Simple Annuity
Simple Annuity
Simple Annuity
Definition of Terms:
In an ordinary annuity, payments are made at the end of each payment inter
Diagram: (R = periodic or regular payment)
4
TERM
R
n1
n(periods)
In an annuity due, payments are made at the beginning of each payment int
Diagram: (R = periodic or regular payment)
R
0
R
n1
n(periods)
TERM
2
h -1
h
h +1
payments for n periods
R
h+2
i. Ordinary
Annuity
Amount of an Ordinary Annuity:
The amount of an ordinaryis the value at the end of the term
is the value on the last payment date
annuity, denoted by S,
is the sum of the accumulated payments
at the end of term.
For instance, to find the amount of a 1000 ordinary annuity, payable annua
for 4 years when money is worth 5%, accumulate the payment of each peri
to the end of 4 years, then add the accumulations.
TERM
1,000
1,000
0
1,000
1,000
2
3
4
1000
= 1000
1000(1+0.05) = 1,050
1000 = 1,102.5
1000 = 1,157.625
S = 4,310.125
R
1
R
2
R
n-2
R
n-1
R
n (periods)
R
R(1+i)
R
R
R
S = sum of the accumulated values of R at the end of the term
S = R + R(1+i) + R+ + R+ R (equation1)
Multiplying equation1 by (1 + i), we get
(1+ i)S = R(1+i) + R+ + R+ R
(equation 2)
Subtracting equation 1 from equation 2, we get
(1+i)S S = R - R
Solving for S: (SEE NEXT SLIDE)
Formula:
For example, to find the present value of an annuity, discount each payment,
add the results, as shown in the diagram.
TERM
1,000
1,000
1,000 1,000
0
1
952.3809 = 1,000
905.0294 = 1,000
863.8376 = 1,000
822.7024 = 1,000
A = 3,545.9505
In deriving
the formula for the present value, we use the fact that A is the pre
value of S due in n periods.
n periods
S
A
0
n1
From the previous formulas, notice that A and S are related to the equations
S=A
A=S
Hence, A = S
=
=
=
The
formulas for the amount S and present value A
of
ordinary annuity:
A=
Note: These formulas are applicable only when the
payment interval is the same as the interest period.
Example 1: Find the amount and the present value
of an ordinary annuity of 250 each quarter
payable for 5 years and 9 months, if money is
worth 12% compounded quarterly.
Example
1: Find the amount and the present value of an ordinary
annuity of
250 each quarter payable for 5 years and 9 months, if money is
worth 12%
compounded quarterly.
Solution:
Given: R = 250
j = 0.12
t = 5 years
m=4
i = j/m = 0.03
n = mt = 23 periods
Find: S and A
= 250(32.452883) = 8,113.22
A = = 250 = 250(16.443608) = 4,110.90
Solution:
Cash price = downpayment + present value of the installment payments (the a
= downpayment + present value of 60 monthly payments at 5,00
Cash price = 30,000 + A
= 30,000 + 5,000
= 30,000 + 5,000(44.955038)
= 254,775.19
If the present value or the amount of an annuity is known, the periodic pay
can be determined by solving the annuity formulas for R. Hence, we have
and
A=
= 476.17
Example 3: A man borrows 10,000. He agrees to pay the principal and inte
paying a sum each year for 4 years. Find his annual payment if he pays inte
8 % compounded annually.
Assignment:
Page: 61
Items: 1, and 8
Solution:
Given: A = 500 200 = 300
=
n=8
= 40.95
i = 24%/12 = 2%
Find: R
Example 3: A man borrows 10,000. He agrees to pay the principal and inte
paying a sum each year for 4 years. Find his annual payment if he pays inte
8 % compounded annually.
Solution:
Given: A = 10,000
=
J = 0.08
= 3,019.21
m=1
t=4
n=4
Find: R
n-2
n1
n(periods)
R
2
n2
n1
n (periods)
The ordinary
0
1
2
n-2
n1
n(periods)
To find S(due) we again produce an ordinary annuity of (n+1) payments of R.
Rs term ends one period after the annuity dues last payment. Diagram of th
ordinary annuity:
R
R
R
-1
n1
R
n (periods)
The ordinary
Formulas:
A(due)= ,
Example 3: Mike deposits 780 at the beginning of each year at 6.5% compo
annually. How much money does he have in 8 years?
Assignment:
Page: 70 , items: 2 and 7; Topics: Find the amount and present value of annu
The length of time for which there are no payments is called the period of
deferment. The first payment is made one period after the period of deferme
Therefore, if an annuity is deferred for 7 periods, the first payment is made o
8th period and if the first payment is made at the end of 15 periods, the annu
is said to be deferred for 14 periods.
no payments
for h periods
0
2
h -1
h
h+1
payments for n periods
R
h+2
n +(h-1) n +h (per
Let A(def)
be the present value of the deferred annuity.
A(def) = (present value of (n + h) payments) (present value of h payments)
=
=
The payment R is:
Example1: Find the present value of 10 payments at 100 each, if the first p
is due at the end of 3.5 years and if money is worth 12% compounded
semi-annually.
Solution:
Given: n = 10, R = 100, h = 6, i = j/m = 0.06, Find : A(def)
Diagram:
h=6
6 payments to be missed
100 100 100 100 100 100 100 100 100 100
0
n = 10
10 payments to be made
A(def) =
= 100
= 100(10.105895 4.9173243)
= 518.86
8(years
Example
2: Find the present value of 10 semi-annual payments of
500 each if the
first is due at the end of 4.5 years and money is worth 10%
converted
semi-annually.
A deferred annuity problem can also be solved as an ordinary
annuity problem.
Discounting can be used to find its present value.
Diagram:
8 payments to be missed
A(def)
A 500 500 500 500 500 500 500
500 500 500
0
1
9(years)
Ordinary annuity of 10
payments
th
Anothersolution:
Given: h = 8, n = 10
A(def) =
= 500
= 500(11.689587 6.4632126)
= 2,613.19
Example 3: A cellphone costs 6,000. A buyer will pay 2,000 now and pa
balance in 8 annual payments. The first payment is due in 5 years. If mon
worth 8% effective, how much is the annual payment?
Example 3: A cellphone costs 6,000. A buyer will pay 2,000 now and pay
balance in 8 annual payments. The first payment is due in 5 years. If mone
worth 8% effective, how much is the annual payment?
Solution:
Given: A(def) = 6,000 2,000 = 4,000
n = 8 (number of payments to be made)
d = 4 (number of payments missed)
i = 0.08
Find: R
h=4
A(def)
0
6
7
8
9
10
11
ordinary annuity of 8 payments
R
12(years)
Anothersolution:
Example 3: A cellphone costs 6,000. A buyer will pay 2,000 now and pay
balance in 8 annual payments. The first payment is due in 5 years. If mone
worth 8% effective, how much is the annual payment?
Solution:
Given: A(def) = 6,000 2,000 = 4,000
n = 8 (number of payments to be made)
d = 4 (number of payments missed)
i = 0.08
Find: R
=
=
=
= 946.98
Example 4: Find the present value of a 200 annuity payable annually for 5 y
but deferred 3 years, if money is worth 7%.
Example 5: A house and lot can be bought for 500,000 downpayment and
quarterly payments of 8,000 each. The first installment is due at the end
5 years and 3 months. If money is worth 12% compounded quarterly, how
is the cash value of the house and lot?