Perpetuities

Kerry Back

Valuing a perpetuity

What is the value of a constant sequence of cash flows that goes on forever?

\[\text{PV} = \frac{c}{1+r} + \frac{c}{(1+r)^2} + \frac{c}{(1+r)^3} + \cdots\]

The value is \(c/r\).
Example: the value of getting $100 forever, starting in one year, when the rate of return is 10%, is $1,000.

Proof 1: sum of geometric series
Proof 2: if you invest $1,000 and earn 10%, you can withdraw $100 in a year without depleting your capital, and continue doing so forever.

Valuing a growing perpetuity

What is the value of a sequence of cash flows that grows (or declines) at a fixed rate forever?

\[\text{PV} = \frac{c}{1+r} + \frac{(1+g)c}{(1+r)^2} + \frac{(1+g)^2c}{(1+r)^3} + \cdots \]

The value is \(c/(r-g)\). Example: the value of getting $100 next year and 8% more each year afterwards, when the rate of return is 10%, is 100/(0.1-0.08) = 5,000.

Proof 1: sum of geometric series
Proof 2: if you invest $5,000 and earn 10%, you can withdraw $100 and have $5,400 remaining (8% more than you started with).
Your withdrawals can grow by 8% each year because your account balance grows by 8% each year.