SCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT
BUSINESS MATHEMATICS / MATHEMATICAL ANALYSIS
Unit Five
Moses Mwale
e-mail: moses.mwale@ictar.ac.zm
BBA 120 Business Mathematics
Contents
Unit 5: Mathematics of Finance
3
5.0 Simple and Compound Interest Calculations .............................................................. 3
5.0.1 Simple interest ................................................................................................ 3
Example ................................................................................................. 3
Practice Exercise.................................................................................... 4
5.0.2 Compound Interest ......................................................................................... 4
Example ................................................................................................. 5
Example ................................................................................................. 5
5.0.3 When interest is compounded several times per year .................................... 6
Example ................................................................................................. 7
5.0.4 Annual percentage rate (APR) or Effective rate ............................................. 8
Example ................................................................................................. 8
Practice Problems .................................................................................. 9
BBA 120 Business Mathematics
Unit 5: Mathematics of Finance
5.0 Simple and Compound Interest
Calculations
Today, businesses and individuals are faced with a bewildering array of
loan facilities and investment opportunities. In this section we explain
how these financial calculations are carried out to enable an informed
choice to be made between the various possibilities available.
We begin by considering what happens when a single lump sum is
invested and show how to calculate the amount accumulated over a
period of time.
5.0.1 Simple interest
Simple interest is a fixed percentage of the principal, 𝑃0 , that is paid to an
investor each year, irrespective of the number of years the principal has
been left on deposit; that is, money invested at simple interest will
increase in value by the same amount each year.
So, if the investor is paid a fixed annual amount, i% of 𝑃0 , then the
amount of simple interest, I, received over t years is given by the formula
𝐼 = 𝑃0 × 𝑖 × 𝑡
Therefore, the total value after t years, 𝑃𝑡 is the principal plus interest and
is given by
𝑃𝑡 = 𝑃0 + (𝑃0 × 𝑖 × 𝑡)
𝑃𝑡 = 𝑃0 (1 + 𝑖𝑡)
When the total value (future value), the interest rate and time are known,
the principal (present value) may be calculated by rewriting formula as
𝑃0 (1 + 𝑖𝑡) = 𝑃𝑡
𝑃0 =
𝑃𝑡
(1 + 𝑖𝑡)
Example
An investment of $3,000 is made at an annual simple interest rate of 5%.
How much additional money must be invested at an annual simple
interest rate of 9% so that the total annual interest earned is 7.5% of the
total investment?
Solution
I
P
i
t
3
4
Unit 5: Mathematics of Finance
first
additional
total
(3,000)(0.05) = 150
3,000
0.05
1
0.09 x
x
0.09
1
(3,000 + x)(0.075)
3,000 + x
0.075
1
First I fill in the P, i, and t columns with the given values.
Then I multiply across the rows (from the right to the left) in order to fill
in the I column.
Then add down the I column to get the equation
150 + 0.09 𝑥 = (3,000 + 𝑥)(0.075).
To find the solution, I would solve for the value of x.
Practice Exercise
1. How long does it take a principal of $25,000 at a simple interest rate of
5% to become $30,000?
2. $45,000 is deposited into a savings account. After one year, 4 months and
20 days it totals $52,500. Calculate the simple interest rate for this
account.
3. Determine the simple interest rate applied to a principal over 20 years if
the total interest paid equals the borrowed principal.
4. How long does it take a principal payment to triple at a simple interst rate
of 6%?
5. Find the total amount of simple interest that is paid over a perod of five
years on a principal of $ 30,000 at a simple interest rate of 6%.
6. Calculate the total worth of an investment after six months with a
principal of $10,000 at a simple interest rate of 3.5%.
5.0.2 Compound Interest
Under compound interest, at the end of each interest period, the interest
earned for that period is added to the principal (the invested amount) so
that it too earns interest over the next interest period. The basic formula
for the value (or compound amount) of an investment after n interest
periods' under compound interest is as follows:
𝑃𝑡 = 𝑃0 (1 + 𝑖)𝑡
BBA 120 Business Mathematics
The compound amount is also called the accumulated amount, and the
difference between the compound amount and the original principal, 𝑃𝑡 −
𝑃0 , is called the compound interest.
Example
Find the value, in 4 years’ time, of $10 000 invested at 5% interest
compounded annually.
Solution
In this problem, 𝑃0 = 10 000, 𝑖 = 5% and 𝑡 = 4, so the formula
𝑃𝑡 = 𝑃0 (1 + 𝑖)𝑡 gives
𝑃𝑡 = 10000(1 + 0.05)4
= 10000(1.05)4
= 12155.0
The compound interest formula derived above involves four variables, r,
t, 𝑃𝑡 and 𝑃0 . Provided that we know any three of these, we can use the
formula to determine the remaining variable. This is illustrated in the
following example.
Example
A principal of $25 000 is invested at 12% interest compounded annually.
After how many years will the investment first exceed $250 000?
Solution
We want to save a total of $250 000 starting with an initial investment of
$25 000. The problem is to determine the number of years required for
this on the assumption that the interest is fixed at 12% throughout this
time. The formula for compound interest is 𝑃𝑡 = 𝑃0 (1 + 𝑖)𝑡
We are given that
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Unit 5: Mathematics of Finance
𝑃0 = 25 000, 𝑃𝑡 = 250 000, 𝑖 = 12
so we need to solve the equation
250 000 = 25 000 (1 + 0.12)𝑡
for n.
One way of doing this would just be to keep on guessing values of n until
we find the one that works.
However, a more mathematical approach is to use logarithms, because
we are being asked to solve an equation in which the unknown occurs as
a power. Following the method described in Section 2.3, we first divide
both sides by 25 000 to get
10 = (1.12)𝑛
Taking logarithms of both sides gives
𝑙𝑜𝑔(10) = 𝑙𝑜𝑔(1.12)𝑛
and if you apply rule 3 of logarithms you get
𝑙𝑜𝑔(10) = 𝑛 𝑙𝑜𝑔(1.12)
Hence
𝑛 =
=
𝑙𝑜𝑔(10)
𝑙𝑜𝑔(1.12)
1
0.049218023
= 20.3
Now we know that n must be a whole number because interest is only
added on at the end of each year. We assume that the first interest
payment occurs exactly 12 months after the initial investment and every
12 months thereafter.
The answer, 20.3, tells us that after only 20 years the amount is less than
$250 000, so we need to wait until 21 years have elapsed before it
exceeds this amount.
5.0.3 When interest is compounded several times per year
So far, it has been assumed that compound interest is compounded once a
year. In reality, interest may be compounded several times per year, for
example it may be compounded daily, weekly, monthly, quarterly, semiannually or continuously.
Each time period is known as a conversion period or interest period. The
number of conversion periods per year is denoted by the symbol, m; the
𝑖
interest rate applied at each conversion is 𝑚.
For example, an investment compounded twelve times per year will have
twelve conversion periods; therefore if a five-year investment was
compounded twelve times annually, then the investment would have sixty
conversion periods; that is,
BBA 120 Business Mathematics
𝑛 =𝑚×𝑡
where n = total number of conversion periods
m = conversion periods per year
t = number of years
The value of the investment at the end of n conversion periods is
𝑃𝑡 = 𝑃0 (1 +
𝑖 𝑛
𝑖 𝑚×𝑡
) = 𝑃0 (1 + )
𝑚
𝑚
Example
£5000 is invested for three years at 8% per annum compounded semiannually.
a) Calculate the total value of the investment.
b) Compare the return on the investment when interest is
compounded annually to that when compounded semi-annually.
c) Calculate the total value of the investment when compounded, (i)
monthly, (ii) daily. Assume all months consist of 365/12 days.
Solution
a) 𝑃0 = £5000, 𝑖 = 0.08, t = 3 and m = 2. Using Compound
interest formula, the total value after three years with n = m x t =
6 conversion periods is calculated as
𝑃3 = 5000 (1 +
0.08 (2)(3)
)
2
= 5000(1 + 0.04)6
= 6326.59
b) The total value of £5000 after three years compounded annually
at 8% interest is
𝑃𝑡 = 5000(1 + 0.08)3
= 6298.56
When the same investment is compounded semi-annually, the
total value is £6326.59, a gain of £28.09 over the value when
compounded annually.
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Unit 5: Mathematics of Finance
c) The value of the investment at the end of three years for monthly
and daily compounding is calculated as
i) Monthly compounding
ii) Daily compounding
𝑖 𝑚𝑡
𝑃3 = 𝑃0 (1 + )
𝑚
𝑃3 = 𝑃0 (1 +
𝑖 𝑚𝑡
)
𝑚
= 5000 (1 +
0.08 (365)(3)
)
365
= 5000 (1 +
0.08 (12)(3)
)
12
= 5000(1.2702)
= 5000(1.2712)
= 63151
= 63151
5.0.4 Annual percentage rate (APR) or Effective rate
Interest rates are usually cited as nominal rates of interest expressed as
per annum figures.
However, as compounding may occur several times during the year with
the nominal rate, the amount owed or accumulated will be different from
that calculated by compounding once a year.
So, a standard measure is needed to compare the amount earned (or
owed) at quoted nominal rates of interest when compounding is carried
out several times per year.
This standard measure is called the annual percentage rate (APR) or
effective annual rate. If interest is compounded once a year at the APR
rate, the investment would yield exactly the same return, 𝑃𝑡 , at the end of
t years, as it would if interest were compounded m times per year at the
nominal rate.
The formula for the annual percentage rate (APR) is given as follows:
𝐴𝑃𝑅 = (1 +
𝑖 𝑚
) −1
𝑚
Example
1. What effective rate is equivalent to a nominal rate of 6% compounded (a)
semi-annually and (b) quarterly?
Solution
a) 𝐴𝑃𝑅 = (1 +
0.06 2
)
2
− 1 = (1.03)2 − 1 = 0.0609 = 6.09%
b) 𝐴𝑃𝑅 = (1 +
0.06 4
)
4
− 1 = (1.015)4 = 0.061364 = 6.14%
This example illustrates that, for a given nominal rate i, the effective rate
increases as the number of interest periods per year (n) increases.
BBA 120 Business Mathematics
2. If an investor has a choice of investing money at 6% compounded daily or
1
6 8 % quarterly, which is the better choice?
Solution
We determine the equivalent effective rate of interest for each nominal
rate and then compare our results.
The respective effective rates of interest are
0.06 365
𝐴𝑃𝑅 = (1 +
)
− 1 ≈ 6.18%
365
And
0.06125 4
𝐴𝑃𝑅 = (1 +
) − 1 ≈ 6.27%
4
Since the second choice gives the higher effective rate, it is the better
choice (in spite of the fact that daily compounding may be
psychologically more appealing).
Practice Problems
1. A bank offers a return of 7% interest compounded annually. Find the
future value of a principal of $4500 after 6 years. What is the overall
percentage rise over this period?
2. Find the future value of $20 000 in 2 years’ time if compounded
quarterly at 8% interest.
3. Midwest Bank offers a return of 5% compounded annually for each
and every year. The rival BFB offers a return of 3% for the first year
and 7% in the second and subsequent years (both compounded
annually). Which bank would you choose to invest in if you decided
to invest a principal for (a) 2 years; (b) 3 years?
4. The value of an asset, currently priced at $100 000, is expected to
increase by 20% a year. (a) Find its value in 10 years’ time. (b) After
how many years will it be worth $1 million?
5. How long will it take for a sum of money to double if it is invested at
5% interest compounded annually?
6. Find the future value of $100 compounded continuously at an
annual rate of 6% for 12 years.
7. How long will it take for a sum of money to triple in value if invested
at an annual rate of 3% compounded continuously?
8. If a piece of machinery depreciates continuously at an annual rate
of 4%, how many years will it take for the value of the machinery to
halve?
9. A department store has its own credit card facilities, for which it
charges interest at a rate of 2% each month. Explain briefly why this
9
10
is not the same as an annual rate of 24%. What is the annual
percentage rate?
10. Determine the APR if the nominal rate is 7% compounded
continuously.
11. Current annual consumption of energy is 78 billion units and this is
expected to rise at a fixed rate of 5.8% each year. The capacity of
the industry to supply energy is currently 104 billion units. (a)
Assuming that the supply remains steady, after how many years will
demand exceed supply? (b) What constant rate of growth of energy
production would be needed to satisfy demand for the next 50
years?