Compound Interest 

Interest is defined as the borrowing cost of money or the income generated on the capital. It is usually calculated in terms of a percentage of the principal amount, which is the original sum borrowed. When one lends money or invests money, the interest becomes the key element to making a profit. There are two key forms of interest: simple interest and compound interest. Here, we will deeply understand compound interest. 

1.0Understanding Compound Interest

Compound interest is the addition of interest upon interest on loans or deposits; it is added based on the principal amount borrowed and the previous period's interests. In short, it's interest on interest, which is the reason the sum grows much faster than that with simple interest, which adds only to the principal amount borrowed.

To understand compound interest, imagine you invest Rs 5000 at an interest rate of 10%. The compound interest for the first year will be Rs 500, and the amount will be 5500. Next year, the interest on the amount will also be 500, but this time, the interest will also be compounded on the interest, meaning the interest on the interest will be 50. Hence, the amount at the end of  2nd year will be 6050.

2.0Compound Interest Formula

The compound interest simple formula is given as: 

Compound Interest (CI) = A - P

Where A is the amount accumulated after n years, the formula to find the amount is: 

$A=P{\left(1+\frac{r}{100}\right)}^{nt}$

Hence, the formula for compound interest will be: 

$CI=P{\left(1+\frac{r}{100}\right)}^{nt}-P$

Here:

• A = The amount of money accumulated after n years, including interest
• P = The principal amount (initial investment)
• r = Annual interest rate (in decimal form)
• n = Number of times the interest is compounded per year
• t = Time for which the money is invested (in years)

3.0Types of Compound Interest

Compound interest is divided into several types based on the amount of time the money is invested, which include: 

• Annually Compound Interest (n = 1): In this type, the interest is calculated and accumulated to the principal once every year. An example of this type is: let you invest Rs 1000 at an annual interest rate of 5%, then you’ll earn Rs 50 in interest at the end of the year. The formula for Annually Compound Interest is: 

$CI=P{\left(1+\frac{r}{100}\right)}^{nt}-P$

• Semi-Annual Compound Interest: The interest is compounded two times per year, meaning every 6 months. This means that the interest is added to the principal twice within the year, so it has a slightly higher effect than annual compounding. The formula for semi-annual compound interest is: 

$CI=P{\left(1+\frac{r/2}{100}\right)}^{2t}-P$

• Quarterly Compound Interest: The interest is compounded four times a year, every 3 months. The rate of the amount is divided by 4, and interest is calculated and added to the principal in quarters or 4 times every year. The formula for Quarterly Compound Interest. 

$CI=P{\left(1+\frac{r/4}{100}\right)}^{4t}-P$

4.0Compound Interest Formula Example

Problem 1:

A person deposits INR 2,000 in a savings account that offers an annual interest rate of 4%, compounded quarterly. Compute compound interest after 1 year?

Solution: According to the question 

Principal (P) = 2000

Annual interest rate (r) = 4%

Time (t) = 1 years 

As the principle is compounded quarterly, hence the amount after 1 year will be: 

$A=P{\left(1+\frac{r/4}{100}\right)}^{4t}$

$A=2000{\left(1+\frac{4/4}{100}\right)}^{4\times 1}$

$A=2000{\left(1+\frac{1}{100}\right)}^4$

$A=2000(1+0.01{)}^4$

$A=2000(1.01{)}^4$

A=20001.041

A=INR 2,082

Problem 2:

A business owner invests INR 50,000 in a certificate of deposit (CD) at an interest rate of 3% per year, compounded annually. What will be the interest earned after 2 years?

Solution: It is given that, 

Principal = 50,000

Rate = 3%

Time = 2 years 

The amount after two years will be, 

$A=P{\left(1+\frac{r}{100}\right)}^{nt}$

$A=50,000{\left(1+\frac{3}{100}\right)}^{1\times 2}$

$A=50,000{\left(1+\frac{3}{100}\right)}^2$

$A=50,000\left(\frac{103\times 103}{100\times 100}\right)$

A=5(10,609)

A=INR 53,045

Compound interest (CI) = A – P 

CI = 53,045-50,000=INR 3,045

Problem 3:

A person invests INR 1,000 in a savings account for 2 years, and after 2 years, the account grows to INR 3,600. If the interest is compounded annually, what is the annual rate of interest?

Solution: Given that, 

Principal = 1000

Amount = 3600

$3600=1000{\left(1+\frac{r}{100}\right)}^{1\times 2}$

$\frac{3600}{1000}={\left(1+r\right)}^2$

$3.6\times 100^2=(1+r{)}^2$

$1+r=0.6\times 100=60$

r=59%

Compound Interest Practice Problems

Q1: You invest INR 5,000 in a savings account that compounds annually at a rate of 6%. After 5 years, the account balance is INR 6,690. How long did the money stay in the account?

Q2: A principal of INR 8,000 is invested for 3 years. The total amount after 3 years is INR 9,720. If the interest is compounded annually, what is the rate of interest?

Q3: A principal of INR 2,500 is invested for 4 years at an interest rate of 5% per year, compounded annually. What is the compound interest earned?