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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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**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**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.**EXAMPLE**Find the amount that results from the investment: $50 invested at 6% compounded monthly after a period of 3 years. $59.83**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**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:**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?**COMPOUNDING PERIODS INCREASING WITHOUT BOUND**As n approaches infinity, it can be shown that the expression is the same as the number e.**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**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**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**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**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**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.**PRESENT VALUE FORMULAS**For continuous compounded interest, P = A e- rt**EXAMPLE**Find the present value of $800 after 3.5 years at 7% compounded monthly. $626.61**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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