# Compound interest - an understanding with examples

**Compound interest** is a fundamental concept in finance that refers to the process of earning interest on both the initial principal (the original amount of money) and on the interest that has already been added to that principal. Unlike simple interest, where interest is only calculated on the initial principal, compound interest allows for the reinvestment of earned interest, resulting in exponential growth of an investment or savings over time.

**How Compound Interest Works**

The basic idea behind compound interest is that, after each compounding period, the interest is calculated on the new total (principal + accumulated interest). Over time, this leads to interest being earned on interest, significantly increasing the total amount of money in an account.

**Formula for Compound Interest**

The formula to calculate compound interest is:

A=P(1+r/n)^nt

where:

1. A is the future value of the investment/loan, including interest.

2. P is the principal investment amount (the initial deposit or loan amount).

3. r is the annual nominal interest rate (as a decimal).

4. n is the number of times that interest is compounded per year.

5. t is the time the money is invested or borrowed for, in years.

__Key Components of Compound Interest__

1. Principal (P): This is the original sum of money put into an investment or loan.

2. Interest Rate (r): This is the rate at which the interest is applied. It is usually expressed as an annual percentage.

3. Compounding Frequency (n): This refers to the number of times interest is added to the principal in a year. Common compounding frequencies include annually (once a year), semi-annually (twice a year), quarterly (four times a year), monthly, or daily.

4. Time (t): This is the length of time for which the money is invested or borrowed. It is typically expressed in years.

**Examples of Compound Interest in Practice**

**Example 1: Annual Compounding**

Suppose you invest ₹10,000 in a savings account that offers an annual interest rate of 5% compounded annually for 3 years.

**· Principal (P)** = ₹10,000

**· Annual interest rate (r)** = 5% = 0.05

**· Compounding frequency (n)** = 1 (compounded annually)

**· Time (t)** = 3 years

Using the compound interest formula:

After 3 years, the investment will grow to approximately **₹11,576.25**. The compound interest earned is:

Compound Interest=A−P=11,576.25−10,000=1,576.25Compound Interest=A−P=11,576.25−10,000=1,576.25

**Example 2: Quarterly Compounding**

Suppose you invest ₹5,000 in a fixed deposit for 2 years at an annual interest rate of 8%, compounded quarterly.

**· Principal (P)** = ₹5,000

**· Annual interest rate (r)** = 8% = 0.08

**· Compounding frequency (n)** = 4 (quarterly)

**· Time (t)** = 2 years

Using the compound interest formula:

Calculating (1.02)^8:

(1.02)^8≈1.17166

A=5,000×1.17166=5,858.30

After 2 years, the investment will grow to approximately **₹5,858.30**. The compound interest earned is:

Compound Interest=A−P=5,858.30−5,000=858.30Compound Interest=A−P=5,858.30−5,000=858.30

**The Power of Compound Interest**

The power of compound interest lies in its ability to generate returns on both the initial principal and the accumulated interest from previous periods. This "interest on interest" effect can cause wealth to grow faster than it would with simple interest.

**Key Takeaways**

**· Time is crucial:** The longer you leave your money invested, the more interest you'll earn, and the more your investment will grow due to the effects of compounding.

**· Frequency of compounding matters:** More frequent compounding periods (like monthly or quarterly) result in more interest being accumulated compared to less frequent compounding (like annually).

**· Higher interest rates accelerate growth:** A higher interest rate will result in a greater accumulation of wealth over time due to compounding.

P×(1+r)^n, is a simplified version of the compound interest formula, typically used when interest is compounded annually. Here:

· P = Principal (initial amount of money)

· r = Annual interest rate (as a decimal)

· n = Number of years

Let's go through an example using this formula in the context :

**Example: Fixed Deposit with Annual Compounding**

**Scenario:**Suppose you invest ₹1,00,000 in a fixed deposit (FD) in an Indian bank that offers an annual interest rate of 7% compounded annually for 5 years.

· P=1,00,000

· r=7%=0.07

· n=5 years

**Calculation:**

**1. Convert the interest rate to a decimal:**

r=0.07

**2. Plug the values into the compound interest formula:**

A=1,00,000×(1+0.07)^5

A=1,00,000×(1+0.07)^5

**3. Calculate** (1.07)^5**:**

(1.07)5≈1.40255(

**4. Calculate the future value** AA**:**

1. A=1,00,000×1.40255=1,40,255

So, after 5 years, the investment will grow to approximately **₹1,40,255**.

**5.Calculate the compound interest earned:**

Compound Interest=A−P=1,40,255−1,00,000=40,255

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