What is the perpetuity formula?

PV = P ÷ r, where P is the constant payment per period and r the discount rate per period. It assumes the first payment arrives one period from now.

What is a perpetuity due?

A perpetuity whose first payment is made today. Its value is P ÷ r + P.

Why is the value of an infinite stream finite?

Each later payment is discounted more heavily. The discounted payments form a geometric series that converges to P ÷ r.

What are real examples of perpetuities?

Fixed-dividend preferred shares, perpetual bonds such as historical British consols, endowment payouts and the terminal value in a DCF model.

Perpetuity Formula and Calculator

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Perpetuity Calculator

A perpetuity is a payment of the same amount every period that continues forever. It sounds theoretical, but it describes preferred shares with fixed dividends, some government consols, endowments that pay out only their income, and the terminal value used in company valuations. Although the payments never stop, their present value is finite because distant payments are discounted almost to nothing.

Formula

PV = Pr (first payment in one period) PV_due = Pr + P

Where

PV: present value of the never-ending payment stream
P: payment per period (constant)
r: discount rate per period, as a decimal

Calculator

Payment per period (P)

Discount rate per period (r) (%)

First payment

Result and step-by-step solution

Present value of the perpetuity

20,000.00

A payment of 1,000.00 every period forever, discounted at 5%.

Convert the rate to a decimal
r = 5% = 0.05
Divide the payment by the rate
PV = 1,000.000.05 = 20,000.00

How to use the formula

The sum of P ÷ (1 + r) + P ÷ (1 + r)² + … simplifies to P ÷ r. So a payment of 1,000 a year at a 5% discount rate is worth 20,000 today: invest 20,000 at 5% and you can withdraw 1,000 a year forever without touching the principal. If the first payment is received today (a perpetuity due), add one payment to get 21,000.

The rate must be positive; at 0% the value would be infinite. If the payment grows at a constant rate each period, use the growing perpetuity formula, which leads to the Gordon growth model for share prices.

Frequently asked questions

What is the perpetuity formula?: PV = P ÷ r, where P is the constant payment per period and r the discount rate per period. It assumes the first payment arrives one period from now.
What is a perpetuity due?: A perpetuity whose first payment is made today. Its value is P ÷ r + P.
Why is the value of an infinite stream finite?: Each later payment is discounted more heavily. The discounted payments form a geometric series that converges to P ÷ r.
What are real examples of perpetuities?: Fixed-dividend preferred shares, perpetual bonds such as historical British consols, endowment payouts and the terminal value in a DCF model.

Last reviewed: September 26, 2026. Calculations run in your browser and were checked against numpy-financial, SciPy and published spreadsheet examples.