Perpetuity

At first, it seems impossible that an infinite series of payments could have a finite value. But because each payment further in the future is worth less in today's terms, the present values eventually become tiny, and their total converges to a fixed amount.

The calculation is simple: the payment each period divided by the discount rate per period. If an investment pays 100 a year forever and the discount rate is 5%, its present value is 100 divided by 0.05, which is 2,000. In other words, if you invested 2,000 at 5%, it would produce 100 a year forever without reducing the original sum.

A growing perpetuity, where payments rise at a constant rate, is valued by dividing the next payment by the discount rate minus the growth rate. This idea is the basis of the Gordon Growth Model and the perpetual growth method for terminal value.

True perpetuities are rare. Some government bonds have been issued without a maturity date, and some preferred shares pay fixed dividends indefinitely. The concept is mainly used as a building block in valuation.