Annuities Formula

If A is given

If S is given

(

[

)

[

]

Find R: (

] p.104

)

[

Find t:

] p.104

(

] p.104

)

Find t: [

] p.106 )

(

[

n = tm

[

]

[

]

(

)

(

)

[

]

[

]

(

)

(

)

[

] (

[

)

] (

If A is given

)

If S is given

Find j: (pp.109-110) (

] p.106 )

(

n = tm

)

)

Find R: [

(

(

Find j: (pp.109-110) )

In quadratic equation form:

√

(

)

(

)

(

)

In quadratic equation form:

√

(

)

Ordinary Annuity – periodic payment is made at the start of each term or period Present Value [

(

)

Annuity Due - periodic payment is made at the beginning of each term or period

]

̈

(

[

(

)

Deferred Annuity-periodic payment does not occur at the start or end of the term or period but delayed to a later date

)

]

̂

[

or ̈

(

(

](

)

or

̂

) p. 112-113

)

(

)

or ̈

(

[

Future Value or Amount [

(

)

] p.104

̈

̈

[

)

(

)(

̈

or ( or

( [

)

](

)

)

̂

]

)

](

)

[

(

)

]

Ordinary Annuity If A is given

Annuity Due

(

[

If ̈ is given

)

̈

]

[

Deferred Annuity

(

)

[

(

)

(

(

)

)

] p.104

(

If ̈ is given ̈

]

)

̈ (

[

] (

[

(

)

)

)

]

[

(

̂ (

)(

)

)

]

](

)

̂

[

(

)

]

Find R (if S is given) [

̈ (

)(

)

)

̈

[ )

](

)

Find t

Find t

)

(

Find R

Find R [

)(

) [

If S is given [

((

] (

)

If A is given ̈

Find t

Find t [

[

]

)

(

[

Find R

Find R

Find R

](

̂

(

(

)

]

)

]

[

̂

(

)

)

]

]

Before the start of the term ( ̂)

ANNUITY’S VALUE AT ANY TIME After the end of the term ( )

At the middle of the term ( )

To find Y: To find X: To find ̂ : 1. Find the amount (S) of 1. Find the amount (S) at 1. Find the present value an ordinary annuity the end of the kth (A) of an ordinary period and the annuity remaining liability after 2. Then accumulate S by the kth period 2. Then discount A by using the formula: 2. Then use the formula: using the formula ̂

(

)

(

)