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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) = $

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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 = $

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EXAMPLE What amount will result from investing $100 at 12% compounded continuously after a period of years. 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

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