Growing Annuity

A growing annuity is a stream of $N$ growing cash flows, paid at regular intervals. It is a growing perpetuity that eventually comes to an end. The following timeline shows a growing annuity with initial cash flow $C$, growing at rate g every period until period $N$:

Conventions:

1. The first cash flow arrives at the end of the first period and
2. The first cash flow is before growth

The last cash flow therefore only reflects $N-1$ periods of growth.

The present value of an $N$-period growing annuity with initial cash flow $C$, growth rate $g$, and interest rate $r$ is given by

$$PV=C\times \frac{1}{r-g}\left(1-{\left(\frac{1+g}{1+r}\right)}^N\right)$$

Because the annuity has a finite number of terms.