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) = $

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 = $

12. 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

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