Languages
Back
Languages
Apply the Annual Compound Interest Formula
> ... Math > Calculating Interest, Excel Functions > Apply the Annual Compound Interest Formula

Introduction

In this lesson, you will learn the concept of compound interest and how to calculate the annual compound interest.

These videos illustrate the lesson material below. Watching the videos is optional.

Introduction to Compound Interest

Simple interest earns a fixed amount of interest based on the original principal amount. Compound interest is calculated by taking the interest earned and adding it to the principal amount for the next interest-earning period of time. Compound interest grows with each compounding period.

Figure 1 is a visual representation of the difference between simple interest and compound interest. Notice that in the simple interest images, the blocks representing interest always stay the same size. However, the blocks representing interest in the compound interest column get bigger with each period. This also makes the principal of the next period grow bigger as well.

m15-02_fig01.png

Figure 1

Look at this piece by piece.

  • In the first period of simple interest, you start out with the initial principal amount. At the end of that period, you have the principal plus a little interest, which is a percentage of the principal, added to it.
m15-02_fig02.png

Figure 2

  • At the beginning of the second period, you start again with the same amount of principal. At the end of the period, you add another little piece of interest that’s the same size as the interest from the first period. Now you have two identical amounts of interest added to the principal.
m15-02_fig03.png

Figure 3

  • For a third period, you start again with the original amount, and at the end of the period this time, add another little chunk of interest that’s the same size as the interest calculated for the previous periods. So, at the end of the third period, you have three little pieces or amounts of interest that are all the same size that have been added to the original amount.
m15-02_fig04.png

Figure 4

Let's use the same format to understand compound interest a little bit better.

  • Compound interest, in its first period, is exactly the same. You start with the principal amount, and then at the end of that period, calculate the interest on the principal as a percentage of the principal and add it to the principal. This gives the principal plus a little interest.
m15-02_fig05.png

Figure 5

  • The calculation changes when you come to the second period. In the second period, you take the total amount that you had at the end of period one, which is the principal plus the interest, and calculate the interest on that entire amount.
m15-02_fig06.png

Figure 6

At the end of period two, the interest that was calculated, which is represented by the green square, is a little bit bigger than the interest that was calculated at the end of period one. This is because this interest at the end of period two is calculated from the principal plus the interest from period one, so it’s being calculated from a slightly bigger amount.

  • You can see this even more obviously in period three. At the beginning of period three, you are using the principal and the interest from periods one and two as the starting amount. This is a much bigger square than you started with in period one, and it is bigger than what you started with in period two. At the end of period three, the interest is calculated on this much bigger amount, so you are going to receive a bigger amount of interest as well.
m15-02_fig07.png

Figure 7

From these examples, it’s easy to see how compounding interest can yield significantly more interest than simple interest. Every compounding period calculates more interest than the period before it.

Calculating Annual Compound Interest


The annual compound interest formula is:

\begin{align*} A=P(1+r)^t \end{align*}

  • A = Amount (ending amount)
  • P = Principal (beginning amount)
  • r = Interest rate when compounding one time per year as a decimal
  • t = Time in years

Example 1
Calculate ending amount where the principal is $300, the interest rate is 8%, and the time is 3 years.

\begin{align*}A&= P(1+r)^t &\color{red}\small\text{Annual compound interest formula}\\\\A& = $300(1 + 0.08)^3 &\color{red}\small\text{Substitute given information}\\\\A& = $300 (1.08)^3 &\color{red}\small\text{Simplify the parenthesis first}\\\\A& = $300(1.2597) &\color{red}\small\text{Evaluate the exponent}\\\\A&= $377.91 &\color{red}\small\text{Multiply last}\end{align*}

The ending amount after 3 years is $377.91.

Things to Remember


  • Simple interest only earns a fixed amount of interest based on the original principal amount.
  • Compound interest is calculated by taking the interest earned and adding it to the principal amount for the next interest earning period of time. 
  • Compound interest grows with each compounding period.

Practice Problems

Use the information given in question 1 to solve questions 2 and 3:

1. If you invest \($100\) in an account that pays \(10\%\) simple interest annually for one year, what is the total amount of money that you would have at the end of one year? (
Solution
x
Solution:
\($100 + $100 \times 10\% = $100 + $100 \times 0.10 = $100 + $10 = $110\)
)2. Suppose that you take the total amount of money that you had after the first year in the previous problem and invest it in the same account for one more year. How much money would you have at the end of the second year? (
Solution
x
Solution:
\($110 + $110 \times 10\% = $110 + $110 \times 0.10 = $110 + $11 = $121\)
)3. Suppose that you take the total amount of money that you had after the second year in the previous problem and invest it in the same account for yet another year. How much money would you have at the end of the third year? (
Solution
x
Solution:
\($121 + $121 \times 10\% = $121 + $121 \times 0.10 = $121 + $12.10 = $133.10\)
)

Use the annual compound interest formula for the following questions:

4. How much will you have at the end of three years if you invest \($100\) in an account that pays \(10\%\) annual interest compounded once a year? (You should get the same answer as in question 3.) (
Solution
x
Solution:
\($100(1 + 10\%)^3 = $100(1+0.1)3 = $100(1.1)^3 = $100(1.331) = $133.10\)
)5. If you invest \($1200\) in an account that pays \(8\%\) interest compounded annually, what is the total amount of money that you would have at the end of three years? (
Solution
x
Solution:
\($1200(1+8\%)^3 = $1200(1+0.08)^3 = $1200(1.08)^3 = $1200(1.259712) = $1511.65\)

Details:
You are investing \($1200\) that pays \(8\%\) compounded annually for 3 years, so you will use the formula:

\({\text{A = P(1+r)}}^{\text{t}}\)

The first thing to do is determine each of the following:

Principle = \({\color{MediumVioletRed}P}\) = \({\color{MediumVioletRed}$1200}\)

Rate = \({\color{DodgerBlue}r}\) = \({\color{DodgerBlue}0.08}\) (you must write it as a decimal for the formula to work)

Time = \({\color{MediumSeaGreen}t}\) = \({\color{MediumSeaGreen}3}\)

Input each value into the formula:

\({\text{A = }{\color{MediumVioletRed}{\text{P}}}(1 + {\color{DodgerBlue}{\text{r}}})}^{\color{MediumSeaGreen}{\text{t}}}\)

\({\text{A = }{\color{MediumVioletRed}{\text{1200}}}(1 + {\color{DodgerBlue}{\text{0.08}}})}^{\color{MediumSeaGreen}{\text{3}}}\)

Add \(1\) to \(0.08\):

\({\text{A = }{\color{MediumVioletRed}{\text{1200}}}({\color{DodgerBlue}{\text{1.08}}})}^{\color{MediumSeaGreen}{\text{3}}}\)

Raise \(1.08\) to the third power:

\({\text{A = }}{\color{MediumVioletRed}{\text{1200}}}({\color{DodgerBlue}{\text{1.259712}}})\)

\({\text{A = }}{\color{MediumVioletRed}{1511.6544}}\)

So you will have \($1511.65\) at the end of 3 years.
)6. If you invest \($1200\) in an account that pays \(9\%\) interest compounded annually, what is the total amount of money that you would have at the end of 25 years? (
Solution
x
Solution:
\($1200(1+9\%)^{25} = $1200(1+0.09)^{25}= $1200(1.09)^{25} = $1200(8.623081) = $10347.70\)

Details:
You are investing \($1200\) that pays \(9\%\) compounded annually for 25 years, so you will use the formula:

\({\text{A = P(1+r)}}^{\text{t}}\)

The first thing to do is determine each of the following:

Principle = \({\color{MediumVioletRed}P}\) = \({\color{MediumVioletRed}$1200}\)

Rate = \({\color{DodgerBlue}r}\) = \({\color{DodgerBlue}0.09}\) (you must write it as a decimal for the formula to work)

Time = \({\color{MediumSeaGreen}t}\) = \({\color{MediumSeaGreen}25}\)

Input each value into the formula:

\({\text{A = }{\color{MediumVioletRed}{\text{P}}}(1 + {\color{DodgerBlue}{\text{r}}})}^{\color{MediumSeaGreen}{\text{t}}}\)

\({\text{A = }{\color{MediumVioletRed}{\text{1200}}}(1 + {\color{DodgerBlue}{\text{0.09}}})}^{\color{MediumSeaGreen}{\text{25}}}\)

Add \(1\) to \(0.09\):

\({\text{A = }{\color{MediumVioletRed}{\text{1200}}}({\color{DodgerBlue}{\text{1.09}}})}^{\color{MediumSeaGreen}{\text{25}}}\)

Raise \(1.09\) to the twenty-fifth power:

\({\text{A = }}{\color{MediumVioletRed}{\text{1200}}}({\color{DodgerBlue}{\text{8.62308}}})\)

Then multiply \(1200\) by \(8.62308\):

\({\text{A = }}{\color{MediumVioletRed}{10347.696792}}\)

So you will have \($10,347.70\) at the end of 25 years.
)

    Need More Help?


    1. Study other Math Lessons in the Resource Center.
    2. Visit the Online Tutoring Resources in the Resource Center.
    3. Contact your Instructor.
    4. If you still need help, Schedule a Tutor.
    Perform Calculations for Simple Interest
    Apply the Compound Interest Formula for monthly Compounding Interest