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Objectives
 Find
future value of investments when interest is
compounded k times per year
 Find future value of investments when interest is
compounded continuously
 Find the present value of an investment
 Use graphing utilities to model investment data
Example
To numerically and graphically view how $1000 grows at
6% interest, compounded annually, do the following:
a. Construct a table that gives the future values
S = (1 + 0.06)t for t = 0, 1, 2, 3, 4, and 5 years.
b. Graph the function for the given values in a.
c. Graph the function as a continues function.
d. Is there another graph that more accurately
represents the amount that would be in the account at
any time during the first 5 years that the money is
invested?
Example (cont)
Solution
a.
b.
Example (cont)
Solution
c.
d.
Future Value of an Investment with Annual
Compounding
If P dollars are invested at an interest rate r per year,
compounded annually, the future value S at the end of t
years is S = P(1 + r)t.
Future Value of an Investment with Periodic
Compounding
If P dollars are invested for t years at the annual interest
rate r, where the interest is compounded n times per
year, then the interest rate per period is r/n, the number
of compounding periods is nt, and the future value that
results is given by
𝑟
𝑆 =𝑃 1+
𝑛
𝑛𝑡
𝑑𝑜𝑙𝑙𝑎𝑟𝑠
Periodic Compounding Interest
Type of Compounding
Number of Periods per Year ( n = ….)
Annually
1
Semi-Annually
2
Quarterly
4
Monthly
12
Daily
365
Hourly
8760
Each Minute
525,600
Example
a. Write the equation that gives the future value of
$1000 invested for t years at 8%, compounded
annually.
b. Write the equation that gives the future value of
$1000 invested for t years at 8%, compounded daily.
c. Graph the equation from parts (a) and (b) on the
same axes, with t between 0 and 30.
d. What is the additional amount of interest earned in 30
years from compounding daily rather than annually?
Example (continued)
Solution
a. P = 1000 and r = 0.08
b. P = 1000, r = 0.08, and k = 365
Example (continued)
Solution
c.
d. Amount compounded annually =
Amount compounded daily=
Interest =
Future Value of an Investment with Continuous
Compounding
If P dollars are invested for t years at an annual interest
rate r, compounded continuously, then the future value
S is given by
S = Pert dollars
Example
a. What is the future value of $3500 invested for 10
years at 9%, compounded continuously?
b. How much interest will be earned on this investment?
Solution
a.
b. Interest =
Present Value
The lump sum that will give future value S in n
compounding periods at rate i per period is the present
value
S
n
P
or,
equivalently,
P

S
(1

i
)
(1  i )n
Example
What lump sum must be invested at 10%, compounded
semiannually, for the investment to grow to $15,000 in 7
years?
Solution
S=
i=
n=
Example
The data in Table give the annual return
on an investment of $1.00 made on
March 31, 1990. The values reflect
reinvestment of all distributions and
changes in net asset value but exclude
sales charges.
a. Use an exponential function to model these data.
b. Use the model to find what the fund would amount to on
March 31, 2001 if $10,000 was invested March 31, 1990
and the fund continued to follow this model after 2000.
c. Is it likely that this fund continued to grow at this rate in
2002?
Example (cont)
Solution
a. Use technology to create an
exponential model for the data.
b. March 31, 2001 is 11 years after March 31, 1990.
Use the equation from part (a) with x = 11.
c.
Assignment
Pg. 378-380
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#17-27 odd
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