Download 3.2 Compound Interest

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 is always greater than or equal to simple
interest in the same time period, given the same annual rate.
 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?
See next slide.
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Solution
 Solution: Using the Future Value with simple interest
formula A = P (1 + rt) we obtain the following amount:
 after one month:
 after two months:
 after three months:
After 12 months, the amount is: ________________________.
With simple interest, the amount after one year would be _______.
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 Future Amount with 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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Alternate Formula: Future Amount
with Compound Interest
 The formula for calculating the Future Amount with Compound
mt
Interest is
r


A  P 1  
 m
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 2a: Find the amount to which $1,500 will grow if
compounded quarterly at 6.75% interest for 10 years.
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Same Problem Using
Simple Interest
Example 2b: Compare your answer from part a) to the amount
you would have if the interest was figured using the simple
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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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
(continued )
Example 5: If the inflation rate remains the same for the next 10
years, what will the house from Example 4 be worth in the
year 2014?
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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?
m
General formula:
r 

APY  1    1
m

Substitute values:
Effective rate is
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