## 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.