5.3
ANNUITY
1

Define ordinary and simple annuity

Find the future and present value

Find the regular periodic payment

Find the interest
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If we wish to invest in “Amanah Saham Bumiputera” with a
fixed installment of RM100 monthly. If the scheme gives
10% interest monthly, try and calculate the amount
accumulated after 10 years.
At the end of the first month, we will have
100(0.1) + 100 = RM A
At the end of the second month, we will have
A (0.1) + 100 = RM B
At the end of the third month, we will have
B (0.1) + 100 = RM C
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Definition - Annuity
• An annuity is a sequence of equal payments
made at equal intervals of time.
• The payments are computed by the compound
interest method such as annually, semiannually,
quarterly or monthly.
• Assume that the first payment is made at the end
of the first interest period.
• Annuities in which payments are made at the
same time the interest is compounded are called
ordinary and simple annuities.
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Future Value of an Ordinary Annuity
The future value of an annuity of R ringgit
per period for n period when the interest
rate is i per period is given by:
 1  i   1 
Sn  R 

i


n
Sn = future value
R = regular or periodic payment
i = interest rate per compounding period
n = number of annuity payments
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Example 1 – future value
Ali has made equal payments of RM100 every 6
months at an interest rate of 5% compounded
semiannually for 5 years. The future value which
is the amount he gets after 5 years is
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Example 2
Lim decides to save RM1000 per month in her
saving account that pays 8% interest p.a
compounded monthly. After making 8 deposits,
how much money does Lim have?
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Example 3
A teenager plans to deposit RM50 in a savings
account at the end of each quarter for the next 6
years. Interest is earned at a rate of 8 percent per
year-compounded quarterly. What should her
account balance be 6 years from now? How much
interest will she earn?
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Example 4 Finding saving amount to achieve
future goal
Suppose you want to buy a house 5 years
from now and you want to estimate that an
initial down payment of RM20,000 will be
required at that time. Suppose a saving
account paying annual interest rate of 6%
p.a compounded annually. How much do
you need to make equal annual end-of-year
deposit into the saving acount to accumulate
the RM20,000 at the end of year 5?
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Present value of an Ordinary Annuity
The present value of an ordinary annuity of R
Ringgit per period for n period when the rate of
return or interest is i per period is given by :
1  1  i 
An  R 
i

n



An = present value
R = regular or periodic payment
i = interest rate per compounding period
n = number of annuity payments
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Example 5 Finding present value
Find the present value which is the amount to be
invested now in order to receive equal payments
of RM100 every 6 months for 5 years.
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Example 6 Finding the amount of payment
of a loan
Lim plans to start up a new business and
he needs to borrow RM100,000. You
propose to pay off the loan quickly by
making 5 equal annual payments. If the
interest rate is 10%p.a compounded
annually, how much is the amount of each
payment?
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Example 7 Finding loan amount and interest
paid
Veni agrees to pay RM 300 per month for
48 months to pay off a car loan. If interest
of 12% per annum is charged monthly, find
a) how much did the car originally cost?
b) how much interest was paid?
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Example 8
Rudy buys a land for RM110,000. He makes 20%
down payment and the balance he takes a loan for
25 years that charges an annual interest rate of 5%
compounded monthly. Find
(a) the monthly payments.
(b) the total amount of interest that will be paid.
(c) the amount of the loan that will have already
paid after 10 years.
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b) Monthly payment of RM514.44 for 25 years yield a
total payment 514.44(25)(12)= RM154,331.77
Thus the total amount of interest = RM154,331.77- 88,000
= RM66,331.77
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1  1  i  n 




c) An  R


i


 12(15) 
 
1  1  0.05 

 

12 
514.44
An=
. 

0.05


12


= 72,444.94
The amount of loan that will have already paid
after 10 years is
RM88,000- RM72,444.94 = RM15,555.05
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Hisham is 20 years away from retiring and
starts saving RM100 a month in an account
paying 6% p.a compounded monthly. When
he retires, he wishes to withdraw a fixed
amount each month for 25 years. What will
the fixed amount be?
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Example 10
Mariam purchased a house by paying RM5,750.32 down
and promised to pay RM811.41 every months for next 15
years. The interest charged is at the rate of 9% compounded
monthly.
a) What was the cash value of the house?
b) If Mariam missed the first 10 payments, what must she
pay at the time the 11th payment is due to bring herself
up to date?
c) After paying for the first 5 years, Mariam wished to
discharge her remaining debt by making single payment
at time when the 61st regular payment was due. What
must she pay in addition to the regular payment then
due?
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Example 11
At the end of every month, Mr. Zaki saves RM200
in an account that pays an annual rate of 10%
compounded monthly. After 3 years, he adds
RM60 to his savings per month. Show that the
total amount after 6 years is RM22,129.17
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