Question

proof PV of a growing perpetuity

Answer 1

A Geometric Progression(GP) is defined as

series A, AR, AR², AR³,..........,ARⁿ.... where first term is A and ratio = R

Sum of this series = A/(1-R) for infinite series of GP

For a growing perpetuity, the cash flow is given as :

C/(1+r), C(1+g)/(1+r)², C(1+g)^2/(1+r)³,...,C(1+g)^(n-1)/(1+r)^n, ...

where C is cash flow assumed to be constant ; g is growth rate for perpetuity ; r is annual rate of return

This is nothing but a Geometric Progression

Using same formula of Geometric Progression with first term A=C/(1+r) and constant ratio R=(1+g)/(1+r)
PV of a growing perpetuity = sum of all the above cash flows  = [C/(1+r)] / [ 1 -{(1+g)/(1+r)}]

= [C/(1+r)] / [ (1+r -1-g)/(1+r)] = [C/(1+r)] / [(r-g)/ (1+r)] = C/(r-g)
=  C/(r -g)

Hence PV of a growing perpetuity = C/(r -g)