Perpetuity

Perpetuity refers to a financial arrangement or cash flow pattern that provides a fixed payment forever. It is a type of annuity with no fixed maturity date. If payments occur at the end of each period, it is called a perpetuity immediate; if the first payment is today, it is a perpetuity due. A growing perpetuity has payments that increase at a constant rate g each period.

Valuation: Let PMT be the payment per period and r the discount rate. For a perpetuity immediate, the present value is PV = PMT / r, assuming r > 0. For a perpetuity due, PV = PMT * (1 + 1/r) (equivalently PMT/r + PMT). For a growing perpetuity with growth rate g, and r > g, PV = PMT / (r - g).

Applications: The concept underpins the valuation of perpetual bonds (consols) and is used in models such as the Gordon growth model for stocks, where P0 = D1 / (r - g) with dividends growing at rate g. In actuarial science and corporate finance, perpetuities are used as simplifying assumptions for endowments or perpetually funded obligations. Limitations include the need for a constant payment, an infinite horizon, and the requirement that r > g; if r <= g, the present value is not finite under the standard formula.

Note: In continuous-time finance, a continuous perpetuity with payment rate c has PV = c / r as well.