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 AF_n

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.

See also

• Annuity
• Annuity formula
• CertFMM
• Discount factor
• Perpetuity factor
• Present value