Perpetuity

What Is a Perpetuity?

A perpetuity is a stream of equal payments that continues forever, with no set end date. It is a special case of an annuity in which the payments never stop. Because there is no final payment and no repayment of principal, a perpetuity has no maturity.

Perpetuities are mainly used to value a steady, ongoing stream of income. Each future payment is worth less today than the same amount received sooner, because money available now can be invested to earn a return. Valuing a perpetuity therefore means discounting every future payment back to the present.

How a Perpetuity Is Valued

When the payments are equal and arrive at the end of each period, the present value of a perpetuity can be calculated with a short formula:

PV = C / r

Here, C is the payment per period and r is the discount rate per period. An infinite number of payments has a finite total value because later payments are discounted so heavily that their contributions shrink toward zero.

If the payments grow at a constant rate g each period, the formula becomes PV = C / (r - g). This version only works when r is greater than g. If g were equal to or greater than r, the total value would have no finite limit.

Example

Suppose a stream pays 100 per year forever and the appropriate discount rate is 5 percent. The present value is 100 / 0.05, which equals 2,000. The result is highly sensitive to the discount rate. At 4 percent, the same stream is worth 2,500.

Where Perpetuities Appear

Few real assets are true perpetuities. Some preferred stock issues have no maturity date and pay fixed dividends, so they are often analyzed as approximate perpetuities. Historically, the British government issued bonds called consols, which paid interest indefinitely without a redemption date. Perpetuity formulas are also used in valuation models. In a discounted cash flow analysis, for example, a growing perpetuity is often used to estimate the value of cash flows beyond the forecast period.

Because the formulas depend on assumed rates, small changes in the discount rate or growth rate can produce large changes in the calculated value, especially when r is close to g.