4COMPOUND 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.
5EXAMPLE Find the amount that results from the investment: $50 invested at 6% compounded monthly after a period of 3 years.$59.83
6COMPARING 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) = $
7COMPARING COMPOUNDING PERIODS Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, and daily will yield the following amounts after 1 year:
8COMPARING 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?
9COMPOUNDING PERIODS INCREASING WITHOUT BOUND As n approaches infinity, it can be shown that the expression is the same as the number e.
10CONTINUOUS COMPOUNDED INTEREST The amount A after t years due to a principal P invested at an annual interest rate r compounded continuously isA = Per t
11COMPARING 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 = $
12EXAMPLEWhat amount will result from investing $100 at 12% compounded continuously after a period of years.A = PertA = 100 e.12(3.75)A = $156.83
13EFFECTIVE RATEEffective 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
14EXAMPLEHow 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.07t16.6 years
15PRESENT VALUEPresent 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.
16PRESENT VALUE FORMULAS For continuous compounded interest,P = A e- rt
17EXAMPLEFind the present value of $800 after 3.5 years at 7% compounded monthly.$626.61
18DOUBLING 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