Perpetuity Calculator

A perpetuity is a financial instrument that pays a fixed (or growing) cash flow at regular intervals indefinitely — in theory, forever. Unlike an ordinary annuity, which terminates after a set number of periods, a perpetuity has no maturity date. The concept sounds counterintuitive, but because of the time value of money, an infinite series of future payments still has a finite present value today.

The most common real-world examples of perpetuities include preferred stock dividends, consol bonds issued by the British government, endowment funds that pay annual grants, and certain scholarships or trust distributions. The Gordon Growth Model — the foundational framework for equity valuation — is mathematically identical to a growing perpetuity formula.

Our perpetuity calculator handles both the simple (fixed-payment) case and the growing-perpetuity case, so you can instantly price a dividend stream, evaluate an endowment, or stress-test valuations across a range of discount rates.

Understanding perpetuity present value is essential for finance students studying the CFA curriculum, investment analysts pricing dividend-paying stocks, and CFOs evaluating long-duration liabilities. The math is elegant: as long as you know the periodic payment and the required rate of return, a single division gives you the asset's intrinsic value.

There are two core perpetuity formulas. The simple perpetuity applies when the payment never changes. The growing perpetuity applies when each payment increases at a constant rate g forever. Both are derived by summing an infinite geometric series and then simplifying.

Simple Perpetuity: The present value equals the payment divided by the discount rate. A $10,000 annual payment at a 5% discount rate is worth $200,000 today — because $200,000 invested at 5% generates exactly $10,000 per year indefinitely.

Growing Perpetuity: When payments grow at a constant rate, the denominator shrinks to (r − g). The same $10,000 payment growing at 2% per year, discounted at 5%, is worth $333,333 — nearly 67% more than the simple case, which illustrates how sensitive perpetuity values are to even small changes in the growth assumption.

One critical constraint: the discount rate r must always exceed the growth rate g. If g ≥ r, the series diverges and the present value is theoretically infinite — a situation that cannot hold in equilibrium.

The perpetuity present value calculator on this page requires three inputs: Payment per Period, Discount Rate, and Growth Rate. All three can be adjusted with sliders for instant, real-time results.

• Payment per Period (PMT): The fixed cash flow paid at the end of each period. For preferred stock, this is the annual dividend. For an endowment, it might be the annual grant. Adjust from $1,000 to $500,000.
• Discount Rate (r): The required rate of return or opportunity cost of capital, expressed as a percentage. This reflects the risk of the cash flow stream. For riskier assets, use a higher discount rate; for government-backed instruments, use a lower one.
• Growth Rate (g): For a simple perpetuity, leave this at 0%. For a growing perpetuity (e.g., a dividend expected to grow in line with inflation), set this to the expected long-run growth rate. The calculator automatically flags and disables the growing-perpetuity result when g ≥ r.

The results section displays the PV of Simple Perpetuity, the PV of Growing Perpetuity, the Effective Yield (which simply equals the payment divided by the simple PV, confirming the discount rate), and a sensitivity table showing how the present value changes across discount rates from 3% to 12%. This sensitivity analysis is especially useful when you are unsure about the exact cost of capital and want to see a range of plausible valuations.

Perpetuity math shows up in more everyday financial contexts than most people realise. Here are the most important use cases where getting the calculation right matters.

Preferred Stock Valuation

Most preferred shares pay a fixed quarterly or annual dividend with no scheduled maturity. A share paying a $5 annual dividend with a required yield of 6% is worth exactly $83.33 using the simple perpetuity formula. If the company's creditworthiness improves and the required yield falls to 5%, that same share is worth $100.00 — a 20% price increase purely from the discount rate moving.

Consol Bonds

The United Kingdom issued perpetual government bonds (Consols) for centuries. A Consol paying £25 per year with a 2.5% yield would trade at £1,000. These instruments are textbook examples used in every introductory finance course precisely because the simple perpetuity formula prices them exactly.

Gordon Growth Model (Equity Valuation)

The Gordon Growth Model — used daily by equity analysts worldwide — is simply PV = D₁ / (r − g), where D₁ is next year's dividend. This is the growing perpetuity formula applied to stocks. If a mature blue-chip company is expected to pay $2.50 in dividends next year, with dividends growing at 3% annually and the required return is 8%, the model gives an intrinsic value of $50 per share.

University Endowments and Scholarships

A donor wanting to fund a $20,000 annual scholarship in perpetuity, assuming a 4% endowment return, needs to contribute $500,000 today. If the scholarship is indexed to inflation at 2.5%, the required donation rises to $20,000 / (0.04 − 0.025) = $1,333,333. The perpetuity calculator makes this sizing exercise instantaneous.

Real Estate Cap Rates

Commercial real estate valuation uses the capitalization rate (cap rate) as the discount rate and net operating income (NOI) as the payment. A property generating $120,000 NOI per year with a 6% cap rate is valued at $2,000,000 — a direct application of the simple perpetuity formula.

Perpetuity valuations are highly sensitive to the discount rate. Because the present value is inversely proportional to r (or r − g), a small change in either rate produces a large change in value. This is why analysts always present a range of values rather than a single point estimate when using perpetuity-based models.

The most common mistake when using a growing perpetuity formula is setting the growth rate too close to the discount rate. As g approaches r, the denominator approaches zero and the present value explodes toward infinity. In practice, long-run sustainable growth rates should rarely exceed 2–3% (roughly nominal GDP growth) for a mature business.

A second common error is confusing real and nominal rates. If you use a nominal discount rate, the growth rate must also be nominal (and vice versa). Mixing real and nominal inputs will produce a systematically incorrect result.

Third, the perpetuity model assumes that the first payment occurs exactly one period from today (an ordinary annuity convention). If the first payment is today (perpetuity due), you would multiply the result by (1 + r) to account for the timing difference.

Finally, remember that no asset truly pays forever. The perpetuity model is a simplifying assumption. When evaluating a business or asset with a finite life, a discounted cash flow model with a terminal value (often modeled as a growing perpetuity) is more appropriate. The calculator's sensitivity table helps you understand the range of values implied by different discount-rate assumptions, which is essential for any rigorous financial analysis.