{ "id": "29af24b4-525a-5a88-b415-44d36e911a9e", "slug": "perpetuity", "term": "Perpetuity", "aliases": [], "category": "Financial Mathematics", "category_slug": "financial-mathematics", "difficulty": "basic", "definition": "A perpetuity is a financial instrument or cash flow stream that pays a fixed (or growing) periodic payment indefinitely, with no maturity or terminal date. Its present value is derived by dividing the periodic payment by the appropriate discount rate, reflecting the time value of money compressed into a single closed-form formula.", "key_takeaways": [ "The present value of a perpetuity is simply C/r, where C is the periodic cash flow and r is the discount rate per period.", "A growing perpetuity (Gordon Growth Model application) has present value C/(r-g), where g is the constant growth rate of the cash flow.", "Despite paying forever, the present value of a perpetuity is finite because future cash flows are discounted increasingly heavily.", "Preferred stock with no maturity and a fixed dividend is the closest real-world approximation to a perpetuity.", "UK Consols (government bonds with no maturity date) are the classic historical example of perpetuities in sovereign debt markets." ], "detailed_explanation": "A perpetuity represents the mathematical limiting case of an annuity as its term approaches infinity. The derivation of the present value formula exploits the convergence of the geometric series: the sum of an infinite series of discounted cash flows, each of equal amount C paid at the end of each period with discount rate r, converges to the finite value C/r when r > 0. This result, known since at least the 18th century, has profound practical applications in valuation.\n\nThe intuition behind the formula is accessible through a simple thought experiment: if a risk-free asset yields 5% annually and you can purchase a perpetual bond paying $50 per year forever, you should pay exactly $1,000 ($50 ÷ 0.05). At any higher price, you earn less than the risk-free rate; at any lower price, an arbitrageur would borrow at the risk-free rate, buy the perpetuity, and earn a riskless profit. This pricing logic underpins the Gordon Growth Model used extensively in equity valuation.\n\nThe growing perpetuity model extends the basic formula to accommodate cash flows that grow at a constant rate g. This is the mathematical foundation of the dividend discount model (DDM) for stocks paying dividends that are expected to grow indefinitely. The formula PV = C/(r-g) requires that r > g, since otherwise the sum of the infinite series diverges—a mathematical impossibility reflected economically in the unsustainability of any company growing faster than the economy indefinitely.\n\nPerpetual bonds exist in modern markets in several forms. Some financial institutions have issued hybrid capital instruments (Additional Tier 1, or AT1 bonds) that function as perpetuities from the issuer's perspective, though they include call provisions that make them behave more like long-dated callable bonds in practice. Many preferred share structures with fixed dividends also approximate perpetuities, and their valuation using the C/r formula is standard in financial analysis.\n\nThe perpetuity concept also appears in real estate valuation through the capitalization rate (cap rate), where property value equals Net Operating Income divided by the cap rate—directly analogous to the perpetuity formula. This approach, while theoretically a perpetuity, implicitly assumes the income stream and terminal value align such that the cap rate properly discounts the entire future cash flow stream.", "example": "A utility company issues preferred stock paying a fixed annual dividend of $4.00 per share forever (no maturity). An investor requires a 6% annual return on investments of similar risk. Using the perpetuity formula, the intrinsic value per share is $4.00 ÷ 0.06 = $66.67. If the stock trades at $60, it offers an implied yield of $4.00 ÷ $60 = 6.67%, above the required return, suggesting the stock is undervalued. Now suppose dividends are expected to grow at 2% annually forever. Applying the growing perpetuity formula: $4.00 ÷ (0.06 - 0.02) = $4.00 ÷ 0.04 = $100.00 per share—a dramatically higher value reflecting the benefit of compounding growth over an infinite horizon.", "formula": "PV = C / r (level perpetuity); PV = C / (r - g) (growing perpetuity, where g < r)", "formula_latex": null, "interactive_type": "calculator", "calculator_id": null, "related_terms": [ "annuity", "bond", "cap", "compound-interest", "continuous-compounding", "convergence", "discount-rate", "dividend", "dividend-discount-model", "equity", "fat-tailed-distribution", "future-value", "gordon-growth-model", "interpolation", "intrinsic-value" ], "backlinks": [ "annuity", "compound-interest", "copula", "dividend-discount-model", "fat-tailed-distribution", "gordon-growth-model", "interpolation", "margin-of-safety", "preferred-stock", "return-on-equity", "sum-of-the-parts-valuation", "sustainable-growth-rate" ], "cross_references": [ "annuity", "bond", "cap", "convergence", "discount-rate", "dividend", "dividend-discount-model", "equity", "gordon-growth-model", "intrinsic-value", "preferred-stock", "present-value", "risk-free-rate", "stock", "terminal-value", "time-value", "time-value-of-money", "yield" ], "tags": [ "level:basic", "cat:financial-mathematics" ], "asset_classes": [], "regulators": [], "see_also": [], "sources": [], "wordcount": 651, "checksum": "b6d456b15aa2a837", "version": "2026.05.03", "license": "CC-BY-4.0", "updated_at": "2026-09-07T02:15:24+00:00", "_links": { "self": "https://hedgefund.wiki/api/v1/terms/perpetuity", "jsonld": "https://hedgefund.wiki/api/v1/terms/perpetuity?format=jsonld", "markdown": "https://hedgefund.wiki/api/v1/terms/perpetuity?format=md", "graph": "https://hedgefund.wiki/api/v1/graph/perpetuity", "category": "https://hedgefund.wiki/api/v1/categories/financial-mathematics", "schema": "https://hedgefund.wiki/schema/term.schema.json", "html": "https://hedgefund.wiki/#/terms/perpetuity" } }