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Section 4A The Power of Compounding

Section 4A The Power of Compounding. Pages 210-222. 4-A. Definitions. The principal in financial formulas ‘ initial amount ’ upon which interest is paid. Simple interest is interest paid only on the original principal, and not on any interest added at later dates.

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Section 4A The Power of Compounding

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  1. Section 4AThe Power of Compounding Pages 210-222

  2. 4-A Definitions • The principal in financial formulas ‘initial amount’ 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.

  3. 4-A Example

  4. Simple Interest – 5.0%

  5. 4-A Example

  6. Compound Interest – 5.0%

  7. Comparing Compound/Simple Interest – 5.0%

  8. Compound Interest – 7%

  9. 4-A General Formual for Compound Interest: Year 1: $1000 + $1000(.05) = $1050 = $1000×(1+.05) Year 2: $1050 + $1050(.05) = $1102.50 = $1050(1+.05) = $1000(1+.05)(1+.05) = $1000(1+.05)2 Year 3: $1102.50+ $1102.50(.05) = $1157.63 = $1102.50(1+.05) = ($1000(1+.05)2)(1+.05) = $1000(1+.05)3 Amount after year t = $1000(1+.05)t

  10. 4-A General Compound Interest Formula A = accumulated balance after t years P = starting principal i = interest rate(written as a decimal) t = number of years

  11. 4-A Suppose an aunt gave $5000 to a child born 3/8/07. The child’s parents promptly invest it in a money market account at 4.91% compounded yearly, and forget about it until the child is 25 years old. How much will the account be worth then? Amount after year 25 = $5000×(1.0491)25 =$5000×(3.314531691...)= $16,572.66

  12. 4-A Suppose you are trying to save today for a $10,000 down payment on a house in ten years. You’ll save in a money market account that pays 4.5% compounded annually (no minimum balance). How much do you need to put in the account now? $10,000 = $P×(1.045)10 so $10,000 = $P (1.045)10 = $6,439.28

  13. Note: 1.04510 = 1.552969422... • $10,000/ 1.5 = $6666.67 • $10,000 / 1.6 = $6250 • $10,000/ 1.55 = $6451.61 • $10,000/ 1.552 = $6443.30 • $10,000 / 1.553 = $6439.15 • $10,000/1.55297 = $6439.27 • $10,000/(1.04510)= $6439.28 • Don’t round in the intermediate steps!!!

  14. 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!

  15. 4-A Example

  16. 4-A Compounding Interest (More than Once a Year) You deposit $5000 in a bank account that pays an APR of 4.5% 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 4.5%/12 = .375%

  17. 4-A General Compound Interest Formula A = accumulated balance after t years P = starting principal i = interest rate (as a decimal) t = number of years

  18. 4-A

  19. 4-A Compound Interest Formulafor 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)

  20. 4-A You deposit $1000 at an APR of 3.5% compounded quarterly. Determine the accumulated balance after 10 years. A = accumulated balance after 1 year P = $1000 APR = 3.50% (as a decimal) = .035 n = 4 Y = 10

  21. 4-A Suppose you are trying to save today for a $10,000 down payment on a house in ten years. You’ll save in a money market account with an APR of 4.5% compounded monthly. How much do you need to put in the account now?

  22. $1000 invested for 1 year at 3.5%

  23. $1000 invested for 20 years at 3.5%

  24. $1000 invested for 1 year at 3.5%

  25. APY = annual percentage yield APY = relative increase over a year Ex: Compound daily for a year: = .03562 ×100% = 3.562%

  26. APR vs APY • APR = annual percentage rate (nominal rate) • APY = annual percentage yield (effective yield) • When compounding annuallyAPR = APY • When compounding more frequently, APY > APR

  27. $1000 invested for 1 year at 3.5%

  28. $1000 invested for 1 year at 3.5%

  29. Leonhard Euler (1707-1783) 4-A 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.

  30. 4-A Compound Interest Formulafor 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 = the special number called Euler’s constant orthe natural number and is an irrational numberapproximately equal to 2.71828…

  31. 4-A Example

  32. 4-A Suppose you have $2000 in an account with an APR of 5.38% compounded continuously. Determine the accumulated balance after 1, 5 and 20 years. Then find the APY for this account. After 1 year:

  33. 4-A Suppose you have $2000 in an account with an APR of 5.38% compounded continuously. Determine the accumulated balance after 1, 5 and 20 years. After 5 years: After 20 years:

  34. 4-A Suppose you have $2000 in an account with an APR of 5.38% compounded continuously. Then find the APY for this account.

  35. 4-A The Power of Compounding Simple InterestCompound InterestOnce a year“n” times a yearContinuously

  36. Homework for Wednesday: Pages 225-226 # 36, 42, 48, 50, 52, 56, 60, 62, 75

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