1. QMT 3301 BUSINESS MATHEMATICS
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CHAPTER 9
COMPOUND INTEREST
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9.1 Time Value of Money
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Money has time value, that is a ringgit today is worth more than a ringgit tomorrow.
Money has time value because of its investment opportunities.
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9.2 Compound Interest
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In compounding, after the interest is calculated, it is then added to the principal and becomes an adjusted principal.
Processes are repeated until the end of the loan or investment term.
Normally used with long-term loan or investment, and the interest is calculated more than once during the loan or investment term.
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The interest earned is called compound interest, and the final sum at the end of the period of borrowing is called the compound amount.
Therefore, compound interest is the difference between the original principal and the compound amount.
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9.3 Some Important Terms
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Some of the common terms used in relation to compound interest are:
1. Original principal
2. Nominal interest rate
3. Interest period or conversion period
4. Frequency of conversions
5. Periodic interest rate
6. Number of interest periods in the investment period
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6. 9.4 Compound Interest Formula
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The method used in finding compound amount at the end of the nth period is as follows:
S = P(1 + i)n
Where:
P = Principal / Present Value
S = Future Value
n = Number of Periods (number of years multiplied by number of times
compounded per year)
i = Interest rate per compound period
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Example 1:
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Find the future value of RM 1000 which was
invested for
a) 4 years at 4% compounded annually,
b) days at 10% compounded daily, and
c) 2 years 3 months at 4% compounded quarterly.
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Solution:
a) S = 1000( )4 = RM
b) S = = RM
360
c) S = = RM 4
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9. QMT 3301 BUSINESS MATHEMATICS
Example 2:
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What is the nominal rate compounded monthly that
will make RM 1000 become RM 2000 in five
years?
Solution:
From S = P(1 + i)n, we get
2000 = k 60 12
2 = 1 + k 60
k = 13.94%
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Example 3:
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How long does it take a sum of money to double
itself at 14% compounded annually?
Solution:
Let the original principal = W
Therefore sum after n years = 2W
From S = P(1 + i)n, we get
2W = W( )n
2 = ( )n
lg 2 = n lg 1.14
n = 5.29 years
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11. 9.5 Effective, Nominal and Equivalent Rates
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Effective rate: Simple rate that will produce the same accumulated amount as the nominal rate that compounded each period after one year.
Nominal rate: Stated annual interest rate at which interest is compounding more than once a year.
Equivalent rate: Two different rates that yield the same value at the end of one year.
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12. 9.6 Relationship Between Effective and Nominal Rates
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9.6 Relationship Between Effective and Nominal Rates
The relationship between the nominal rate and effective rate is derived as follows:
Assume a sum RM P is invested for one year. Then the future value after one year:
a) At r% effective = P(1 + r)
b) At k% compounded m times a year
= P(1 + k/m)m
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Example 1:
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Find the effective rate which is equivalent to 16%
compounded semi-annually.
Solution:
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Example 2:
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Find the nominal rate, compounded monthly which
is equivalent to 9% effective rate.
Solution:
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15. 9.7 Relationship Between Two Nominal Rates
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The relationship between two nominal rates is given as follows:
(1 + k/m)m = (1 + K/M)M
Where:
k and K are two different annual rates with respectively two different frequencies of conversions, m and M.
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Example :
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Find k% compounded quarterly which is
equivalent to 6% compounded monthly.
Solution:
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9.8 Present Value
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Present value or discounted value is the value which will yield the sum (S) after certain time and at a specific interest rate.
We can find present value by transposing the formula as follows:
S = P(1 + i)n
P = S
(1 + i)n
OR P = S(1 + i)-n
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Example :
A debt of RM 3000 will mature in three years’ time.
Find
a) The present value of this debt,
b) The value of this debt at the end of the first year,
Assuming money is worth 14% compounded semi
annually.
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Solution:
a) From P = S(1 + i)-n, we get
P = 2
= RM
b) P = S(1 + i)-n
P =
= RM
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9.9 Equation of Value
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An equation that expresses the equivalence of two sets of obligations at a focal date.
In other words, it expresses the following:
What is owed = What is owned at the focal date OR
What is given = What is received at the focal date
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Example :
A debt of RM 7000 matures at the end of the
second year and another of RM 8000 at the end of
six years. If the debtor wishes to pay his debts by
making two equal payments at the end of the
fourth year and the seventh year, what are these
payments assuming money is worth 6%
compounded semi-annually?
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Solution:
years
X X
Focal date
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Let the payment be RM X each.
Formulating the equation of value at the focal date
as shown, we get
What is owed = What is owned
X(1 + 3%)6 + X = 7000(1 + 3%) (1 + 3%)2
X = RM
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24. 9.10 Continuous Compounding
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We have been discussing compounding of interest on discrete time intervals (daily, monthly, etc).
If compounding of interest is done on a continuous basis, then we will have a different picture of the future value as shown below:
Discrete compounding
Continuous compounding
Future value
Future value
Time
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Time

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The future value of a sum of money compounded continuously is given by:
S = Peit
Where:
S = Future value
P = Original principal
e = …
i = Continuous compounding rate
t = Time in years
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Example :
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Find the accumulated value of RM 1000 for six
months at 10% compounded continuously.
Solution:
From S = Peit, we get
S = 1000[e10% x 0.5]
= RM
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