College Algebra Tutorial 61

College Algebra
Tutorial 61: Compound Interest: Future Value

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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

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!!!

Tutorial

Exponent Key
on Calculator

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:
^ Base raised to an exponent key:

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.

The carrot top key is most frequently found on graphing calculators but can be found on other types of calculators.
On most graphing calculators your exponent key is the carrot top key: ^. If you have the carrot top key let's practice taking 15 and raising it to the 5th power. To do this you would type in 15^5 and press your enter or = key. If you got 759375, 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 look like OR are most common in business and scientific calculators, but can be found on other types of calculators.
On most business and scientific calculators, the exponent key looks like or or very similar to this . So check for this key. If you have this key let's practice taking 15 and raising it to the 5th power. In this situation, you first type in your base then you activate your exponent key and then you type in your exponent and then press enter or equal. Go ahead and try it out by finding 15 raised to the 5th power. Type in 15, then press or and then type in 5 and press the enter or equal key. You should have gotten 759375 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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Using the Calculator,
in general

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

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

compound

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

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.

notebook 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.

*Compound interest formula

*Plug in appropriate values

*Calculate inside ( )
*Raise ( ) to the 108th power

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.

*Compound interest formula

*Plug in appropriate values

*Calculate inside ( )
*Raise ( ) to the 572nd power

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

Effective Rate

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

= effective rate
r = nominal (annual) rate
n = number of compound periods per year

You will use this formula when you are trying to find the effective rate.

notebook 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:

*Effective rate formula

*Plug in appropriate values

*Calculate inside ( )

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:

*Effective rate formula

*Plug in appropriate values

*Calculate inside ( )
*Raise ( ) to the 2nd power

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:

*Effective rate formula

*Plug in appropriate values

*Calculate inside ( )
*Raise ( ) to the 365th power

This means to get the same interest on something that is 12% compounded daily would have an effective rate of 12.747%.

notebook 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:

*Compound interest formula

*Plug in appropriate values

*Calculate inside ( )
*Raise ( ) to the 7th power

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.

Base e

e is approximately 2.718281828...

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.

e Key
on Calculator

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 ^ e raised to an exponent key:
(1 key)

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

Note that some of the letters used in this formula may look different than the one in your book.

Compounded continuously means that it is compounded at every instant of time.

notebook Example 5: Find the compound amount and compound interest if $4000 is invested for 6 years and interest is compounded continuously at 9%.

*Principal is $4000
*9% = .09

*6 years

*Compound Continuous Formula
*Plug in values given above

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.

1a. $2000 for 12 months at an effective rate of 5%.
(answer/discussion to 1a)

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?

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

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