What it means
The idea sounds theoretical, but perpetuities show up in ordinary finance more often than people expect. Certain government bonds have historically paid a fixed coupon with no maturity date, and preference shares often promise a fixed dividend for as long as the company exists.
Ground rents on very long leases behave the same way. The mathematics works because of discounting.
A payment of $50,000 due in sixty years is worth almost nothing today at any sensible discount rate, so adding more and more distant payments barely moves the total. The infinite series therefore settles on a tidy number rather than running away to infinity.
The most common business use is not a literal forever-payment at all, it is the terminal value in a valuation model. Analysts forecast cash flows in detail for five or ten years and then assume the business keeps generating cash indefinitely after that, capturing the remainder as a growing perpetuity.
In many discounted cash flow models this terminal value is more than half the total valuation. The growing perpetuity variant assumes the payment rises by a fixed percentage each year, which is more realistic for a trading business than a flat payment.
The formula divides by the discount rate minus the growth rate, and that small denominator is where valuations go wrong. If the assumed growth rate creeps close to the discount rate the value explodes, which is why analysts usually cap long-run growth near expected inflation or the growth rate of the wider economy.
One timing detail catches people out. A standard perpetuity assumes the first payment arrives exactly one period from now, so if a payment is due immediately you are dealing with a perpetuity due, and the value is the ordinary figure multiplied by one plus the discount rate.
Getting this wrong produces small but entirely avoidable valuation errors.
In practice
Real-world examples
Example 1
A utility issues preference shares paying a fixed dividend of $6 per share every year with no redemption date. An investor who requires a 7.5% return values each share at $6 / 0.075 = $80, and would only pay more if required returns fell.
Example 2
A charity wants to fund a $25,000 scholarship every year indefinitely. With an endowment expected to earn 5% a year after costs and inflation, the fund needs $25,000 / 0.05 = $500,000 before it can make the first award.
Example 3
An analyst valuing a logistics firm forecasts free cash flow of $9,000,000 in year six, growing at 2% a year thereafter, using a 10% cost of capital. The terminal value at the end of year five is $9,000,000 / (0.10 - 0.02) = $112,500,000, which then needs discounting back to today.
Think of it
Imagine owning a magic fruit tree that drops exactly ten apples on your lawn every single autumn, year after year, with no final season.
Formula
Calculation
Present value of a perpetuity = Annual Cash Flow / Discount Rate
Present value of a growing perpetuity = Next Year's Cash Flow / (Discount Rate - Growth Rate)
Suppose a property investor is offered a ground rent that pays $50,000 every year forever, and the investor requires an 8% return.
Present value = $50,000 / 0.08 = $625,000.
Now assume the same rent is index-linked and rises 3% a year, with the first $50,000 arriving in twelve months.
Present value = $50,000 / (0.08 - 0.03) = $50,000 / 0.05 = $1,000,000.
The 3% growth assumption alone adds $1,000,000 - $625,000 = $375,000 to the value, which shows how sensitive the answer is to the gap between the discount rate and the growth rate.Case study
Seen in the real world
Harborlight Mills is an illustrative family business used to show how perpetuity maths drives a negotiation. The company owned a royalty agreement that paid it $120,000 a year, with no end date, from a licensee that used one of its old processes. A private investor offered to buy the royalty outright.
The investor's opening offer was based on a 12% required return, giving a value of $120,000 / 0.12 = $1,000,000. The family's adviser argued that the payer was a long-established business with a strong record, so the appropriate return was closer to 10%, giving $120,000 / 0.10 = $1,200,000. The whole $200,000 gap came from two percentage points of perceived risk, not from any disagreement about the cash flows.
In this fictional negotiation the parties settled at $1,100,000, an implied discount rate of roughly 10.9%. The useful lesson is that when a stream is treated as a perpetuity, the discount rate is doing almost all of the work, so that is where the argument should focus.
Watch out
Common mistakes
- Assuming an infinite stream of payments must be worth an infinite amount, when discounting makes distant payments contribute almost nothing to the total.
- Setting the long-run growth rate close to, or above, the discount rate in a growing perpetuity, which produces a value that is either absurdly large or negative and meaningless.
- Using this year's cash flow instead of next year's in the growing perpetuity formula, which understates the value by one year of growth.
Questions
People also ask
Why does terminal value matter so much in a valuation?
Because it captures every year beyond the explicit forecast, and in a typical discounted cash flow model it commonly accounts for well over half of the total value.
What is the difference between a perpetuity and an annuity?
An annuity pays for a fixed number of periods and then stops, while a perpetuity has no end date, which is why the annuity formula is more involved.
Can a real business really be valued as a perpetuity?
Not literally, but it is a reasonable modelling shortcut for a mature business with stable cash flows, provided the growth assumption stays modest and the result is sense-checked against comparable company multiples.