COMPOUND INTEREST

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SECTION 4.7

• COMPOUND INTEREST

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TERMINOLOGY
Principal Total amount borrowed. Interest Money
paid for the use of money. Rate of
Interest Amount (expressed as a percent)
charged for the use of the principal.
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SIMPLE INTEREST FORMULA
I Prt
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COMPOUND INTEREST FORMULA
Where A is the amount due in t years and P is the
principal amount borrowed at an annual interest
rate r compounded n times per year.
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EXAMPLE
Find the amount that results from the
investment 50 invested at 6 compounded monthly
after a period of 3 years.
59.83
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COMPARING COMPOUNDING PERIODS
Investing 1,000 at a rate of 10 compounded
annually, quarterly, monthly, and daily will
yield the following amounts after 1 year A P(1
r) 1,000(1 .1) 1100.00
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COMPARING COMPOUNDING PERIODS
Investing 1,000 at a rate of 10 compounded
annually, quarterly, monthly, and daily will
yield the following amounts after 1 year
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COMPARING COMPOUNDING PERIODS
The amount increases the more frequently the
interest is compounded. Question What would
happen if the number of compounding periods were
increased without bound?
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COMPOUNDING PERIODS INCREASING WITHOUT BOUND
As n approaches infinity, it can be shown that
the expression is the same as the number e.
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CONTINUOUS COMPOUNDED INTEREST
The amount A after t years due to a principal P
invested at an annual interest rate r compounded
continuously is A Per t
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COMPARING COMPOUNDING PERIODS
Investing 1,000 at a rate of 10 compounded
daily yields
Investing 1,000 at a rate of 10 compounded
continuously yields A 1000 e.1 1105.17
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EXAMPLE
A Pert A 100 e.12(3.75) A 156.83
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EFFECTIVE RATE

• Effective Rate is the interest rate that would
  have to be applied on a simple interest
  investment in order for the interest earned to be
  the same as it would be on a compound interest
  investment.
• See the table on Page 405

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EXAMPLE

• How many years will it take for an initial
  investment of 25,000 to grow to 80,000? Assume
  a rate of interest of 7 compounded continuously.
• 80,000 25,000 e.07t
• 16.6 years

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PRESENT VALUE

• Present Value is the principal required on an
  investment today in order for the investment to
  grow to an amount A by the end of a specified
  time period.

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PRESENT VALUE FORMULAS
For continuous compounded interest, P A e- rt
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EXAMPLE
Find the present value of 800 after 3.5 years at
7 compounded monthly.
626.61
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DOUBLING AN INVESTMENT
How long does it take an investment to double in
value if it is invested at 10 per annum
compounded monthly? Compounded continuously?
6.9 years
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• CONCLUSION OF SECTION 4.7