What Is a Perpetuity? Definition, Formula, and Examples

A perpetuity represents a financial concept where a stream of cash flows continues indefinitely. This idea is fundamental for understanding the present value of long-term income streams. It helps in valuing assets or investments that are expected to generate payments far into the future.

Defining Perpetuity

A perpetuity is an infinite series of identical cash flows occurring at regular intervals, continuing forever. This distinguishes it from an annuity, which has a finite duration. While “forever” is theoretical, it serves as a practical assumption in financial models for assets with very long or indeterminate lives. Its valuation depends on a constant discount rate, reflecting the time value of money and the risk of future payments. This concept determines the present value of an ongoing stream of payments, even though the total sum would theoretically be infinite.

Types of Perpetuities

Two primary types of perpetuities are recognized in financial analysis. An ordinary perpetuity, also known as a constant or zero-growth perpetuity, involves fixed, equal payments that continue indefinitely. For example, an investment promising $100 annually forever is an ordinary perpetuity.

A growing perpetuity accounts for payments that increase at a constant rate over time. Each subsequent payment is larger than the last by a consistent percentage. For instance, if an investment pays $100 in the first year, and payments grow by 2% each year thereafter, this exemplifies a growing perpetuity. This growth factor reflects scenarios where cash flows rise consistently due to factors like inflation or business expansion.

Calculating Perpetuity Value

Calculating the present value of a perpetuity involves specific formulas that discount the infinite stream of future cash flows to today’s value. For an ordinary perpetuity, with constant payments, the present value (PV) is determined by dividing the cash flow (C) by the discount rate (r). The formula is: PV = C / r. This calculation reveals the worth of a perpetual stream of fixed payments in today’s dollars, considering the opportunity cost of capital.

For a growing perpetuity, where payments increase at a constant rate, the formula adjusts. The present value (PV) is calculated by dividing the cash flow of the first period (CF1) by the difference between the discount rate (r) and the constant growth rate (g). The formula is: PV = CF1 / (r – g). A crucial condition is that the discount rate must be greater than the growth rate; otherwise, the present value would theoretically be infinite. This method values a stream of payments that continues indefinitely and increases over time.

Common Applications of Perpetuities

The concept of perpetuities finds several practical applications in finance and investment analysis. Preferred stock, for instance, often pays fixed dividends indefinitely, closely approximating an ordinary perpetuity. As long as the issuing company remains operational and declares dividends, preferred stockholders expect a continuous income stream.

Historically, government-issued perpetual bonds, known as Consols, promised interest payments forever without returning the principal. While some consols, like those issued by the UK government, have been redeemed, their structure exemplified the perpetuity concept. Large endowments, such as those for universities or charitable organizations, are managed to generate a perpetual income stream for operations, relying on the principle of perpetuity to ensure long-term funding. Perpetuities are also a component in business valuation models, particularly in determining the terminal value within a discounted cash flow (DCF) analysis, which represents the value of a company’s cash flows beyond a specific forecast period.