Annuities - Hand Out

8/7/2019 Annuities - Hand Out

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ANNUITIES

Present Value (A) - the sum of the present values of the payments.- the Value of the Annuity at the beginning of its term.

- the Sum of the discounted values of the payments of the annuityat the beginning of its term.

( )[ ]no ir m

R A

+= 11

The Amount of the Annuity (S)- the sum of the compound amounts on hand at the end of the term

if each payment accumulates until then from the date when itwas due.

- the value of the Annuity at the end of its term.

- the accumulation of the annuity- the sum of the accumulated values of the same payments at theend of the term.

( )[ ]11 += no ir m

RS

( ) niS A += 1 ( )ni AS 1 +=

Annuity a series of periodic payments (usually equal) made at the regular intervals of time.

Example: installment payments, monthly rentals, and life insurance premiums

Payment interval (pi) period of time between consecutive payments

(m = 1, m = 2, m = 4, m = 12)

Term of annuity the time from the beginning of the first payment interval to the end of the payment interval

Types of Annuity

A. Annuity Certain an annuity payable for a definite duration not dependent on some outside

contingency begins and ends on a definite or fixed date

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B. Annuity Uncertain or Contingent Annuity an annuity payable for an indefinite duration in which the beginning or termination

is dependent on some certain event. Annuitys first and last payment or both depends upon some events (e.g., pension,

life insurance policy)

Kinds of Annuity Certain

1. Simple Annuity an annuity whose interest conversion period (m) is equal or thesame as the payment interval (m = pi)

2. General Annuity an annuity whose interest conversion period (m) is unequal or notthe same as the payment interval.

Classification of Simple Annuity

1. Ordinary Annuity (A o) annuity in which the periodic payment (R) is made at theend of each payment interval.

2. Annuity Due (A due) an annuity in which the periodic payment (R) is made at the beginning of each payment interval.

3. Deferred Annuity (A def ) an annuity in which the periodic payment (R) is neither made at the beginning nor at the end of each paymentinterval, but at some later date.

Note for analyzing annuity problems:

1. If the payments follow the single sum, it is an A problem; if the payments precede the single sum, it is an S problem.

2. Generally, problems that involve expenses or cash equivalent are A problem, and problems that involve income are S problems.



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1. Ordinary Annuity

( )[ ]no ir m

R A

+= 11 ( )[ ]11 += no ir m

RS

S

x x x x x0 1 2 3 4 5

A

Given: R = P100 r = 8% t = 5 years m = 1

The first payment (R) is due at the end of the first year.

( )[ ]=+= 5081108

1100 .

.o A

( )[ ]=+= 1081081

1005

..oS



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2. Annuity Due

( ) ( )[ ] Rir m

R A ndue ++= 11 1 ( ) ( )[ ] Ri

r m

RS ndue +=+ 11 1

S

x x x x x 0-1 0 1 2 3 4 5

A

The first payment (R) is due at the beginning of the first payment interval.In this case, the 1 st payment is due at the beginning of the first year interval.

( ) ( )[ ] 100 081108

1100 15 ++= i Adue ..

( ) ( )[ ] 100 1081081

100 15

+=+

..dueS



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3. Deferred Annuity

( ) ( )( ) 1 1 1 1m mn h h A R i R idef r r

+ = + +

( )( )

( )

1 1 1 1n hm h

A R i idef r

+ = + +

where h is the number of deferred payments

( )[ ]11 +== nodef ir m

RS S

S

0 0 x x x x x

0 1 2 3 4 5 6 7

h

n + h

A

Additional given: First payment due at the end of the 3 rd year

Five payments (n)

( ) ( ) ( )5 21 2100 1 1 .08 1 1 .08.08

Adef + = + + =

( )[ ]=+== 108108

1100 5 ..odef S S


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