perpetuity
/ per-pe-TOO-i-tee /
What is an endless stream of payments worth — a sum that pays you a fixed amount every year forever, never stopping? It sounds like it should be worth an infinite amount, but it is not. Because each payment is pushed further into the future and discounted more heavily, the contributions shrink fast enough that their total settles on a finite, surprisingly simple value. Such a never-ending level annuity is a perpetuity.
The present value of a perpetuity-immediate paying 1 at the end of each period forever, at interest rate i, is simply 1 divided by i. The perpetuity-due, which pays at the start of each period, is worth 1 divided by d. The logic is clean: the present value of an n-payment annuity is (1 minus v to the n) over i, and as n grows without bound v to the n goes to zero, leaving 1 over i. So a perpetuity of 1 per year at 5 percent is worth 20; at 4 percent it is worth 25.
Perpetuities are not just a textbook curiosity. The British government once issued literal perpetual bonds called consols, and the perpetuity formula is the engine behind the Gordon growth model used to value shares with steady dividends. For actuaries it is a sanity-check tool and a building block: any long annuity can be approximated as the difference of two perpetuities. It also delivers a memorable lesson — the value of a perpetual income is inversely proportional to the interest rate, so as rates fall, perpetual obligations balloon in value.
A charity wants to fund a 10,000-dollar annual scholarship forever. At a 4 percent perpetual yield it must set aside 10,000 divided by 0.04, which is 250,000 dollars today.
An infinite stream has a finite value: 1 per period forever is worth exactly 1 over i.
The simple 1-over-i formula assumes a constant interest rate forever; if rates can change or if payments grow, the value can be very different and is not simply 1 over i.