A perpetuity is a stream of cash flows that continues without an end date. In the standard case the payment is level, and because each later payment is discounted more heavily than the one before it, the endless stream still has a finite present value, and that value has a closed form, PV = C / r, where C is the constant payment received each period and r is the discount rate for one period of the same length. Property law uses the same word for a different idea, and the rule against perpetuities is the limit it sets on that one.
The Formula and Its Terms
The closed form takes two inputs and no horizon. The constant perpetuity formula is PV = C / Rs, where PV is the price of the preferred stock, C is the constant dividend, and Rs is the required rate of return (OpenStax, Principles of Finance, 8.1 Perpetuities). Rs is the r of the general form above, the return the market requires on money of that risk over one period, and PV is the value of the stream, which in this case is the price of the share. Preferred stock is only the textbook illustration, and nothing in the arithmetic is specific to it, because C is any level payment received each period.
The two inputs have to share a period. An annual payment goes with an annual rate and a quarterly payment with a quarterly rate, and mixing the two gives an answer that is wrong by a multiple rather than by a rounding. What the formula does not carry is a horizon, because a perpetuity does not have one. Where r comes from is a separate question, answered by cost of capital and, for the equity leg of that blend, the CAPM equation.
Why a Stream With No End Has a Finite Value
The payments do not stop, but their present values shrink toward nothing. Each one is divided by a larger power of one plus the rate, so the contribution of each successive payment falls geometrically, and a geometric series of shrinking amounts converges on a total rather than running away.
The scale of that shrinking is easy to underestimate. To grow to one dollar 70 years from now at 5% a year takes $0.0329, and over 100 years it takes 0.76 cents, so the distant payments become negligible and stop having a significant effect on today’s value (OpenStax, Principles of Finance, 8.1 Perpetuities).
That is the exact point where this parts company with an ordinary net present value calculation. The NPV summation runs from the first period to a final period n and stops there, so the answer is built by adding up a finite list of discounted flows. A perpetuity has no final period, so the summation is replaced by the closed form and the single division by r does that work instead.
The Growing Perpetuity
A payment that rises at a constant rate each period is a growing perpetuity, and the growth rate moves into the denominator: PV = C / (Rs - G), where G is the expected rate of growth (OpenStax, Principles of Finance, 8.1 Perpetuities). The formula only says anything while growth stays below the discount rate. Set the two equal and the denominator goes to zero; set growth higher and the answer comes back negative, which is arithmetic rather than a valuation.
That constraint has an economic version as well as a mathematical one. No firm can grow forever at a rate higher than the growth rate of the economy in which it operates, so the constant growth rate cannot be greater than the overall growth rate of that economy, and a rule of thumb is that the stable growth rate should not exceed the riskless rate used in the valuation (Aswath Damodaran, Closure in Valuation: Estimating Terminal Value, Stern School of Business).
Of all the inputs into a discounted cash flow valuation model, which prices a business by discounting the cash flows it is forecast to produce and then adding a terminal value for everything beyond the forecast period, none can affect the value more than the stable growth rate. Small changes in it can change the terminal value significantly, the effect gets larger as the growth rate approaches the discount rate used in the estimation, and analysts often use it to alter a valuation to reflect their biases (Aswath Damodaran, Closure in Valuation: Estimating Terminal Value, Stern School of Business).
A Worked Example
The inputs below are stated for the example rather than drawn from a real engagement.
Constant perpetuity
Level payment each year, first one a year from now, C 12,000 stated input
Discount rate per year, r 8% stated input
PV = 12,000 / 0.08 = 150,000
Growing perpetuity
Payment one year from now, C 12,000 stated input
Growth per year, g 3% stated input
Discount rate per year, r 8% stated input
PV = 12,000 / (0.08 - 0.03) = 240,000The payment never changed between the two. Only the denominator did, and the answer moved with it.
Where a CPA Firm Meets a Perpetuity
It turns up in three places in a US firm’s work.
The terminal value of a discounted cash flow model is the common one. A DCF forecasts explicit cash flows for a handful of years and then has to account for everything after that, and the terminal value can be found by assuming the firm’s assets are liquidated in the terminal year, by applying a multiple to earnings, revenues or book value, or by assuming a growth rate that can be sustained in perpetuity, which allows the value of all cash flows beyond that point to be estimated as a terminal value for a going concern (Aswath Damodaran, Closure in Valuation: Estimating Terminal Value, Stern School of Business). Only the last of those runs on a perpetuity. What the resulting figure actually means depends on the standard of value the engagement was written to, which valuation sets out.
An instrument with no maturity date is the second. Most preferred stock issues carry a fixed and predetermined dividend rate, and because such shares are created with the intention of continuity the holder can theoretically expect that income stream in perpetuity, so where the dividend is declared and paid the price of one share is that annual dividend divided by the required rate of return (OpenStax, Principles of Finance, 8.1 Perpetuities). The balance sheet questions that come with the class sit in preferred stock.
Nonprofit work is the third, and there the word is a restriction rather than a valuation. A gift held in perpetuity is a permanent endowment, and endowment fund covers how that restriction is written, what an organization may spend, and what releases the money.
Perpetuity in Property Law Is a Different Question
The finance sense asks what an endless payment stream is worth today. The property law sense asks how long the law will let a future interest sit unvested before it fails, which is a question about title, not about price.
The common law rule against perpetuities states that if an interest in real property does not vest within 21 years of a life in being at the creation of the interest, that interest in land is not good, and the test is applied when the interest is created, because the transfer of the right to land violates the rule if there is even a remote possibility that no life in being would acquire the interest in land. Many jurisdictions have modified the common law rule and some have abolished it altogether (Cornell Legal Information Institute, rule against perpetuities).
The statutory replacements matter in trust and estate work because they set the actual deadline. Under Virginia’s enactment of the Uniform Statutory Rule Against Perpetuities, a nonvested property interest is invalid unless, when created, it is certain to vest or terminate no later than 21 years after the death of an individual then alive, or it in fact vests or terminates within 90 years after its creation. For a nonvested interest in personal property held in trust created on or after July 1, 2024, the statute substitutes 1,000 years for 90 years throughout sections 55.1-124 through 55.1-129, and that substitution does not extend to a nonvested property interest in real property held in trust (Code of Virginia, section 55.1-124).
Common Questions
In the finance sense it means the payments carry no end date, not that the far end of the stream matters much to the value. In property law it means close to the opposite, because the rule against perpetuities exists to stop an interest in land from staying unresolved indefinitely.
Usually one of two things. It describes a payment stream with no end date, valued as PV = C / r when the payment is level, most often the terminal-value step of a valuation or a security with no maturity date. It also appears in gift and trust instruments as a restriction, where holding something in perpetuity means the principal is never to be spent.
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