy in which the business operates, currently around 4% for the United States. For terminal-value modeling specifically, see our terminal value calculator, which uses the same growing perpetuity formula inside a multi-stage DCF framework.
Perpetuity in Stock Valuation
The perpetuity formula is the mathematical backbone of equity valuation. A stock's intrinsic value equals the present value of all future dividends, and when those dividends grow at a constant long-run rate, the entire valuation collapses to the growing perpetuity formula: P = D₁ / (r − g). This is why the perpetuity value calculator doubles as a basic stock valuation tool; the math is identical, only the labels change.
As a perpetuity formula calculator, it works whenever a company's payout policy is mature enough to model as a single steady growth rate. The perpetuity approach works best for mature dividend payers with stable payout policies: regulated utilities, consumer staples, large banks, and Dividend Aristocrats. For these businesses, a single growth rate plausibly captures decades of future distributions. It works less well for growth companies, cyclicals, or buyback-heavy firms, for those, a multi-stage model that combines an explicit forecast horizon with a perpetuity tail produces more defensible answers. Our discounted cash flow calculator implements that two-stage approach for free cash flow, and the dividend-focused equivalent lives in our dividend discount model calculator.
One subtle but important point in stock valuation is that the perpetuity formula uses next year's dividend, not today's. That is why the equation is written with D₁ rather than D₀. If you only know the trailing-twelve-month dividend D₀, multiply by (1 + g) before applying the formula. Most professional valuation reports show this adjustment explicitly because forgetting it understates intrinsic value by exactly the growth rate. Browse the full library of equity valuation tools on our investing calculators hub to see how each one builds on the perpetuity foundation.
Perpetuity in Real Estate: NOI and Cap Rates
Commercial real estate valuation relies on the perpetuity formula more openly than most other parts of finance. The standard direct-capitalization approach values a property as Value = NOI / Cap Rate, where NOI is net operating income and the cap rate is the market-required yield. This is exactly the same equation as PV = C / r, the cap rate is just the real-estate-industry name for a discount rate applied to a stabilized rental stream that is implicitly assumed to continue forever.
Real estate investors sometimes call this a perpetuity calculator for rental income rather than a cap-rate model, though the underlying math never changes. Real estate professionals use cap rates because they encode market expectations about risk, growth, and required return in a single number that can be compared across properties. A Class-A office building in a prime market might trade at a 5% cap rate, implying a value of 20× annual NOI; a Class-B industrial property in a secondary market might trade at 8%, implying 12.5× NOI. The perpetuity value calculatorcan be used directly on any of these scenarios by entering the property's NOI as the annual cash flow and the cap rate as the discount rate.
When rents are expected to grow at a steady rate, real estate practitioners use a growing perpetuity: Value = NOI₁ / (cap rate − g). This is mathematically identical to the Gordon Growth Model and inherits the same constraint, the cap rate must exceed g. Combining the growing perpetuity logic with explicit short-horizon cash flow forecasts produces a discounted cash flow valuation, which is the gold-standard approach for non-stabilized properties or those undergoing repositioning. The perpetuity formula is therefore the connective tissue that links direct capitalization, DCF, and equity valuation across the entire investing toolkit. Whether you reach for it as a perpetuity PV calculator or a perpetuity formula calculator, the underlying arithmetic never changes.