Perpetuity Calculator

Calculate the present value of infinite periodic cash flows

Understanding Perpetuity: The Infinite Cash Flow Stream

A perpetuity is a financial instrument that pays a constant stream of identical cash flows with no end date. Unlike bonds or loans that have a maturity date, perpetuities continue indefinitely. The concept of perpetuity is fundamental in finance, investment analysis, and business valuation, providing a theoretical framework for understanding the present value of infinite cash flows.

What is a Perpetuity?

A perpetuity is a type of annuity that continues forever, making regular payments at fixed intervals without termination. The most common examples include preferred stocks with fixed dividends, consol bonds (issued by the British government), and certain types of real estate investments where land generates perpetual rental income.

The value of a perpetuity is derived from the fact that even though payments continue indefinitely, their present value converges to a finite amount due to the time value of money. Each future payment is discounted, and payments far in the future contribute negligibly to the present value.

Types of Perpetuities

• Standard (Level) Perpetuity: Pays the same fixed amount at regular intervals forever. This is the simplest form where each payment is identical.
• Growing Perpetuity: Payments increase at a constant growth rate each period. This type is commonly used in stock valuation models like the Gordon Growth Model.
• Delayed Perpetuity: Payments begin at some future date rather than immediately, requiring adjustment for the delay period.
• Decreasing Perpetuity: Payments decline at a constant rate (negative growth), though this is less common in practical applications.

Perpetuity Formulas

Standard Perpetuity Formula

PV = C / r

Where:
PV = Present Value of the perpetuity
C = Periodic cash payment
r = Discount rate (per period)

This elegant formula shows that the present value of a perpetuity is simply the periodic payment divided by the discount rate. For example, a perpetuity paying $1,000 annually with a 5% discount rate has a present value of $1,000 / 0.05 = $20,000.

Growing Perpetuity Formula

PV = C / (r – g)

Where:
PV = Present Value of the growing perpetuity
C = Initial periodic payment
r = Discount rate (per period)
g = Growth rate (per period)

Important: r must be greater than g for the formula to work

The growing perpetuity formula accounts for payments that increase over time. This is particularly useful in stock valuation, where dividends are expected to grow with company earnings. The constraint that r > g ensures the present value converges to a finite number.

Real-World Applications of Perpetuity Calculations

1. Stock Valuation

The Gordon Growth Model uses the growing perpetuity formula to value stocks based on their expected future dividends. If a company pays annual dividends of $2 per share, the dividend is expected to grow at 3% annually, and the required rate of return is 8%, the stock's intrinsic value would be:

Example:
Initial Dividend (C) = $2.00
Discount Rate (r) = 8% = 0.08
Growth Rate (g) = 3% = 0.03

Stock Value = $2.00 / (0.08 – 0.03) = $2.00 / 0.05 = $40.00

2. Real Estate Valuation

Commercial properties generating rental income can be valued as perpetuities. If a property generates $50,000 in annual net operating income and the market capitalization rate is 6%, the property value using perpetuity logic would be:

Example:
Annual Net Operating Income (C) = $50,000
Capitalization Rate (r) = 6% = 0.06

Property Value = $50,000 / 0.06 = $833,333.33

3. Preferred Stock Valuation

Preferred stocks typically pay fixed dividends indefinitely, making them true perpetuities. A preferred stock paying $5 quarterly dividends with a required return of 8% annually (2% quarterly) would be valued as:

Example:
Quarterly Dividend (C) = $5.00
Quarterly Discount Rate (r) = 2% = 0.02

Preferred Stock Value = $5.00 / 0.02 = $250.00

4. Endowment Funds

Universities and charitable organizations use perpetuity calculations to determine how much principal is needed to generate a desired annual payout forever. If an organization wants to fund a $100,000 annual scholarship in perpetuity, assuming a 4% annual return:

Example:
Desired Annual Payment (C) = $100,000
Expected Return Rate (r) = 4% = 0.04

Required Endowment = $100,000 / 0.04 = $2,500,000

Key Factors Affecting Perpetuity Value

Discount Rate Sensitivity

The present value of a perpetuity is extremely sensitive to changes in the discount rate. Small changes in the rate can result in large changes in present value. For a $1,000 annual perpetuity:

• At 4% discount rate: PV = $1,000 / 0.04 = $25,000
• At 5% discount rate: PV = $1,000 / 0.05 = $20,000
• At 6% discount rate: PV = $1,000 / 0.06 = $16,667

A 1% increase in the discount rate (from 5% to 6%) reduces the perpetuity value by approximately 16.7%.

Growth Rate Impact

In growing perpetuities, the difference between the discount rate and growth rate determines the value. As the growth rate approaches the discount rate, the present value increases dramatically. Consider a $1,000 initial payment with a 7% discount rate:

• No growth (g = 0%): PV = $1,000 / 0.07 = $14,286
• 2% growth: PV = $1,000 / (0.07 – 0.02) = $20,000
• 4% growth: PV = $1,000 / (0.07 – 0.04) = $33,333
• 6% growth: PV = $1,000 / (0.07 – 0.06) = $100,000

Payment Frequency

More frequent payments increase the present value of a perpetuity when comparing the same annual total. This occurs because earlier payments have less time to discount. However, the discount rate must be adjusted to match the payment frequency.

Limitations and Considerations

Theoretical vs. Practical Application

While perpetuities provide elegant mathematical solutions, true perpetuities are rare in practice. Most "perpetual" instruments have some form of call provision, redemption clause, or finite expected life. The perpetuity model serves as an approximation for very long-lived cash flow streams.

Assumption of Constant Rates

Perpetuity formulas assume constant discount and growth rates, which rarely hold in reality. Economic conditions, company performance, and market rates fluctuate over time. Sensitivity analysis and scenario planning are essential when using perpetuity valuations for decision-making.

Growth Rate Constraints

For growing perpetuities, the growth rate must be less than the discount rate. If growth equals or exceeds the discount rate, the formula produces infinite or negative values, which are economically meaningless. In practice, sustainable growth rates are typically much lower than discount rates.

Advanced Perpetuity Concepts

Perpetuity with Delayed Start

When perpetuity payments begin at some future date, the present value must be discounted back to today. If payments begin in year 3, first calculate the perpetuity value at year 2, then discount that value back to present:

PV = [C / r] / (1 + r)^(n-1)

Where n is the year the first payment occurs

Multiple Growth Stages

Some valuations use a two-stage model: high growth for a finite period, then perpetual growth at a lower rate. This is common in startup valuations where rapid initial growth eventually stabilizes to sustainable long-term rates.

Inflation-Adjusted Perpetuities

Real (inflation-adjusted) perpetuities maintain purchasing power over time. The growth rate in these models represents real growth plus inflation, ensuring the payment stream keeps pace with price increases.

Practical Tips for Using Perpetuity Calculations

Selecting Appropriate Discount Rates

• Risk-free rate: Use government bond yields for low-risk perpetuities
• Required return: For equities, use CAPM or similar models to determine risk-adjusted rates
• Opportunity cost: Consider alternative investments with similar risk profiles
• Market rates: Observe actual transaction prices for comparable perpetual instruments

Estimating Growth Rates

• Historical growth analysis (use caution, past ≠ future)
• Industry and economic growth projections
• Company-specific factors (competitive position, market share)
• Conservative assumptions (sustainability over infinite horizon)

Validation and Sensitivity Testing

Always perform sensitivity analysis by varying key assumptions. Calculate present values across a range of discount and growth rates to understand the valuation's sensitivity. Compare perpetuity valuations with other methods (DCF, comparables) to ensure reasonableness.

Common Mistakes to Avoid

• Mismatched periods: Ensure payment frequency, discount rate, and growth rate all use the same time period
• Unrealistic growth assumptions: Growth rates exceeding long-term economic growth are usually unsustainable
• Ignoring risk: Higher-risk perpetuities require higher discount rates
• Nominal vs. real rates: Be consistent in using either nominal or inflation-adjusted figures
• Forgetting tax implications: After-tax cash flows require after-tax discount rates

Conclusion

Perpetuity calculations provide a powerful tool for valuing infinite cash flow streams, with applications spanning stock valuation, real estate analysis, and endowment planning. While the mathematics is straightforward, proper application requires careful consideration of discount rates, growth assumptions, and real-world constraints. By understanding both the theoretical foundations and practical limitations of perpetuity models, you can make more informed financial decisions and valuations.

Use this perpetuity calculator to quickly determine present values under various scenarios, but always supplement quantitative analysis with qualitative assessment and professional judgment. The true value of any perpetual cash flow stream lies not just in the formula, but in the quality of the assumptions that drive it.