How an endless stream collapses into one present value
The trick behind every perpetuity is that money received in the distant future is worth very little today. Each payment is discounted by an extra factor of (1 + r) for every period you wait, so the contributions shrink geometrically and their infinite sum converges to a finite total. For a level stream that total is simply the payment divided by the rate; for a stream that grows at a constant rate g you subtract that growth from the discount rate in the denominator. The growing case applied to a company’s dividends is exactly thedividend discount model, which prices a share as the present value of an ever-growing dividend perpetuity.
The perpetuity formulas
PV = C / r
PV = C / ( r − g )
where C = the recurring payment, r = the per-period discount rate, and g = the constant growth rate of the payment. The first formula values a level perpetuity; the second agrowing perpetuity and requires r > g to converge.
What makes this calculator different
- Level and growing in one place. Switch between a flat perpetuity (PV = C / r) and a constantly growing one (PV = C / (r − g)) without leaving the page or re-deriving the formula.
- Ordinary and due timing. Choose whether the first payment arrives at the end of the period (ordinary) or at the start (perpetuity-due), and the present value adjusts accordingly.
- It flags the undefined case. When the growth rate meets or exceeds the discount rate (r ≤ g) the series diverges, so the tool tells you the value is undefined rather than returning a misleading number.
- It shows the implied yield. Enter a price and a payment and the calculator backs out the rate the perpetuity is offering, the same way a consol or preferred share is quoted.
Frequently asked questions
What is a perpetuity and what is its formula?+
A perpetuity is a stream of equal cash payments that continues forever, with no end date. Its present value is given by the simple formula PV = C / r, where C is the recurring payment and r is the per-period discount rate. So a £100 annual payment discounted at 5% is worth £100 / 0.05 = £2,000 today. The result is finite even though the payments never stop, because each future payment is worth a little less than the one before once discounted.
What is a growing perpetuity?+
A growing perpetuity is a never-ending stream of payments that increases at a constant rate g every period rather than staying flat. Its present value is PV = C / (r − g), where C is the first payment, r is the discount rate, and g is the growth rate. For example, a first payment of £100 growing at 2% and discounted at 5% is worth £100 / (0.05 − 0.02) = £3,333. This formula is the engine behind the Gordon growth model used to value dividend-paying shares and DCF terminal values.
How can an infinite stream of payments have a finite value?+
The key is discounting. A payment received far in the future is worth far less today, and that value shrinks geometrically the further out you go. When you add up an infinite series of payments that each shrink by a constant factor, the total converges to a finite number rather than growing without bound. Mathematically the sum of C/(1+r) + C/(1+r)² + … is a convergent geometric series equal to C / r, so the distant payments contribute almost nothing and the whole stream has a definite price.
Why must the discount rate be greater than the growth rate?+
For a growing perpetuity the formula PV = C / (r − g) only makes sense when r is strictly greater than g. If g equalled r the denominator would be zero and the value would be infinite; if g exceeded r the payments would grow faster than they are discounted, so the series diverges and no finite present value exists. Economically, a cash stream cannot grow forever faster than the rate at which money is discounted, so r > g is a required condition. This calculator flags the r ≤ g case as undefined rather than returning a misleading figure.
What are real-world examples of perpetuities?+
The classic example is the UK consol, a government bond that paid a fixed coupon indefinitely with no maturity date, valued exactly with PV = C / r. Preferred shares behave similarly, paying a fixed dividend in perpetuity, so their price is often estimated the same way. The growing-perpetuity formula appears in equity valuation as the Gordon growth model and in discounted-cash-flow analysis as the terminal value, which captures all cash flows beyond the explicit forecast horizon. In each case a long-lived or endless stream is collapsed into a single present value.
Disclaimer: This calculator is foreducation and illustration only. The perpetuity model rests on simplifying assumptions — payments that truly continue forever and a constant discount (and growth) rate — that rarely hold exactly in practice. Its output is not a valuation or a tradeable quote, and nothing here is investment, tax, or trading advice.