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

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

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

CONCLUSION OF SECTION 4.7