A perpetuity is one of the simplest — and most useful — building blocks in finance: a cash flow that repeats forever. Despite the infinite time horizon, the present value of a perpetuity is a single, finite number, and the formula behind it underpins everything from valuing preferred stock to estimating terminal value in a discounted cash flow model.
Why an Infinite Cash Flow Has a Finite Value
It seems counterintuitive that a payment continuing forever could be worth a finite amount today, but the math works because each future payment is discounted more heavily than the last. As the number of periods grows toward infinity, the present value of each additional payment shrinks toward zero, so the sum of all discounted payments converges to C / r rather than growing without bound. This is the same discounting logic used for any multi-period cash flow — a perpetuity is simply the limiting case where the number of periods becomes infinite.
Level vs. Growing Perpetuities
A level perpetuity pays the same fixed amount every period, valued with PV = C / r. A growing perpetuity increases the payment by a constant rate g each period, valued with PV = C / (r - g). The growing version requires the discount rate to exceed the growth rate — if g were to reach or exceed r, the discounted value of each successive payment would stop shrinking, and the sum would no longer converge. In practice, this means the growth-rate assumption in any growing-perpetuity model should represent a long-run, sustainable rate, not a short-term burst of growth.
Real-World Uses: Consols, Preferred Stock, and Terminal Value
Historically, the UK government issued consols — perpetual bonds with no maturity date that paid a fixed coupon forever — as the textbook example of a level perpetuity. Preferred stock, which typically pays a fixed dividend indefinitely, is valued the same way. The growing-perpetuity formula is even more widely used: it is the basis of the Gordon Growth Model for valuing common stock from expected dividends, and it is the standard method for estimating the terminal value in a multi-year discounted cash flow (DCF) analysis, where cash flows beyond the explicit forecast period are assumed to grow at a constant long-run rate.
Limitations of the Perpetuity Model
The perpetuity formula assumes a constant discount rate and a constant (or constantly growing) payment forever — assumptions that rarely hold exactly in the real world. Interest rates change, growth rates fluctuate, and few cash flow streams genuinely have no end date. In practice, perpetuities are used as simplifying approximations: a reasonable long-run average rate and growth assumption, applied to a cash flow that is expected to continue for a very long but not necessarily infinite time. Small changes in the assumed growth rate can have an outsized effect on the result, especially as g approaches r, so it is worth stress-testing the growth assumption before relying on the output for a major decision.