Introductory Mathematics
& Statistics for Business
Chapter 5
Simple interest
Copyright  2010 McGraw-Hill Australia Pty Ltd
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5-1
Learning Objectives
• Understand the effect of inflation on interest rate
levels
• Perform calculations involving simple interest
• Manipulate the simple interest formula
• Distinguish between, and calculate, flat and
effective rates of interest
• Estimate the effective rate of interest
Copyright  2010 McGraw-Hill Australia Pty Ltd
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5-2
5.1 Introduction
•
Interest denotes any income received from capital that has
been invested, or the amount of money paid for the use of
borrowed money
•
The actual amount of money that has been lent or invested, or
the current balance of the original amount, is called the
principal (or capital).
•
The amount of money received as interest depends on three
factors:
1. The principal or capital
2. The rate of interest
3. The duration of the debt or investment
Copyright  2010 McGraw-Hill Australia Pty Ltd
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Introduction (cont…)
•
The amount of a loan or investment before interest is added on
(or a discount is taken off) is called the face value
•
The actual interest (or return) expressed as a percentage of the
price paid is called the yield.
•
The actual rate of interest charged by lending institutions
depends on many factors, including
– current and expected inflation rates,
– competition between financial institutions,
– the size of Australia’s budget deficit
– the amount of the principal
Copyright  2010 McGraw-Hill Australia Pty Ltd
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Introduction (cont…)
• The effect of inflation on the level of interest rates
– According to monetary theory, price levels in general will
rise if the money supply grows more quickly than the total
output of the economy.
– Inflation may be defined as persistent increase in the
general level of prices and is normally accompanied by high
interest rates
– Those who borrow money are able to pay the higher interest
rate if inflation enables them to raise their selling price to
pay for it. In this way, borrowers themselves are helping to
create an inflationary spiral in which inflation and interest
rates continually push each other higher.
Copyright  2010 McGraw-Hill Australia Pty Ltd
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Introduction (cont…)
•
One of the factors affecting mortgage rates of interest is the cost
of goods and services. That is, inflation is going to reduce the
value of the money and mortgage rates will rise
•
Many factors affect interest rates, including
– high unemployment rates
– overseas interest rates,
– balance of payments figures,
– deregulation of the financial system,
– the amount held in deposits and especially
– government policy
Copyright  2010 McGraw-Hill Australia Pty Ltd
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5-6
5.2 Calculation of simple interest
•
Simple interest, is the interest calculated, based on the original
principal during the entire period at the stated rate of interest.
•
Given
P = principal
R = rate of interest
T = time (in years)
I = total simple interest
S = amount or maturity value of the principal
•
Simple interest:
I  PR  T
Copyright  2010 McGraw-Hill Australia Pty Ltd
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Calculation of simple interest (cont…)
• The maturity value of the principal is:
S  P1 RT 
• What amount of principal will give a certain maturity value.
S
P
1  RT
• When P is calculated from S, the value of P is called the
present value of S or the discounted value of S.
Copyright  2010 McGraw-Hill Australia Pty Ltd
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Calculation of simple interest (cont…)
Example
Calculate the simple interest earned on $15 000 deposited at
a rate of 1.5% per annum for a period of 4 years and 3
months
Solution
Here
So
T  4.25
I  PR  T
 $1500  0.015  4.25
 $956.25
Hence, the total simple interest earned is $956.25.
Copyright  2010 McGraw-Hill Australia Pty Ltd
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5-9
Calculation of simple interest (cont…)
Example
A bank customer invests $5000 for 18 months at a simple
interest rate of 9.5% per annum. Find the maturity value of the
principal
Solution
P= $5000, R = 0.095 and T = 1.5
S  P1  RT 
 $50001  0.095  1.5
 $50001.1425
 $5712.50
Hence, the maturity value of the principal is $5712.50
Copyright  2010 McGraw-Hill Australia Pty Ltd
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Calculation of simple interest (cont…)
Example
An investor wishes to have an amount of $4000 on maturity in 5
years time. A simple interest rate of 12.0% per annum is offered.
How much money should be invested now in order to achieve
this aim?
Solution
S= $4000, R = 0.12 and T = 5
S
P
1  RT
$4000

1  0.12  5
 $2500
Hence, the investor should invest $2500 now in order to have
$4000 at the end of 5 years.
Copyright  2010 McGraw-Hill Australia Pty Ltd
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5-11
Calculation of simple interest (cont…)
• Minimum Monthly Balance
– Many financial institutions in Australia pay simple interest based
on the minimum monthly balance in an account each month.
– That is, if a sum of money is deposited and withdrawn during the
same month, it will not attract any interest.
Use
I  P R  T
Where
P = minimum monthly balance
R = rate
T
1
12
Copyright  2010 McGraw-Hill Australia Pty Ltd
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Calculation of simple interest (cont…)
Example
A customer opens a savings account on 15 October 2009 with a
sum of $1000. The bank pays simple interest on the account at
a rate of 2.5% per annum based on the minimum monthly
balance. Calculate the interest due for November
Solution
I  PRT
 $1000  0.025 
1
12
 $2.08
Hence the interest due for November is $2.08
Copyright  2010 McGraw-Hill Australia Pty Ltd
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5-13
5.3 Effective rate of interest
•
This is an interest rate that is always based on the original
principal, even though the amount of the loan is reducing.
•
For every flat rate of interest their will be an equivalent effective
rate. Its approximate value is given by:
2N  R
E 
N  1
Where:
E = effective rate
R = flat rate
N = number of repayments
Copyright  2010 McGraw-Hill Australia Pty Ltd
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Effective rate of interest (cont…)
Example
A customer takes out a $15 000 personal loan from the bank to
be repaid monthly over 5 years. The bank has quoted a flat rate
of interest of 9.5%. Calculate the equivalent effective rate of
interest
Solution
N  5  12  60, R  0.095
2N  R
N1
2  60  0.095

60  1
 0.1869
E
Hence, the effective rate of interest is 18.69%.
Copyright  2010 McGraw-Hill Australia Pty Ltd
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Effective rate of interest (cont…)
To find the flat rate of interest that is equivalent to a
given effective rate

N  1  E
R
2N
Where:
E = effective rate
R = flat rate
N = number of repayments
Copyright  2010 McGraw-Hill Australia Pty Ltd
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Effective rate of interest (cont…)
Example
A credit union is offering home equity loans to be repaid over 20
years with monthly repayments and an effective rate of interest
of 11.5%. Calculate the equivalent flat rate of interest
Solution
N  20  12  240, E  0.115
R
N  1  E
2N
240  1  0.115

2  240
 0.05774
Hence, the equivalent flat rate of interest is 5.774%.
Copyright  2010 McGraw-Hill Australia Pty Ltd
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Summary
• We looked at understanding the effect of inflation
on interest rate levels
• We perform calculations involving simple interest
• Also looked at how to manipulate the simple
interest formula
• Looked at the difference between, and calculate,
flat and effective rates of interest
• Lastly estimated the effective rate of interest
Copyright  2010 McGraw-Hill Australia Pty Ltd
PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e
5-18