## Formula

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

**Variables used in the growing annuity payment formula PV**

PV = Present Value

Pmt = Initial payment at the end of period one

i = Discount rate

n = Number of periods

g = Growth rate

## Use

This growing annuity payment formula PV calculates the initial annuity payment required to provide a given value today PV (present value) 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 PV

### Initial Payment

The initial payment is made at the end of period one. Suppose an investment provides 12 (n) regular periodic payments growing at a rate of 2% (g) each period. Assuming a discount rate of 5% (i), and a required present value of 5,000 (PV), what is the amount of the first (initial) payment required.

The initial payment amount is given by the growing annuity payment formula PV as follows:

Pmt = PV x (i - g) / (1 - (1 + g)^{n}x (1 + i)^{-n}) Pmt = 5,000 x (5% - 2%) / (1 - (1 + 2%)^{12}x (1 + 5%)^{-12}) Pmt = 510.56

### 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 = 8 (periods between 1 and 9) Pmt(n) = Pmt(1) x (1 + g)^{n}Pmt(9) = 510.56 x (1 + 2%)^{8}Pmt(9) = 598.20

At a growth rate of 2% (g), the initial payment at the end of period one of 510.56, would have grown into a payment 598.20 by the end of period nine.

The growing annuity payment formula PV is one of many growing annuity formulas used in time value of money calculations, discover another at the link below.