Download Continuous Compound Interest

SECTION
6.3
Compound Interest and
Continuous Growth
Copyright © Cengage Learning. All rights reserved.
Learning Objectives
1 Use the compound interest formula to calculate
the future value of an investment
2 Construct and use continuous growth models
3 Use exponential models to predict and interpret
unknown results
2
Compound Interest
3
Compound Interest
4
Example 1 – Compound Interest
Consider the following situation. In April 2007, Heritage
Bank offered a 1-year certificate of deposit (CD) with a
minimum investment of $1000 at a nominal interest rate of
4.42% compounded quarterly. (Nominal means “in name
only.”) What is the value of the CD after 1 year?
5
Compound Interest
We can now find the future value of the CD after 1 year by
using a quarterly growth factor of 1.01105 (an increase of
1.105%) as shown in Table 6.17.
Table 6.17
6
Compound Interest
Note that the interest payments continue to increase
throughout the year.
This is due to compound interest, which occurs when
interest is paid on the initial amount invested as well as all
previously earned interest.
7
The Compound Interest Formula
Table 6.19 shows compounding frequencies and
associated formulas for compound interest.
Table 6.19
8
Example 2 – Comparing Future Values
In March 2007, MetLife Bank advertised a 5-year CD with a
4.64% nominal interest rate compounded daily.
Assuming someone could continue to invest their money at
this rate and compounding frequency up until retirement at
age 65, compare the future value of a $5000 investment of
a 25-year-old to the future value of a $10,000 investment
for a 45-year-old.
Assume no additional deposits or withdrawals are made.
9
Example 1 – Solution
The 25-year-old will be accumulating interest 365 days a
year for 40 years (the time until she retires), while the
45-year-old will be accumulating interest 365 days a year
for 20 years (the time until he retires). Using the compound
interest formula, we have
10
Example 1 – Solution
cont’d
Deducting the amount of the initial investment, we see that
the 25-year-old earned $26,986.69 in interest while the
45-year-old earned $15,292.96 in interest.
11
The Compound Interest Formula
Compound Interest as an Exponential Function
Since compound interest causes an account to grow by a
fixed percent each year, we may categorize the compound
interest formula as an exponential function.
First, consider the following property of exponents.
12
The Compound Interest Formula
From this property we can rewrite the compound interest
formula as follows.
If we then let
can rewrite the compound interest
formula in the form y = abx.
A = Pbt
13
The Compound Interest Formula
The annual percentage yield (APY) is the actual amount
of interest earned during the year and takes into account
not only the nominal rate but also the compounding
frequency.
The Truth in Savings Act of 1991 requires banks to publish
the APY for all savings accounts and CDs so consumers
can more easily compare offers from different banks and
make informed financial decisions.
14
The Compound Interest Formula
15
Example 3 – Modeling Compound Interest with Exponential Functions
An investor makes a $3000 initial deposit in a CD with a
3.68% interest rate compounded weekly.
a. Use the compound interest formula to find an
exponential function to model the future value of the
investment.
b. Determine the annual percentage yield for the
investment.
16
Example 3(a) – Solution
So
models the future value of the CD.
17
Example 3(b) – Solution
cont’d
From the exponential model, we can
see the annual growth factor is 1.0375
and the corresponding annual
percentage yield is 3.75% (1.0375 – 1).
18
Continuous Growth Factor
19
Example 4 – Finding the Continuous Growth Factor
Using the compound interest formula, examine
what happens to $1 invested in an account with a
100% interest rate if interest is compounded an
increasing number of times per year. Discuss what
you discover.
Solution:
Based on the examples we have seen thus far, it
appears that the value of the account will increase
as we increase the compounding frequency.
20
Example 4 – Solution
cont’d
To see whether compounding every hour, minute, or
second will dramatically increase the account value,
consider Table 6.20.
Table 6.20
21
Example 4 – Solution
cont’d
The table reveals an interesting result. It appears the
account will never be worth more than $2.72 after 1 year,
and the maximum annual growth (APY) will never be more
than 171.8% (a growth factor of approximately 2.718).
This number ( 2.718) is so important in mathematics
that it has its own name: e.
22
The Compound Interest Formula
We represent the notion of infinitely large n by using the
term continuous compounding.
23
Example 5 – Continuous Compound Interest
Find the future value after 4 years of $2000 invested in an
account with a 5.3% nominal interest rate compounded
continuously.
Solution:
Using the formula for continuous compound interest, we
have
24
Example 5 – Solution
cont’d
After 4 years, the account will be worth $2472.30.
25
e: The Natural Number
26
Converting from General Form to Using e
Consider the following situation. The total annual healthrelated costs in the United States in billions of dollars can
be modeled by the function
where t is the number of years since 1960.
27
e: The Natural Number
The function H is similar to continuous compound interest,
so we could use an exponential function involving the
number e to model the annual health-related expenditures.
Since we are not dealing with an interest-bearing account,
we will use the more generalized form of f(x) = aekx instead
of the more specific A = Pert.
By finding the value of k such that ek = b, we can rewrite
the exponential function using the number e.
28
e: The Natural Number
We see that k = 0.09649, so e0.09649 = 1.1013. Thus, we
rewrite H as
The functions
and
are equivalent, provided we disregard round-off error.
29
Example 6 – Converting f(x) = aekx to f(x) = abx
Rewrite the function f(x) = 1000e0.07x in the form f(x) = abx.
Solution:
Since b = ek, we can find the growth factor b by evaluating
e0.07.
30
Continuous Growth (Decay) Rate
31
Continuous Growth (Decay) Rate
We know that functions of the form f(x) = abx exhibit
exponential growth when b > 1 and exponential decay
when b < 1. Since b = ek, what effect does k have on the
value of b?
Let’s consider the results when k = 0.1 and when k = –0.1.
It appears that f(x) = aekx has exponential growth when
k > 0 and exponential decay when k < 0.
32
Continuous Growth (Decay) Rate
33