1. Copyright © 2008 Pearson Education, Inc. Slide 4-1 Unit 4B The Power of Compounding

2. Copyright © 2008 Pearson Education, Inc. Slide 4-2 Definitions The principal in financial formulas is the balance upon which interest is paid. Simple interest is interest paid only on the original principal, and not on any interest added at later dates. Compound interest is interest paid on both the original principal and on all interest that has been added to the original principal. 4-B

3. Accumulated Balance Simple Interest (not in text) Copyright © 2008 Pearson Education, Inc. Slide 4-3 A = accumulated balance after Y years P = starting principal APR = annual percentage rate (as a decimal) Y = number of years (may be a fraction)

4. Copyright © 2008 Pearson Education, Inc. Slide 4-4 Compound Interest Paid, n times per year, Formula 4-B A = accumulated balance after Y years P = starting principal APR = annual percentage rate (as a decimal) Y = number of years (may be a fraction) n = number of compoundings per year

5. Copyright © 2008 Pearson Education, Inc. Slide 4-5 Simple and Compound Interest Compare the growth in a $100 investment for 5 years at 10% simple interest per year and at 10% interest compounded annually. The compound interest account earns $11.05 more than the simple interest account. 4-B

6. Comparing Simple and Compound Interest Simple Interest is linear growth with respect to time (Years). Compound Interest is exponential growth with respect to time (Years). Copyright © 2008 Pearson Education, Inc. Slide 4-6

7. Practice: 1) Compute the accumulated balance of $40,000 invested in both simple and compound interest with an APR of 7% for 25 years. Assume one compounding per year when calculating the accumulated balance for compound interest. 2) Compute the accumulated balance of $15,000 invested in an account earning compound interest that is compounded quarterly with a 5.6% APR for 20 years. Copyright © 2008 Pearson Education, Inc. Slide 4-7

8. More Practice 3) Re-due question #2 but this time have interest compounded only once per year. Which compounding earns you more money, annual or quarterly compounding? Copyright © 2008 Pearson Education, Inc. Slide 4-8

9. Copyright © 2008 Pearson Education, Inc. Slide 4-9 Compound Interest Show how quarterly compounding affects a $1000 invested at 8% per year. 4-B

10. Copyright © 2008 Pearson Education, Inc. Slide 4-10 APR vs. APY 4-B APR = annual percentage rate APY = annual percentage yield (also known as effective yield) APY = APR if interest is compounded annually. APY > APR if interest is compounded more than once a year. APY = relative increase = (over one year) absolute increase starting principal

11. Example: Find the annual percentage yield (APY) for a bank offering an APR of 4.25% compounded monthly. As the number of compoundings increases the APY increases. Let’s explore whether this increase is without bound; Or is there some upper limit to the APY as the number of compoundings per year increases. Copyright © 2008 Pearson Education, Inc. Slide 4-11

12. Copyright © 2008 Pearson Education, Inc. Slide 4-12 Increasing the Number of Compoundings per Year: Show how different compounding periods affect the APY for an APR of 8%. 4-B

13. Copyright © 2008 Pearson Education, Inc. Slide 4-13 Continuous Compounding 4-B

14. Copyright © 2008 Pearson Education, Inc. Slide 4-14 Compound Interest Formula for Continuous Compounding 4-B P = starting principal A = accumulated balance after Y years e = the special number called Euler’s constant and is an irrational number approximately equal to 2.71828 Y = number of years (may be a fraction) APR = annual percentage rate (as a decimal)

15. Example: Calculate the accumulated balance after 5 years for an account with a starting principal of $10,000 that is invested in an account with an APR of 8% compounded continuously. Copyright © 2008 Pearson Education, Inc. Slide 4-15

16. Copyright © 2008 Pearson Education, Inc. Slide 4-16 Definitions Present value is the original principal. Future value is the accumulated amount. 4-B

17. Example: Suppose you want to have a $100,000 college fund in 18 years for an heir. How much will you need to deposit now in an account that earns 4%, compounded daily. Assume that you will make no deposits into the account after the initial deposit. Round answers to the nearest dollar. Copyright © 2008 Pearson Education, Inc. Slide 4-17