Section 4A The Power of Compounding Pages

A true story July 18, 1461 King Edward IV borrowed equivalent of $384 from New college of Oxford July 18, 1461 King Edward IV borrowed equivalent of $384 from New college of Oxford King repaid $160 but not remaining $224 King repaid $160 but not remaining $224 Debt forgotten for 535 years Debt forgotten for 535 years 1996 New college contacted Queen 1996 New college contacted Queen Repayment with 4% interest- $290 billion Repayment with 4% interest- $290 billion

By the way Suggested compromise at 2% only $8.9 million Suggested compromise at 2% only $8.9 million Queen has not paid Queen has not paid

Definitions The (PV) in financial formulas ‘initial amount’ upon which interest is paid. The principal (PV) 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 4-A

Simple Interest – 2.0% PrincipalTime (years) Interest Paid Total $10000$0$1000 $10001$20$1020 $10002$20$1040 $10003$20$1060 $10004$20$1080 $10005$20$1100

Compound Interest – 2.0% PrincipalTime (years) Interest Paid Total $10000$0$1000 $10001$20$1020 $10202$20.40$ $ $20.81$ $ $21.22$ $ $21.65$

Compound Interest – 2.0% PrincipalTime (years) Interest Paid Total Compound (FV) Total Simple $10000$0$1000$1000 $10001$20.00$1020$1020 $10202$20.40$ $1040 $ $20.81$ $1060 $ $21.22$ $1080 $ $21.65$ $1100

Compound Interest – 2.75% PrincipalTime (years) Interest Paid Total Compound $10000$0$1000 $10001$27.5$ $ $28.26$ $ $29.03$

Calculations: 4-A Year 1: $ $1000(.0275) = $  = $1000  ( ) Year 2: $ $ (.0275) = $  = $  ( )  = $1000  ( )  ( )  = $1000  ( ) 2  Year 3: $ $  (.0275) = $  = $  ( )  = ($1000  ( ) 2 )  ( )  = $1000  ( ) 3 Amount after year t = $1000( ) t

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

King’s debt Using the formula Using the formula $224 X (1+.04) 535 is approximately$2.9 x10 11 $224 X (1+.02) 535 is approximately$8.9 x 10 6

4-A Suppose an aunt gave $5000 to a child born 3/8/05. The child’s parents promptly invest it in a money market account at 3.25% 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 × ( ) 25 = $11,122.99

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 2.75% compounded annually (no minimum balance). How much do you need to put in the account now? $10,000 = $P × ( ) 10 so $10,000 = $P (1.0275) 10 = $

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

Example 4-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)

$1000 invested at 3.5% compounded quarterly for one year 4-A A = accumulated balance after 1 year P = $1000 APR = 3.50% (as a decimal) =.035 n = 4 Y = 1

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

$1000 invested for 10 years at 3.5% CompoundedFormulaTotal Annually (yearly) (yearly)$ quarterly$ monthly$ daily$

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

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$ %

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

$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 3.4% 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 3.4% 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 3.4% compounded continuously. Then find the APY for this account.

Homework for Wednesday: Pages # 34, 42, 48, 50, 54, 56, 62, 76