Section 4A The Power of Compounding

1
Section 4AThe Power of Compounding

• Pages 210-222

2
A true story

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

3
By the way

• Suggested compromise at 2 only 8.9 million
• Queen has not paid

4
Definitions
4-A

• The principal (PV) 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 simple interest
4-A
6
(No Transcript)
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Simple Interest 2.0
Principal Time (years) Interest Paid Total
1000 0 0 1000
1000 1 20 1020
1000 2 20 1040
1000 3 20 1060
1000 4 20 1080
1000 5 20 1100
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Compound Interest 2.0
Principal Time (years) Interest Paid Total
1000 0 0 1000
1000 1 20 1020
1020 2 20.40 1040.40
1040.40 3 20.81 1061.21
1061.21 4 21.22 1082.43
1082.43 5 21.65 1104.08
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Compound Interest 2.0
Principal Time (years) Interest Paid Total Compound(FV) Total Simple
1000 0 0 1000 1000
1000 1 20.00 1020 1020
1020 2 20.40 1040.40 1040
1040.40 3 20.81 1061.21 1060
1061.21 4 21.22 1082.43 1080
1082.43 5 21.65 1104.08 1100
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Compound Interest 2.75
Principal Time (years) Interest Paid Total Compound
1000 0 0 1000
1000 1 27.5 1027.5
1027.50 2 28.26 1055.76
1055.76 3 29.03 1084.79
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Calculations
4-A
Year 1 1000 1000(.0275) 1027.50
1000?(1.0275) Year 2 1027.50
1027.50(.0275) 1055.76 1027.50?(1.0275)
1000?(1.0275)?(1.0275)
1000?(1.0275)2 Year 3 1055.76
1055.76?(.0275) 1084.79
1055.76?(1.0275) (1000?(1.0275)2)?(1.0275
) 1000?(1.0275)3 Amount after year t
1000(1.0275)t
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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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Kings debt

• Using the formula

224 X (1.04)535 is approximately2.9 x1011
224 X (1.02)535 is approximately8.9 x 106
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4-A
Suppose an aunt gave 5000 to a child born
3/8/05. The childs 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(1.0325)25
11,122.99
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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
2.75 compounded annually (no minimum balance).
How much do you need to put in the account now?
10,000 P(1.0275)10 so 10,000
P (1.0275)10
7623.98
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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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Example
4-A
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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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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
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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 10 years at 3.5
Compounded Formula Total
Annually (yearly) 1410.60
quarterly 1416.91
monthly 1418.34
daily 1419.04
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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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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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APY

• APY relative increase over a year
• Ex Compound daily for a year
• .03562 100
• 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 3.4 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 3.4 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 3.4 compounded continuously. Then find the
APY for this account.
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• Homework for Wednesday
• Pages 225-226
• 34, 42, 48, 50, 54, 56, 62, 76