Chapter 9 | Compound Interest
9.1 |
Introduction
Compound interest
Compound interest is a procedure where interest is calculated periodically (at regular
intervals, known as compounding periods) and reinvested to earn interest at the end of each
compounding period during the term; i.e., interest is added to the principal to earn interest.
This is different from simple interest where interest is calculated only on the original amount
(principal) for the entire term.
is a procedure where
interest is added to the
principal at the end of each
compounding period to earn
interest in the subsequent
compounding period.
10% compounded daily
Value of Investment (in dollars)
$10,000
10% compounded annually
$7387
$6727
p
Ex
$3000
on
t
en
G
i al
wt
ro
h
10% p.a. simple interest
rowth
Linear G
$1000
Original Amount (Principal)
Time (in years)
Exhibit 9.1(a): Difference Between Simple Interest and Compound Interest at Different
Compounding Frequencies
In compound interest, the interest earned in the previous compounding period also earns
interest in subsequent compounding periods. For any given interest rate, the shorter the
compounding period, the greater the interest earned because the number of compoundings
during the term will be more. Generally, the compounding period can be one year (annually),
six months (semi-annually), three months (quarterly), one month (monthly), or one day (daily).
Consider an example of $1000 invested or borrowed at 10% compounded annually for four
years. Interest earned in the 1st year is added to the principal at the end of the 1st year. In
the 2nd year, interest is calculated on the sum of the principal and interest earned. This is
continued similarly at the end of each year.
Exhibit 9.1(b): Interest Calculated Upon Interest
The value at the:
Beginning (initial value) = $1000.00
End of the 1st year
= $1000(1.10)
End of the 2nd year
= $1000(1.10)(1.10) = 1000(1.10)2 = $1210.00
End of the 3rd year
= $1000(1.10)2 (1.10) = 1000(1.10)3 = $1331.00
End of the 4th year
= $1000(1.10)3 (1.10) = 1000(1.10)4 = $1464.10
= $1100.00
297
298
Chapter 9 | Compound Interest
Now consider an investment of 'PV' compounded 'n' times at the rate of 'i'
The value of the initial investment = PV­
The value at the end of the 1 compounding period = PV(1 + i)1
The value at the end of the 2nd compounding period = PV(1 + i)2
The value at the end of the 3rd compounding period = PV(1 + i)3
st
The value at the end of the nth compounding period = PV(1 + i)n
If 'FV' is the value at the end of the nth compounding period, we obtain the following formula for
Future Value:
Formula 9.1(a)
Future Value
FV = PV(1 + i)
Future Value (FV ):
the accumulated
value or maturity
value of an
investment or loan.
n
Formula 9.1(a) is the basic compound interest formula where:
■■
■■
■■
■■
'FV' is the future value (accumulated value or maturity value)
'PV' is the present value (discounted value or principal amount)
'i' is the interest rate per compounding period
'n' is the total number of compounding periods during the term
Using the Formula 9.1(a), FV = PV(1 + i)n and dividing both sides by (1 + i)n, we obtain the following formula
for Present Value:
Formula 9.1(b)
Present Value (PV ):
the principal amount
or discounted value
of an investment or
loan.
Formula 9.1(c)
Present Value
PV =
or
PV = FV(1 + i)
-n
Similar to simple interest, the amount of compound interest 'I', is the difference between the Future
Value and the Present Value, which gives us the formula:
Amount of Compound Interest
I = FV - PV
Notation
Compounding
frequency (m):
the number of times
interest is compounded
every year.
Compounding period:
the time interval between
two successive interest
calculation dates.
m = compounding frequency: it is the number of times interest is compounded every year.
That is, the number of compouding periods in one year. It usually appears after the word
'compounding' or 'compounded'.
For example, if the interest is compounded quarterly, then the compounding frequency (m) = 4.
Compounding period (or interest period): it is the period of time between the compounding of interest.
For example, if the interest is compounded semi-annually, then the compounding period is 'semiannual'. It usually appears after the word 'compounded' or 'compounding'.
Length of compounding period: it is the time interval between successive interest calculation dates. For
example, if the interest is compounded monthly, then the length of the compounding period is every month.
Chapter 9 | Compound Interest
Table 9.1
Compounding Frequencies for Common Compounding Periods
Compounding Period
Length of
Compounding Period
Compounding
Frequency (m)
Annually
Every year
1
Semi-annually
Every 6 months
2
Quarterly
Every 3 months
4
Monthly
Every month
12
Daily
Every day
365
If the compounding
period is quarterly
(every three months),
then m = 4 because
there are four 3-month
periods in a year.
Note: The length of the compounding period may be more often than 'every day' (daily compounding).
For example, it may be every hour, every minute, every second, or continuous. With continuous
compounding, the compounding frequency, 'm', becomes very large.
In this textbook, we will only be using annual, semi-annual, quarterly, monthly, and daily
compounding periods.
Nominal interest
rate ( j ): the quoted
interest rate per
annum.
j = nominal interest rate: it is the quoted or stated interest rate per annum on which the
compound interest calculation is based for a given compounding period. It is the rate (expressed as
a percent) that usually precedes the word 'compounding' or 'compounded'.
For example, 6% compounded monthly: j = 6% = 0.06
i = periodic interest rate: it is the interest rate for a given compounding period and is calculated
as follows:
Nominal Interest Rate
Periodic Interest Rate =
Compounding Frequency
This is given by the formula,
Formula 9.1(d)
The nominal interest rate ( j ) is always
provided with a compounding frequency
(m) and can easily be remembered as: 'j'
compounded 'm' times.
Periodic Interest Rate
i=
Periodic interest
rate (i ): the interest
rate for a given
compounding
period.
For example, 6% compounded monthly: i =
j
= 0.06 = 0.005
12
m
n = total number of compounding periods during the term: it is calculated as follows:
Total Number of Compounding Periods = Compounding Frequency # Time in Years
This is given by the formula,
Formula 9.1(e)
299
Number of Compounding Periods
n=m#t
Where 't' is the time period in years.
For example,
■■
■■
6% compounded semi-annually for two years, the number of compounding periods,
n=m#t=2#2=4
5% compounded quarterly for 18 months, the number of compounding periods,
n = m # t = 4 # 18 = 6
12
300
Chapter 9 | Compound Interest
Time period (t):
the period of time, in
years, during which
interest is calculated.
Example 9.1(a)
t = time period: it is the period of time (in years) during which interest is calculated.
The time period in the previous example, t = 18 months = 18 = 1.5 years.
12
Identifying and Calculating Compound Interest Terms
An investment is growing at 6% compounded monthly for ten years.
(i) Identify the nominal interest rate, compounding frequency, and time period.
(ii) Calculate the periodic interest rate and number of compounding periods.
Solution
(i) 6% compounded monthly for 10 years
j = 0.06
m = 12
t = 10
Therefore, the nominal interest rate is 6%, compounding frequency is 12, and time period
is 10 years.
j
= 0.06 = 0.005 = 0.5%
12
m
Number of compounding periods: n = m # t = 12 # 10 = 120 compounding periods
Therefore, the periodic interest rate is 0.5% per month, and the number of compounding
periods is 120.
(ii) Periodic interest rate: i =
Example 9.1(b)
Identifying and Calculating Compound Interest Terms When Time Period is in Months
A loan is issued at 4.32% compounded quarterly for nine months.
(i) Identify the nominal interest rate, compounding frequency, and time period in years.
(ii) Calculate the periodic interest rate and number of compounding periods.
Solution
(i) 4.32% compounded quarterly for 9 months
j = 0.0432
m=4
t=
9
years
12
Therefore, the nominal interest rate is 4.32%, compounding frequency is 4, and time
9
years.
period is
12
j
(ii) Periodic interest rate: i =
= 0.0432 = 1.08%
4
m
9
=3
Number of compounding periods: n = m # t = 4 #
12
Therefore, the periodic interest rate is 1.08% per quarter and the number of compounding
periods is three.
Example 9.1(c)
Identifying and Calculating Compound Interest Terms When Time Period is in Years
and Months
An investment is earning 8.2% compounded quarterly for 5 years and 3 months.
(i) Identify the nominal interest rate, compounding frequency, and time period in years.
(ii) Calculate the periodic interest rate and number of compounding periods.
Chapter 9 | Compound Interest
Solution
(i) 8.2% compounded quarterly for 5 years and 3 months
j = 0.082
3
t = `5 + 12 j years = 5.25 years
m=4
Therefore, the nominal interest rate is 8.2%, compounding frequency is 4, and time period
is 5.25 years.
j
(ii) Periodic interest rate: i =
= 0.082 = 0.0205 = 2.05%
4
m
Number of compounding periods: n = m # t = 4 # 5.25 = 21 compounding periods
Therefore, the periodic interest rate is 2.05% per quarter and the number of compounding
periods is 21.
9.1 |
Exercises Answers to the odd-numbered problems are available at the end of the textbook
For the following problems, express the answers rounded to two decimal places, wherever applicable.
Calculate the missing values for Problems 1 and 2:
1.
Nominal Interest Rate,
Compounding Frequency, and
Time Period
Number of
Compounding
Periods per Year
(m)
Periodic
Interest Rate
(i)
Term
(in years)
(t)
Number of
Compounding
Periods for the
Term (n)
a.
5% compounded semi-annually
for 2 years
?
?
?
?
b.
11.4% compounded quarterly
for 1 year and 6 months
?
?
?
?
c.
?% compounded monthly
for 1 year and 6 months
?
0.70%
?
?
d.
?% compounded daily
for 146 days
?
0.02%
?
?
Nominal Interest Rate,
Compounding Frequency, and
Time Period
Number of
Compounding
Periods per Year
(m)
Periodic
Interest Rate
(i)
Term
(in years)
(t)
Number of
Compounding
Periods for the
Term (n)
a.
11.40% compounded monthly
for 15 months
?
?
?
?
b.
10.95% compounded daily
for 292 days
?
?
?
?
c.
?% compounded semi-annually
for 4 21 years
?
2.775%
?
?
d.
?% compounded quarterly
for 1 year and 9 months
?
0.98%
?
?
2.
3. A bank released a new credit card that charges interest at a rate of 1.35% p.m. Calculate the nominal interest rate
compounded monthly for the credit card.
4. Calculate the nominal interest rate compounded monthly for a credit card that has a periodic interest rate of 1.1%
p.m.
301
302
Chapter 9 | Compound Interest
5. If the nominal interest rate of an investment is 5% and the periodic interest rate is 1.25%, calculate the compounding
frequency.
6. Rose invested her savings in a bank account that was giving her a nominal interest rate of 2.86% and a periodic
interest rate of 1.43%. Calculate the compounding frequency.
7. Liana’s investment has a time period of 3 years and 8 months and a monthly compounding frequency. Calculate the
number of compounding periods during the term of her investment.
8. Jonathan received a mortgage that was compounded semi-annually for a period of 15 years. Calculate the number
of compounding periods during the term of this mortgage.
9. Lila receives a personal loan that has 66 compoundings for a period of 16 years and 6 months. Calculate the
compounding frequency of the loan.
10. If Carrey invested money for 1 year and 3 months in a low-risk investment vehicle that had 15 compounding periods
over the term, calculate the compounding frequency of the investment.
11. The periodic interest rate and nominal interest rate of a loan are 2.25% and 4.5%, respectively. Calculate the
number of compounding periods if the loan is for a period of 8 years?
12. Ramya deposits her savings in an investment that has nominal and periodic interest rates of 4.48% and 2.24%,
respectively. If she invests this money for a period of two years, calculate the number of compounding periods
during the term.
13. Calculate the nominal interest rate of a loan that has 16 compounding periods over four years and a periodic
interest rate of 0.5%.
14. George deposits his money for nine years in a savings account that has a periodic interest rate of 1.1%. Calculate
the nominal interest rate of the account if there are 18 compounding periods during the period of investment.
9.2 |
Calculating Future Value (FV )
The following examples will illustrate future value calculations using the future value formula,
Formula 9.1(a) :
n
FV = PV(1 + i)
Example 9.2(a)
Calculating the Future Value and Interest Earned
Hassan Furnishings Inc. invested $40,000 for two years at 3.5% compounded annually.
(i) Calculate the accumulated value of the investment at the end of two years.
(ii) Calculate the compound interest earned on this amount.
Solution
Although $42,849 is
numerically different from
$40,000, they are economically
equal in value, considering
the time value of money; i.e.,
money grows with interest
over time.
(i) FV = PV (1 + i)n = 40,000 (1 + 0.035)2 = 40,000 (1.071225) = $42,849.00
Therefore, the accumulated value of the investment at the end of 2 years would be $42,849.00.
(ii) Compound interest, I = FV - PV = 42,849.00 - 40,000.00 = $2849.00
Therefore, the compound interest earned was $2849.00.