1. Chapter 8 Application of Exponential Equations: Compound Interest

2. Recall the function for compound interest: P is the principal amount r is the annual interest rate m is the number of times interest is compounded per year t is the number of years

3. Compound Interest PeriodInterest Credited Times Credited per year Rate per compounding period Annualyear1 R Semiannual6 months2 Quarterlyquarter4 Monthlymonth12

4. Suppose $1000 is invested at 6% for 1 year. P = 1000 r =.06 t = 1 year If interest is compounded annually (m = 1), then the amount in the account at the end of the year is A = P(1 + r/m) mt = 1000(1 +.06/1) (1)(1) = 1060

5. If interest is compounded quarterly, then the amount in the account at the end of the year is A = P(1 + r/n) nt = 1000(1 +.06/4) (4)(1) = 1061.36

6. The following table contains the results for different compounding periods

7. When the formula A = Pe rt is used to calculate the compound amount, we say that the interest is compounded continuously. Now, when $1000 is invested at 6% for 1 year with the interest compounded continuously, we have A = 1000e.06(1) which is approximately 1061.84

8. In many computations it is simpler to use the formula for interest compounded continuously as an approximation to ordinary compound interest.

9. Problem One thousand dollars is invested at 5% interest compounded continuously. a.Give the formula for A(t), the compounded amount after t years. b.How much will be in the account after 6 years? c.How long is required to double the initial investment?

10. One thousand dollars is invested at 5% interest compounded continuously.

12. Problem Pablo Picasso’s “The Dream” was purchased in 1941 for a war distressed price of $7000. The painting was sold in 1997 for $48.4 million, the second highest price ever paid for a Picasso painting at auction. What rate of interest compounded continuously did the investment earn?

13. Pablo Picasso’s “The Dream” was purchased in 1941 for a war distressed price of $7000. The painting was sold in 1997 for $48.4 million, the second highest price ever paid for a Picasso painting at auction. What rate of interest compounded continuously did the investment earn?

14. If P dollars are invested today, the formula A = Pe rt gives the value of this investment after t years (assuming continuously compounded interest). P is called the present value of the amount A to be received in t years. If we solve for P in terms of A, we obtain

15. Find the present value of $5000 to be received in 2 years if the money can be invested at 12% compounded continuously.