## The Perpetuity Formula

A perpetuity or infinite annuity is an asset that is expected to give a steady stream of cash flows forever. In essence, it is an annuity whose cash flows never end. Although, perpetuities are somewhat rare, they do exist in some instances. However, understanding a perpetuity as a never-ending annuity helps gain a better understanding of the nature and valuation of annuities with different numbers of periods or payments.

As the length of an annuity grows, the contribution of each successive payment to the present value of the asset becomes less and less. In other words, the first payment of an annuity contributes more to the present value than does the second which contributes more than the fiftieth and so on. In fact, payments that are expected far into the future contribute so little to the present value that they can often be ignored.

The present value of an annuity can be calculated with the following formula:

PVA_{n} = [CF / r] – [CF / (r * (1 + r)^{n}]

where CF is a repeating cash flow, r is the interest rate, and n is the number of periods.

As the number of periods, n, becomes large, the second term on the right side of the equation becomes so small that it eventually goes to zero as it approaches infinity. For those with knowledge of calculus, this concept is elementary. As the right term goes to zero we are left only with the following:

[CF / r]

This formula, therefore, represents the value of a perpetuity. Recall that a perpetuity is nothing more than an annuity that has unlimited, or infinite, pay periods.

Suppose that a perpetuity is expected to pay $1,000 a year at 10% interest. What is the present value of this asset? In other words, what amount would an investor accept today as compensation for selling the perpetuity? Using the perpetuity formula, we have:

Present Value_{Perpetuity} = CF / r = $1,000 / 0.1 = $10,000

An investor would be indifferent with keeping the perpetuity (or infinite annuity) or selling it for $10,000 today. Notice that since the payments never end, the value of the perpetuity will always be $10,000.

Annuities with unusually long expected cash flows can be treated as perpetuities because cash flows far into the future have little effect on the present value of the asset. With sufficiently long payments out in the future, a perpetuity is a reasonable present value estimate of an annuity.