Perpetuity

The present value of a perpetuity is the sum of an infinite geometric series: each payment is divided by one plus the discount rate raised to a progressively higher power. Because every term is a constant fraction of the one before it, and that fraction is smaller than one whenever the discount rate is positive, the series converges. The closed-form result is the payment divided by the rate.

The intuition is that discounting compounds against distance. At a 5 percent rate, a payment received in year 100 is worth less than one percent of its face amount, and the cash flows beyond that contribute a rounding error to the total. Value accumulates quickly in the early years and then flattens. The only case in which the value genuinely explodes is a zero or negative discount rate, which is why a strictly positive rate is a condition of the formula rather than a convenience.

The standard formula assumes payments at the end of each period and returns a value dated one full period before the first cash flow. A perpetuity-due, with the first payment today, is worth the ordinary result multiplied by one plus the rate. A deferred perpetuity, where payments begin several periods out, is handled in two steps: value the stream one period before its first payment, then discount that single figure back to today.

If payments grow at a constant rate, the value becomes the first payment divided by the difference between the discount rate and the growth rate. This growing perpetuity underpins the dividend discount model. It is valid only when growth is strictly below the discount rate; a growth rate that meets or exceeds the discount rate produces a negative or undefined result, signalling that the assumption of perpetual growth at that pace is untenable.

Non-callable fixed-rate preferred stock is the closest live instrument: a fixed dividend, no maturity, valued as the dividend divided by the required return. Analysts invert the same relation to quote a preferred issue's yield from its market price. Undated sovereign debt behaves identically while it remains outstanding.

The formula matters far more as a building block than as a bond pricer. Terminal value in a discounted cash flow model is usually a growing perpetuity attached to the end of the explicit forecast horizon, and it frequently supplies the majority of the estimated enterprise value. Commercial real estate capitalisation rates are perpetuity arithmetic: dividing stabilised net operating income by a cap rate is dividing a level payment by a discount rate. Endowment spending rules and the residual income model rest on the same logic.

Expect to move fluently in both directions: from payment and rate to value, and from observed price and payment to the implied discount rate. Match the payment frequency to the rate. A quarterly dividend must be divided by a quarterly rate, obtained from the stated annual rate and its compounding convention, not by an annual rate.

Two composite skills recur. First, decomposing an annuity as the difference between two perpetuities, one starting immediately and one deferred to the annuity's end, which is often faster than the annuity formula for awkward horizons. Second, correctly dating the answer, since most valuation errors in this area are off-by-one-period timing slips rather than algebraic mistakes. Sketching a timeline and marking the valuation date before computing removes almost all of them.