# Growing annuity factor

*Financial maths*

(GAF).

Growing annuity factors are used to calculate present values of growing annuities.

The simplest type of growing annuity is a finite series of growing future cash flows, starting exactly one period into the future, and growing at a constant percentage rate per period.

## Contents

## Present value calculations

A growing annuity factor can be used to calculate the total present value of a growing annuity.

The Growing Annuity Factor is the sum of the adjusted Discount factors for maturities 1 to n inclusive, when the cost of capital is the same for all relevant maturities.

The discount factors need to be adjusted because of the growth of the cash flows.

By analogy with the simple annuity factor abbreviated as AF(n,r) *or* AF_{n,r}

the growing annuity factor can be abbreviated as GAF(n,r,g) *or* AF_{n,r,g}

### Present value

The present value of the growing annuity is calculated from the Growing Annuity Factor (GAF) as:

= GAF x Time 1 cash flow.

**Example 1: Present value calculation**

Annuity factor = 1.842.

Time 1 cash flow = $10m.

Present value is:

= AF x Time 1 cash flow

= 1.842 x 10

= $**18.42**m

### Growing annuity factor calculation

The growing annuity factor for 'n' periods at a periodic yield of 'r' and a periodic growth rate of 'g' is calculated as:

AF(n,r,g)

= **(**1 / (r-g) **)**

- x
**(**1 - ( (1+g) / (1+r) )^{n}**)**

- x

Where

n = number of periods

r = periodic cost of capital

g = periodic growth rate from Time 1 period in the future

**Example 2: Growing annuity factor calculation**

When the periodic cost of capital (r) = 6%

growth rate (g) per period = 1%

and the number of periods in the total time under review (n) = 2

The growing annuity factor is:

= **(**1 / (r-g) **)**

- x
**(**1 - ( (1+g) / (1+r) )^{n}**)**

- x

= **(**1 / (0.06 - 0.01) **)**

- x
**(**1 - ( (1.01) / (1.06) )^{2}**)**

- x

= **(**20**)**

- x
**(**0.0921**)**

- x

= **1.842**