Perpetuity Calculator · Present Value of Perpetual Payments

Perpetuity Calculator

Value a never-ending stream of payments · Regular or growing · Payment & rate solver.

Currency
What do you want to calculate?
Regular perpetuity: PV = PMT ÷ r. A payment that never ends, discounted at a constant rate.
Inputs
$
% / yr
Formula: PV = PMT ÷ r  ·  For growing: PV = PMT ÷ (r − g), requires r > g.
Result
📊 Ready
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📊 Present Value
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💰 Payment / Year
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📈 Discount Rate
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🎯 PV / PMT Multiple
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Perpetuity vs Finite Annuity (How Much Value Comes From "Forever")
Years to reach 90% of perpetuity value —
Years to reach 95% of perpetuity value —
Years to reach 99% of perpetuity value —
Value from first 30 years —
Value from years 31 to ∞ —
💡 Interpretation
Enter values to see the perpetuity analysis.
Perpetuity vs N-Year Annuity
DurationPresent Value% of PerpetuityRemaining Tail Value

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Image of Faiq Ur Rahman, CEO & Founder Toolraxy

Faiq Ur Rahman

Founder & CEO, Toolraxy

Faiq Ur Rahman is a web designer, digital product developer, and founder of Toolraxy, a growing platform of web-based calculators and utility tools. He specializes in building structured, user-friendly tools focused on health, finance, productivity, and everyday problem-solving.

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Introduction

A payment that continues forever sounds like it should be worth an infinite amount, but the math says otherwise. Distant future dollars are discounted so heavily that their contribution to present value collapses toward nothing. This perpetuity calculator makes that intuition concrete. Enter a payment stream and a discount rate, and the tool produces a finite present value, the PV-to-payment multiple, and a breakdown showing what fraction of the value comes from the first thirty years versus everything after. It’s used by analysts building terminal value estimates in discounted cash flow models, investors pricing preferred stock or perpetual bonds, and anyone working through endowment or scholarship fund sizing.

 

How to Use the Perpetuity Calculator

  1. Pick the currency your payment stream is denominated in the symbol carries through to every money output.

  2. Choose a mode: Regular PV, Growing PV, Payment, or Required Rate.

  3. In Regular PV mode, enter the annual payment and the discount rate.

  4. For Growing PV, add the growth rate the payments rise at each year. The discount rate must exceed the growth rate.

  5. In Payment mode, enter a lump sum and a discount rate to see what annual payment it can sustain forever.

  6. In Required Rate mode, provide the payment and the present value to back out the implied discount rate.

  7. Click a quick-example button, preferred stock, UK Consol, endowment, growing perpetuity, ground rent, or scholarship fund to load a realistic scenario.

  8. Review the PV/PMT multiple and scroll to the comparison chart to see what fraction of the perpetuity value comes from finite horizons.

 

How the Perpetuity Calculator Formula Works

Four related formulas drive the calculator, one for each mode.

Formula: Regular perpetuity — PV = PMT ÷ r

Formula: Growing perpetuity — PV = PMT ÷ (r − g)

Formula: Solve for payment — PMT = PV × r

Formula: Solve for required rate — r = PMT ÷ PV

Where PMT is the payment per year, r is the annual discount rate as a decimal, and g is the annual growth rate of the payments.

The multiple PV ÷ PMT is simply 1 ÷ r, which is why a 5% discount rate produces a 20× multiple and a 4% rate produces a 25× multiple. Lower rates create higher multiples — the same relationship that explains why bond prices rise when yields fall.

The growing version requires r > g. If the growth rate equals or exceeds the discount rate, the denominator collapses to zero or goes negative, and the present value diverges to infinity. The tool flags this case explicitly rather than returning a nonsensical number. A growing payment stream that outpaces its discount rate has no finite value, because each successive payment is worth more in present-value terms than the one before it.

Rate and payment inputs are treated as positive figures. The comparison table uses the standard annuity formula PV = PMT × (1 − (1 + r)^−n) ÷ r to compute finite-horizon values side by side with the perpetuity. Years-to-percentage milestones are derived from −ln(1 − target) ÷ ln(1 + r), which gives the holding period needed to capture a given share of the perpetuity’s total value.

 

Worked Example

Suppose a preferred stock pays a $4,000 annual dividend and your required return is 5.5%. You want to know what the dividend stream is worth today and how that value breaks down across time.

Present value:
PV = 4,000 ÷ 0.055 = $72,727

Multiple: 72,727 ÷ 4,000 = 18.18×

The chart compares this against finite annuities of the same payment:

  • 10-year annuity: 4,000 × (1 − 1.055^−10) ÷ 0.055 ≈ $30,153 (41.5% of perpetuity)

  • 30-year annuity: 4,000 × (1 − 1.055^−30) ÷ 0.055 ≈ $58,612 (80.6%)

  • 50-year annuity: 4,000 × (1 − 1.055^−50) ÷ 0.055 ≈ $67,605 (93.0%)

  • 100-year annuity: 4,000 × (1 − 1.055^−100) ÷ 0.055 ≈ $72,183 (99.3%)

The first thirty years capture 80.6% of the perpetuity’s value. Add another twenty years and you reach 93%. The remaining 7% comes from years 51 through infinity, a span that never ends but contributes only about $5,100 in present-value terms. At a 5.5% discount rate, it takes roughly 40 years to capture 90% of the value, 53 years for 95%, and about 82 years for 99%.

The takeaway: an infinite payment stream behaves, in valuation terms, almost like a very long finite annuity. The practical implication is that terminal value estimates in DCF models are not especially sensitive to the “forever” assumption they’re sensitive to the discount rate you pick.

Frequently Asked Questions

What exactly is a perpetuity?

A perpetuity is a series of equal payments that continues forever, with no end date. A perpetual government bond like a UK Consol pays interest indefinitely; a scholarship fund pays out a fixed amount each year in perpetuity. Despite the infinite time span, the present value is finite as long as the discount rate is positive.

 

Why does a perpetuity have a finite value if it pays forever?

Because the present value of a payment shrinks with time. A dollar paid in fifty years is worth very little today at any reasonable discount rate. When you sum the present values of every future payment, the series converges to a finite number,  PMT ÷ r  because the terms get smaller fast enough.

 

How does a growing perpetuity differ from a regular one?

In a growing perpetuity, the payment increases each year at a fixed growth rate. The formula becomes PV = PMT ÷ (r − g), which requires the discount rate to exceed the growth rate. This model is used for dividend discount analysis and terminal value calculations where the cash flow is expected to grow, not stay flat.

 

What happens if the growth rate exceeds the discount rate?

The present value becomes infinite. This is mathematically correct but financially meaningless, no real asset is worth an infinite amount. In practice, a growth rate at or above the discount rate signals an unrealistic assumption, and the model should be recalibrated with more conservative inputs.

 

What’s a realistic discount rate for a perpetuity calculation?

For a government-backed payment stream, 3% to 5% is common. For corporate preferred stock or mature dividends, 5% to 8% is typical. For terminal value calculations in a DCF model, 8% to 12% is a common band. The rate should reflect the riskiness and stability of the underlying cash flow.

 

Why does the PV/PMT multiple matter?

The multiple is simply 1 ÷ r, and it tells you how many years’ worth of payments equal the present value. At a 5% discount rate, the multiple is 20×, meaning the entire perpetuity is worth twenty years of payments. At 10%, the multiple drops to 10×. Comparing the multiple across rate scenarios shows how sensitive the valuation is to the discount assumption.

 

How much of a perpetuity’s value comes from the first thirty years?

At a 5% discount rate, the first thirty years capture roughly 77% of the total value. At 3%, the figure is closer to 60%. At 10%, it climbs to about 94%. The pattern is consistent: lower discount rates stretch the value further into the future, while higher rates concentrate it in the near term.

 

Can a perpetuity be used to value a business?

It’s used for the terminal value component of a DCF, not for the entire business. After an explicit forecast period of five to ten years, analysts assume the business generates a growing perpetuity of free cash flows into the indefinite future. That terminal value is then discounted back to the present along with the explicit period cash flows.

 

What’s the difference between a perpetuity and an annuity?

An annuity runs for a fixed number of periods, twenty years, thirty years, until death. A perpetuity never ends. The present value formula is nearly identical: the annuity has an extra term, (1 − (1 + r)^−n), which vanishes as n grows. This is why a very long annuity (100+ years) looks numerically similar to a perpetuity.

 

How is the payment for a perpetuity calculated from a lump sum?

Divide the lump sum by the PV/PMT multiple, which equals multiplying the lump sum by the discount rate. On $500,000 at a 5% rate, the sustainable annual payment is $25,000. On the same sum at 4%, it’s $20,000. The relationship is linear in the rate.

 

Why does the calculator flag unusual rates?

If the rate is below zero or above 30%, the tool flags it because these inputs are outside the range where a perpetuity valuation is meaningful. Negative rates imply nonsensical results, and rates above 30% produce multiples so small that the perpetuity behaves almost like a one-year payment.

 

Can I use this for preferred stock valuation?

Yes. Preferred stock with a fixed dividend and no maturity date behaves like a perpetuity. Enter the annual dividend as PMT and your required return as the rate. The output is the fair value of the preferred share today. Note that the model assumes the dividend never changes, if the dividend is expected to grow, switch to Growing PV mode.

Financial Disclaimer

This perpetuity calculator is an educational tool and does not constitute investment, tax, or financial advice. The model assumes a constant payment (or a constantly growing one) and a constant discount rate over an infinite horizon, both are simplifications. Real cash flows fluctuate, and real discount rates change over time. Valuations are highly sensitive to the inputs, particularly the discount rate and any assumed growth rate. Consult a qualified financial professional before making investment decisions based on perpetuity valuations.

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