Perpetuity Calculator
What an income that never stops is worth today, with or without growth, and what capital a given income needs.
Results
What this tool does
Some payments have no end date: a ground rent, a preference dividend, an endowment, the income you hope a retirement pot will throw off indefinitely. Surprisingly, an endless stream has a perfectly finite value, because each payment further away counts for less. Divide the payment by the rate and you have it. If the payments grow — a dividend that rises with inflation, say — subtract the growth from the rate first, which is the Gordon model. This page also runs it backwards: tell it what capital you have and it says what income that capital can sustain.
Formula
PV = PMT ÷ (i − g) · the payment is the one at the end of the first period (Gordon model)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
pmt | Payment each period | — |
rate | Discount rate | % |
growth | Growth rate | % |
freq | Payments per year | — |
timing | When each payment lands | — |
capital | Capital you have | — |
PV | Capital the income is worth | — |
IY | Income per year | — |
IC | Income that capital gives per period | — |
IA | Income that capital gives per year | — |
NR | Rate minus growth | % |
PR | Rate per period | % |
MU | Times the payment | x |
FY | First year yield | % |
Worked example
- Payment each period1000
- Discount rate5 %
- Growth rate0 %
- Payments per year1
- When each payment landsordinary
- Capital you have0
- Capital the income is worth20,000.00
- Income per year1000.00
- Income that capital gives per period0.00
- Income that capital gives per year0.00
- Rate minus growth5.0000 %
- Rate per period5.000000 %
- Times the payment20.00 x
- First year yield5.0000 %
Limitations
- This is an informational calculator, not personalised financial advice. Rates, fees, taxes and contract conditions vary between institutions and countries.
- The tool works with whatever currency you use for the inputs; it does not convert between currencies.
Frequently asked questions
How can an endless stream have a finite value?
Because each payment is worth less than the one before it, and the amounts shrink faster than they accumulate. The payment arriving in year fifty, discounted at five per cent, is worth under nine per cent of its face amount; the one in year two hundred is worth about six thousandths of one per cent. Add up the whole infinite list and it converges — to exactly the payment divided by the rate. This is the same reason a half plus a quarter plus an eighth never passes one.
Why must the rate be above the growth?
Because if the payments grow at least as fast as they are discounted, the sum never converges — mathematically the answer is infinite, which is a sign the model has been pushed past where it makes sense. In practice this matters a great deal: the Gordon model's answer is very sensitive when the two numbers are close. At a rate of eight and growth of seven, the value is a hundred times the payment; move growth to seven and a half and it becomes two hundred. A valuation that rests on that gap rests on almost nothing.