Section 4A The Power of Compounding Pages

4-A

Example

The Power of Compounding Simple Interest Compound Interest Once a year More than once a year Continuously 4-A

Definitions The in financial formulas is the initial amount upon which interest is paid. The principal in financial formulas is the initial amount upon which interest is paid. is interest paid only on the original principal, and not on any interest added at later dates. Simple interest is interest paid only on the original principal, and not on any interest added at later dates. is interest paid on both the original principal and on all interest that has been added to the original principal. Compound interest is interest paid on both the original principal and on all interest that has been added to the original principal. 4-A

35/225 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. PrincipalTime (years) Interest PaidTotal $5000$0$500 $5001(500x.05)=$25$525 $5002$25$550 $5003$25$575 $5004$25$600 $5005$25$625 4-A

PrincipalTime (years) Interest PaidTotal $5000$0$500 $5001 (500x.05) = $25 $525 $5252 (525 x.05)= $26.25 $ $ ( x.05)= $27.56 $ $ ( x.05) = $28.94 $ $ ( x.05) = $30.39 $ /225 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. 4-A

Time (years) Total Simple Total Compound 0$500$500 1$525$525 2$550$ $575$ $600$ $625$ /225 Compare Yancy’s and Samantha’s balances over a 5 year period. The POWER OF COMPOUNDING! 4-A

A general formula for compound interest 4-A Year 1: new balance is 5% more than old balance Year1 = Year0 + 5% of Year0 Year1 = 105% of Year0 = 1.05 x Year0 Year 2: new balance is 5% more than old balance Year2 = Year1 + 5% of Year1 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 = Year2 + 5% of Year2 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 0$ x 500 = $525 2 (1.05) 2 x 500 = $ (1.05) 3 x 500 = $ (1.05) 4 x 500 = $ (1.05) 5 x 500 = $ (1.05) 10 x 500 = $ /225 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. 4-A

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

4-A ex4/216 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 ( ) 50 = $ Compound Interest (for interest paid once a year)

4-A pg 221 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 7% compounded annually? Planning Ahead with Compound Interest A = P x (1 + i ) T = P x ( ) /(1.07) 18 = P $29,586 = P

Example 4-A

Compounding Interest (More than Once a Year) ex5/218 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% 4-A

TimeAccumulated Value 0 m$ m x m (1.0025) 2 x m (1.0025) 3 x m (1.0025) 4 x m (1.0025) 5 x m (1.0025) 6 x m (1.0025) 7 x m (1.0025) 8 x m (1.0025) 9 x m (1.0025) 10 x m (1.0025) 11 x yr = 12 m (1.0025) 12 x 5000 = $ yr = 24 m (1.0025) 24 x 5000 = $ yr = 60 m (1.0025) 60 x 5000 = $ A

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

45/225 You deposit $5000 at an APR of 5.6% compounded quarterly. Determine the accumulated balance after 20 years. A = 5000 x (1.014) 80 = 5000 x 3.04 = $15, A

Ex9/222 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 10% and monthly compounding? = P x (1.0083) = P x /5.962 = P $ = P 4-A

ex6’/219 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? CompoundedFormulaTotal yearly yearly$1035 quarterly$ monthly$ daily$ A

Definition The is the actual percentage by which a balance increases in one year. The annual percentage yield(APY) is the actual percentage by which a balance increases in one year. 4-A

APY calculations for $1000 invested for 1 year at 3.5% CompoundedTotal Annual Percentage Yield annually$ %* quarterly$ % monthly$ % daily$ % * (1035 – 1000) / (1000) 4-A

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

Euler’s Constant e 4-A 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 ( )

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

Example 4-A

59/225 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. APY = ( ) / (5000) = = 6.716% = = 6.716% = $ = $ = $

The Power of Compounding Simple Interest Compound Interest Once a year More than once a year Continuously 4-A A = P x (1 + APR ) T

More Practice 39/225 45/225 51/225 55/225 63/225 65/225 4-A

Homework Pages # 36, 42, 48, 50, 52, 56, 60, 62, 66