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# Compounding|Definition & Meaning

## Definition

**Compounding** is the process of earning on the initial investment and the **interest** credited to the **principal investment**. It has the ability to make investments grow at a **larger scale** with the passing of time. It can be applied on a** loan** or on an initial **investment**/**deposit** in an account.

**Compounding** refers to the **exponential growth** of money as shown in **figure 1**. The exponential **graph** shows that after reaching a certain point, the y-values **drastically increase** which refers to the **substantial** amount of earnings.

## Types of Interest

We normally encounter **two types** of interest: **simple** and **compounded**. To understand **compounded** interest comprehensively, one must know about **simple** interest.

### Simple Interest

**Simple interest** is the process of applying **interest** on a loan or a deposit over a period of time.

Unlike **compounded** interest, simple interest applies only to the** principal** investment and not the **interest** accumulated over **time**.

**Figure 2** shows an example of simple interest.

The simple interest in **figure 2** can be **calculated** using the **formula** given below for **simple interest**.

**A = P(1 + rt)**

Where **P, A,** **r, **and **t** are respectively the** principal investment**, the **final amount**, the **annual interest rate**, and the** time passed** since the initial investment (in years).

From **figure 2:**

**P = $1000 , r = 10% = 0.1 , t = 3 years**

So, the **final amount** will be:

A = P(1 + rt)

A = 1000[1 + 0.1(3)]

A = 1000[1 + 0.3]

A = 1000(1.3)

**A = $1300**

### Compounded Interest

Compounded Interest is earning **interest** on the **initial** investment and on the previously **accumulated interest**.

It means that as **time** goes by, the **principal** investment becomes the **previous** principal plus the **interest** added to it, on which the interest is **added** again and again.

**Figure 3** shows the same example as in **figure 2** for **compounded interest**.

Compared to simple interest, **compounded** interest **yields more** earnings over the same time period hence it is **more potent** in financial dealings.

### Periodic Compounded Interest

**Periodic Compounding** refers to interest compounded over a **period** of time. The **formula** for periodic compounded interest is given as follows:

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

Where **A** and **P** are the** final** and **principal base** amounts, and **R**, **n**, and **t** are respectively the **interest rate**, the number of times interest is **received** in a year, and the **time elapsed** (in years).

The **interest rate R** is usually given in **percentage**. It is divided by **100** to get the rate in **decimal** which is plugged into the equation.

From **figure 3**, the compounded interest can be calculated as follows:

\[ A = 1000 { \Big[ 1 + \frac{0.1}{1} \Big] }^{1(3)} \]

A = 1000 $[ 1 + 0.1 ]^{3}$

A = 1000 $[ 1.1 ]^{3}$

A = 1000 (1.331)

A = 1000 (1.331)

**A = 1331**

So, the final earnings amount to **$1330**.

## The Role of Time in Interest Compounding

Time plays an **essential **role while dealing with interests. For compounded interest, it serves as a key pillar for **enhancing** the **investment** to a greater extent.

**Figure 4** shows the importance of time in **compounded interest**.

With **twice** the time gone by, the **final** earnings become approximately twice with **compounded** interest. Hence more time produces **more gains** in investment.

## Value of ānā

The value of **n** changes with the compounded time period associated with the **annual interest** explained as follows:

### Monthly

For monthly compounded interest, the value of ā**n**ā is **12** as the interest received would be twelve times a **year**.

If the** 24%** annual interest rate is **compounded monthly**, the extra money received every** month** would be according to 24/12 = **2%** interest rate.

### Weekly

For compounded interest applied weekly, ā**n**ā will be **52** as there are fifty-two weeks in a **year**.

### Daily

The value of ā**n**ā will be 365 for **daily** compounding as there are **365** days in a year.

### Quarterly

The number of times interest is received in a year **quarterly** would be **four times**, so the value of ā**n**ā would be **4**.

### Semi-Annually

Semi-annually means **half a year**. The value of ā**n**ā would be **2** for interest compounded semi-annually.

### Annually

For annually, ā**n**ā is **1** as annually means once a year as in **figure 3**.

The value of ā**n**ā only **divides** the annual interest over the specific period which is** compounded**.

**Figure 5** shows a **12% **annual **interest** division over different time periods. The interest rate **decreases** with the increased period divisions **annually**.

## Continuously Compounded Interest

For continuously compounded interest, the following **formula** is used:

A = P. $e^{Rt}$

Notice that this formula does not have the ā**n**ā factor as it is not for **annual compounding** over a period.

This shows constant **compounding** as the **exponential** function keeps on increasing until specified by the time **t** in **years**.

## Interest Compounding Example

Ali deposits **$40,000** in his savings account with a **10%** annual interest rate compounded **semi-annually**.

How much money will be accumulated in the account after **20 years** at this interest rate?

### Solution

The word ā**semi-annually**ā in the problem refers to the formula for **periodic** compounded interest. The **formula** is given as:

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

Here, the initial **principal** amount** P** deposited in the account is **$40,000**.

The interest rate **R** is **10%** or 10/100 =** 0.1** and the number of times the **interest** is received in a year is semi-annually, so ā**n**ā is **2**.

The** time t** passed on the initial deposit is **20 years**. Placing the values in the **formula** gives:

\[ A = 40,000 { \Big[ 1 + \frac{0.1}{2} \Big] }^{2(20)} \]

A = 40,000 $[ 1 + 0.05 ]^{40}$

A = 40,000 $[ 1.05 ]^{40}$

A = 40,000(7.039988)

**A = 281,599.5**

The **final** amount after **20 years** compounded semi-annually at the **10%** annual interest rate will be approximately **$281,600**.

*All the images are created using GeoGebra.*