Question

Let PV be the present value of a growing perpetuity (the ‘time 1 perpetuity’) with an initial payment of C beginning one period from now and a growth rate of g. If we move all the cash flows back in time one period, the present value becomes PV*(1+r) Note that this is the present value of a growing perpetuity with an initial payment of C beginning today (‘time 0 perpetuity’).

Question: How do the cash flows of the time 1 perpetuity compare to those of the time 0 perpetuity from time 1 on?

Answer 1

The general PV formula for perpetual growing periodic cash flows of say C beginning one period from now is given by

PV =C / r-g where r is the discount rate and g is the periodic growth rate of cash flows

If the cash flows start immediately (i.e each cash flow is preponed by one period) it essentially means that each cash flow C is being discounted for one period less than what it was being done previously.

This implies that Equation (A) becomes PV x (1+r) = [Cx (1+r) / r-g]

The cash flows of time 1 perpetuity will be similar to those of time 0 perpetuity in terms of their individual magnitude. However, each cash flow C of time 0 perpetuity will be discounted for one time period less as compared to time 1 perpetuity.