ANNUITY
An annuity is a series of equal payments
occurring at equal periods of time.
P = value or sum of money at present
F = value or sum 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
ORDINARY ANNUITY
Type of annuity where the 1st payment happen at
the end of the first period
Present Worth
(1 + 𝑖)! −1
𝑃=𝐴
(1 + 𝑖)! 𝑖
Future Worth
(1 + 𝑖)! −1
𝐹=𝐴
𝑖
1. You deposit $4000 at the end of each year to your saving
account which is compounded yearly at 4% annual percent
yield. What is the accumulated total in your account at the
end of 20-th year ?
A=4000 i = 4% n=20
"[(%&')! )%]
F=
'
+,,,[(%&,.,+)"# )%]
= $119,112.31
,.,+
2. Find the annual payment to extinguish a debt of P100,000
payable for 6 years at 12% interest annually.
P=100000 A=? i = 4% n=20
(%&')! )%
𝑃=𝐴 (%&')! '
(%&,.%.)$ )%
100000 =𝐴 = P34,322.57
%&,.%. $ (,.%.)
3. What annuity is required over 12 years to equate to a future
amount of P 200,000? i= 8% effective.
F=200000 A=? i = 8% n=12
"[(%&')! )%]
F=
'
"[(%&,.,/)%" )%]
200000 = ,.,/
= P10,539
4. A 5-year ordinary annuity has a present value of $1,000. If
the interest rate is 8 percent, what’s the amount of each annuity
payment?
P=1000 A=? i = 8% n=5
(%&')! )%
𝑃=𝐴 (%&')! '
(%&,.,/)& )%
1000 =𝐴 = $250.46
%&,.,/ & (,.,/)
5. A 5-year ordinary annuity has periodic cash flows of $100
each year. If the interest rate is 8 percent, the present value of
this annuity is closest to which of the following?
A=100 i = 8% n=5
(%&')! )%
𝑃=𝐴 (%&')! '
(%&,.,/)& )%
=100 = $329.27
%&,.,/ & (,.,/)
ANNUITY DUE
ANNUITY DUE
Type of annuity where the 1st payment happen at the
beginning of the first period.
Present Worth
1 − (1 + 𝑖)!(#!$)
𝑃=𝐴 +1
𝑖
Future Worth
(1 + 𝑖)#&$ −1
𝐹=𝐴 −1
𝑖
1. A man bought an equipment costing P60,000 payable quarterly for 3 years, each
installment payable at the beginning of each period. The rate of interest is 24%
compounded quarterly, what is the amount of each payment?
Given: #."( )(%")%)
%)(%& )
60,000= 𝐴 (
#."( +1
P = P60,000 (
n=3
r = 24% or 0.24
m=4 A = P6,751.53
A=?
2. What is the future worth of P1,000 yearly deposited at 2% annual interest for 10 years
if the payments are made at the beginning of each year.
Given:
(%&,.,.)(%#,%) ) %
F = 1,000 ,.,.
−1
A = P1,000
i = 2% or 0.02
n = 10 F = P11,168.72
F=?
3. What is the present worth of P1,000 monthly deposit at 2% per month for one year if
the payments are made at the beginning of each month?
Given:
%)(%&,.,.))(%")%)
P = 1,000 ,.,.
+1
A = P1,000
i = 2% or 0.02
n=1 P = P10,786.85
P=?
4. What is the future worth of P500 semi-annual deposit at 5% compounded semi-
annually for 2 years if the payments are made at the beginning of each semi-annual
period.
Given: #.#& (,%
(%& ) )%
F = 500 "
#.#& −1
A = P500 "
r = 5% or 0.05
n=2
m=2 F = P2,128.16
F=?
5. Suppose you want to save money for your child’s college expenses. You suppose to
deposit $1,000 at the beginning of each year, for 18 years, at an interest rate of 5%. How
much is available for your child when he/she starts school?
Given:
(%&,.,0)%-,% ) %
F = 1000 −1
,.,0
A = $1,000
i = 5% or 0.05
n = 18 F = $29,539
F=?
DEFERRED
ANNUITY
Deferred Annuity is also ordinary annuity but the payment of
the first amount is deferred a certain number of periods
after the first.
WHAT IS DEFERRED
Present Value - is the current value of a future sum of money
ANNUITY? or stream of cash flows given a specified rate of return.
Future Value - Future value (FV) is the value of a current asset
at a future date based on an assumed growth rate.
EXAMP
LE
Problem 1
Mr. Erick wants to receive a consistent amount from a bank that
offers 6% interest to be compounded annually. If he will delay
his withdrawal for the first 5 years, how much should he
deposit now to receive Php 100,000 every year for the next 10
years after the period of deferral?
Given:
r=0.06 i= 0.06
m= 1 PV= P549,988.52
n=10
k=5
A= 100,000
t= 10
EXAMP
LE
Problem 2
Mariel purchased a smart television set through the credit cooperative of
their company. The cooperative provides an option for a deferred
payment. Mariel decided to pay after 2 months of purchase. Her monthly 2
payment is computed as P3,800 payable in 12 months. How much is the
cash value of the television set if the interest rate is 12% convertible
monthly?
Given:
R=3800
r=0.12
t= 1
m=12
n= (12)(1) = 12
P= P42,345.84
k=1
j=
EXAMP
LE
Problem 3
Mr. Quijano decided to sell their farm and to deposit the
fund in a bank. After computing the interest, they learned
6 7
that they may withdraw P480,000 yearly for 8 years
starting at the end of 6 years when it is time for him to
retire. How much is the fund deposited if the interest rate
is 5% converted annually?
Given:
R=480,000
r=0.05
t= 8
m=1
P= P2,430,766.23
n= (1)(8) = 8
k=5
j=
EXAMP
LE
Problem 4
18 19 20 21
A parent on the day a child is born wishes to determine what 17
lump sum would have to be paid into an account bearing
interest at 5% compounded annually, in order to withdraw
P20,000 each on the child's 18th, 19th, 20th and 21st birthdays. 20k 20k 20k 20k
How much is the lump sum amount?
Given:
A= 20,000
r=5% or 0.05
m=1
n=4 P= P30,941.73
k=17
i=
EXAMP
LE
Problem 5
P
An air conditioning company offers a deferred payment options
1 year 2 year
for a split type air conditioner with monthly payments of P3,000
for 2 years. The payments will begin after 4 months. How much
is the cash price of the air conditioner if the interest rate is 8% 3K 3K
compounded monthly?
Given:
A= 3000
r=8% or 0.08
m=12
n=21 P= P57,450.31
k=3
i=
