How a perpetuity is priced
By Jude Wallis
A perpetuity pays forever, and it still has a finite price. Divide the annual payment by the discount rate: $1,000 a year at 5 percent is worth $20,000 today. Payments far enough out discount to almost nothing, so the infinite sum converges.
Perpetuity value
$20,000.00
$1,000.00 a year discounted at 5.00%.
- Value
- $20,000.00
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In short
- Value is payment divided by rate: $1,000 over 0.05 is $20,000.
- The sum is infinite in length but finite in value, because each payment is discounted more heavily than the last.
- Raising the rate cuts the value hard: $2,500 a year at 8 percent is $31,250, only a little more than a smaller payment at a lower rate.
- The identity is the backbone of the Gordon growth model and of terminal values in a discounted cash flow.
Why an endless stream has a finite price
Discounting shrinks distant money geometrically. A payment 60 years out at 5 percent is worth a small fraction of its face amount today, and one 200 years out is worth almost nothing at all. Add every one of them and the total converges rather than running away.
Where it converges is payment divided by rate. At $1,000 a year and 5 percent that is $20,000, which also has a plain reading: it is the amount that, earning 5 percent, would throw off $1,000 a year without ever being touched.
The rate is the whole valuation
With only two inputs, one of them a payment you can observe, the discount rate carries every judgement in the calculation. Halve the rate and the value doubles. That sensitivity is why perpetuity-based valuations are argued about at the level of the rate rather than the arithmetic.
It also means the rate has to match the risk of the payment. A government-backed stream and a tenant's rent do not deserve the same rate, and using one rate for both prices them as though they carried the same chance of stopping. Present value is the general form this identity sits inside.
Growing perpetuities and terminal values
Most real streams are expected to grow. The standard adjustment subtracts growth from the discount rate in the denominator, which is the Gordon growth model, and it is the same identity with one extra term. How the Gordon growth model works sets it out.
The same shape appears at the end of a discounted cash flow, where the years past the forecast horizon are collapsed into a terminal value. That terminal figure is frequently the majority of a valuation, which means a perpetuity assumption made in one line is doing most of the work. The DCF calculator shows the split.
Where the model applies
Use it for streams with no scheduled end and a stable character: a ground rent, a preferred dividend, a licence fee, the tail of a valuation. For anything with a maturity date, discount the actual payments instead, which is what the annuity present value calculator does. Read the two together in annuity present against future value. The perpetuity calculator prices the endless case from the two inputs. This is educational material, not financial advice.
Worked examples
\$1,000 a year discounted at 5 percent
A stream pays $1,000 a year forever, and the appropriate discount rate is 5 percent. What is it worth today?
- Divide the annual payment by the rate: 1,000 divided by 0.05.
- That gives $20,000, which is also the sum a 5 percent return would need to produce $1,000 a year indefinitely.
The perpetuity is worth $20,000 today.
A bigger payment at a higher rate
A second stream pays $2,500 a year and is discounted at 8 percent.
- Divide 2,500 by 0.08.
- The value is $31,250.
$31,250. The payment is two and a half times larger, but the higher rate takes most of that advantage back.
Common questions
How can an infinite stream be worth a finite amount?
Because each payment is discounted more than the one before, so the series converges to payment divided by rate.
What if the payment grows?
Subtract the growth rate from the discount rate in the denominator. That is the Gordon growth model.
What happens as the rate approaches zero?
The value runs away towards infinity, which is a warning that a near-zero discount rate is not a usable input here.
Is this financial advice?
No. It is educational material about a valuation identity.
Keep reading
This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.