# Logic and derivation - Perpetual growing annuity

Present value (PV) of perpetual growing annuity is calculated as

PV = Annual cash flow at the end of first year / (Required rate of return – Growth rate)

Please let me know in simple words the logic behind deducting growth rate in the denominator

Also please let me know how the formula is derived.

P_0 = \frac{D_1}{1 + r} + \frac{D_2}{\left(1 + r\right)^2} + \frac{D_3}{\left(1 + r\right)^3} + \cdots
P_0 = \frac{D_1}{1 + r} + \frac{D_1\left(1 + g\right)}{\left(1 + r\right)^2} + \frac{D_1\left(1 + g\right)^2}{\left(1 + r\right)^3} + \cdots
\left(\frac{1 + g}{1 + r}\right)P_0 = \frac{D_1\left(1 + g\right)}{\left(1 + r\right)^2} + \frac{D_1\left(1 + g\right)^2}{\left(1 + r\right)^3} + \frac{D_1\left(1 + g\right)^3}{\left(1 + r\right)^4} + \cdots

Subtracting the third equation from the second:

P_0 - \left(\frac{1 + g}{1 + r}\right)P_0 = \frac{D_1}{1 + r}
\left(1 - \frac{1 + g}{1 + r}\right)P_0 = \frac{D_1}{1 + r}
\left(\frac{1 + r}{1 + r} - \frac{1 + g}{1 + r}\right)P_0 = \frac{D_1}{1 + r}
\left(\frac{1 + r - 1 - g}{1 + r}\right)P_0 = \frac{D_1}{1 + r}
\left(\frac{r - g}{1 + r}\right)P_0 = \frac{D_1}{1 + r}
\left(r - g\right)P_0 = D_1
P_0 = \frac{D_1}{r - g}

Thank you. It is clear now.

My pleasure.