5.1 Simple and Compound Interest
Question 1: What is simple interest?
Question 2: What is compound interest?
Question 3: What is an effective interest rate?
Question 4: What is continuous compound interest?
Businesses operate with borrowed money. When a business needs order inventory or
expand, it may borrow the money needed for the expansion. The borrower will be
charged interest for the opportunity to use the money. The interest on the loan is
typically charged at some percentage of the amount borrowed called the interest rate.
Not only do businesses borrow, but banks may also borrow money from other banks or
even individuals. For instance, you may invest money with a bank and receive interest
from the bank. Most consumers borrow money regularly using credit cards. If the
balance is not paid off when the lending period is through, you must pay interest to the
credit card company for the privilege of borrowing the money.
In this section, we’ll examine several type of interest. Simple interest is interest where a
fixed amount is paid based on the amount borrowed and the length of time the money is
borrowed. In compound interest, interest accumulates according to the amount
borrowed over time and any interest that has accumulated during that period of time.
Both types of interest are used extensively in business and finance.
1
Question 1: What is simple interest?
In business, individuals or companies often borrow money or assets. The lender
charges a fee for the use of the assets. Interest is the fee the lender charges for the use
of the money. The amount borrowed is the principal or present value of the loan.
Simple interest is interest computed on the original principal only. If the present value
PV, in dollars, earns interest at a rate of r for t years, then the interest is
I  PV rt
The future value (also called the accumulated amount or maturity value) is the sum of
the principal and the interest. This is the amount the present value grows to after the
present value and interest are added.
Simple Interest
The future value FV at a simple interest rate r per year is
FV  PV  PV rt
 PV 1  rt 
where PV is the present value that is deposited for t years.
The interest rate r is the decimal form of the interest rate written as a percentage. This
means an interest rate of 4% per year is equivalent to r  0.04 .
In this text, we use the variable names commonly used in finance textbooks. Instead of
writing the present value as the single letter P, we use two letters, PV. Be very careful to
interpret this as a single variable and not a product of P and V. Similarly, the future
value is written FV. This set of letters represents a single quantity, not a product of F
and V. This allows us to use groups of letters to represent quantities that suggest their
meaning.
2
If we know two of the quantities in this formula, we can solve for the other quantity. This
formula is also used to calculate simple interest paid on investments or deposits at a
bank. In these cases, we think of the deposits or investment as a loan to the bank with
the interest paid to the depositor.
Example 1
Simple Interest
An investment pays simple interest of 4% per year. An investor deposits
$500 in this investment and makes no withdrawals for 5 years.
a. How much interest does the investment earn over the five-year
period?
Solution Use I  PV rt to compute the interest,
I  500  0.04  5
Set PV = 500, r = 0.04, and t = 5
 100
b. What is the future value of the investment in 5 years?
Solution The future value is computed using FV  PV 1  rt  ,
A  500 1  0.04  5 
Set PV = 500, r = 0.04, and t = 5
 600
c.
Find an expression for the future value if the deposit accumulates
interest for t years. Assume no withdrawals over the period.
Solution In this part, the time t is variable,
FV  500 1  0.04t 
 500  20t
3
This relationship corresponds to a linear function of t. The vertical
intercept is 500 and the slope is 20. This tells us that the initial
investment is $500 and the accumulated amount increases by $20 per
year.
Figure 1 – The linear function describing the accumulated amount in Example
1c.
Example 2
Simple Interest
A small payday loan company offers a simple interest loan to a
customer. They will loan the customer $750. The customer promises to
repay the company $808 in two weeks. What is the annual interest rate
for this loan?
Solution Since there are 52 weeks in a year, the length of this loan is
2
52
years. Use the information in the problem in the simple interest formula,
FV  PV 1  rt  , to solve for the rate r:
4
808  750 1  r  522 
52
2
Set FV = 808, PV = 750 and t 
808
750
 1  522 r
Divide both sides by 750
58
750

Subtract 1 or
2
52
r
58
 750
r
750
750
2
52
from both sides
Multiply both sides by
52
2
r  2.01
This decimal corresponds to an interest rate of 201% per year. Because
of such high rates, many states are passing legislation to limit the
interest rates that pay day loan companies charge.
5
Question 2: What is compound interest?
In a loan or investment earning compound interest, interest is periodically added to the
present value. This additional amount earns interest. In other words, the interest earns
interest.
Let us illustrate this process with a concrete example. Suppose we deposit $500 in an
account that earns interest at a rate of 4% compounded annually. This rate is the
nominal or stated rate. By saying that interest is compounding annually, we mean that
interest is added to the principal at the end of each year.
For instance, we use the simple interest formula, FV  PV 1  rt  , to compute the future
value at the end of the first year,
FV  500(1  0.04 1)
 500 1.04 
 520
To find the future value at the end of the second year, we let the present value be the
future value from the end of the first year in the simple interest formula,
A  500 1  0.04 11  0.04 1

future value
from first year
 500 1.04 
2
 540.80
Since the present value in this amount includes the interest from the first year, the
interest from the first year is earning interest. This is the effect of compounding.
To find the future value at the end of the third year, we let the future value at the end of
the second year be the present value in the simple interest formula,
6
A  500 1  0.04 1 1  0.04 1

2
future value
from first two years
 500 1.04 
3
 562.43
Let us summarize these amounts in a table.
End of the
Calculation for the
Future Value
Future Value
First Year
500 (1.04)
520
Second Year
500 1.04 
540.80
Third Year
500 1.04 
2
3
562.43
The middle column establishes a simple pattern. At the end of each year, the future
value is equal to the present value times several factors of 1.04. These factors
correspond to the compounding of interest.
In general, if interest is compounded annually, then the future value is
FV  PV 1  r  t
where PV is the principal, r is the nominal rate and t is the time in years.
If interests compounds more than once a year, finding the future value is more
challenging. It is more likely that interest is compounded quarterly (4 times a year),
monthly (12 times a year) or daily (365 times a year). The length of time between which
interest is earned is the conversion period. The length of time over which the loan or
investment earns interest is the term. To account for compounding over shorter
conversion periods, we need more factors in the expression for the future value.
However, in each of these factors we only earn a fraction of the interest rate.
7
For instance, suppose deposit $500 in an account earning 4% compounded quarterly.
To calculate the future value, we multiply the principal by a factor corresponding to onefourth of the interest rate each quarter. The future value after one quarter is
FV  500(1  0.011)
 500 1.01
 505
After two quarters, the future value contains two factors corresponding to one percent
interest per quarter,
FV  500 1  0.01 11  0.011
 500 1.01
2
 510.05
Continue this pattern for twelve conversion periods (twelve quarters or three years)
gives
FV  500 1.01  563.41
12
If we compare this expression to the expression for compounding quarterly,
A  500 1.04  , we note several differences. When we compound quarterly, we get four
3
times as many factors in the future value. This is due to the fact compounding quarterly
means we need four times as many factors. When we compound quarterly, each factor
utilizes a rate that is one-fourth the rate for compounding annually.
8
Compound Interest
The future value FV of the present value PV compounded
over n conversion periods at an interest rate of i per period is
FV  PV 1  i 
n
where
i
r
nominal rate
,

m number of conversion periods in a year
and
n  mt   number of conversion periods in a year  term in years  .
You may also see compound interest computed from the formula A  P 1  mr 
mt
. This is
the exact same formula as the one above except the present value is called the
principal P and the future value is called the accumulated amount A.
Example 3
Compound Interest
A customer deposits $5000 in an account that earns 1% annual interest
compounded monthly. If the customer makes no further deposits or
withdrawals from the account, how much will be in the account in five
years?
Solution To utilize the compound interest formula, FV  PV 1  i  , we
n
must find the present value PV, the interest rate per conversion period i,
and the number of conversion periods n. The present value or principal
9
is the amount of the original deposit so P  5000 . The account earns 1%
annual interest, compounded monthly. This means the account earns
i
0.01
12
percent per month
over each conversion period. Since the interest is compounded monthly
over 5 years, there are n  12  5 or 60 conversion periods during the
time this money is deposited. The future value is
60
 0.01 
FV  5000 1 
  5256.25 dollars
12 

If the future value, interest rate, and number of conversion periods is known, we can
solve for the present value in FV  PV 1  i  . In problems like this, we want to know
n
what amount should we start with to grow to a known future value.
Example 4
Present Value
A couple needs $25,000 for a large purchase in five years. How much
must be deposited now in an account earning 2% annual interest
compounded quarterly to accumulate this amount? Assume no further
deposits or withdrawals during this time period.
Solution To find the amount needed today, we must find the present
value of $25,000. The interest for each conversion period is
i
0.02
percent per period
4
The account earns interest over a total of 4  5 or 20 conversion periods.
Substitute these values into the compound interest formula,
FV  PV 1  i  , and solve for PV:
n
10
25000  PV 1  0.005 
25000
1  0.005
20
 PV
20
Substitute
FV  25, 000 , i  0.005 ,and n  20
Divide both sides by
1  0.005
20
22626.57  PV
We round the present value in the last step to two decimal places. This
ensures the value is accurate to the nearest cent. If the couple invests
$22,626.57 for five years, it will grow to $25,000 at this interest rate.
11
Question 3: What is an effective interest rate?
The amount of interest compounded depends on several factors. The nominal rate r
and the number of conversion periods m both influence the future value over a
predetermined time period. A savings account earning a higher nominal rate over fewer
conversion periods might have the same future value as another savings account with a
lower nominal rate and a higher number of conversion periods. To help us compare
nominal interest rates, we use the effective interest rate. The effective interest rate is
the simple interest rate that leads to the same future value in one year as the nominal
interest rate compounded m times per year.
The effective interest rate is
m
r

re  1    1
 m
where r is the nominal interest rate, and m is the number of
conversion periods per year.
Another name for the effective interest rate is the annual percentage yield or APR.
Example 5
Best Investment
An investor has the opportunity to invest in one of two opportunities.
The first opportunity is a certificate of deposit (CD) earning 1.140%
compounded daily. The second opportunity is an investment yielding a
dividend of 1.141% compounded quarterly. Which investment is best?
Solution The better investment is the one with the higher effective
interest rate. The nominal rate for the CD is r  0.01140 . Interest is
earned on a daily basis so m  365 . This gives an effective rate of
12
 0.01140 
re   1 

365 

365
 1  0.01147
For the other investment, r  0.01141 and m  4 . The effective rate for
this investment is
4
 0.01141 
re   1 
  1  0.01146
4 

The effective rate for the CD, 1.1147%, is higher than the effective rate
for the investment, 1.1146%. Because of this, the CD is the better
investment.
By law, the effective rate of interest is shown in all transactions involving interest
charges. The APR is always prevalent in advertisements, such as the one below for
five-year CD rates from Bankrate.com on December 29, 2011.
Institution
APR
Rate
Minimum Deposit
Bank of America
1.1965%
1.19
Compounded monthly
$1000
We can also use the APR to compute accumulated amounts. Suppose we want to
compute the future value from depositing $1000 in the Bank of America five year CD.
We could calculate the future value using the rate,
 0.0119 
FV  1000  1 

12 

512
 1061.27
Alternatively, we compute the future value using the APR and compound annually,
13
FV  1000 1  0.011965   1061.27
5
This gives us another way of computing accumulated amounts.
The future value FV compounded at an effective interest
rate (APR) of re is
FV  PV 1  re 
t
where PV is the present vvalue or principal, and t is the
term in years.
Since the APR is always shown in financial transactions, this formula allows us to
compute accumulated amounts from the APR.
We can also use the compound interest formula to find the rate at which an amount
grows. In this case, we think of PV as the original amount and FV as the amount it
grows to.
Example 6
Growth of Ticket Prices
In 2000, the average price of a movies theater ticket was $5.39. In 2010,
the average price increased to $7.89. At what annual percentage rate
did prices increase over the period from 2000 to 2010 on average?
Source: National Association of Theater Owners
Solution The original price in 2000 is $ 5.39. This price grows in ten
years to $7.89. Use these values in FV  PV 1  re  to find the effective
t
rate
re :
14
7.89  5.39 1  re 
10
7.89
10
 1  re 
5.39
To solve for
re , remove the tenth power by raising both sides of the
equation to the one-tenth power.
1
1
10 10
 7.89  10 

r
1


e 
 5.39 

 7.89 
 5.39 
1
10
 1  re
Multiply exponents,
10  110  1
Subtract 1 from both sides
1
10
 7.89 
 5.39   1  re
0.0388  re
Over this period, the price of tickets increased by an average of 3.88%
per year.
15
Question 4: What is continuous compound interest?
As the frequency of compounding increases, the effective interest rate also increases.
We can see this by computing the effective interest rate at a specific nominal rate, say
r  0.1 .
Frequency
Number of conversion
periods per year m
Effective interest rate
m
1  0.1m   1
annually
1
0.100000
semiannually
2
0.102500
quarterly
4
0.103813
monthly
12
0.104713
daily
365
0.105156
hourly
8760
0.105170
every minute
525,600
0.105171
As the number of conversion periods per year increases, the effective interest rate gets
closer and closer to 0.105171.
In fact, it is possible to show that the effective interest rate gets closer and closer to the
value e0.1  1 as the frequency of computing increases. If this is done at a nominal rate of
r  0.1 , the accumulated amount is
A  P 1  re 
t
 P 1  e0.1  1
 P e

0.1 t
t
Set
re  e0.1  1
Simplify using the fact that
a 
m n
 am n
 P e0.1t
16
In general, as the frequency of compounding increases, the effective interest rate gets
closer and closer to e r  1 . We can express this symbolically by writing,
m
r

r
1    1  e  1 as m  
 m
Think of the symbol  as meaning “approaches”.
Larger and larger values of m mean that we are compounding interest more and more
frequently. When this happens, we say that the interest is term compounded
continuously.
The future value FV of the present value PV compounded
continuously at a nominal interest rate of r per period is
FV  PV er t
where t is the time in years.
Like the compound interest formula, this formula may also be written in several
equivalent forms. In a biological context, the size of a population P with an initial amount
of P0 growing at a continuous rate of r % per year over t years grows according to
P  P0 ert In some business applications, an original amount of money or principal P grows to an
accumulated amount A at a continuous rate of r % per year over t years according to
A  P ert In each of these applications, some quantity is growing at a continuous rate r. The
original amount of the quantity is multiplied by a factor of ert to yield the amount of the
quantity at some later time.
17
Example 7
Continuous Interest
Third Federal Savings and Loan offers a CD that earns 1.79%
compounded quarterly (on February 3, 2012). If $5000 is invested in the
CD, how much more money would be in the account in 5 years if the
interest is compounded continuously versus quarterly?
Solution The future value with interest compounded quarterly is
 0.0179 
FV  5000 1 

4 

45
 5467.05
The future value with interest compounded continuously is
FV  5000e0.01795  5468.14
The future value with continuous interest is greater than the future value
with interest compounded quarterly by
5468.14  5467.05  1.09
In general, compounding some amount continuously will always yield a larger amount
than compounding the same amount at the same rate a finite number of times per year.
The greater number of times the amount is compounded in a year, the closer the future
value will be to the future value compounded continuously.
18