Transcript Section 3B Putting Numbers in Perspective

Section 4B
The Power of
Compounding
Pages 228-246
4-B
The Power of Compounding
Simple Interest
Compound Interest
Once a year
“n” times a year
Continuously
4-B
Definitions/p229



The principal in financial formulas is the balance
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.
4-B
45/243 Yancy invests $500 in an account that
earns simple interest at an annual rate of 5% per
year. Make a table that shows the performance of
this investment for 5 years.
Principal
Interest Paid
Total
$500
Time
(years)
0
$0
$500
$500
1
(500x.05)=$25
$525
$500
2
$25
$550
$500
3
$25
$575
$500
4
$25
$600
$500
5
$25
$625
$500
10
$500 + $25 x 10 = $750
4-B
Simple Interest Formula
(for interest paid once a year)
A = P + (i x P) x T
A
P
i
T
=
=
=
=
accumulated balance after T years
starting principal
interest rate (as a decimal)
number of years
Practice 43/243
4-B
45/243 Samantha invests $500 in an account with
annual compounding at a rate of 5% per year.
Make a table that shows the performance of this
investment for 5 years.
Principal
Interest Paid
Total
$500
Time
(years)
0
$0
$500
$500
1
(500x.05) = $25
$525
$525
2
(525 x .05)= $26.25
$551.25
$551.25
3
(551.25 x .05)= $27.56
$578.81
$578.81
4
(578.81 x .05) = $28.94 $607.75
$607.75
5
(607.75 x .05) = $30.39 $638.14
4-B
45/243 Compare Yancy’s and Samantha’s
balances over a 5 year period.
Time
(years)
0
Total
Simple
$500
Total
Compound
$500
1
$525
$525
2
$550
$551.25
3
$575
$578.81
4
$600
$607.75
5
$625
$638.14
The POWER OF COMPOUNDING!
A general formula for compound interest
Year 1: new balance is 5% more than old balance
Year1 = 105% of Year0 = 1.05 x Year0
Year 2: new balance is 5% more than old balance
Year2 = 105% of Year1
Year2 = 1.05 x Year1
Year2 = 1.05 x (1.05 x Year0) = (1.05)2 x Year0
Year 3: new balance is 5% more than old balance
Year3 = 105% of Year2
Year3 = 1.05 x Year2
Year3 = 1.05 x (1.05)2 x Year0 = (1.05)3 x Year0
Balance after year T is (1.05)T x Year0
4-B
4-B
45/243 Samantha invests $500 in an account with
annual compounding at a rate of 5% per year.
Make a table that shows the performance of this
investment for 5 years.
Time (years)
Accumulated Value
0
$500
1
1.05 x 500 = $525
2
(1.05)2 x 500 = $551.25
3
(1.05)3 x 500 = $578.81
4
(1.05)4 x 500 = $607.75
5
(1.05)5 x 500 = $638.15
10
(1.05)10x 500 = $814.45
4-B
Compound Interest Formula
(for interest paid once a year)
A = P x (1 + i )
A
P
i
T
=
=
=
=
T
accumulated balance after T years
starting principal
interest rate (as a decimal)
number of years
4-B
Compound Interest
(for interest paid once a year)
ex4/234 Your grandfather put $100 under the
mattress 50 years ago. If he had instead invested
it in a bank account paying 3.5% interest (roughly
the average US rate of inflation) compounded
yearly, how much would it be worth today?
A = P x (1 + i )
T
A = 100 x (1 + .035 )
= $558.49
50
4-B
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!
4-B
Planning Ahead with
Compound Interest
8/241 Suppose you have a new baby and want to
make sure that you’ll have $100,000 for his or her
college education in 18 years. How much should
you deposit now at an interest rate of 5%
compounded annually?
A = P x (1 + i )
T
100000 = P x (1 + .05 )
100000/(1.05)18 = P
$41,552 = P
18
4-B
Compounding Interest
(More than Once a Year)
ex5/235 You deposit $5000 in a bank
account that pays an APR of 3% 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 3%/12 = .25%
4-B
Time
Accumulated Value
0m
$5000
1m
1.0025x 5000
2m
(1.0025)2 x 5000
3m
(1.0025)3 x 5000
4m
(1.0025)4 x 5000
5m
(1.0025)5 x 5000
6m
(1.0025)6 x 5000
7m
(1.0025)7x 5000
8m
(1.0025)8 x 5000
9m
(1.0025)9 x 5000
10 m
(1.0025)10 x 5000
11 m
(1.0025)11 x 5000
1 yr = 12 m
(1.0025)12x 5000 = $5152.08
2 yr = 24 m
(1.0025)24x 5000 = $5308.79
5 yr = 60 m
(1.0025)60x 5000 = $5808.08
4-B
Compound Interest Formula
(Interest Paid
n Times per Year)
A = P  (1+
A
P
APR
n
Y
=
=
=
=
=
APR ( n  Y)
n
)
accumulated balance after Y years
starting principal
annual percentage rate (as a decimal)
number of compounding periods per year
number of years (may be a fraction)
4-B
55/244 You deposit $15000 at an APR of 5.6%
compounded quarterly. Determine the
accumulated balance after 20 years.
A = P  (1+
APR ( n  Y)
n
A = 15000  (1+
)
.056 ( 4  20)
4
A = 15000 x (1.014)80
= 15000 x 3.04
= $45,617.10
)
4-B
Ex9/241 Suppose you have a new baby and want
to make sure that you’ll have $100,000 for his or
her college education in 18 years. How much
should you deposit now in an investment with an
APR of 7% and monthly compounding?
A = P  (1+
APR ( n  Y)
n
100000 = P  (1+
)
.07 ( 12  18)
12
)
100000 = P x (1.0058)216
100000 = P x 3.513
100000/3.513 = P
$28,469.43 = P
ex6’/237 You have $1000 to invest for a year in
an account with APR of 3.5%. Should you
choose yearly, quarterly, monthly or daily
compounding?
Compounded
yearly
quarterly
monthly
daily
Formula
Total
(1)
.035 
1000 1 +

1 





(4)
.035 
1000 1 +

4 





(12)
.035 
1000 1 +

12 










1000 1 +
.035
365
(365)





$1035
$1035.46
$1035.57
$1035.62
4-B
4-B
Euler’s Constant e
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.
( n  1)
A = $1  (1+ 1.0
)
n
n = number of compoundings
1 = year
4 = quarters
12 = months
365 = days
365•24 = hours
365•24•60 = minutes
365•24•60•60 = seconds
infinite number of compoundings
A = accumulation
2.0
2.44140625
2.6130352902236
2.7145674820245
2.7181266906312
2.7182792153963
2.7182824725426
e  2.71828182846
Leonhard Euler
(1707-1783)
4-B
Compound Interest Formula
(Continuous Compounding)
(
APR

Y
)
A=P  e
A = accumulated balance after Y years
P = principal
APR = annual percentage rate (as a decimal)
Y = number of years (may be a fraction)
e = Euler’s constant or the natural number
-an irrational number approximately equal
to 2.71828…
4-B
69/244 Suppose you have $2500 in an account with an
APR of 6.5% compounded continuously. Determine the
accumulated balance after 1, 5 and 20 years.
(.0651)
A = 2500  e
(.065  5)
= $2667.90
A = 2500  e
= $3460.07
A = 2500  e(.065  20)
= $9173.24
4-B
Definition

The annual percentage yield(APY) is the
actual percentage by which a balance increases in
one year.
value after 1 year - principal amount
APY =
principal amount
This is a relative change calculation
4-B
APY calculations for
$1000 invested for 1 year at 3.5%
Compounded
annually
Total
$1035
Annual Percentage
Yield
3.5%*
quarterly $1035.46
3.546%
monthly $1035.57
3.557%
daily $1035.62
3.562%
* (1035 – 1000) / (1000)
4-B
69/244 Suppose you have $5000 in an account with an
APR of 6.5% compounded continuously. Determine the
accumulated balance after 1, 5 and 20 years. Then find
the APY for this account.
(.0651)
A = 5000  e
(.065  5)
= $5335.80
A = 5000  e
= $6920.15
A = 5000  e(.065  20)
= $18346.48
APY = (5335.80 - 5000) / (5000)
= .06716 = 6.716%
4-B
APR vs APY
When compounding annually
APR = APY
When compounding more frequently,
APY > APR
4-B
The Power of Compounding
Simple Interest
A = P + (i x P) x T
Compound Interest
Once a year
A = P x (1 + APR )
T
“n” times a year
A = P  (1+
APR ( n  Y)
n
)
Continuously
(
APR

Y
)
A=P  e
More Practice
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4-B
Homework
Pages 242-246
# 46, 52, 58, 60, 62, 66, 72, 76