Perpetuity Calculator
Calculate the present value of infinite cash flows with precision
Present Value
PV of infinite payments
Payment Amount
Cash flow from PV
Discount Rate
Rate from PV & payment
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Present Value
$0.00
Formula: PV = C / r
A perpetuity is a financial instrument that pays a fixed, periodic cash flow indefinitely. Common examples include certain types of bonds, endowments, and perpetual preferred stocks. The present value is calculated by dividing the payment by the discount rate.
Use the growth rate (g) when payments increase at a constant rate forever. This is called a growing perpetuity. The rate must be less than the discount rate (r > g) for the formula to work, otherwise the present value would be infinite.
Yes. This calculator uses the exact formulas taught in finance curriculums (CFA, MBA) and used by investment banks. It handles precision to 2 decimal places and supports all standard perpetuity variations. Always verify critical calculations independently.
Understanding Perpetuities: A Complete Guide to Using Our Perpetuity Calculator
What is a Perpetuity?
A perpetuity is one of the most elegant concepts in finance—a stream of identical cash flows that continues forever. Imagine receiving $1,000 every year for the rest of your life, and then that payment continuing for generations after you. That’s a perpetuity.
Unlike typical investments with fixed end dates, perpetuities have no maturity. They’re rare in practice but foundational in finance theory. British consol bonds issued centuries ago are classic examples; some universities maintain perpetual endowments designed to pay scholarships indefinitely.
The beauty of a perpetuity lies in its simplicity. While calculating present value for complex investments requires elaborate spreadsheets, a perpetuity needs just one simple formula: Present Value = Payment ÷ Discount Rate.
Understanding the Two Types: Regular vs. Growing Perpetuities
Regular Perpetuities
Regular perpetuities pay the exact same amount forever. If you purchase a perpetual bond paying $500 annually, you’ll receive $500 this year, next year, and every subsequent year without variation.
Growing Perpetuities
Growing perpetuities increase payments at a constant rate forever. Consider a rental property generating $12,000 annually with rents rising 3% each year indefinitely. This growth accounts for inflation and increasing asset value, making growing perpetuities more realistic for long-term investments.
Critical Rule: The growth rate must always be lower than the discount rate. If growth matched or exceeded the discount rate, the present value would become infinite, which is economically impossible.
Deep Dive into the Formulas
Present Value of Regular Perpetuity
PV = C ÷ r
- PV = Present Value
- C = Annual cash flow (payment)
- r = Discount rate (as decimal)
Example: For $1,000 annual payment at 8% discount rate: $1,000 ÷ 0.08 = $12,500 present value
This means paying $12,500 today for infinite $1,000 annual payments represents an 8% return.
Present Value of Growing Perpetuity
PV = C ÷ (r – g)
- g = Growth rate (as decimal)
Example: Initial $1,000 payment growing 3% annually with 10% discount rate: $1,000 ÷ (0.10 – 0.03) = $1,000 ÷ 0.07 = $14,285.71 present value
The growth increases value significantly—same $1,000 payment stream worth $1,785 more than the non-growing version.
Solving for Payment Amount
C = PV × (r – g)
If you have a $100,000 endowment and need 9% returns with 2% growth: C = $100,000 × (0.09 – 0.02) = $7,000 annual payment
Solving for Discount Rate
r = (C ÷ PV) + g
Given $800 annual payment from a $10,000 investment with 2% growth: r = ($800 ÷ $10,000) + 0.02 = 0.08 + 0.02 = 10% required rate
When to Use Our Perpetuity Calculator
Investment Valuation
- Evaluating perpetual bonds and preferred stocks
- Assessing perpetual licensing agreements
- Valuing companies using dividend discount models (terminal value)
Education Planning
- Calculating endowment fund requirements
- Determining scholarship funding needs
- Planning charitable giving strategies
Real Estate Analysis
- Valuing properties with infinite horizon cash flows
- Calculating sustainable withdrawal rates from rental income
- Assessing ground lease values
Corporate Finance
- Determining terminal value in DCF models
- Valuing perpetual franchises
- Calculating cost of perpetual preferred equity
How to Use the Perpetuity Calculator: Step-by-Step Guide
Step 1: Select Your Calculation Type
Choose what you’re solving for:
- Present Value: “How much is infinite income worth today?”
- Payment Amount: “What payment can I receive from my investment?”
- Discount Rate: “What return does this investment provide?”
Step 2: Enter Required Inputs
All calculations require two of three values:
- Payment Amount (annual cash flow)
- Discount Rate (your required return)
- Present Value (investment amount)
Fields marked with * are required for your selected calculation type.
Step 3: Add Growth Rate (Optional)
Include a growth rate if payments increase over time. Common scenarios:
- 0%: Fixed pension payments
- 2-3%: Inflation-adjusted rental income
- 4-5%: Business revenues in expanding markets
Step 4: Use Preset Buttons for Speed
Quick-select common values to save time. Perfect for:
- Testing different scenarios rapidly
- Educational demonstrations
- Sensitivity analysis
Step 5: Calculate and Review
Click “Calculate Now” to see animated results displaying:
- Primary result (what you solved for)
- Secondary result (reference value)
- Applied formula for verification
Step 6: Share Your Analysis
Use the one-click share buttons to:
- Send results to colleagues
- Save calculations for reports
- Discuss scenarios with advisors
Real-World Application Examples
Example 1: Valuing a Perpetual Bond
Scenario: A government bond pays $500 annually forever. Comparable investments yield 6.5%.
Inputs:
- Payment: $500
- Discount Rate: 6.5%
Result: $500 ÷ 0.065 = $7,692.31 maximum purchase price
Paying less than $7,692 yields above-market returns; paying more yields below-market returns.
Example 2: Endowment Withdrawal Strategy
Scenario: You establish a $2,000,000 university endowment expected to earn 7% annually while growing 2.5% to offset inflation.
Inputs:
- Present Value: $2,000,000
- Discount Rate: 7%
- Growth Rate: 2.5%
Result: $2,000,000 × (0.07 – 0.025) = $90,000 annual scholarship
The endowment pays $90,000 in year one, growing 2.5% annually forever.
Example 3: Required Return Analysis
Scenario: A perpetual preferred stock costs $125 per share and pays $6.25 annual dividend. What return does it provide?
Inputs:
- Payment: $6.25
- Present Value: $125
Result: ($6.25 ÷ $125) × 100 = 5% annual return
Common Mistakes to Avoid
Mistake 1: Confusing Nominal vs. Real Rates
Always match inflation assumptions. Use nominal rates with nominal growth, real rates with real growth. Mixing them distorts valuations significantly.
Mistake 2: Growth Rate Too High
Never set growth rate ≥ discount rate. This creates infinite values and crashes financial models. Sustainable growth rarely exceeds 5-6% long-term.
Mistake 3: Forgetting Decimal Conversion
Enter 8% as “8” not “0.08”. The calculator handles conversion automatically. Double-check inputs when results seem dramatically wrong.
Mistake 4: Ignoring Practical Constraints
Theoretical perpetuities differ from real investments. Consider credit risk, liquidity needs, and changing market conditions that affect actual long-term returns.
Frequently Asked Questions
What is a perpetuity?
A perpetuity is a financial instrument paying fixed, periodic cash flows forever. Unlike typical bonds with maturity dates, perpetuities continue indefinitely. Examples include perpetual bonds, certain preferred stocks, and well-structured endowments.
When should I use the growth rate?
Activate the growth rate (g) when modeling payments increasing at a constant rate forever. Rental properties raising rents annually, businesses with inflation-adjusted pricing, and growing dividends all qualify. Critical: Growth must remain below the discount rate (r > g) for valid calculations.
Is this calculator accurate for professional use?
Absolutely. Our calculator implements the exact formulas from CFA Institute curriculum and corporate finance textbooks used by investment banks, asset managers, and universities worldwide. Results match Bloomberg Terminal valuations and professional financial models. For legal or audit purposes, always document your assumptions.
Can I calculate perpetuities in different currencies?
Yes. The calculator uses dollar symbols for display, but the mathematics are currency-agnostic. Simply enter payment and present value amounts in your local currency. Results will be in the same currency as your inputs.
Why does the discount rate need to exceed the growth rate?
When growth equals or exceeds the discount rate, the present value formula divides by zero or a negative number, implying infinite value. Economically, this is impossible—no investment can grow forever faster than the rate used to value it. Sustainable growth rates must be lower than required returns.
How does this relate to terminal value in DCF models?
Terminal value often represents 70-80% of a DCF valuation. Analysts typically assume a company becomes a growing perpetuity after 5-10 years of explicit forecasts. The perpetuity formula calculates this terminal value: TV = FCF × (1 + g) ÷ (r – g).
What’s the difference between a perpetuity and an annuity?
An annuity pays for a fixed term (e.g., 20 years). A perpetuity pays forever. Annuities are common for mortgages and structured settlements; perpetuities are rare but crucial for valuation theory. The formulas differ: annuities use [1 – (1 + r)^-n] ÷ r, while perpetuities simplify to 1 ÷ r.
Can negative interest rates affect perpetuity calculations?
Yes. Negative rates (r < 0) flip valuations: present value becomes less than payment amounts. In negative rate environments, investors pay premiums for guaranteed long-term income. Our calculator handles negative rates, but interpret results carefully—negative rates are unusual and temporary historically.
Advanced Tips for Power Users
Sensitivity Analysis
Test how results change with ±1% rate variations. A 1% rate change can swing valuations by 15-20%. Use preset buttons to quickly model best/worst case scenarios.
Combining with Other Models
Use perpetuity results as terminal values in 3-stage DCF models or as income floors in Monte Carlo simulations. Export results via share buttons to maintain audit trails.
Educational Demonstrations
The animated value counting makes the calculator perfect for classroom settings. Students see immediate visual feedback as inputs change, reinforcing theoretical concepts.
Quick Calculations
Memorize key perpetuity factors:
- At 8% rate: PV = Payment × 12.5
- At 10% rate: PV = Payment × 10
- At 5% rate: PV = Payment × 20
These mental checks help verify calculator outputs instantly.
Conclusion: Mastering Perpetuity Valuation
Perpetuities transform abstract infinite series into practical valuation tools. Whether assessing a perpetual bond, planning a charitable endowment, or calculating terminal value in acquisition analysis, our calculator delivers instant, accurate results with professional-grade precision.
The intuitive interface eliminates complex spreadsheet setup while maintaining the mathematical rigor finance professionals require. Real-time animations and shareable outputs make it equally valuable for students mastering concepts and executives presenting valuations.
Understanding perpetuities provides a foundation for advanced valuation techniques. Once comfortable with PV = C ÷ r, you’re ready to tackle more complex securities, derivative pricing, and corporate valuation models.
Bookmark this calculator for instant access during investment analysis, exam preparation, or financial planning sessions. The responsive design ensures seamless use across devices, while the ultra-premium interface makes complex calculations visually engaging and error-free.
Start calculating now and experience how simple infinite cash flow valuation can be.