Transcript 5.3: Compound Interest

Mathematics of Finance
5.3 Compound Interest
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Objectives:


Find the compound interest and compound
amount
Apply the concept of compounding period to
find




compound amount
time and interest rate
present value
effective rate
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Definition - Compound Interest
If the interest due at the end of each
payment period is added to the principal,
so that the interest computed for the
next payment period is based on this new
amount of old principal plus the interest,
then the interest is said to have been
compounded. Compounding the interest
means earning interest on interest and
we call the result as compound interest.
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Terms for payment period
Annually
Once per year
Semiannually
Twice per year
Quarterly
4 times per year
Monthly
12 times per year
Daily
365 times per year
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Simple interest VS Compound interest
Simple interest
-is the interest earned
only on the original
principal amount invested
compound interest
-is calculated based on
both the original principal
and the interest
reinvested from prior
periods.
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Assume that we have deposited RM8,000 in a
credit union, which pays interest of 8% per yearcompounded quarterly. Assume that we want to
determine the amount of money we will have on
deposit at the end of one year if all interest is left
in the savings account.
At the end of the first quarter interest is computed as
I1 = (RM8,000)(0.08)(0.25)
= RM160.00
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Note that t was defined as 0.25 year. With the
interest left in the account, the principal on which
interest is earned in the second quarter is the
original principal plus the RM160 in interest
earned during the first quarter, or
P2 = P1 + I1 = RM 8160
Interest earned during the second quarter is
I2 = (RM8,160)(0.08)(0. 25) = RM163.20
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Table below summarizes the computations for the
four quarters. Note that for each quarter the
accumulated principal plus interest to as the
compound amount. Notice that the total interest
earned during the four quarters equals RM659.46
Quarter
Principal ( P )
(RM)
Interest ( I )
(RM)
Compound amount ( S = P + I )
1
8,000.00
160.00
8,000.00 + 160.00 = RM8,160.00
2
8,160.00
163.20
8,160.00 + 163.20 = RM8,323.20
3
8,323.20
166.46
8,323.20 + 166.46 = RM8,489.66
4
8,489.66
169.79
8,489.66 + 169.79 = RM8,659.46
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In this example, simple interest for the year would
have been equal to
I = (8,000)(0.08)(1)
= RM640.00
The difference between the simple interest and the
compound interest is
659.46 – 640.00 = RM19.46
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The accumulated value for compound interest is
greater than that of simple interest.
Accumulated Value
Compound interest
Simple interest
Time
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Compound amount
The compound amount or future value or accumulated
value is the final sum at the end of the period.
For example, RM5,000 invested in a bank at the rate
of 10%, compounded yearly. The first year
accumulated value is:
S1  P(1  i)
= 5,000(1 + 0.1)
= RM5,500.00
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For the second year, the calculation of the future value
(S2) is
S 2  S1 (1  i)
 P(1  i)1  i 
= 5,500(1+0.1)
= RM6,050
or
S 2  P(1  i) 2
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The S2 amount is used as a principal to calculate the
accumulated value for the third year.
S 3  S 2 (1  i)
 P1  i 1  i 1  i 
= 6,050(1+0.1)
= RM6,655 or
S 3  P(1  i)
3
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Hence, the compound amount (future value) for
the n-th period may be expressed as:
S  P 1 i 

Where

S
P
i
j
m
n
t
n
j
i ,
m
n  tm
= compound amount
= original principal
= the yearly interest rate
= nominal rate
= frequency
= number of compounding period
= time or terms in years
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A long - term investment of RM 250,000 has
been made by a small company. The interest
rate is 12 percent per year, and the interest is
compounded quarterly. What will the value of
the investment be after 8 years? How much
interest will be earned?
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Suppose that RM 1,000 is invested in a bank,
which earns interest at a rate of 8% per year
compounded annually. What will the account
balance be after 10 years?
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Sara decided to invest RM1000 in a saving
account at an annual interest rate of 10%.
What is the amount after 5 years if the
compounding takes place:
a) annually
b) monthly
c) daily
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Example 4
Khalidah deposited RM500 in a saving account for
eleven years. The interest rate is 6% compounded
semiannually for the five years and 8%
compounded quarterly for the next six years. What
is the amount in the saving account at the end of
the 11th year?
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Present Value
Present value starts with the future (amount) and tries to
calculate its worth in the present (now), at nth
interest period.
From the relationship,
S = P( 1 + i )n
The present value (P ) is obtained as follows
P
S
1  i 
n
or
P = S( 1 + i )-n
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Example 5
Computing present value
How much money should be invested now
at 8% per annum so that after 2 years the
amount will be RM10,000 when interest is
compounded
a) annually
b) monthly
c) daily
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Example 6
A young man has recently received an
inheritance of RM 200,000. He wants to take a
portion of his inheritance and invest it for his
later years. His goal is accumulated
RM300,000 in 15 years. How much of the
inheritance should be invested if the money
will earn 12% per year compounded
semiannually? How much interest will be
earned over the 15 years?
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Example 7
How much should Aimen invest now to get
RM3,216.90 in 6 years if the interest rate is
8% compounded quarterly.
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Example 8 Finding the period
Min has been saving up to buy a Tanjung Print
Co. The total cost will be RM10 million. Min
currently has RM2.3 million. If Min can earn
5% p.a. compounded annually on his money,
how long will Min has to wait?
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How long will it take to increase RM 1,800 to
RM 3,002 at the rate of 9% per year
compounded every three months?
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A person wishes to invest RM 10,000 and wants
the investment to grow to RM 20,000 over the
next 10 years. At what annual interest rate would
the RM 10,000 have to be invested for this
growth to occur, assuming annual compounding?
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Example 11
Rahman invests RM30,000 in a fund that pays a
compound interest at a rate of 6.5% per year
compounded monthly.
(a) How long does it take for Rahman’s
investment to triple?
(b) Find the interest earned after 5 years.
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For compound interest, if the interest is
compounded m times per year at an
annual rate i, the effective rate is the
annual interest rate that yields the same
interest after one year.
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Example 12 - Effective rate
Calculate the effective rate associated
with an 8% annual rate when interest is
compounded
a) annually
b) quarterly
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Find the effective rate for:
(a) 16% compounded quarterly,
(b) 15% compounded monthly.
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