What Is a Perpetuity? Formula and Examples

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A perpetuity is a stream of equal cash payments, made at regular intervals, that never ends. Because there is no maturity date, the holder never gets principal back; the value lies entirely in the payments. Those payments can be added up into a single present value because each one is discounted for time: for a level perpetuity paying C per period at a discount rate r, the present value is C ÷ r.

How a Perpetuity Works

An endless series of payments has a finite value because of the time value of money. A dollar received next year is worth less than a dollar today, and a dollar received in 50 years is worth far less. Each payment adds less present value than the one before, so the total converges to a finite number.

Here is how fast that happens. At a 5% discount rate, the payments in the first 20 years of a level perpetuity account for about 62% of its total present value (1 − 1 ÷ 1.05²⁰). The first 50 years account for roughly 91%. Payments centuries away contribute almost nothing, which is why the infinite sum settles at C ÷ r.

Perpetuity Formula

The present value of a perpetuity is the sum of every discounted payment:

PV = C/(1 + r) + C/(1 + r)² + C/(1 + r)³ + …

This is a geometric series, and it simplifies to:

PV = C ÷ r

where:

• PV = present value
• C = cash flow per period
• r = discount rate per period

The formula assumes the first payment arrives one period from today. This standard form is sometimes called an ordinary perpetuity or a perpetuity-immediate.

Hypothetical example: An instrument pays $1,000 a year forever, and the investor requires a 5% annual return. Its present value is $1,000 ÷ 0.05 = $20,000. Paying more than $20,000 means accepting a return below 5%. Paying less means earning more.

Matching the rate to the payment frequency

The rate must match the payment interval. For monthly payments, divide by a monthly rate: a hypothetical $100 monthly payment discounted at 6% a year (0.5% a month) is worth $100 ÷ 0.005 = $20,000.

Perpetuity due

If the first payment arrives today rather than in one period, the stream is a perpetuity due. Its value is the immediate payment plus an ordinary perpetuity: PV = C + C ÷ r, or equivalently C × (1 + r) ÷ r. Using the $1,000 and 5% example, that is $21,000.

Growing Perpetuity Formula

A level perpetuity loses purchasing power if prices rise. A growing perpetuity increases each payment by a constant rate, which can offset some or all of that erosion, depending on how the growth rate compares with actual inflation.

The formula subtracts the growth rate from the discount rate:

PV = C₁ ÷ (r − g)

where:

• C₁ = the payment one period from now (not the payment just received)
• r = discount rate
• g = constant growth rate of the payments

Hypothetical example: The first payment, due in one year, is $1,000. Payments grow 2% a year, and the discount rate is 5%. The present value is $1,000 ÷ (0.05 − 0.02) = $33,333, well above the $20,000 value of the level version.

Two conditions matter. First, r must exceed g. If growth equals or exceeds the discount rate, the payments do not shrink in present value terms and the formula produces no meaningful result. Second, use C₁, not the most recent payment C₀. If the latest payment was $1,000, the next one is $1,000 × (1 + g).

Decreasing and arithmetic perpetuities

A payment that shrinks by a constant percentage d each period is a decreasing perpetuity. Its value is C₁ ÷ (r + d), the growing formula with a negative g.

An arithmetic perpetuity grows by a fixed dollar amount G rather than a fixed percentage. The payments are C, C + G, C + 2G, and so on, starting in one period. Its value is:

PV = C ÷ r + G ÷ r²

For example, payments of $1,000 that rise by $50 a year, discounted at 5%, are worth $20,000 + $50 ÷ 0.0025 = $40,000.

Where Perpetuities Appear

British consols

The best-known historical example is the British consol, short for “consolidated annuities.” The British government began issuing consols in 1751. They paid interest indefinitely and had no maturity date. The UK government redeemed the last of them in 2015.

Perpetual securities today

Few instruments are true perpetuities in practice. Some bonds are issued without a maturity date, but they usually give the issuer the right to call them back, which can end the payment stream. Preferred stock with a fixed dividend and no maturity is often valued as a perpetuity, as dividend ÷ required return.

Terminal value in valuation

The most common use of the perpetuity formula is in corporate valuation. In a discounted cash flow (DCF) model, a company is valued as a going concern, meaning it is assumed to operate indefinitely. Analysts forecast cash flows year by year for a set horizon, often five to ten years, and then treat everything after that as a growing perpetuity. This is the terminal value:

Terminal value = CFₙ × (1 + g) ÷ (r − g)

The discount rate r is typically the company’s cost of capital, such as its weighted average cost of capital (WACC).

Hypothetical example: A company is projected to generate $100,000 of cash flow in year 10. Its cost of capital is 8%, and its long-term growth rate is 3%.

1. Next year’s cash flow: $100,000 × 1.03 = $103,000.
2. Terminal value at the end of year 10: $103,000 ÷ (0.08 − 0.03) = $2,060,000.
3. Value today: $2,060,000 ÷ 1.08¹⁰ ≈ $954,000.

Step 3 is easy to skip. The perpetuity formula gives the value as of the date one period before the first payment it covers, here the end of year 10. To add it to the other discounted cash flows in a net present value or enterprise value calculation, it must be discounted back to today.

Limitations of a Perpetuity

Sensitivity to the discount rate. Because r sits alone in the denominator, small rate changes move the value sharply. Raising the discount rate on the $1,000 perpetuity from 5% to 6% cuts its value from $20,000 to about $16,667, a drop of roughly 17%. A level perpetuity has a Macaulay duration of (1 + r) ÷ r, about 21 years at 5%, which makes it highly sensitive to interest rates.

Sensitivity to the growth assumption. In the growing formula, the gap between r and g drives the result. In the terminal value example, raising g from 3% to 4% increases the year-10 value from $2.06 million to $2.6 million. Because terminal value often makes up a large share of a DCF valuation, this single input can dominate the conclusion.

Constant rates forever. The formulas assume that r and g never change. Real businesses do not grow at a steady rate, and required returns shift with markets.

Real-world payment risk. The formula treats every payment as certain. An issuer can default, call the security or stop paying a dividend, and none of these possibilities appears in C ÷ r unless they are built into the discount rate.

Perpetuity vs. Annuity

Both pay a series of cash flows at regular intervals. The difference is the end date. An annuity has a fixed number of payments, and its present value is:

PV = (C ÷ r) × [1 − 1 ÷ (1 + r)ⁿ]

As the number of periods n grows, the bracketed term approaches 1, and the annuity formula becomes C ÷ r. A perpetuity is the limit of an annuity with an infinite term, which is why it is sometimes called a perpetual annuity.

FAQs

The Bottom Line

The perpetuity formula matters most to analysts building DCF models and to investors valuing fixed-dividend preferred stock or perpetual bonds. It turns an endless payment stream into one present value, but the result swings sharply with small changes in the discount or growth rate. Treat it as an estimate built on constant assumptions, and account for call, default and dividend risk through the discount rate.