Section 4B The Power of Compounding Pages 228-246

The Power of Compounding 4-B The Power of Compounding Simple Interest Compound Interest Once a year “n” times a year Continuously

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.

Principal Time (years) Interest Paid Total $500 $0 1 (500x.05)=$25 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. Principal Time (years) Interest Paid Total $500 $0 1 (500x.05)=$25 $525 2 $25 $550 3 $575 4 $600 5 $625 10 $500 + $25 x 10 = $750

Simple Interest Formula (for interest paid once a year) 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

Principal Time (years) Interest Paid Total $500 $0 1 (500x.05) = $25 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. Principal Time (years) Interest Paid Total $500 $0 1 (500x.05) = $25 $525 2 (525 x .05)= $26.25 $551.25 3 (551.25 x .05)= $27.56 $578.81 4 (578.81 x .05) = $28.94 $607.75 5 (607.75 x .05) = $30.39 $638.14

45/243 Compare Yancy’s and Samantha’s balances over a 5 year period. Time (years) Total Simple Compound $500 1 $525 2 $550 $551.25 3 $575 $578.81 4 $600 $607.75 5 $625 $638.14 The POWER OF COMPOUNDING!

A general formula for compound interest 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

Time (years) Accumulated Value $500 1 1.05 x 500 = $525 2 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. Time (years) Accumulated Value $500 1 1.05 x 500 = $525 2 (1.05)2 x 500 = $551.25 3 (1.05)3 x 500 = $578.81 4 (1.05)4 x 500 = $607.75 5 (1.05)5 x 500 = $638.15 10 (1.05)10x 500 = $814.45

Compound Interest Formula (for interest paid once a year) 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

Compound Interest (for interest paid once a year) 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

WOW! The Power of Compounding 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!

Planning Ahead with Compound Interest 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

Compounding Interest (More than Once a Year) 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%

Time Accumulated Value 0 m $5000 1 m 1.0025x 5000 2 m (1.0025)2 x 5000 4-B Time Accumulated Value 0 m $5000 1 m 1.0025x 5000 2 m (1.0025)2 x 5000 3 m (1.0025)3 x 5000 4 m (1.0025)4 x 5000 5 m (1.0025)5 x 5000 6 m (1.0025)6 x 5000 7 m (1.0025)7x 5000 8 m (1.0025)8 x 5000 9 m (1.0025)9 x 5000 10 m (1.0025)10 x 5000 11 m (1.0025)11 x 5000 1 yr = 12 m (1.0025)12x 5000 = $5152.08 2 yr = 24 m (1.0025)24x 5000 = $5308.79 5 yr = 60 m (1.0025)60x 5000 = $5808.08

Compound Interest Formula (Interest Paid n Times per Year) 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)

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

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

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? Compounded Formula Total yearly $1035 quarterly $1035.46 monthly $1035.57 daily $1035.62

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. Leonhard Euler (1707-1783) The main message of these values might be that the differences between daily compounding and continuous compounding are minimal.

Compound Interest Formula (Continuous Compounding) 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…

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

This is a relative change calculation 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

APY calculations for $1000 invested for 1 year at 3.5% 4-B APY calculations for $1000 invested for 1 year at 3.5% Compounded Total Annual Percentage Yield annually $1035 3.5%* quarterly $1035.46 3.546% monthly $1035.57 3.557% daily $1035.62 3.562% * (1035 – 1000) / (1000)

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%

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

The Power of Compounding 4-B The Power of Compounding A = P + (i x P) x T Simple Interest Compound Interest Once a year “n” times a year Continuously A = P x (1 + APR ) T

4-B More Practice 49/244 55/244 61/244 65/244 73/244 75/244

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