Pmt = FV x (i - g) / ( (1 + i)n - (1 + g)n )
FV = Future Value
Pmt = Initial payment at the end of period one
i = Discount rate
n = Number of periods
g = Growth rate
This growing annuity payment formula FV calculates the initial annuity payment required to provide a given future value (FV) using a growing annuity. The growing annuity payment formula assumes payments are made at the end of each period for n periods and are growing or declining at a constant rate g, and a discount rate i is applied.
Example Using Growing Annuity Payment Formula FV
The initial payment is made at the end of period one. Suppose an investment requires 24 (n) regular periodic payments growing at a rate of 4% (g) each period. Assuming a discount rate of 6% (i), and a required future value of 8,000 (FV), what is the amount of the first (initial) payment required.
The initial payment amount is given by the growing annuity payment formula FV as follows:
Pmt = FV x (i - g) / ( (1 + i)n - (1 + g)n ) Pmt = 8000 x (6% - 4%) / ( (1 + 6%)24 - (1 + 4%)24 ) Pmt = 107.70
Later Period Payments
The initial payment calculated above is made at the end of period one. The value of the payment at the end of any other period is calculated by compounding the initial payment forward at the growth rate (g) using the future value of a single payment formula. For example, if the amount of the payment at the end of period nine is required, then this would be calculated as follows:
FV = PV x (1 + i)n FV = Pmt (9) PV = Pmt (1) n = 15 (periods between 1 and 16) Pmt(n) = Pmt(1) x (1 + g)n Pmt(16) = 107.70 x (1 + 4%)15 Pmt(16) = 193.96
At a growth rate of 2% (g), the initial payment at the end of period one of 107.70, would have grown into a payment 193.96 by the end of period sixteen.
The growing annuity payment formula FV is one of many growing annuity formulas used in time value of money calculations, discover another at the link below.
About the Author
Chartered accountant Michael Brown is the founder and CEO of Double Entry Bookkeeping. He has worked as an accountant and consultant for more than 25 years in all types of industries. He has been the CFO or controller of both small and medium sized companies and has run small businesses of his own. He has been a manager and an auditor with Deloitte, a big 4 accountancy firm, and holds a BSc from Loughborough University.