Compound Interest

When you deposit money in a savings account, you expected it to gain interest over time. The money will continue to gain interest until you decide to make a withdrawal. It is like an accumulation of interest.

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Meaning of compound interest

Compound interest is the accumulation or addition of interest to a principal amount.

The idea is that the interest gained on the principal amount is reinvested, and future interest is added to the principal amount plus the earlier interest, where the principal amount is the original amount of money that was invested. This continues until the period elapses. Take a look at the compound interest graph below.

Compound interest graph

Compound interest graph

From the compound interest graph we can see that as time increases, the money also increases. This is the whole idea of compound interest.

When solving questions related to compound interest, we are actually asked to find the amount of money that is obtained or earned over a particular period of time as a result of the rate of compound interest added.

To calculate the compound interest, we should know:

• the principal or initial amount.
• the percentage rate of the compound interest on the principal amount.
• the time which is the period that the money will be withdrawn or stop gaining interest.

Calculating compound interest: Formula

There are two ways to calculate compound interest. You can calculate using a table, and you can calculate using the compound interest formula.

Using the formula of compound interest

The compound interest formula is given by:

$FinalAmount=Principal×{\left(multiplier\right)}^{n}$

where,

$\begin{array}{rcl}Multiplier& =& 1+rate\\ n& =& timeperiod\end{array}$

Therefore,

$FinalAmount=Principal×{\left(1+rate\right)}^{timeperiod}$

The result obtained from the compound interest formula is the amount of money earned or gained after interest has been added over time.

The multiplier is the sum of one and the percentage interest rate.

Using the compound interest table

For each year, we calculate the money to be held until the time elapses. In order to do so, we follow the following steps.

1. Draw a table with two columns, one for 'amount' and another for 'rate in percentage'.
2. Write the principal amount on the first row under the amount and multiply it by the percentage rate under interest.
3. Add the principal amount to the interest on the second row under amount and multiply by the percentage rate on that row under interest.
4. Repeat step 3 until the time elapses.

Calculating the compound interest using the table will take longer than it would if you used the compound interest formula.

In the next section, we will learn how to calculate compound interest using both methods.

Compound interest examples

Let’s take some examples using both the compound interest formula and the table.

If you deposit £4000 in a bank for three years and you are paid 4% interest per annum. How much will you have at the end of the 3 years?

Solution

Let's try to solve this problem using the table first and then we will try the formula. From the steps above, we know we have to draw a two-column table.

 Amount Percentage Rate 4% 1st year – $£4000$ This is the principal or initial amount you deposited in the bank. $\frac{4}{100}×4000=£160$This means you will get $£160$extra in the first year. 2nd year – $4000+160=£4160$ You are starting the second year with the principal amount and the interest gained. $\frac{4}{100}×4160=£166.4$ This means you will get $£166.4$ extra by the end of the second year. 3rd year – $4160+166.4=£4326.4$You are starting the third year with the amount earned in the second year plus the interest gained; this is the amount of money you will have after 3 years. $\frac{4}{100}×4326.4=£173.056$ This means you will get $£173.056$ extra by the end of the third year; this is the interest gained at the end of 3 years. $4326.4+173.056=£4499.456$This is the amount of money you will have after 3 years. $160+166.4+173.056=£499.455$ This is the interest gained at the end of 3 years.

So, at the end of the 3 years, you will have $£4499.456$.

That was quite a long process but we can get to our answer quicker using the compound interest formula. The formula is below.

$FinalAmount=Principal×{\left(1+rate\right)}^{n}$

From the given, we know that

$Principal=£4000\phantom{\rule{0ex}{0ex}}Rate=4%\phantom{\rule{0ex}{0ex}}n=3$

We will now substitute the values in the formula.

$FinalAmount=4000×{\left(1+\frac{4}{100}\right)}^{3}\phantom{\rule{0ex}{0ex}}=4000×{\left(1+0.04\right)}^{3}\phantom{\rule{0ex}{0ex}}=4000×{\left(1.04\right)}^{3}\phantom{\rule{0ex}{0ex}}=4000×1.124864\phantom{\rule{0ex}{0ex}}=£4499.456$

You can see that the same answer was gotten using the table and the formula.

Let's take another example.

Jane deposits £800 in a bank paying 1% compound interest per annum. What will she have after two years? Use the table and the formula for the compound interest calculation for the following question.

Solution

First of all, state the information given:

$Principal=£800\phantom{\rule{0ex}{0ex}}Rate=1%\phantom{\rule{0ex}{0ex}}n=2years\phantom{\rule{0ex}{0ex}}Finalamount=?$

 Amount Percentage Rate - 1% 1st year – $£800$This is the initial money she deposited at the bank. $\frac{1}{100}×800=£8$This means at the end of the first year, Jane will have extra money. 2nd year – $800+8=£808$ The second year will start with the initial amount plus the interest gained in the first year. $\frac{1}{100}×808=£8.08$This means that at the end of the second year, Jane will have $£8.08$ extra money. After 2 years, Jane will have $808+8.08=£816.08$ The total interest earned after 2 years is $8+8.08=£16.08$

According to the table, by the end of 2 years, Jane will have $£816.08$.

Let's now use the formula. The compound interest formula is given by:

$FinalAmount=Principal×{\left(1+rate\right)}^{n}$

We will now substitute our values in the formula to get:

$FinalAmount=800×{\left(1+\frac{1}{100}\right)}^{2}\phantom{\rule{0ex}{0ex}}=800×{\left(1+0.01\right)}^{2}\phantom{\rule{0ex}{0ex}}=800×{\left(1.01\right)}^{2}\phantom{\rule{0ex}{0ex}}=800×1.0201\phantom{\rule{0ex}{0ex}}=£816.08$

You can see that the same answer was obtained using the table and the formula.

Let's take one more example.

Ben takes a loan of £15000, and the bank charges him 10% compound interest per annum. If Ben does not pay off the loan in four years, how much does Ben owe the bank?

Solution

Let’s state the information given.

$Principal=£15000\phantom{\rule{0ex}{0ex}}Rate=10%\phantom{\rule{0ex}{0ex}}Numberofyears\left(n\right)=4years\phantom{\rule{0ex}{0ex}}Amountowedafter4years=?$

The amount owed after four years is the final amount and we will get it using the compound interest formula. The compound interest formula is given by:

$FinalAmount=Principal×{\left(1+rate\right)}^{n}$

We will substitute our values in the formula:

$Finalamount=15000×{\left(1+\frac{10}{100}\right)}^{4}\phantom{\rule{0ex}{0ex}}=15000×{\left(1+0.1\right)}^{4}\phantom{\rule{0ex}{0ex}}=15000×{\left(1.1\right)}^{4}\phantom{\rule{0ex}{0ex}}=15000×1.4641\phantom{\rule{0ex}{0ex}}=£21961.5$

Ben will owe the bank $£21961.5$by the end of the 4 years.

Difference between simple and compound interest

Aside from compound interest, there is also what is called simple interest.

The significant difference between simple interest and compound interest is that simple interest has to do with one-time interest on the principal amount while compound interest has to do with an accumulation of interest on the principal amount over a period of time. To find out more about simple interest, check out our article on Simple Interest.

Compound Interest - Key takeaways

• Compound interest is the accumulation or addition of interest to a principal amount.
• There are two ways to calculate compound interest. You can calculate it using a table or by using the compound interest formula.
• The compound interest formula is: $FinalAmount=Principal×{\left(1+rate\right)}^{n}$.
• Aside from compound interest, there is also what is called simple interest. The significant difference between simple and compound interest is that simple interest has to do with one-time interest on the principal amount while compound interest has to do with an accumulation of interest on the principal amount over a period of time.

Flashcards in Compound Interest 6

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What is compound interest with examples?

Compound interest is the accumulation or addition of interest to a principal amount. An example is when money is deposited in a savings account. The money is expected to gain interest continuously over time.

What is compound interest?

Compound interest is the accumulation or addition of interest to a principal amount.

How do you calculate compound interest?

Compound interest can be calculated by using the compound interest formula or by using the table method.

What is the difference between compound interest and simple interest?

The significant difference between simple and compound interest is that simple interest has to do with one-time interest on the principal amount while compound interest has to do with an accumulation of interest on the principal amount over a period of time.

What is the formula for compound interest?

The compound interest formula is:

Final Amount = Principal x (1 + rate)^n

where n is the time period.

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