How this instrument works
A perpetuity is a payment that repeats every period with no final date — the historical example is a UK consol, a bond London issued in the 18th century that paid a fixed coupon indefinitely and carried no maturity requiring the principal to be redeemed. Each payment in the stream is discounted by one extra factor of (1 + r) compared with the payment ahead of it, so payments far enough out shrink toward zero even though the stream itself never stops — an infinite run of ever-smaller discounted terms still totals a finite figure: the payment size divided by the rate.
The identical math shows up wherever someone prices an income stream expected to continue indefinitely rather than for a fixed number of years. A preferred-stock investor values the shares by treating the stated dividend as a payment with no scheduled end. A commercial-real-estate appraiser dividing a property's net operating income by a cap rate is running this exact formula, even if the word perpetuity never appears on the worksheet. An endowment officer figuring out how much principal must be invested to fund a fixed annual payout forever runs the same division in reverse.
The formula only holds for a payment that is flat and genuinely open-ended — a stream expected to grow with inflation or revenue needs the growing-perpetuity version, payment divided by rate minus growth, not this one, and setting growth equal to or above the rate makes that variant break down entirely. Because there is no period count anywhere in the formula, the whole answer rests on the discount rate alone; a small change in Discount rate, % moves the result far more than the same-sized change would move a mortgage payment or a bounded annuity.
- Enter the fixed amount received every period into Payment per period, $.
- Set Discount rate, % to the rate you want applied to each period's wait — it must stay above zero.
- Read Present value of the perpetuity, $ — the lump sum today equivalent to the endless stream.
- Raise or lower Discount rate, % on its own to see how far the value swings with no period count to soften it.
Worked example — a $1,000 payment that never stops
Set Payment per period, $ to 1,000 and Discount rate, % to 5. The decimal rate is 0.05, and Present value of the perpetuity, $ reads exactly $20,000.00 — the lump sum today that is worth the same as receiving $1,000 every year, forever, at that rate.
Raise Discount rate, % to 10 with the payment unchanged and present value falls to $10,000.00, exactly half — because there is no n in this formula, doubling the rate always exactly halves the price, a cleaner relationship than a bounded annuity or a mortgage ever shows, where the period count also matters.
Questions
What exactly is a perpetuity?
It is a payment that repeats every period with no scheduled final payment — unlike a loan or a bounded annuity, the stream is defined to go on forever. The UK's 18th-century consol bonds are the textbook example: they paid interest indefinitely and were never required to be redeemed for principal.
Why does an infinite stream of payments add up to a finite value?
Because each payment is discounted by one more factor of (1 plus the rate) than the payment before it, so payments far enough in the future are worth almost nothing today even though they keep arriving. The infinite geometric series of shrinking terms still sums to a finite total, and that sum works out to exactly payment divided by rate.
How is a perpetuity different from an annuity?
An annuity pays for a set number of periods and its formula includes that count; a perpetuity formula has no period count at all. Stretch the annuity formula's number of payments toward infinity and it collapses into payment divided by rate — a perpetuity is the limiting case of an annuity that never ends, not a separate kind of arithmetic.
Who actually prices something as a perpetuity?
A preferred-stock investor treats the stated dividend as continuing indefinitely and values the shares by this formula. A commercial-real-estate appraiser dividing net operating income by a cap rate is running the identical division. An endowment or foundation officer sizing how much principal must be invested to sustain a fixed annual payout forever runs the same formula in reverse.
What if the payment is expected to grow every period?
Then this formula understates the value — use the growing-perpetuity version instead, payment divided by rate minus growth rate, often called the Gordon growth model when applied to dividends. That version requires the discount rate to stay above the growth rate; set growth equal to or above the rate and the formula turns negative or breaks down entirely.
Why won't the instrument accept a discount rate of zero?
Dividing by a zero rate is undefined, and the real-world meaning matches the math: with no discounting at all, an endless stream of positive payments would be worth an infinite sum today, which cannot be priced. Discount rate, % must stay above zero for the same reason a bond's yield cannot be pushed to negative infinity.
References
Read this first: This instrument shows arithmetic, not advice. Real offers add fees, taxes and terms that vary by lender and place — verify the figures against your actual paperwork before deciding anything.