Contents

# Interest|Definition & Meaning

## Definition

**Interest** is the **additional amount** on top of the **principal amount** that a financial institution or a borrower **pays to** the depositor or lender at a **specific rate**. It is the price of borrowing money, paid or charged over some** period of time**. It is usually expressed as a **percentage**.

**Figure 1** demonstrates the concept of **interest**.

Sara took **100 dollars** from Ali at **10%** annual interest. After a year, Sara had to pay Ali **$100**, and **$10** which is the **10%** interest. So, she had to pay **$110**.

**Interest** refers to the concept that money is **not free** to **borrow**. The more **time** passes on the loan or deposit, the more **interest** accumulates on it.

The interest on the **initial** amount **borrowed** or **lent** depends upon the interest rate, the **principal** balance, the total **time** for which it is deposited or lent, and the **compounding** frequency.

## Interest Rate

The interest is usually written as a **percentage** known as the interest **rate**. It is the rate of interest applied to the **initial** principal amount a **borrower** takes from a **lender** due some **time**.

The interest rate can be **fixed** or **variable**.

### Fixed Interest Rate

A loan with a fixed interest rate has the **same** interest rate throughout the life of the **loan**. It is usually **higher** than the variable interest rate.

### Variable Interest Rate

A variable interest rate **varies** over the time of the loan. It starts with a percentage usually **smaller** than the **fixed** interest rate and **increases **or** decreases** with time. It depends upon an **index** that fluctuates with time.

**Figure 2** shows the comparison of **fixed** and **variable** interest rates through a graph.

## Important Terminologies Related to Interest

In **economics** and finance, some critical **terminologies** related to interest are described below.

### Investment

An investment is an **amount** of **money** given to a bank or **financial** institution to gain some **profit** on it. It is the first deposit on which **interest** is added later as a profit.

### Borrower

A borrower is a person who **borrows** money from someone to **return** it. The money is taken on the condition of returning the **initial** amount with a pre-determined **interest**.

### Lender

A lender is a person who **loans** money to a **person** or a **company** with the presumption that the borrower will **repay** the funds. It can also be a financial institution such as a **bank** that wants repayment with **interest**.

### Principal Amount

The **principal** is the **initial** amount borrowed from a lender on which the **interest** is applied. This term is used while dealing with **simple** or **compound** interest. It can be a loan or an investment in a bank.

It is also known as the “**principal sum**”.

**Figure 3** shows the demonstration of these **terms** related to **interest**.

## Types of Interest

The **two** types of **interest** are explained as follows:

### Simple Interest

Simple interest involves applying **interest** on the **principal** amount paid after the due course of **time**.

The **borrower** has to pay the principal amount plus the interest on the principal and so on. The **formula** for simple interest is as follows:

A = P(1 + rt)

Where **A** is the **final** amount after **time** **t** has passed and **P** is the **principal** amount. The time** t** passed on the principal sum is in** years**.

**Annual** interest rate is the interest rate applied after a **year** on the loan **borrowed** or investment **deposited** and is denoted by **r**. This **percentage** is converted into **decimal** before placing in the equation by **dividing** it by **100**.

**Figure 4** shows an example of **simple** interest with an annual **interest** rate of **5%** on a principal of** $10,000** over **3** years.

If the period extends to **9 years** in the above **example**, the **final** amount can be calculated as follows:

A = 10,000[1 + (0.05)(9)]

A = 10,000[1 + 0.45]

A = 10,000[1.45]

**A = $14,500**

### Compound Interest

**Compounded** interest involves having interest on the **principal** amount and the **interest** accumulated previously. It refers to the concept of “**interest on interest**” unlike simple interest.

The **principal** now becomes the **previous** principal plus the **interest** earned on which more interest is applied.

**Periodic** compounding deals with **compounded** interest over a period. The compound interest can be calculated by using the **formula**:

\[ A = P { \Big[ 1 + \frac{R}{n} \Big] }^{nt} \]

Where **P** is the **principal** sum,** A** is the **final** amount, **R** is the **interest** rate and **t** is the **time** passed in years.

The **number** of times interest is received in a **year** is “**n**”. Its value depends upon the **period** the annual **interest** is compounded. **Table 1** shows the value of **n** for the annual interest **compounded** over different periods.

Compounded Time Periods | Value of “n” |

Annually | 1 |

Semi-annually | 2 |

Quarterly | 4 |

Monthly | 12 |

Weekly | 52 |

Daily | 365 |

For **example**, if a **10%** annual interest is compounded **semi-annually**, the interest received after **half** a **year** will be according to **10/2 = 5%** interest rate.

**Figure 5** shows the example of figure 4 for **compound** interest.

For **9 years**, the amount will accumulate to

\[ A = P { \Big[ 1 + \frac{R}{n} \Big] }^{nt} \]

\[ A = 10,000 { \Big[ 1 + \frac{0.05}{1} \Big] }^{(1)9} \]

A = 10,000 $[ 1 + 0.05 ]^9$

A = 10,000 $[ 1.05 ]^9$

A = 10,000(1.5513)

**A = $15,513**

## Comparison of Simple and Compound Interest

From the same period of 9 years with a principal of **$10,000** at a **5%** annual **interest**, the **final** amount from receiving **simple** interest is **$14,500** and from **compound** interest is **$15,513**.

**Compound** interest refers to **exponential** growth whereas **simple** interest applies **linear** growth. After a certain point, the **money** from **compound** interest **overgrows** increasing from the amount received from simple interest.

## Example Problems Based on Interest

### Example 1 – Simple Interest

Alina invests **$6000** in a bank at an **8%** annual **interest** rate. How much money she will receive after **14 years**?

### Solution

From the given** data**,

P = $6000, r = 8% = 0.08, t = 14 years

The **final** amount **A** will be:

A = P(1 + rt)

A = 6000[1 + (0.08)14]

A = 6000[1 +1.12]

A = 6000[2.12]

**A = $12,720**

### Example 2 – Compound Interest

Simon deposits **$50,000** in savings account with a **16%** annual **interest** rate compounded **quarterly**. What will be the total amount in the account after **14 years**?

### Solution

The **formula** for periodic **compounding** is given as

\[ A = P { \Big[ 1 + \frac{R}{n} \Big] }^{nt} \]

Here

P = $50,000, R = 16% = 0.16, t = 14 years

As the annual **interest** is compounded **quarterly**, so **n = 4**.

Placing the **values** in the equation gives

\[ A = 50,000 { \Big[ 1 + \frac{0.16}{4} \Big] }^{4(14)} \]

A = 50,000 $[ 1 + 0.04 ]^{56}$

A = 50,000 $[ 1.04 ]^{56}$

A = 50,000(8.9922)

**A = $449,610**

So, the amount after **14** years will be **$449,610**.

*All the images are created using Geogebra.*