Section 4A The Power of Compounding Pages

Definitions The in financial formulas ‘initial amount’ upon which interest is paid. The principal in financial formulas ‘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

Example

Simple Interest – 5.0% PrincipalTime (years) Interest PaidTotal $10000$0$1000 $10001 $1000×.05=$50 $1050 $10002$50$1100 $10003$50$1150 $10004$50$1200 $10005$50$1250 $100010$1000+ $50×10 = $1500

Example 4-A

Compound Interest – 5.0% PrincipalTime (years) Interest PaidTotal $10000$0$1000 $10001$50$1050 $10502$52.50$ $ $55.13$ $ $57.88$ $ $60.78$

Comparing Compound/Simple Interest – 5.0% PrincipalTime (years) Interest Paid Total Compound Total Simple $10000$0$1000$1000 $10001$50.00$1050$1050 $10502$52.50$ $1100 $ $55.13$ $1150 $ $57.88$ $1200 $ $60.78$ $1250

Compound Interest – 7% PrincipalTime (years) Interest Paid Total Compound $10000$0$1000 $10001$70$1070 $10702$74.90$ $ $80.14$ $ $85.75$

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

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

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×( ) = $16,572.66

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

Note: = Note: = $10,000/ 1.5 = $ $10,000/ 1.5 = $ $10,000 / 1.6 = $6250 $10,000 / 1.6 = $6250 $10,000/ 1.55 = $ $10,000/ 1.55 = $ $10,000/ = $ $10,000/ = $ $10,000 / = $ $10,000 / = $ $10,000/ = $ $10,000/ = $ $10,000/( ) = $ $10,000/( ) = $ Don’t round in the intermediate steps!!! Don’t round in the intermediate steps!!!

The Power of Compounding On July 18, 1461, King Edward IV of England borrows the equivalent of $384 from New College of Oxford. WOW! 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.

Example 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% 4-A

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

TimeAccumulated Value 0 months$ month × $ months ( ) 2 × $ months (1.0375) 3 × $ months ( ) 4 × $ months ( ) 5 × $ months ( ) 6 × $ months ( ) 7 × $ months ( ) 8 × $ months ( ) 9 × $ months ( ) 10 × $ months ( ) 11 × $ yr = 12 m ( ) 12 × $5000 = $ yr = 24 m ( ) 24 × $5000 = $ yr = 60 m ( ) 60 × $5000 = $ A

Compound Interest Formula for Interest Paid n Times per Year 4-A 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)

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

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?

$1000 invested for 1 year at 3.5% CompoundedFormulaTotal Annually (yearly) (yearly)$1035 quarterly$ monthly$ daily$

$1000 invested for 20 years at 3.5% CompoundedFormulaTotal Annually (yearly) (yearly)$ quarterly$ monthly$ daily$

$1000 invested for 1 year at 3.5% CompoundedTotal Annual Percentage Yield annually$1035 quarterly$ monthly$ daily$

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

APR vs APY APR = annual percentage rate (nominal rate) APR = annual percentage rate (nominal rate) APY = annual percentage yield APY = annual percentage yield (effective yield) When compounding annually APR = APY When compounding annually APR = APY When compounding more frequently, APY > APR When compounding more frequently, APY > APR

$1000 invested for 1 year at 3.5% CompoundedTotalAnnual Percentage Yield annually$ % quarterly$ % monthly$ % daily$ %

$1000 invested for 1 year at 3.5% CompoundedTotal annually$1035 quarterly$ monthly$ daily$ Twice daily $ continuously$

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

Example 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:

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:

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

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

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