College Algebra
Tutorial 61: Compound Interest: Future Value
 Learning Objectives
After completing this tutorial, you should be able to:
- Set up and solve a compound interest problem given a finite number of compound periods.
- Set up and solve a continuous compound interest problem.
- Set up and solve an effective rate problem
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Introduction
In this tutorial, we will be looking at applications that deal with something we can all relate to
- MONEY, MONEY, MONEY!! Sounds good, huh? Specifically, we will be learning how to find the future value of an account that is either compounded for a finite number of times or continuously. Even if you are
not
a business related major, a lot of these applications can be used with
your own finances.
It is to your benefit to step through the examples on the
page with
your calculator to make sure that you understand how to work the
problems.
Let's have some fun working with money!!!
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Tutorial
Exponent Key
on Calculator
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Before we get going with the application problems, I wanted
to make
sure that everyone knew how to use the exponent key on their
calculator.
Since there are a lot of different calculators I will be go over the
more
common ones. At this point you need to check to make sure that
you
know how to use your exponent key, because we will be using it heavily
throughout this as well as the next unit.
The two main exponent keys found on
calculators
are (you will only have one of these):
Carrot top key:
^
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Base raised to an exponent
key:
OR 
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Check to see if you have one of the two main type of
exponent keys
(you won’t have both). If you don’t have one check the
other.
If you don’t see either, look in the reference manual that came with
the
calculator to see which key it is or email me.
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Using the Calculator,
in general
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As mentioned above there are a lot of different types of
calculators
out there.
Graphing Calculators
Most graphing calculators allow you to put in the whole formula before
you press enter. In fact you are able to see it all. If you
are going to plug in the whole formula at one time, just make sure you
are careful. Pay special attention to putting in the parenthesis
in the right place.
Business and scientific calculators:
On most business and scientific calculators you will have to put the
formula in part by part. Work your way inside out of the
parenthesis. DO
NOT round until you are at the end. As you go step by step,
don't
erase what you have on your calculator screen, but use it in the next
step,
so you will have have the full decimal number. The examples are
set
up to show you how to piece it together - it goes step by step.
All calculators:
DO NOT round until you get to the final answer. You
will
note on a lot of the examples that I put dots after my numbers that
would
keep going on an on if I had more space on my calculator. Keep in
mind that your calculator may have fewer or more spaces than my
calculator
does - so your calculator may have a slightly different answer
than
mine due to rounding. It should be very close though.
Make sure that you go through these
examples
with your calculator to check to make sure you are entering in
everything
ok. If you are having problems, either check your reference book
that came with your calculator or ask me about it. |
Compound interest means that at the end of each interest
period
the interest earned for that period is added to the previous principle
(the invested amount) so that, it too, earns interest over the next
interest
period.
In other words this is an accumulative interest. |
Compound Interest
Formula
where,
S = compound amount (future value)
P = Principal (starting or present
value)
r = nominal rate (annual %
rate)
n = the number of compound periods
per
year
t = number of years
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Note that some of the letters used in this formula may look
different than
the one in your book.
In some books they use A or Fv instead of S. Just note that this formula is set up to find the future value of the accumulated amount of an account where the interest is compounded, whether you call it S, A, or Fv.
Also, in some books, an i is used instead of r/n. I like to write the formula out using r/n because it helps to remind us that we need to divide the periodic rate by the number of compound periods per year.
 Example
1: Find the a) compound amount AND b) the
compound
interest for the given investment and rate.
$7000 for 9 years at an annual rate of 8% compounded
monthly.
Note that the steps shown in all of
these examples
go with how to put it piece by piece into a business or scientific
calculator.
If you have a graphing calculator, you can chose to do it this way
(piece
by piece) or you can plug the whole formula in and then press enter to
get your final answer. |
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*Compound interest formula
*Plug in appropriate values
*Calculate inside ( )
*Raise ( ) to the 108th power
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So the compound AMOUNT would be $14346.71
The compound amount is the total amount that is in the
account.
How do you think we are going to get the interest?? Well we have
the principle which is the beginning amount and we have the compound
amount
which is the end result. Looks like, if we take the difference of
the two, that will give us how much interest was earned from beginning
to end. What do you think?
Compound amount - principle:
14346.71 - 7000 = 7346.71
So our compound interest is $7346.71.
Wow, our money doubled and then some - of course it compounded
108 times. |
Example
2: Find the a) compound amount AND b) the compound
interest
for the given investment and rate. $15,400 for 11 years at a annual rate of 7.5% compounded
weekly. |
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*Compound interest formula
*Plug in appropriate values
*Calculate inside ( )
*Raise ( ) to the 572nd power
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The compound AMOUNT for this problem is $35120.08.
As above, we will calculate the compound interest by taking
the difference
between the compound amount and principle:
35120.08 - 15400 = 19720.08
The compound interest is $19720.08 |
This is the rate of simple interest earned over a period
of 1 year (in other words, how much it would be if it were compounded annually). |
Effective Rate
Formula
 =
effective rate
r = nominal (annual) rate
n = number of compound periods per
year
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You will use this formula when you are trying to find the
effective
rate.
Example
3: Find the effective rate of interest (rounded to 3
decimal
places) that is equivalent to a nominal rate of 12% compounded
a) yearly
b) semiannually
c) daily |
a) Let's look for the
effective rate
equivalent to the nominal rate of 12% compounded yearly:
Since effective rate is equivalent to compounding annually,
I’m guessing
that we are going to come out with 12% here. It is yearly, so it
looks like n in this problem is going to
be
1. Let’s put it into the formula and check it out: |
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*Effective rate formula
*Plug in appropriate values
*Calculate inside ( )
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This means the equivalent effective rate to compounding it
annually
at 12% is also 12%, which is what we predicted earlier.
b) Let’s see what equivalent
effective
rate we get when we compound this semiannually: |
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*Effective rate formula
*Plug in appropriate values
*Calculate inside ( )
*Raise ( ) to the 2nd power
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This means to get the same interest on something the is
12% compounded
semiannually would have an effective rate of 12.36%.
c) Let’s see what equivalent
effective
rate we get to compounded this daily: |
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*Effective rate formula
*Plug in appropriate values
*Calculate inside ( )
*Raise ( ) to the 365th power
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This means to get the same interest on something that is
12% compounded
daily would have an effective rate of 12.747%.
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Example
4: Find the a) compound amount and b) compound
interest
for the given investment and rate:
$10,000 for 7 years at an effective rate of 7%. |
Since we are wanting the compound amount we are back to the
first formula
we were using earlier. We also need to know the nominal rate. Remember
that effective rate is equivalent to being compounded annually.
Putting this together we have: |
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*Compound interest formula
*Plug in appropriate values
*Calculate inside ( )
*Raise ( ) to the 7th power
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So our compound amount is $16057.82
Taking the difference between the compound amount and the
principle
we get 16057.82 - 10000 = 6057.82.
Therefore, our compound interest is $6057.82. |
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Base e
e is approximately
2.718281828...
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The exponential function with base e is
called the natural exponential function.
e is has a value attached it
(similar to
pi). e is approximately
2.718281828...
This base is used in economic analysis and problems involving
natural
growth and decay. |
I want to make sure that everyone knows how to use the e key on their calculator. Since there are a lot of different
calculators
I will be go over the more common ones. At this point you need to
check to make sure that you know how to use this key, because we will
be
using it heavily.
The two main e keys found on calculators are (you will only have one of these):
e key and
the Carrot top key:
(2 keys)
e and then ^
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e raised
to
an exponent key:
(1 key)
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Check to see if you have one of the two main type of e keys (you won’t have both). If you don’t have one check the
other. If you don’t see either, look in the reference manual that
came with the calculator to see which key it is or email me.
Having to use the e key with the carrot top key is most frequently found on graphing
calculators
but can be found on other types of calculators.
On most graphing calculators in order to raise e to a power you must press the e key
first,
then press your exponent key ^, and then enter in your
exponent.
If you have the e key (with no exponent
showing)
and the carrot top key let's practice taking e and
raising it to the 5th power. To do this you would press the e key, then press the exponent key ^, and then type in 5. It
should look like this on your screen: e^5. If you got
148.41316..., you entered it in correctly. If not, try again. If
you still can't get it, either look in your reference manual that
came with the calculator or email me and I will try to help you.
The key that looks like is most common in business and scientific calculators, but can be found
on other types of calculators.
On most business and scientific calculators, the e function key looks like or very similar to this. So check for this key - note that
the difference between this and the one above is that this key has a
variable
exponent showing on the key - the above key only has an e (no
exponent. If you have this key let's practice taking e and raising it to the 5th power. In this situation, you
first
type in your exponent and then you activate your key. Go ahead and try it out by finding e raised to the 5th power. Type in 5 and then press .
You should have gotten 148.41316... as your answer. If not, try
again. If
you still can't get it, either look in your reference manual that came
with the calculator or email me and I will try to help you. |
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Compound Amount Under
Continuous Interest
S = compound amount (future value)
P = Principal (starting or present
value)
r = nominal rate (annual %
rate)
t = number of years
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Note that some of the letters used in this formula may look
different than
the one in your book.
In some books they use A or Fv instead of S. Just note that this formula is set up to find the future value of the accumulated amount of an account where the interest is compounded, whether you call it S, A, or Fv.
Compounded continuously means that it
is compounded
at every instant of time.
Example
5: Find the compound amount and compound
interest
if $4000 is invested for 6 years and interest is compounded
continuously
at 9%. |
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*Principal is $4000
*9% = .09
*6 years
*Compound Continuous Formula
*Plug in values given above
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The compound amount is $6864.03.
Don’t forget, the problem also asked for the compound
interest.
Like before, we will have to take the difference between the
compound
amount and principle.
Compound interest = 6864.03 - 4000 = $2864.03 |
Practice Problems
These are practice problems to help bring you to the next level.
It will allow you to check and see if you have an understanding of these
types of problems. Math works just like anything
else, if you want to get good at it, then you need to practice it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice. To get the most out of these, you should work the problem out on
your own and then check your answer by clicking on the link for the answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer. |
 Practice
Problem 1a: Find the a) compound amount and b) the compound interest for the
given
investment and rate.
Practice
Problem 2a: Find the compound amount for the given problem.
2a. A $5000 certificate of deposit is purchased for
$5000 and
is held for six years. If the certificate earns an effective rate
of 5 1/4 % what is it worth at the end of that period?
(answer/discussion
to 2a) |
Practice
Problem 3a: Solve the given problem.
3a. If an investor has a choice of investing money at
6% compounded
daily or 6 1/8 % compounded quarterly, which is the better
choice?
(answer/discussion
to 3a) |
Need Extra Help on these Topics?
Last revised on October 7, 2011 by Kim Seward.
All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.
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