Transcript Learning Objectives for Section 3.2 Compound Interest After this lecture, you should be able to  Compute compound interest.  Compute the annual percentage.

Learning Objectives for Section 3.2
Compound Interest
After this lecture, you should be able to
 Compute compound interest.
 Compute the annual percentage yield of a compound
interest investment.
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Compound Interest
 Compound interest: Interest paid on interest reinvested.
 Compound interest grows faster than simple interest.
 Annual nominal rates: How interest rates are generally quoted
annual nominal rate
 Rate per compounding period: # of compounding periods per year
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Compounding Periods
The number of compounding periods per year (m):
 If the interest is compounded annually, then m = _______
 If the interest is compounded semiannually, then m = _______
 If the interest is compounded quarterly, then m = _______
 If the interest is compounded monthly, then m = _______
 If the interest is compounded daily, then m = _______
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Example
Example 1: Suppose a principal of $1 was invested in an
account paying 6% annual interest compounded monthly. How
much would be in the account after one year?
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Solution
 Solution: Using the Future Value using simple interest
formula A = P (1 + rt) we obtain:
 0.06 
 amount after one month
1 1 
  11  0.005   (1.005)
12 

 after two months
 0.06 
1.005 1 
  1(1.005)(1.005)  1.005
12


 after three months
2
 0.06 
2
3
1.0052 1 
  1.005 1.005   1.005
12 

After 12 months, the amount is (1.005)12 = 1.0616778.
With simple interest, the amount after one year would be 1.06.
The difference becomes more noticeable after several years.
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Graphical Illustration of
Compound Interest
The growth of $1 at 6% interest compounded monthly compared
to 6% simple interest over a 15-year period.
Dollars
The blue curve refers to the
$1 invested at 6% simple
interest.
The red curve refers to the
$1 at 6% being compounded
monthly.
Time (in years)
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General Formula: Compound Interest
 The formula for calculating the Amount using Compound
Interest is
mt
r

A  P 1  
 m
Where
A is the future amount,
P is the principal,
r is the annual interest rate as a decimal,
m is the number of compounding periods in one year, and
t is the total number of years.
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Simplified Formula: Compound Interest
 The formula for calculating the Amount: Compound Interest is
r

A  P 1  
 m
mt
r
To simplify the formula, let i 
m
and
n  mt
We now have,
A  P 1  i 
n
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Example
Example 2: Find the amount to which $1,500 will grow if
compounded quarterly at 6.75% interest for 10 years. Then,
compare it to the amount if the interest was figured using the
simple interest formula.
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Example
 Find the amount to which $1500 will grow if compounded
quarterly at 6.75% interest for 10 years.
n
 Solution: Use
A  P 1  i 
10(4)
 0.0675 
A  1,500 1 

4


A  $2,929.50
 Helpful hint: Be sure to do the arithmetic using the rules for
order of operations.
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Same Problem Using
Simple Interest
Using the simple interest formula, the amount to which $1500
will grow at an interest of 6.75% for 10 years is given by
A = P (1 + rt)
= 1,500(1 + 0.0675(10)) = $2,512.50
which is more than $400 less than the amount earned using the
compound interest formula.
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Changing the number of
compounding periods per year
Example 3: To what amount will $1,500 grow if
compounded daily at 6.75% interest for 10 years?
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Changing the number of
compounding periods per year
To what amount will $1,500 grow if compounded daily at
6.75% interest for 10 years?
10(365)
0.0675 

A  1500 1 

365


Solution:
= $2,945.87
This is about $15.00 more than compounding $1,500 quarterly
at 6.75% interest.
Since there are 365 days in year (leap years excluded), the
number of compounding periods is now 365. We divide the
annual rate of interest by 365. Notice, too, that the number of
compounding periods in 10 years is 10(365)= 3650.
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Effect of Increasing the
Number of Compounding Periods
 If the number of compounding periods per year is increased
while the principal, annual rate of interest and total number of
years remain the same, the future amount of money will
increase slightly.
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Computing the Inflation Rate
Example 4: Suppose a house that was worth $68,000 in 1987 is
worth $104,000 in 2004. Assuming a constant rate of inflation
from 1987 to 2004, what is the inflation rate?
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Computing the Inflation Rate
Solution
 Suppose a house that was worth
$68,000 in 1987 is worth
$104,000 in 2004. Assuming a
constant rate of inflation from
1987 to 2004, what is the inflation
rate?
1. Substitute in compound interest
formula.
2. Divide both sides by 68,000
3. Take the 17th root of both sides
of equation
4. Subtract 1 from both sides to
solve for r.
 Solution:
104,000  68,000 1  r  
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104,000
17
 1  r  
68,000
104,000
17
 (1  r ) 
68,000
104,000
17
 1  r  0.0253
68,000
r = 2.53%
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Computing the Inflation Rate
(continued )
Example 5: If the inflation rate remains the same for the next 10
years, what will the house be worth in the year 2014?
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Computing the Inflation Rate
(continued )
 If the inflation rate remains
the same for the next 10
years, what will the house
be worth in the year 2014?
 Solution: From 1987 to
2014 is a period of 27 years.
If the inflation rate stays the
same over that period, r =
0.0253. Substituting into the
compound interest formula,
we have
A  68,000(1  0.0253)  $133,501
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Example
Example 6: If $20,000 is invested at 4% compounded monthly,
what is the amount after a) 5 years b) 8 years?
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Which is Better?
Example 7: Which is the better investment and why: 8%
compounded quarterly or 8.3% compounded annually?
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Inflation
Example 8: If the inflation rate averages 4% per year compounded
annually for the next 5 years, what will a car costing $17,000
now cost 5 years from now?
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Investing
Example 9: How long does it take for a $4,800 investment at 8%
compounded monthly to be worth more than a $5,000 investment
at 5% compounded monthly?
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Annual Percentage Yield
 The simple interest rate that will produce the same amount as a
given compound interest rate in 1 year is called the annual
percentage yield (APY). To find the APY, proceed as follows:
Amount at simple interest APY after one year
= Amount at compound interest after one year
m
r

P(1  APY )  P 1   
 m
m
r

1  APY  1   
 m
m
r

APY  1    1
 m
This is also called
the effective rate.
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Annual Percentage Yield
Example
What is the annual percentage yield for money that is invested at
6% compounded monthly?
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Annual Percentage Yield
Example
What is the annual percentage yield for money that is invested at
6% compounded monthly?
m
General formula:
Substitute values:
r

APY  1    1
 m
12
 0.06 
APY  1 
  1  0.06168
12 

Effective rate is 0.06168 = 6.168
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