## Formula

PV = Pmt x (1 - (1 + g)^{n}x (1 + i)^{-n}) / (i - g)

**Variables used in the annuity formula**

PV = Present Value

Pmt = Periodic payment

i = Discount rate

g = Growth rate

n = Number of periods

## Use

The present value of growing annuity formula shows the value today of series of periodic payments which are growing or declining at a constant rate (g) each period. The payments are made at the end of each period for n periods, and a discount rate i is applied.

A growing annuity is sometimes referred to as an increasing annuity or graduated annuity.

The formula discounts the value of each payment back to its value at the start of period 1 (present value).

When using the formula, the discount rate (i) should not be equal to the growth rate (g).

## Present Value of a Growing Annuity Formula Example

If a payment of 8,000 is received at the end of period 1 and grows at a rate of 3% for each subsequent period for a total of 10 periods, and the discount rate is 6%, then the value of the payments today is given by the present value of a growing annuity formula as follows:

PV = Pmt x (1 - (1 + g)^{n}x (1 + i)^{-n}) / (i - g) PV = 8,000 x (1 - (1 + 3%)^{10}x (1 + 6%)^{-10}) / (6% - 3%) PV = 66,550.43

## Present Value of a Growing Annuity Formula if i = g

The above formula will not work when the discount rate (i) is the same as the growth rate (g). In this situation, the formula shown below should be substituted.

PV = Pmt x n / (1 + i)

**Variables used in the annuity formula**

PV = Present Value

Pmt = Periodic payment

i = Discount rate

n = Number of periods

For example, if a payment of 8,000 is received at the end of period 1 and grows at a rate of 3% for each subsequent period for a total of 10 periods, and the discount rate is also 3%, then the value of the payments today is given by the present value of a growing annuity formula as follows:

PV = Pmt x n / (1 + i) PV = 8,000 x 10 / (1 + 3%) PV = 77,669.90

The present value of a growing annuity formula is one of many 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.