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

= $18.33m

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

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):

AF₂ = DF₁ + DF₂

= 1.06^-1 + 1.06^-2

= 0.9434 + 0.8900

= 1.833

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
• Financial maths
• Growing annuity factor
• Perpetuity factor
• Present value
• Instalment
• Equated instalment
• Principal

Student article

Ever decreasing circles - using annuity factors to unlock circularity in loan instalments, The Treasurer