Checklist
Notes
Part I
Find the simple interest on a principal.
Find a compounded interest on a principal.
Part II
Use the compound interest formula.
Compare interest growth rates.
APR vs. APY
Continuous compounding.
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Assignment
Notes
Assignment:
1. p 223 Quick Quiz
2. (Part I) p 224 Excercises 43, 45, 47
3. (Part II) p 225 Exercises 49, 51, 53, 55, 59, 60, 65,
68, 71, 73
4. For an algebra review, p 224 Exercises 15 - 42.
Practice using your calculator!
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Key Words - Part II
Notes
The Compound Interest Formula for Interest
Paid Once a Year
The Compound Interest Formula for Interest
Paid More than Once a Year
The Compound Interest Formula for
Continuously Compounded Interest
Annual Percentage Rate The interest rate when
interest is compounded just once a year.
Annual Percentage Yield The relative increase of a
balance over one year.
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Compounding Interest Once A Year
Notes
The balance at the end of year 1 was the starting principal
times 1.05:
$1000 × 1.05 = $1050.
The balance at the end of year 2 was the year 1 ending
balance times 1.05:
$1050 × 1.05 = $1000 × (1.05)2 = $1102.50.
And the third year ending balance is the year 2 ending
balance times 1.05:
$1102.50 × 1.05 = $1000 × (1.05)3 = $1157.63.
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Compounding Interest Once A Year
Notes
Continuing this pattern, we can generalize the ending
balance after N years is the starting principal times 1.05
raised to the Nth power. For instance, after N = 20
years, the balance is
$1000 × (1.05)20 = $2653.30.
Example Use your calculator to find the accumulated
balance after 15 years.
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The Compound Interest Formula
Notes
The Compound Interest Formula
(For Interest Paid Once A Year)
A = P × (1 + APR)Y
(1)
A is the accumalated balance after Y years
P is the starting principal
APR is the annual percentage rate (as a decimal)
Y is the number of years
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The Compound Interest Formula
Notes
Suppose you invest $100 in two accounts. Both pay 10%
per year, but one pays simple interest and one pays
compound interest. Make a table that shows the growth
of each account over a 5-year period.
Use the compound interest formula for the results for
the second account.
Make a chart showing the growth of each account.
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The Compound Interest Formula
Notes
Suppose you deposit $100 into a bank that pays
compound interest at an APR of 10%.
Now assume it pays the interest quarterly. Then the
quarterly interest rate is APR / 4, or 10%/4 = 2.5%.
The table below shows the account growth each quarter.
N quarters
1st Q
2nd Q
3rd Q
4th Q
(Math 1030)
Interest Paid
New Balance
2.5% × $100.00 = $2.50
$102.50
2.5% × $102.50 = $2.56
$105.06
2.5% × $105.06 = $2.63
$107.69
2.5% × $107.69 = $2.69
$110.38
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The Compound Interest Formula
Notes
The Compound Interest Formula
(For Interest Paid More Than Once A Year)
(nY )
APR
A=P × 1+
n
(2)
A is the accumalated balance after Y years
P is the starting principal
APR is the annual percentage rate (as a decimal)
n is the number of compounding periods per year
Y is the number of years
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The Compound Interest Formula
Notes
You invest $3,000 for 15 years with an APR of 4.5% and
monthly.
How much money will accumulate?
Compare this amount to the amount you would have if
interest were paid only once each year.
Note the APR is the same in both cases. One account
pays more, though!
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APR vs. APY
Notes
The previous example compares compounding once a year
with compounding monthly. For the first year. the
interest earned is $135 and $137.82 respectively. The
APR for both are the same (4.5%), but the effective yield
of compounding monthly is higher. This relative increase
is APY:
$137.82
= 0.04594 = 4.594%
$3000
Then over one year,
APY =
(Math 1030)
absolute increase
.
starting principal
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APR vs. APY
Notes
Example You deposit $1000 into an account with APR =
8%. Find the annual percentage yield with monthly
compounding and daily compounding.
Method: Find the accumulated balance at the end of one
year. Then take the absolute and relative differences with
starting principal as the reference value.
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Continuous Compounding
Notes
What if the interest were compounded more often?
(Every hour or minute or second!) The previous example
gives APR = 8% (compounded annually), APY = 8.3%
(compounded monthly), APY = 8.328% (compounded
daily). The APY increases, but the change gets smaller as
the interest compounds more frequently.
If we can compound infinitely many times per year, or
continuously, this would not grow above a certain amount.
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Continuous Compounding
Notes
n
APY
n
APY
1 8.000%
500 8.3280%
12 8.2999 %
1000 8.3284 %
365 8.3278 % 10,000 8.3287%
The Continuous Compounding Formula
A = P × e APR×Y
(3)
e is a special number with a value e ≈ 2.71828.
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Using ”backwards” Compounding Interest
Notes
Suppose you want to create a college fund for a newborn.
How much should you invest now to have a future value
of $100,000 in an account promises APR = 3%,
compounded monthly, in 18 years?
(n×Y )
APR
A=P × 1+
n
(n×Y
)
Divide both sides by 1 + APR
, interchange the left
n
and right sides,
A
P=
(n×Y )
1 + APR
n
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