Annuity

ANNUITIES

An annuity is a series of equal payments occurring at equal periods of time

Annuities occur in the following instances:
1. Payment of a debt by a series of equal payments at equal interval of time. This
occurs when goods are brought on the installment plan, the payments for w/c are
usually of equal amounts paid periodically, usually monthly.

2. Accumulation of a certain amount by setting equal amounts periodically. This
occurs when a person saves equal amounts and deposits these periodically in a
bank; when equal amounts are set aside at equal intervals of time to take care of the
depreciation of equipment & to provide for their replacement at a definite future time.

3. Substitution of a series of equal amounts periodically in lieu of a lump sum at
retirement of an individual.





Types of Annuities:
An ordinary annuity is one where the equal payments are made at the end of each payment
period starting from the first period.

A deferred annuity is one where the payment of the first amount is deferred a certain number
of periods after the first.

An annuity due is one where the payments are made at the start of each period, beginning
from the first period.

A perpetuity is an annuity where the payment periods extend forever or in w/c the periodic
payments continue indefinitely.


Symbols & their meaning
P = value of money at present
F = value of money at some future time
A = a series of periodic, equal amounts of money
n = number of interest periods
i = interest rate per interest period



ORDINARY ANNUITY
Finding P when A is given


A
A
A
A
A

P
0
1 2
3
n-1 n
A(P/F,i%,1)
A(P/F,i%,2)
A(P/F,i%,3)
A(P/F,i%,n-1)
A(P/F,i%,n)

P = A 1 – (1 + i)
-n
= A ( 1 + i)
n

- 1
i i ( 1 + i)
n


The quantity in brackets is called “uniform series present
worth factor” and is designated by the functional symbol
P/A,i%,n, read as “P given A at i percent in n interest
periods.” And can be expressed as
P = A(P/A,i%,n)



Finding F when A is given

0 1
2
3
A A
A
A A
n-1 n
F
A(F/P,i%,1)
A(F/P,i%,n-3)
A(F/P,i%,n-2)
A(F/P,i%,n-1)



(1+i)
n
- 1
F = A
i
The quantity in brackets is called the “uniform series compound
amount Factor” and is designated by the functional symbol
F/A,i%,n, read as“F given A at I percent in n interest periods. And
can be written as


F = A (F/A,i%,n)



Finding A when P is given
i
1-(1+i)
-n

The quantity in brackets is called the “capital recovery factor.” It is denoted by the
Functional symbol A/P,i%,n w/c is read as “A given P at i percent in n interest
periods.” Hence, A = P(A/P, i%,n)

Finding A When F is Given
A = F
(1+i)
n
-1
i
The quantity in brackets is called the “sinking fund factor.” It is denoted by
the Functional symbol A/F,i%,n w/c is read as “A given F at i percent in n
interest Periods.” Hence,

A = F(A/F,i%,n)
A = P


0 1 2
3 9 10
F
P10,000
P10,000
P10,000
P10,000
P10,000
1. What are the present worth and the accumulated amount of a 10-year
annuity paying P10,000 at the end of each year, w/ interest at 15%
compounded annually?

Solution:

A = P10,000 n = 10 i = 15%




P = A(P/A,i%,n)
[1-(1+i)
-n
]

= A i
= P10,000 [1 – (1.15)
-10
]
0.15
P = P50,187.68626

F = A(F/A,i%,n)
= A [(1+i)
n
- 1

i
= 10,000 [(1.15)
10
-1]
0.15
= P203,037.1824
2.What is the present worth of P500 deposited at the end of every three
months for 6 years if the interest rate is 12% compounded semi-
annually?

Solution:
Solving for the interest rate per quarter,
(1+i/4)
4
– 1 = (1 + 0.12/2)
2
– 1
i/4 = 0.0296 or 2.96%
i = 0.1182520564
P = A (P/A,2.96%,24)

A[1-(1+i)
-n
] i = .1182520564/4 = 0.0296
= i
= P500 [1-(1+0.0296)
-24
]

0.0296
= P500 (17.0087)
= P8,504




SAMPLE PROBLEMS ON ORDINARY ANNUITIES
1. Ria rose borrowed P50,000.00 fr. SSS in the form of calamity loan, w/
int.@8% comp. quarterly payable in equal quarterly installments for 10yrs. Find
the quarterly payments.
2. A manufacturing firm wishes to give each 80 employees a holiday bonus.
How much is needed to invest monthly for a yr. @ 12% nominal interest rate,
comp. mo., so that each employee will receive a P2,000.00 bonus?
3. A man paid a 10% down payment of P200,000.00 for a house & lot & agreed
to pay the balance on monthly installments for 5yrs. @ an int. rate of 15%
compounded mo. What was the mo. Inst. In pesos?
4. Money borrowed today is to be paid in 6 equal payments @ the end of 6
quarters. If the int. is 12% comp. quarterly, how much was initially borrowed if
quarterly payments is P2,000.00?
5. Mr. Robles plans a deposit of P500.00 @ the end of each month for 10yrs. @
12% annual int., comp. mo. The amt. that will be available in two yrs. is?

DEFERRED ANNUITY
A deferred annuity is one where the first payment is made several
periods after the beginning of the annuity.


m periods
n periods
Deferred periods
Ordinary annuity
periods
m
n
0
1
2
3
0’
1
2
3
4
A
A
A
A
A
FORMULA:

P = A 1 – (1 + i)
–n
( 1 + i)
-m

i

Sample problem
1. A new generator has just been installed. It is expected that there will
be no maintenance charges until the end of the 6
th
year, when P300
will be spent at the end of each successive year until the generator is
scrapped at the end of its fourteenth year of service. What sum of
money set aside at the time of installation of the generator at 6% will
take care of all maintenance expenses for the generator?
2. A parent wishes to develop a fund for a new born child’s college
education. The fund is to payp50,000.00 on the 18
th
, 19
th
, 20th & 21
st

birthdays of the child. The fund will be built up by the deposit of
fixed sum on the child’s first to seventeenth birthday’s.
3. A man loans P187,400.00 from a bank w/ int. at 5%
compounded annually. He agrees to pay to pay his obligations
by paying 8 equal annual payments, the first being due at the
end of 10yrs. Find the annual payments.
4. A house & lot can be acquired with a down payment of
P500,000.00 and a yearly payment of P100,000.00 at the end
of each year for a period of 10yrs., starting at the end of 5 yrs.
from the date of purchase. If money is worth 14%
compounded annually, what is the cash price of the property?
5. If money is worth 5% compounded semi-annually, find the
present value of a sequence of 12 semi-annual payments of
P500.00 each, the first of which is due at the end of 4 ½ years.
it is anannuity where payments are made indefinitely or forever






P = A/i
PERPETUITY
P
0
1
2
3 4 5
n =∞

A
A A A
A
SAMPLE PROBLEM:
1. A wealthy man donated a certain amount of money in a bank at a rate of 12% compounded
annually to b able to pay the following scholarship awards; P4,000 per year for the first 5 yrs.; P6000
per yr. for the next 5 yrs. and P9,000 per year on the years thereafter. Find the amount of money
deposited by the man.





1
2 3 4
5 6
7
8 9 10 11 12
13
14
P
O

Ordinary annuity
P4,000 ea.
P
d

deferred
Annuity
P6,000 ea.
P
F

perpetuity
P
P

m=5
n=5
The equation of value at 0 is,
P = P
O
+ P
d
+ P
F
0
’

P9,000 ea.
- It is the annuity where the payment started at the beginning of the annuity periods.

FINDING P WHEN A IS GIVEN


0 1 2 3 4 n-1 n

A A A A A A

P = A 1 – (1 + i )
– (n-1)
+
i

F = A ( 1 + i )

(n+1)
- 1 1
i


ANNUITY DUE
1
A man bought a car costing P 450,000 payable in 5 years at a rate of
24% compounded semi-annually in installment basis. If each semi-
annual payment is payable at the beginning of each period,
determine the amount of each payment?