Section 4B The Power of Compounding

1. Section 4BThe Power of Compounding Pages 228-246

2. 4-B The Power of Compounding Simple InterestCompound InterestOnce a year “n” times a yearContinuously

3. 4-B Definitions/p229 • 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. 4-B 45/243 Yancy invests $500 in an account that earns simple interest at an annual rate of 5% per year. Make a table that shows the performance of this investment for 5 years.

5. 4-B Simple Interest Formula(for interest paid once a year) A = P + (i x P) x T A = accumulated balance after T years P = starting principal i = interest rate (as a decimal) T = number of years Practice 43/243

6. 4-B 45/243 Samantha invests $500 in an account with annual compounding at a rate of 5% per year. Make a table that shows the performance of this investment for 5 years.

7. 4-B 45/243 Compare Yancy’s and Samantha’s balances over a 5 year period. The POWER OF COMPOUNDING!

8. 4-B A general formula for compound interest Year 1: new balance is 5% more than old balance Year1 = 105% of Year0 = 1.05 x Year0 Year 2: new balance is 5% more than old balance Year2 = 105% of Year1 Year2 = 1.05 x Year1 Year2 = 1.05 x (1.05 x Year0) = (1.05)2 x Year0 Year 3: new balance is 5% more than old balance Year3 = 105% of Year2 Year3 = 1.05 x Year2 Year3 = 1.05 x (1.05)2 x Year0 = (1.05)3 x Year0 Balance after year T is (1.05)T x Year0

9. 4-B 45/243 Samantha invests $500 in an account with annual compounding at a rate of 5% per year. Make a table that shows the performance of this investment for 5 years.

10. 4-B Compound Interest Formula(for interest paid once a year) A = P x (1 + i ) T A = accumulated balance after T years P = starting principal i = interest rate (as a decimal) T = number of years

11. 4-B Compound Interest(for interest paid once a year) ex4/234 Your grandfather put $100 under the mattress 50 years ago. If he had instead invested it in a bank account paying 3.5% interest (roughly the average US rate of inflation) compounded yearly, how much would it be worth today? A = P x (1 + i ) T A = 100 x (1 + .035 ) 50 = $558.49

12. 4-B The Power of Compounding On July 18, 1461, King Edward IV of England borrows the equivalent of $384 from New College of Oxford. The King soon paid back $160 but never repaid the remaining $224. This debt was forgotten for 535 years. In 1996, a New College administrator rediscovered the debt and asked for repayment of $290,000,000,000 based on an interest rate of 4% per year. WOW!

13. 4-B Planning Ahead with Compound Interest 8/241 Suppose you have a new baby and want to make sure that you’ll have $100,000 for his or her college education in 18 years. How much should you deposit now at an interest rate of 5% compounded annually? A = P x (1 + i ) T 100000 = P x (1 + .05 ) 18 100000/(1.05)18 = P $41,552 = P

14. 4-B Compounding Interest (More than Once a Year) ex5/235 You deposit $5000 in a bank account that pays an APR of 3% and compounds interest monthly. How much money will you have after 1 year? 2 years? 5 years? APR is annual percentage rate APR of 3% means monthly rate is 3%/12 = .25%

15. 4-B

16. 4-B Compound Interest Formula(Interest Paid n Times per Year) A = accumulated balance after Y years P = starting principal APR = annual percentage rate (as a decimal) n = number of compounding periods per year Y = number of years (may be a fraction)

17. 4-B 55/244 You deposit $15000 at an APR of 5.6% compounded quarterly. Determine the accumulated balance after 20 years. A = 15000 x (1.014)80 = 15000 x 3.04 = $45,617.10

18. 4-B Ex9/241 Suppose you have a new baby and want to make sure that you’ll have $100,000 for his or her college education in 18 years. How much should you deposit now in an investment with an APR of 7% and monthly compounding? 100000 = P x (1.0058)216 100000 = P x 3.513 100000/3.513 = P $28,469.43 = P

19. 4-B ex6’/237 You have $1000 to invest for a year in an account with APR of 3.5%. Should you choose yearly, quarterly, monthly or daily compounding?

20. Leonhard Euler (1707-1783) 4-B Euler’s Constant e Investing $1 at a 100% APR for one year, the following table of amounts — based on number of compounding periods — shows us the evolution from discrete compounding to continuous compounding.

21. 4-B Compound Interest Formula(Continuous Compounding) A = accumulated balance after Y years P = principal APR = annual percentage rate (as a decimal) Y = number of years (may be a fraction) e = Euler’s constant or the natural number -an irrational number approximately equal to 2.71828…

22. 4-B 69/244 Suppose you have $2500 in an account with an APR of 6.5% compounded continuously. Determine the accumulated balance after 1, 5 and 20 years. = $2667.90 = $3460.07 = $9173.24

23. 4-B Definition • The annual percentage yield(APY) is the actual percentage by which a balance increases in one year. This is a relative change calculation

24. 4-B APY calculations for$1000 invested for 1 year at 3.5% * (1035 – 1000) / (1000)

25. 4-B 69/244 Suppose you have $5000 in an account with an APR of 6.5% compounded continuously. Determine the accumulated balance after 1, 5 and 20 years. Then find the APY for this account. = $5335.80 = $6920.15 = $18346.48 APY = (5335.80 - 5000) / (5000) = .06716 = 6.716%

26. 4-B APR vs APY When compounding annually APR = APY When compounding more frequently, APY > APR

27. 4-B The Power of Compounding A = P + (i x P) x T Simple InterestCompound InterestOnce a year “n” times a yearContinuously A = P x (1 + APR ) T

28. 4-B More Practice 49/244 55/24461/24465/24473/24475/244

29. Homework Pages 242-246 # 46, 52, 58, 60, 62, 66, 72, 76