Perpetuity Calculator

Find the present value of a perpetuity - a fixed or growing cash flow that continues forever - using the discount rate.

Advanced payment frequency and valuation timing

Growing and inverse payment input

Matching rate and dollar basis

Finite display horizon (number of payments)

Payment frequency

Additional delay before the stream (years)

Level Perpetuity Inputs

First actual receipt (C1, per payment)

One actual payment per selected period, continuing indefinitely.

Discount Rate (r)

Effective annual required return; converted to the selected payment period.

Result

Present Value

Enter a payment and discount rate to see your result.

PV = C1 / r | PV = -

Growing Perpetuity Inputs

First actual receipt (C1, per payment)

First actual receipt. Later payments use the converted period growth rate.

Discount Rate (r)

Growth Rate (g)

Effective annual growth quote; converted to period growth. Must be less than the annual discount rate.

Result

Present Value

Enter a payment, discount rate, and growth rate to see your result.

PV = C1 / (r - g) | PV = -

Solve For

Present Value (PV)

First actual receipt (C1, per payment)

Discount Rate (r)

Growth Rate (g) - Optional

Leave at 0 to solve a level perpetuity.

Result

Present Value

Choose what to solve for, then fill in the other two values.

PV = C1 / (r - g)

Finite receipts and the remaining present-value tail

PV of displayed payments

Remaining discounted tail

Infinite present value

Payment times and present-value table

Finite table values do not replace the infinite valuation. Unrepresentable future receipts display N/A; discounted values retain their own numerical range.

Actual receipt times and time-zero present values (USD)
Payment period	Years from valuation t=0	Payment	Discounted payment	Cumulative PV	Remaining PV tail

Infinite value with additional precision: USD.

Compare your own discount and growth pairs

Editable assumptions, not forecasts. Invalid pairs have no finite valuation or ranking.

Comparison pair 1Effective annual discount rate (%)Effective annual payment growth (%)

Comparison pair 2Effective annual discount rate (%)Effective annual payment growth (%)

Matched payment, timing and dollar-basis assumptions
Pair	Annual discount	Annual growth	Status	First receipt	Infinite PV	PV of first N payments	Remaining PV tail

The CSV includes each valid pair's complete finite payment ledger and preserves invalid convergence status.

Include my comparison rate/growth pairs

4 min read 4 steps 7 terms 3 examples 6 FAQs PV = C1 / r

Receipts continue indefinitely under fixed matching effective annual assumptions.

YMYL Reviewed by Calculover Editorial Review · Updated 2026-08-11 · 2 sources

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Walk-through

How to Use This Calculator

4 steps

Choose a Tab

Use Present Value for a level (non-growing) perpetuity, Growing Perpetuity for a payment that increases every year, or Solve to work backward from a present value.

Enter the Payment and Discount Rate

Enter C1 per payment or choose C0. Advanced sets frequency and extra delay; results show the first receipt time and conversion from matching effective annual rates.

Add a Growth Rate If Needed

Annual default: C1 is the first receipt at the end of year 1. Rates r and g are matching annual decimals. Require r>g, r>−1 and g≥−1.

Read the Result

The nominal/real selector labels the annual rate and dollar basis; it does not convert inflation. Use matching inputs.

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Reference

Formula & Methodology

3 formulas

Level Perpetuity

PV = C1 / r

Default annual, no-delay convention. Advanced timing uses the converted period rates and discounts the stream back to t=0. C1 is the first receipt at the end of year 1. Rates r and g are matching annual decimals. Require r>g, r>−1 and g≥−1. A zero stream has zero value under this convergent convention; zero inputs can leave inverse solutions indeterminate. Entered payment is C1, the receipt at the end of year 1.

Periodic and deferred growing value

rₚ=(1+R)^(1/f)−1; gₚ=(1+G)^(1/f)−1; PV₀=C1/(rₚ−gₚ)×(1+R)^(−D)

Annual no-delay inverse formulas

C1 = PV × (r − g); C0 = C1 / (1 + g); r = C1 / PV + g

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YMYL

Trust, Methodology & Sources

Reviewed by Calculover Editorial Review · Updated 2026-08-11 · 2 sources

Editorial accountability

Author: Quinn Sumner - Editor

Owner: Quinn Sumner - Editorial owner

Reviewer: Calculover Editorial Review - Source and limitation review

Last reviewed: 2026-08-11

Last verified: 2026-08-11

Data effective: 2026-08-11

Methodology

Rates are effective annual quotes on the chosen nominal or real basis. C1 is one actual payment, not an annual total. With f payments/year and D additional years of delay, the first receipt is at D+1/f years; value is at t=0. Equal periods are modeled, not calendar dates. C0 grows once before that first receipt; no extra waiting-period growth or inflation conversion is inferred. Supported delay: 0–1000 years. Default annual, no-delay convention. Advanced timing uses the converted period rates and discounts the stream back to t=0. With annual payments and no delay, the first payment occurs at the end of year 1. Rates r and g are matching annual decimals. Require r>g, r>−1 and g≥−1. A zero stream has zero value under this convergent convention; zero inputs can leave inverse solutions indeterminate. The nominal/real selector labels the annual rate and dollar basis; it does not convert inflation. Use matching inputs. The displayed 1–200 payment horizon only divides the infinite PV into a finite subtotal and an analytical remaining tail. Unrepresentable values remain unavailable.

Assumption: Perpetuity Calculator assumes the entered payment, discount rate, and (where applicable) growth rate remain constant indefinitely; it does not model changing rates or payment schedules over time.

Assumption: The model assumes no default or credit risk on the underlying cash flow stream.

Assumption: Default annual, no-delay convention. Advanced timing uses the converted period rates and discounts the stream back to t=0. With annual payments and no delay, the first payment occurs at the end of year 1. Rates r and g are matching annual decimals. Require r>g, r>−1 and g≥−1. A zero stream has zero value under this convergent convention; zero inputs can leave inverse solutions indeterminate.

Limitations & guidance

Perpetuity Calculator produces a point-in-time estimate based on the inputs provided and does not project how discount rates or growth rates might change over time.

Real cash flow streams rarely continue literally forever; the perpetuity model is a simplifying approximation best suited to very long-duration or theoretically infinite payment streams.

This calculator is an educational tool designed to illustrate the perpetuity present-value formula and does not constitute investment advice, a financial recommendation, or a solicitation to buy or sell any security. Real cash flows carry default, reinvestment, and inflation risk that the simple perpetuity model does not capture. Consult a licensed Certified Financial Planner (CFP), Chartered Financial Analyst (CFA), or corporate finance professional before making valuation or investment decisions.

Professional guidance: Perpetuity Calculator is for educational purposes only and does not constitute investment, tax, accounting, or legal advice. Consult a licensed Certified Financial Planner (CFP), Chartered Financial Analyst (CFA), or corporate finance professional before making valuation or investment decisions.

Primary sources

Gordon, M.J. & Shapiro, E. (1956), Capital Equipment Analysis: The Required Rate of Profit - Management Science

Time Value of Money - CFA Institute

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Glossary

Key Terms Explained

7 terms

Perpetuity A series of equal (or steadily growing) cash flows that continues forever, with no maturity date or final payment.

Present Value The value today of a future cash flow (or stream of cash flows), found by discounting each payment back to the present using a required rate of return.

Discount Rate The annual rate of return used to convert future cash flows into today's dollars. A higher discount rate produces a lower present value.

Growing Perpetuity Annual default: C1 is the first receipt at the end of year 1. Rates r and g are matching annual decimals. Require r>g, r>−1 and g≥−1.

Gordon Growth Model The application of the growing-perpetuity formula to stock valuation, treating a company's dividend stream as a perpetuity that grows at a constant long-run rate.

Consol A perpetual bond, historically issued by the UK government, that pays a fixed coupon forever with no repayment of principal — the classic real-world example of a level perpetuity.

Cash Flow Choose C1 to enter the first actual payment, or choose C0 to enter its reference amount one payment period earlier. The payment-period growth rate is applied once to C0 to obtain C1. There is no payment at time zero.

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Scenarios

Real-World Examples

3 worked examples

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Level Perpetuity

Valuing a $1,000/year cash flow at 5%

Payment (C) $1,000 Discount Rate (r) 5%

PV = $1,000 / 0.05 = $20,000. A payment of $1,000 every year, forever, is worth $20,000 today at a 5% discount rate — the classic level-perpetuity result used to check any perpetuity calculation.

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Growing Perpetuity

Valuing a dividend that grows 2% a year at a 5% discount rate

Payment (C) $1,000 Discount Rate (r) 5% Growth Rate (g) 2%

PV = $1,000 / (0.05 - 0.02) = $33,333.33. Because the payment grows each year, the present value is substantially higher than the level-perpetuity case — this is the Gordon Growth Model used to value dividend-paying stocks.

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Solve for Payment

Finding the payment needed to reach a $20,000 present value

Present Value (PV) $20,000 Discount Rate (r) 5% Growth Rate (g) 0%

C = $20,000 x 0.05 = $1,000. Rearranging the formula shows that a $20,000 present value at a 5% discount rate implies a level payment of $1,000 per year — useful for checking what income stream a lump sum could support.

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Reference

Cite This Calculator

APA & MLA

Use either format to cite this calculator in a paper, report, or resource list.

APA

MLA

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Deep Dive

Understanding Perpetuities: Valuing Cash Flows That Never End

4 chapters · 4 min read

Receipts continue indefinitely under fixed matching effective annual assumptions. The displayed payments and remaining analytical tail are parts of the infinite time-zero value, not investment advice.

Why an Infinite Cash Flow Has a Finite Value

Default annual, no-delay convention. Advanced timing uses the converted period rates and discounts the stream back to t=0.

C1 is the first receipt at the end of year 1. Rates r and g are matching annual decimals. Require r>g, r>−1 and g≥−1. A zero stream has zero value under this convergent convention; zero inputs can leave inverse solutions indeterminate.

Entered payment is C1, the receipt at the end of year 1.

Level vs. Growing Perpetuities

Default annual, no-delay convention. Advanced timing uses the converted period rates and discounts the stream back to t=0.

Entered payment is C0, a reference before the first receipt. C1 = C0 × (1 + g); no time-zero receipt is added.

Real-World Uses: Consols, Preferred Stock, and Terminal Value

Default annual, no-delay convention. Advanced timing uses the converted period rates and discounts the stream back to t=0.

Annual end-year receipts continue indefinitely under fixed matching rate assumptions. The finite displayed subtotal is separate from infinite present value; this is not an investment recommendation.

The nominal/real selector labels the annual rate and dollar basis; it does not convert inflation. Use matching inputs. The displayed 1–200 payment horizon only divides the infinite PV into a finite subtotal and an analytical remaining tail. Unrepresentable values remain unavailable.

Limitations of the Perpetuity Model

Default annual, no-delay convention. Advanced timing uses the converted period rates and discounts the stream back to t=0.

A small r−g spread makes the indefinite value highly sensitive. No arbitrary cap is applied.

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Questions

Frequently Asked Questions

6 questions

What is the formula for a perpetuity?+

For a level perpetuity with annual payments and no delay, PV = C1 / r. C1 is the payment at the end of year 1, and r is the effective annual discount rate. For a growing stream, use the growing-perpetuity formula and choose whether to enter C1 or its one-period-earlier reference C0.

What is the formula for a growing perpetuity?+

Annual default: PV = C1 / (r − g). C1 is the first receipt at the end of year 1. Rates r and g are matching annual decimals. Require r>g, r>−1 and g≥−1. A zero stream has zero value under this convergent convention; zero inputs can leave inverse solutions indeterminate.

Does the discount rate have to exceed the growth rate?+

Annual default: C1 is the first receipt at the end of year 1. Rates r and g are matching annual decimals. Require r>g, r>−1 and g≥−1. A zero stream has zero value under this convergent convention; zero inputs can leave inverse solutions indeterminate.

What is a consol?+

A consol is a perpetual bond, historically issued by the UK government, that pays a fixed coupon forever with no maturity date or repayment of principal — the classic textbook example of a level perpetuity.

How is a perpetuity used to value stocks?+

The Gordon Growth Model treats a stock's expected dividend stream as a growing perpetuity, using PV = D1 / (r - g) to estimate intrinsic value from next year's expected dividend, the required return, and the long-run dividend growth rate.

What units does this calculator use?+

Annual default: The nominal/real selector labels the annual rate and dollar basis; it does not convert inflation. Use matching inputs. The displayed 1–200 payment horizon only divides the infinite PV into a finite subtotal and an analytical remaining tail. Unrepresentable values remain unavailable.

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Perpetuity Calculator

Finite receipts and the remaining present-value tail

Compare your own discount and growth pairs

How to Use This Calculator

Choose a Tab

Enter the Payment and Discount Rate

Add a Growth Rate If Needed

Read the Result

Formula & Methodology

Trust, Methodology & Sources

Key Terms Explained

Real-World Examples

Cite This Calculator

Understanding Perpetuities: Valuing Cash Flows That Never End

Why an Infinite Cash Flow Has a Finite Value

Level vs. Growing Perpetuities

Real-World Uses: Consols, Preferred Stock, and Terminal Value

Limitations of the Perpetuity Model

Frequently Asked Questions

Get a branded PDF of your results

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