Perpetuity Calculator
A perpetuity is a stream of equal (or steadily growing) payments that continues forever. This calculator finds the present value of both regular and growing perpetuities, and it can also reverse-solve for the payment, discount rate, or growth rate when you know the other values. Results include step-by-step math, a present-value sensitivity chart, and a reference table of common discount-rate benchmarks.
Formula
Worked example
A preferred share pays a $50 annual dividend forever. With a 5% discount rate, PV = 50 / 0.05 = $1,000. If the dividend instead grows at 2% per year, PV = 50 / (0.05 - 0.02) = 50 / 0.03 = $1,667.
What is a perpetuity?
A perpetuity is a financial instrument that pays a fixed (or growing) amount at regular intervals and never stops. Unlike an annuity, which has a defined end date, a perpetuity has no maturity. Real-world examples include certain government bonds (the UK issued "Consols" that paid interest indefinitely), preferred stock that pays a fixed dividend forever, and endowment funds structured to pay out income in perpetuity. The concept is also used in valuation models such as the Gordon Growth Model, which treats a stock as a growing perpetuity of dividends.
Regular vs growing perpetuity
A regular (or flat) perpetuity pays the same amount every period: PV = D / r, where D is the payment and r is the discount rate. A growing perpetuity adds a constant growth rate g, so each payment is (1 + g) times the previous one: PV = D / (r - g). The denominator r - g is called the effective spread. When the growth rate approaches the discount rate, the present value approaches infinity, because the future payments grow so quickly they are almost as valuable as those received today. The growth rate must always be strictly less than the discount rate for the formula to give a finite positive answer.
Reverse-solving: payment, discount rate, or growth rate
This calculator solves for any one of the four variables when you know the other three. To find the implied discount rate of a preferred share trading at $1,200 that pays $60 per year, set "Solve for" to "Discount rate" and enter the payment and price - the result is 60 / 1,200 = 5%. To find the implied growth rate priced into a stock, set "Solve for" to "Growth rate", enter the current price (PV), the next expected dividend (D), and your required return (r). The answer g = r - D/PV is the growth rate the market is implicitly pricing in.
Present value sensitivity to discount rate
Perpetuity values are highly sensitive to the discount rate, far more so than fixed-term bonds or annuities, because the cash flows go on forever. A small drop in the rate causes a large rise in value. For a $100 annual payment, a rate of 5% gives PV = $2,000, but a rate of 4% gives PV = $2,500 - a 25% increase from just one percentage point. The chart above illustrates this relationship across a range of rates. This sensitivity is important when valuing preferred shares or endowments: a change in the required return assumption has an outsized effect on the fair value.
Common discount rate benchmarks
| Asset / context | Typical discount rate | Notes |
|---|---|---|
| Government bond (long-term) | 3% - 5% | Risk-free rate in stable economies |
| Investment-grade corporate bond | 4% - 7% | Adds credit spread to risk-free rate |
| Preferred stock | 5% - 8% | Fixed dividend, senior to common equity |
| Stable blue-chip equity | 7% - 10% | Gordon Growth Model WACC range |
| Growth stock / emerging market | 10% - 15% | Higher risk premium applied |
| Private equity / real estate | 8% - 14% | Illiquidity premium included |
Typical discount rates used for perpetuity valuation, by asset class. Actual rates depend on market conditions, credit risk, and required return.
Frequently asked questions
What is the formula for a regular perpetuity?
PV = D / r, where PV is the present value, D is the payment per period, and r is the discount rate expressed as a decimal (e.g. 5% becomes 0.05). Rearranging gives D = PV x r (to find the payment) and r = D / PV (to find the implied rate).
What is the formula for a growing perpetuity?
PV = D / (r - g), where g is the constant growth rate of each payment. This requires g to be strictly less than r. When g equals r the denominator is zero and the present value is undefined (infinite). The formula is the basis of the Gordon Growth Model used to value dividend-paying stocks.
Why must the growth rate be less than the discount rate?
If g equals r, the denominator r - g equals zero, making PV infinite. If g exceeds r, each future payment grows faster than the discount factor shrinks it, so the sum of discounted payments never converges - it blows up to infinity. Only when g is strictly below r does the infinite series converge to the finite value D / (r - g).
Can a perpetuity actually last forever?
In practice, very few instruments last forever - companies can go bankrupt, governments can default, and endowments can be depleted. But the perpetuity model is useful whenever the time horizon is long enough that the terminal value is negligible. A 30-year annuity discounted at 6% has a present value within about 18% of a true perpetuity, so for long-horizon analysis the simpler perpetuity formula is often a good approximation.
How is a perpetuity different from an annuity?
An annuity pays for a fixed number of periods and then stops. A perpetuity pays forever. The present value of an annuity uses the formula PV = D x [1 - (1+r)^(-n)] / r, where n is the number of periods. As n approaches infinity, that formula approaches D / r - the perpetuity formula. So a perpetuity is simply the limiting case of an annuity with an infinite term.