# Annuity factor

*Financial maths*.

(AF).

Annuity factors are used to calculate present values of annuities, and equated instalments.

The simplest type of annuity is a finite series of identical future cash flows, starting exactly one period into the future.

## Present value calculations

An annuity factor can be used to calculate the total present value of a simple fixed annuity.

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

Commonly abbreviated as AF(n,r) *or* AF_{n,r}

Sometimes also known as the Present Value Interest Factor of an Annuity (PVIFA).

### Present value

The present value of the annuity is calculated from the Annuity Factor (AF) as:

= AF x Time 1 cash flow.

**Example 1: Present value calculation**

The Annuity factor = 1.833.

Time 1 cash flow = $10m. The Present value is:

= AF x Time 1 cash flow

= 1.833 x 10

= $**18.33**m

1.833 is the Annuity factor for 2 periods, at a rate of 6% per period, as we'll see in Example 2 below.

### Annuity factor calculation

The annuity factor for 'n' periods at a periodic yield of 'r' is calculated as:

AF(n,r) = (1 - (1 + r)^{-n} ) / r

Where

n = number of periods

r = periodic cost of capital.

**Example 2: Annuity factor calculation**

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

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

The Annuity factor is:

= (1 - (1 + r)^{-n} ) / r

= (1 - 1.06^{-2} ) / r

= **1.833**

This figure is also the sum of the related Discount Factors (DF):

AF_{2} = DF_{1} + DF_{2}

= 1.06^{-1} + 1.06^{-2}

= 0.9434 + 0.8900

= **1.833**

### Alternative notation

(1 + r)^{-n} can also be written as:

1 / (1 + r)^{n}

Using this notation, the annuity factor can also be written as:

AF(n,r) = (1 - (1 / (1 + r)^{n} ) ) / r

Annuity Factors (AF) can also be considered as a combination of a Discount Factor (DF) and a Perpetuity Factor (AF):

AF = (1 - DF) x PF

## Equated instalments

Annuity factors are also used to calculate equated loan instalments.

For a loan drawn down in full at the start, the equated loan instalment is given by:

Instalment = Principal / Annuity factor

**Example 3: Loan instalment**

$20m is borrowed at an annual interest rate of 6%.

The loan is to be repaid in two equal annual instalments, starting one year from now.

The annuity factor is 1.833 (as before).

The loan instalment is:

20 / 1.833

= **$10.9m**

The Annuity Factor is sometimes also known as the *Annuity formula*.

An annuity factor is a special case of a cumulative discount factor (CumDF).

## See also

- Annuity
- Annuity formula
- Cumulative Discount Factor
- Discount factor
- Equated instalment
- Financial maths
- Growing annuity factor
- Instalment
- Perpetuity factor
- Present value
- Principal