What this calculator does
A stream of payments continuing forever has a finite value, which surprises people the first time they meet it. Because later payments are discounted more heavily, the sum converges rather than running away.
Growth changes that dramatically. A $5,000 annual payment at a 6 per cent discount rate is worth $83,333 flat, but growing at 4 per cent it is worth $250,000, three times as much, because the growth eats most of the discount rate.
The formula
A level perpetuity is the payment divided by the discount rate. A growing one divides by the discount rate less the growth rate, which requires the discount rate to be the larger of the two.
| Term | Meaning |
|---|---|
| Perpetuity | A cash flow that continues indefinitely with no end date. |
| Growing perpetuity | The same, but with the payment increasing at a constant rate each period. |
| Terminal value | The most common practical use, capping off a discounted cash flow model. |
The inputs explained
| Field | What to enter |
|---|---|
| Cash flow next period ($) | The cash flow expected in the next period. |
| Discount rate (%) | The discount rate, which must exceed the growth rate. |
| Perpetual growth rate (%) | The perpetual growth rate. Use 0 for a level perpetuity. |
When to use it
Setting a terminal value
Discounted cash flow models typically forecast a few years then use a growing perpetuity for everything after.
Valuing a ground rent or consol
Some instruments genuinely have no maturity, and this is the correct way to value them.
Checking a dividend valuation
The Gordon growth model is a growing perpetuity applied to dividends.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How much value does perpetual growth add?
The same payment under three growth assumptions.
| Growth rate | Growing perpetuity value | Value added by growth |
|---|---|---|
| 0% | $83,333.33 | $0.00 |
| 2% | $125,000.00 | $41,666.67 |
| 4% | $250,000.00 | $166,666.67 |
Questions
How can an infinite stream have a finite value?
Because each payment is discounted more heavily than the last, so the terms shrink geometrically. A payment a hundred years out contributes almost nothing at any realistic discount rate, and the series converges.
Why must the discount rate exceed the growth rate?
Because otherwise the payments grow at least as fast as they are discounted and the sum diverges. It is also economically sensible: nothing can grow faster than the economy forever.
Why is the terminal value so large in DCF models?
Exactly because of this sensitivity. The terminal value often accounts for the majority of a valuation, and it rests on a perpetual growth assumption that cannot be verified. That is worth knowing when reading someone else's model.
Do real perpetuities exist?
A few. British consols paid interest with no maturity for centuries before being redeemed, and some ground rents and preference shares have no end date. The concept is used far more often as a modelling device than as a real instrument.
For applying the same maths to dividends, see the dividend discount model calculator. For discounting a finite stream instead, see the present value calculator.