Annuity factor
Financial maths.
(AF).
Annuity factors are used to calculate present values of annuities, and equated instalments.
Present value calculations
An annuity factor is a method for calculating the total present value of a simple fixed annuity.
Mathematically, the Annuity Factor is the cumulative Discount factor 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
Annuity factor = 1.833.
Time 1 cash flow = $10m.
Present value is:
= AF x Time 1 cash flow
= 1.833 x 10
= $18.33m
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
When the periodic cost of capital (r) = 6%,
and the number of periods in the total time under review (n) = 2.
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:
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
$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.
