Annuity factor

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Financial maths.


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 AFn,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 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


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

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

Student article

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