Section 4A The Power of Compounding

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Section 4AThe Power of Compounding

• Pages 210-222

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Definitions
4-A

• 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.

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Example
4-A
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Simple Interest 5.0
Principal Time (years) Interest Paid Total
1000 0 0 1000
1000 1 1000.0550 1050
1000 2 50 1100
1000 3 50 1150
1000 4 50 1200
1000 5 50 1250
1000 10 1000 5010 1500 1000 5010 1500
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Example
4-A
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Compound Interest 5.0
Principal Time (years) Interest Paid Total
1000 0 0 1000
1000 1 50 1050
1050 2 52.50 1102.50
1102.50 3 55.13 1157.63
1157.63 4 57.88 1215.51
1215.51 5 60.78 1276.29
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Comparing Compound/Simple Interest 5.0
Principal Time (years) Interest Paid Total Compound Total Simple
1000 0 0 1000 1000
1000 1 50.00 1050 1050
1050 2 52.50 1102.50 1100
1102.5 3 55.13 1157.63 1150
1157.63 4 57.88 1215.51 1200
1215.51 5 60.78 1276.29 1250
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Compound Interest 7
Principal Time (years) Interest Paid Total Compound
1000 0 0 1000
1000 1 70 1070
1070 2 74.90 1144.90
1144.90 3 80.14 1225.04
1225.04 4 85.75 1310.79
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General Formual for Compound Interest
4-A
Year 1 1000 1000(.05) 1050
1000(1.05) Year 2 1050 1050(.05)
1102.50 1050?(1.05) 1000?(1.05)?(1.0
5) 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 Amou
nt after year t 1000(1.05)t
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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
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4-A
Suppose an aunt gave 5000 to a child born
3/8/07. The childs 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
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4-A
Suppose you are trying to save today for a
10,000 down payment on a house in ten years.
Youll 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
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• 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
• Dont round in the intermediate steps!!!

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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!
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Example
4-A
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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
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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
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4-A
Time Accumulated Value
0 months 5000
1 month 1.00375 5000
2 months (1.00375)2 5000
3 months (1.0375)3 5000
4 months (1.00375)4 5000
5 months (1.00375)5 5000
6 months (1.00375)6 5000
7 months (1.00375)7 5000
8 months (1.00375)8 5000
9 months (1.00375)9 5000
10 months (1.00725)10 5000
11 months (1.00725)11 5000
1 yr 12 m (1.00375)12 5000 5229.70
2 yr 24 m (1.00375)24 5000 5469.95
5 yr 60 m (1.00375)60 5000 6258.98
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Compound Interest Formulafor 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)
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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
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4-A
Suppose you are trying to save today for a
10,000 down payment on a house in ten years.
Youll 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?
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1000 invested for 1 year at 3.5
Compounded Formula Total
Annually (yearly) 1035
quarterly 1035.46
monthly 1035.57
daily 1035.62
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1000 invested for 20 years at 3.5
Compounded Formula Total
Annually (yearly) 1989.79
quarterly 2007.63
monthly 2011.70
daily 2013.69
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1000 invested for 1 year at 3.5
Compounded Total Annual Percentage Yield
annually 1035
quarterly 1035.46
monthly 1035.57
daily 1035.62
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APY annual percentage yield

• APY relative increase over a year
• Ex Compound daily for a year
• .03562 100
• 3.562

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APR vs APY

• APR annual percentage rate (nominal rate)
• APY annual percentage yield
• (effective yield)
• When compounding annually APR APY
• When compounding more frequently, APY
  gt APR

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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
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1000 invested for 1 year at 3.5
Compounded Total
annually 1035
quarterly 1035.46
monthly 1035.57
daily 1035.617971
Twice daily 1035.61884
continuously 1035.619709
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Eulers 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.

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Compound Interest Formulafor Continuous
Compounding
4-A
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 Eulers
constant orthe natural number and is an
irrational numberapproximately equal to 2.71828
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Example
4-A
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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
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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
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4-A
Suppose you have 2000 in an account with an APR
of 5.38 compounded continuously. Then find the
APY for this account.
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4-A
The Power of Compounding
Simple InterestCompound Interest Once a
year n times a year Continuously
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• Homework for Wednesday
• Pages 225-226
• 36, 42, 48, 50, 52, 56, 60, 62, 75