Formula and Use
The present value of a growing annuity due formula shows the value today of series of periodic payments which are growing or declining at a constant rate (g) each period. Additionally, the payments are made for n periods and a discount rate i is applied.
It is important to realize that for an annuity due the payments are made at the start of each period. A growing annuity due is sometimes referred to as an increasing annuity due or graduated annuity due.
In the formula PV is the present value of the annuity due, PMT is the amount of each payment, i is the discount rate per period, g is the growth rate applied to each payment, and n is the number of periods.
The formula discounts the value of each payment back to its value at the start of period 1 (present value). It is important to realize that when using the formula, the discount rate (i) should be greater than the growth rate (g).
Present Value of a Growing Annuity Due Formula Examples
Example 1
To illustrate, suppose a payment of 8,000 is received at the start of period 1. The payments grow at a rate of 3% for each subsequent period for a total of 10 periods. Additionally, the discount rate is 6%. The value of the payments today is given by the present value of a growing annuity due formula as follows:
Pmt = 8,000 n = 10 periods g = 3% i = 6% PV = Pmt x (1 + i) x (1 - (1 + g)n x (1 + i)-n ) / (i - g) PV = 8,000 x (1 + 6%) x (1 - (1 + 3%)10 x (1 + 6%)-10 ) / (6% - 3%) PV = 70,543.46 periods
Example 2
To further illustrate, suppose a payment of 5,000 is received at the start of year 1. The payments grow at an annual rate of 4% for each subsequent year for a total of 5 years. Additionally, the discount rate is 5%. The value of the payments today is given by the PV of a growing annuity due formula as follows:
Pmt = 5,000 n = 5 years g = 4% i = 5% PV = Pmt x (1 + i) x (1 - (1 + g)n x (1 + i)-n ) / (i - g) PV = 5,000 x (1 + 5%) x (1 - (1 + 4%)5 x (1 + 5%)-5 ) / (5% - 4%) PV = 24,528.32 years
The PV of a growing annuity due formula is one of many time value of money formulas. 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 and has built financial models for 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 degree from Loughborough University.