Question

Consider an ordinary annuity with growing cash flows. The annuity’s first cash flow is given by C1>0, the periodic rate of growth of the cash flows is fixed at g, and the periodic discount rate equals r, where 0<g<r<∞. Please, use the equation for the PV of growing perpetuity (i.e. PV=C1/(r-g)) to derive the equation for the present value of a growing ordinary annuity with t payments, where t is an integer greater than 2. Simplify your answer as much as possible (collect terms!). Show all substitutions and manipulations to receive credit.

Answer 1

First Cash Flow = C1, Periodic Growth Rate = g and Discount Rate = r

The Present Value of growing perpetuity that begins at t=1 with cash flow C1 is given by PV1 = C1 / (r-g) - (A)

As the perpetuity grows at a rate of g% per annum, the cash flow at t=4 would be worth = C1 x (1+g)^(3). Now if we start considering cash flows from this point onward (t=4), thereby ignoring the cash flows at t=1, t=2 and t=3, we have the total present value of this perpetuity (the one assumed to begin at t=4 with first cash flow being C1 x (1+g)^(3)) given by:

PV' = [C1(1+g)^(3)] / [r-g] x [1/(1+r)^(3)] - (B)

Now if we subtract (A) from (B), we get the expression for the present value of the cash flows between t=1 and t=3 which is in fact the present value of an ordinary annuity, three years long and growing at a constant growth rate of g.

Therefore, PV of three years long ordinary annuity = PV - PV' = C1 / (r-g) - [C1(1+g)^(3)] / [r-g] x [1/(1+r)^(3)] = C1/(r-g) x [1-{(1+g)^(3) / (1+r)^(3)}]

Hence, for any time period T, the PV of an ordinary constant growth annuity can be determined as:

PV of Annuity = PV of perpetuity beginning at t=1 - PV of perpetuity beginning at t=T+1