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# SECTION 4.7

SECTION 4.7. COMPOUND INTEREST. 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. SIMPLE INTEREST FORMULA. I = Prt. COMPOUND INTEREST FORMULA.

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## SECTION 4.7

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1. SECTION 4.7 • COMPOUND INTEREST

2. 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.

3. SIMPLE INTEREST FORMULA I = Prt

4. 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.

5. EXAMPLE Find the amount that results from the investment: \$50 invested at 6% compounded monthly after a period of 3 years. \$59.83

6. 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

7. 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:

8. 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?

9. COMPOUNDING PERIODS INCREASING WITHOUT BOUND As n approaches infinity, it can be shown that the expression is the same as the number e.

10. 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

11. 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

12. What amount will result from investing \$100 at 12% compounded continuously after a period of years. EXAMPLE A = Pert A = 100 e.12(3.75) A = \$156.83

13. 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

14. 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

15. 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.

16. PRESENT VALUE FORMULAS For continuous compounded interest, P = A e- rt

17. EXAMPLE Find the present value of \$800 after 3.5 years at 7% compounded monthly. \$626.61

18. 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

19. CONCLUSION OF SECTION 4.7

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