Annuity factor
Financial maths.
(AF).
The purpose of Annuity factors is to answer questions of the type:
"What is the value today (at Time 0) of a promise to receive $10m at Time 1 year (one year into the future) and a further $10m every year until the end of a predetermined fixed future period."
An annuity factor is a method for calculating the total present value of a simple fixed annuity.
Such an annuity is a finite series of fixed future cash flows, the first cash flow being at Time 1 period hence, and the last one being at Time n periods hence.
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
Also known as the Present Value Interest Factor of an Annuity (PVIFA).
Present value calculation
The present value of the annuity is calculated from the Annuity Factor (AF) as:
= AF x Time 1 cash flow.
The Time 1 cash flow being the cash flow which occurs one period into the future.
Today being Time 0.
Example
For example, when the Annuity factor in relation to two fixed cash flows at Time 1 and Time 2 = 1.833
and the Time 1 period hence cash flow = $10m, then:
Present value = AF x Time 1 cash flow
= 1.833 x $10m
= $18.33m
Annuity factor calculation
The annuity factor for 'n' periods at a periodic yield of 'r' is calculated as:
AF(n,r) = 1/r x [1-(1+r)-n]
where
n = number of periods, and
r = periodic cost of capital.
Example
For example, when the periodic cost of capital (r) = 6% and the number of periods in the total time under review (n) = 2, then:
Annuity factor = 1/r x [1-(1+r)-n]
= 1/0.06 x [1-(1 + 0.06)-2]
= 1.833
This figure is also the sum of the two related Discount Factors:
AF2 = DF1 + DF2
= 1.06-1 + 1.06-2
= 0.9434 + 0.8900
= 1.833
The Annuity Factor is sometimes also known as the Annuity formula.
