## How To Calculate Simple Interest and Compound Interest

When money is deposited into a bank account, such a savings account, the bank uses the money and makes a profit. In return, the bank pays an agreed sum of money together with the initial deposit to the depositor. That sum of money paid to the depositor on top of the their deposit is called the interest.

**Interest** is the sum of money paid for the use of borrowed money.

For example, when you deposit money in your savings account, the following happens:

- the bank "borrows" your money and invests it in other businesses.
- the bank makes a profit.
- after a stipulated amount of time, the bank gives you a certain amount of money as interest in return for using your money.
- you then withdraw more money than you initially deposited in the bank.

Another example is when you borrow money from the bank. Money borrowed from the bank is generally know as a loan. After an agreed amount of time, when you return the loan, you pay an additional amount of interest to pay the bank for using their money.

There are basically two types of interest:

- Simple interest
- Compound interest

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## Simple Interest

Simple interest is calculated as an agreed percentage of the initial money borrowed, lent or invested. The initial amount of money is called the principal, capital or present value.

### Simple interest formula

The simple interest formula is:

$latex \displaystyle I = PRT$

Whereby:

- I is the interest to be paid out.
- P is the principal or the initial amount of money.
- R is the interest rate per annum, as a percentage.
- T is the time duration of the loan or investment, expressed in years.

After T years, the total amount of money to be paid out will be equal to:

$latex \displaystyle A = P + I$

$latex \displaystyle A = P + PRT$

$latex \displaystyle A = P(1 + RT)$

### Simple interest example

Juliet invested $10 000 in a bank at 15% p.a. simple interest. Calculate the total amount of money she will get after 10 years.

#### Solution

First we calculate the interest she will get after 10 years.

- P = $10 000
- R = 15% = 0.15
- T = 10 years

$latex \displaystyle I = PRT$

$latex \displaystyle I = 10000 \times 0.15 \times 10$

Interest = $15 000

So the total amount of money she will get after 10 years is:

$latex \displaystyle A = P + I$

Amount = 10 000 + 15 000

= $25 000

## Compound Interest

Compound interest is calculated as an agreed percentage of the existing money on regular basis.

### Compound Interest formula

$latex \displaystyle A = P(1 + R)^T$

Whereby:

- A is the future value of the amount
- P is the principal or the present value
- R is the interest rate per annum
- T is the time in years

### Compound interest example

Sydney invested $10 000 in a bank at 15% p.a. compound interest. Calculate the total amount of money he will get after 10 years.

#### Solution

- P = $10 000
- R = 15% = 0.15
- T = 10 years

$latex \displaystyle A = P(1 + R)^T$

$latex \displaystyle A = 10000(1 + 0.5)^{10}$

Amount = $576 650

## Simple Interest vs Compound Interest

Simple interest is paid out as a percentage of the original investment whilst compound interest is paid out as a percentage of newly updated amount. To make it simple let us compare how simple interest and compound interest work side by side.

Let us say $5 000 is invested at 15% p.a interest rate

Period | Amount based on Simple interest | Amount based on Compound interest |
---|---|---|

Initially | $5 000 | $5 000 |

End of year 1 | A = 5 000 + 0.15(5 000) = $ 5 750 | A = 5 000 + 0.15(5 000) = $ 5 750 |

End of year 2 | A = 5 750 + 0.15(5 000) = $ 6 500 | A = 5 750 + 0.15(5 750) = $ 6 613 |

End of year 3 | A = 6 500 + 0.15(5 000) = $ 7 250 | A = 6 613 + 0.15(6 613) = $ 7 605 |

As you can see from the table above:

- simple interest is always a percentage of the original value.
- compound interest is a percentage of the opening value that year.

### Simple Interest vs Compound Interest Example

Mr Matava has $25 000 he wishes to invest for 5 years. The bank offers two investment options:

- at 10% p.a. compound interest
- at 12% p.a. simple interest

As his financial advisor, choose the best investment option for him.

#### Solution

At 10% p.a. compound interest:

$latex \displaystyle A = P(1 + R)^T$

$latex \displaystyle A = 25000(1 + 0.1)^5$

Amount = $40 262.75

At 15% p.a simple interest:

$latex \displaystyle A = P(1 + RT)$

$latex \displaystyle A = 25000(1 + 0.12 \times 5)$

A = $40 000

The best option for him is the best option for him. Though it has a lower interest rate, the amount after 5 years is higher.

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