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Annuity factors are used to calculate present values of annuities, and equated instalments.


The purpose of Annuity factors is to answer questions of the type:
The simplest type of annuity is a finite series of identical future cash flows, starting exactly one period into the future.


"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]].
== Present value calculations ==


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.
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 factor]]s 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<SUB>n</SUB>


Commonly abbreviated as AF(n,r) ''or'' AF<SUB>n,r</SUB>


Also known as the Present Value Interest Factor of an Annuity (PVIFA).


Sometimes also known as the Present Value Interest Factor of an Annuity (PVIFA).




=== Present value calculation ===
 
=== Present value ===


The [[present value]] of the annuity is calculated from the Annuity Factor (AF) as:
The [[present value]] of the annuity is calculated from the Annuity Factor (AF) as:
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The Time 1 cash flow being the cash flow which occurs one period into the future.
<span style="color:#4B0082">'''Example 1: Present value calculation'''</span>


Today being Time 0.
The Annuity factor = 1.833.  


Time 1 cash flow = $10m.
The Present value is:


'''''Example'''''
= AF x Time 1 cash flow


For example, when the Annuity factor in relation to two fixed cash flows at Time 1 and Time 2 = 1.833  
= 1.833 x 10


and the Time 1 period hence cash flow = $10m, then:
= $'''18.33'''m


Present value = AF x Time 1 cash flow
= 1.833 x $10m
= '''$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.




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The annuity factor for 'n' periods at a periodic yield of 'r' is calculated as:
The annuity factor for 'n' periods at a periodic yield of 'r' is calculated as:


AF(n,r) = 1/r x [1-(1+r)<sup>-n</sup>]
AF(n,r) = (1 - (1 + r)<sup>-n</sup> ) / r




Where


where
n = number of periods


n = number of periods, ''and''
r = periodic cost of capital.


r = periodic cost of capital.


<span style="color:#4B0082">'''Example 2: Annuity factor calculation'''</span>
When the periodic cost of capital (r) = 6%,


'''''Example'''''
and the number of periods in the total time under review (n) = 2.


For example, when the periodic cost of capital (r) = 6% and the number of periods in the total time under review (n) = 2, then:
The Annuity factor is:


Annuity factor = 1/r x [1-(1+r)<sup>-n</sup>]
= (1 - (1 + r)<sup>-n</sup> ) / r


= 1/0.06 x [1-(1 + 0.06)<sup>-2</sup>]
= (1 - 1.06<sup>-2</sup> ) / r


= '''1.833'''
= '''1.833'''




This figure is also the sum of the two related Discount Factors:
This figure is also the sum of the related Discount Factors (DF):


AF<sub>2</sub> = DF<sub>1</sub> + DF<sub>2</sub>
AF<sub>2</sub> = DF<sub>1</sub> + DF<sub>2</sub>
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= 0.9434 + 0.8900
= 0.9434 + 0.8900


= 1.833
= '''1.833'''
 
 
=== Alternative notation ===
 
(1 + r)<sup>-n</sup> can also be written as:
 
1 / (1 + r)<sup>n</sup>
 
 
Using this notation, the annuity factor can also be written as:
 
AF(n,r) = (1 - (1 / (1 + r)<sup>n</sup> ) ) / 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
 
 
<span style="color:#4B0082">'''Example 3: Loan instalment'''</span>
 
$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 Annuity Factor is sometimes also known as the Annuity formula.
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]]).




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* [[Annuity]]
* [[Annuity]]
* [[Annuity formula]]
* [[Annuity formula]]
* [[CertFMM]]
* [[Cumulative Discount Factor]]
* [[Discount factor]]
* [[Discount factor]]
* [[Equated instalment]]
* [[Financial maths]]
* [[Growing annuity factor]]
* [[Instalment]]
* [[Perpetuity factor]]
* [[Perpetuity factor]]
* [[Present value]]
* [[Present value]]
* [[Principal]]
== Student article ==
[[Media:2014_11_Nov_-_Ever_deceasing_circles.pdf| Ever decreasing circles - using annuity factors to unlock circularity in loan instalments, The Treasurer]]


[[Category:Financial_management]]
[[Category:Financial_management]]

Latest revision as of 21:42, 28 October 2021

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 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


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


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

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