Transcript QUADRATIC FUNCTIONS

SECTION 4.7
COMPOUND INTEREST
TERMINOLOGY
Principal:
Total amount borrowed.
Interest:
Money paid for the use of
money.
Rate of Interest:
Amount (expressed as a
percent) charged for the
use of the principal.
SIMPLE INTEREST
FORMULA
I = Prt
COMPOUND INTEREST
FORMULA
r

A  P 1  
n

nt
Where A is the amount due in t years
and P is the principal amount borrowed
at an annual interest rate r compounded
n times per year.
EXAMPLE
Find the amount that results from the
investment:
$50 invested at 6% compounded
monthly after a period of 3 years.
.06

A  50  1 

12 

12(3)
$59.83
COMPARING
COMPOUNDING PERIODS
Investing $1,000 at a rate of 10%
compounded annually, quarterly,
monthly, and daily will yield the
following amounts after 1 year:
A = P(1 + r) = 1,000(1 + .1) = $1100.00
.1

A  1000 1  
4

4
 $1103.81
COMPARING
COMPOUNDING PERIODS
Investing $1,000 at a rate of 10%
compounded annually, quarterly,
monthly, and daily will yield the
following amounts after 1 year:
.1 

A  1000 1 

12 

12
.1 

A  1000 1 

365

 $1104.71
365
 $1105.16
COMPARING
COMPOUNDING PERIODS
The amount increases the more
frequently the interest is
compounded.
Question:
What would happen if the number
of compounding periods were
increased without bound?
COMPOUNDING PERIODS
INCREASING WITHOUT BOUND
r

A  P 1  
n

n
As n approaches infinity, it can be
shown that the expression is the same
as the number e.
CONTINUOUS
COMPOUNDED INTEREST
The amount A after t years due to a
principal P invested at an annual
interest rate r compounded
continuously is
A = Per t
COMPARING
COMPOUNDING PERIODS
Investing $1,000 at a rate of 10%
compounded daily yields :
.1 

A  1000 1 

365

365
 $1105.16
Investing $1,000 at a rate of 10%
compounded continuously yields :
A = 1000 e.1 = $1105.17
EXAMPLE
What amount will result from investing
$100 at 12% compounded continuously
after a period of 3 3 years.
4
A = Pert
A = 100 e.12(3.75)
A = $156.83
EFFECTIVE RATE
Effective Rate is the interest rate that
would have to be applied on a simple
interest investment in order for the
interest earned to be the same as it
would be on a compound interest
investment.
See the table on Page 405
EXAMPLE
How many years will it take for an
initial investment of $25,000 to
grow to $80,000? Assume a rate
of interest of 7% compounded
continuously.
80,000 = 25,000 e.07t
16.6 years
PRESENT VALUE
Present Value is the principal
required on an investment today in
order for the investment to grow to
an amount A by the end of a
specified time period.
PRESENT VALUE
FORMULAS
r

P  A 1  
n

-n t
For continuous compounded interest,
P = A e- rt
EXAMPLE
Find the present value of $800 after 3.5
years at 7% compounded monthly.
.07

P  800 1 

12 

$626.61
- 12(3.5)
DOUBLING AN
INVESTMENT
How long does it take an investment
to double in value if it is invested at
10% per annum compounded
monthly? Compounded
continuously?
.1 

2P  P  1 

12 

12 t
6.9 years
CONCLUSION OF SECTION 4.7