Perpetuity Calculator

A perpetuity is a financial instrument that pays a fixed amount of money at regular intervals forever (in perpetuity). Unlike bonds with maturity dates, perpetuities have no end date. KEY CHARACTERISTICS: • Fixed periodic payments (typically annual) • Payments continue indefinitely • Principal is never repaid • Value derived entirely from cash flows REAL-WORLD EXAMPLES: • Consols (British government bonds) • Preferred stock dividends • Endowment fund distributions • Ground rent leases Perpetuities are primarily theoretical tools used in finance to value infinite cash flows and establish baseline valuations for long-term investments.

The basic perpetuity formula is elegantly simple: PV = PMT ÷ r WHERE: • PV = Present Value (what the perpetuity is worth today) • PMT = Periodic Payment amount • r = Discount Rate (as a decimal) EXAMPLE CALCULATION: • Annual Payment: $5,000 • Discount Rate: 5% (0.05) • PV = $5,000 ÷ 0.05 = $100,000 This means an investment paying $5,000 annually forever is worth $100,000 today at a 5% discount rate. The formula works because as payments extend into infinity, the present value of distant payments approaches zero, creating a finite sum.

A growing perpetuity is a series of infinite cash flows that grow at a constant rate each period. GROWING PERPETUITY FORMULA: PV = CF₁ ÷ (r - g) WHERE: • PV = Present Value • CF₁ = First period cash flow • r = Discount rate • g = Growth rate (must be less than r) EXAMPLE: • First year dividend: $3 per share • Discount rate: 8% • Growth rate: 3% • PV = $3 ÷ (0.08 - 0.03) = $60 per share IMPORTANT CONSTRAINT: The growth rate (g) MUST be less than the discount rate (r). If g ≥ r, the formula produces an undefined or infinite result. Growing perpetuities are commonly used in the Gordon Growth Model for stock valuation.

The difference lies in WHEN payments occur: ORDINARY PERPETUITY (End of Period): • Payments occur at the END of each period • Formula: PV = PMT ÷ r • Example: Receive $1,000 at end of each year PERPETUITY DUE (Beginning of Period): • Payments occur at the BEGINNING of each period • Formula: PV = PMT × (1 + 1/r) or PMT × (1 + r) / r • Example: Receive $1,000 at start of each year VALUE DIFFERENCE: Perpetuity Due is always worth MORE than Ordinary Perpetuity because you receive each payment one period sooner. • Ordinary PV: $1,000 ÷ 0.05 = $20,000 • Due PV: $1,000 × (1 + 1/0.05) = $21,000 The due version is worth exactly one additional payment more.

A deferred perpetuity is a series of infinite payments that begin after a waiting period (deferral period). DEFERRED PERPETUITY FORMULA: PV = (PMT ÷ r) × (1 + r)^(-n) WHERE: • PMT = Payment amount • r = Discount rate • n = Number of deferral periods CALCULATION STEPS: 1. Calculate ordinary perpetuity value: PMT ÷ r 2. Discount this value back n periods: × (1+r)^(-n) EXAMPLE: • Annual payment: $10,000 • Discount rate: 6% • Deferral: 5 years • Step 1: $10,000 ÷ 0.06 = $166,667 • Step 2: $166,667 × (1.06)^(-5) = $124,576 APPLICATIONS: • Retirement planning (payments start at age 65) • Trust fund distributions • Delayed pension benefits

Selecting the appropriate discount rate is crucial because perpetuity values are highly sensitive to this input. GENERAL GUIDELINES BY ASSET TYPE: | Investment Type | Typical Rate | Risk Level | |-----------------|--------------|------------| | Government bonds | 2-4% | Low | | Investment-grade corporate | 4-6% | Moderate | | Dividend stocks | 5-8% | Moderate-High | | Real estate | 6-10% | Variable | | Emerging markets | 10-15% | High | KEY CONSIDERATIONS: • Risk-Free Rate: Start with Treasury yields • Risk Premium: Add based on investment risk • Inflation: Use real vs nominal rates appropriately • Market Conditions: Adjust for current environment IMPORTANT: A 1% change in discount rate can change perpetuity value by 20-30%. For example: • At 5%: PV = $1,000 ÷ 0.05 = $20,000 • At 4%: PV = $1,000 ÷ 0.04 = $25,000 (+25%)

The Gordon Growth Model (also called Dividend Discount Model) uses growing perpetuity to value stocks based on expected dividends. GORDON GROWTH MODEL: Stock Value = D₁ ÷ (r - g) WHERE: • D₁ = Expected dividend next year • r = Required rate of return • g = Dividend growth rate EXAMPLE: • Current dividend (D₀): $2.00 • Growth rate: 4% • Required return: 10% • D₁ = $2.00 × 1.04 = $2.08 • Stock Value = $2.08 ÷ (0.10 - 0.04) = $34.67 ASSUMPTIONS: • Dividends grow at constant rate forever • Growth rate is less than required return • Company continues paying dividends indefinitely LIMITATIONS: • Only works for dividend-paying companies • Assumes constant, sustainable growth • Highly sensitive to r and g estimates

While true perpetuities are rare, several financial instruments closely resemble them: GOVERNMENT PERPETUITIES: • British Consols (issued 1751, consolidated 1923) • War bonds (some still outstanding) • Historical Dutch water board bonds CORPORATE/FINANCIAL: • Preferred Stock: Fixed dividend payments, no maturity • Some REIT distributions • Perpetual bonds (issued by banks/corporations) REAL ESTATE: • Ground leases (99+ year terms) • Hereditary land rights • Long-term easements ENDOWMENTS: • University endowment funds • Charitable foundation payouts • Museum permanent funds LEGAL ARRANGEMENTS: • Trusts with perpetual beneficiaries • Family office distributions Most "perpetuities" have call provisions or practical limits, but they're valued using perpetuity formulas for practical purposes.

The perpetuity formula is foundational in finance for several reasons: THEORETICAL IMPORTANCE: • Simplest form of infinite series valuation • Building block for annuity formulas • Basis for many valuation models PRACTICAL APPLICATIONS: 1. Terminal Value in DCF Analysis • Value of business beyond forecast period • Often 60-80% of total company value 2. Preferred Stock Valuation • Fixed dividends = perpetuity payments • Value = Dividend ÷ Required Return 3. Real Estate Valuation • Capitalization rate method uses perpetuity logic • Property Value = NOI ÷ Cap Rate 4. Bond Valuation Theory • Consols priced as true perpetuities • Long-term bonds approximate perpetuities 5. Company Valuation • Gordon Growth Model • Multi-stage dividend models The perpetuity formula demonstrates the TIME VALUE OF MONEY: infinite future payments have finite present value.

Payment frequency significantly impacts perpetuity present value through compound interest effects. FREQUENCY ADJUSTMENTS: For non-annual payments, adjust both payment amount and rate: • Periodic Payment = Annual Payment ÷ Periods per Year • Periodic Rate = Annual Rate ÷ Periods per Year EXAMPLE (Annual vs Monthly): Annual Payment: $12,000, Rate: 6% ANNUAL: • PV = $12,000 ÷ 0.06 = $200,000 MONTHLY: • Monthly Payment = $12,000 ÷ 12 = $1,000 • Monthly Rate = 0.06 ÷ 12 = 0.005 • PV = $1,000 ÷ 0.005 = $200,000 EFFECTIVE ANNUAL RATE: More frequent compounding increases effective rate: • Annual: 6.00% • Quarterly: 6.14% • Monthly: 6.17% KEY INSIGHT: Same total annual payment has slightly higher PV with more frequent payments due to time value of receiving money sooner.

While powerful, perpetuity models have important limitations: THEORETICAL LIMITATIONS: • Nothing truly lasts forever • Constant growth assumption rarely holds • Discount rates change over time • Ignores inflation uncertainty PRACTICAL LIMITATIONS: 1. Interest Rate Sensitivity • Small rate changes = large value changes • 1% rate change can mean 20%+ value swing 2. Growth Rate Constraints • g must be less than r (for growing perpetuity) • Companies rarely sustain constant growth 3. Company/Asset Survival • Businesses fail, policies change • Legal and regulatory risks • Technology disruption BEST PRACTICES: • Use conservative discount rates • Test sensitivity to assumptions • Consider multiple scenarios • Apply appropriate risk premiums • Update valuations regularly REMEMBER: Perpetuity formulas provide estimates, not guarantees. They're best used as one tool among many in investment analysis.

Perpetuity concepts help answer: "How much do I need to never run out of money?" BASIC RETIREMENT PERPETUITY: If you want $50,000/year forever at 4% return: PV = $50,000 ÷ 0.04 = $1,250,000 needed WITH INFLATION PROTECTION (Growing Perpetuity): For $50,000/year growing at 2% inflation, with 6% return: PV = $50,000 ÷ (0.06 - 0.02) = $1,250,000 DEFERRED PERPETUITY FOR RETIREMENT: Age 40, retiring at 65, want $60,000/year, 5% return: • Ordinary PV at 65: $60,000 ÷ 0.05 = $1,200,000 • PV today: $1,200,000 × (1.05)^(-25) = $354,000 4% RULE CONNECTION: The "4% rule" is essentially perpetuity thinking: • Safe withdrawal = 4% of portfolio • Implies portfolio = 25× annual spending • Same as PV = PMT ÷ 0.04 REALITY CHECK: • Life expectancy isn't infinite • Markets are volatile • Healthcare costs increase • Consider Social Security, pensions • Annuities for guaranteed income