Compound Interest

Compound interest has a long history, dating back to ancient Mesopotamia where Babylonians recorded and calculated interest growth on mathematical tablets for financial transactions. By the 4th century BC, Aristotle criticized compound interest as unnatural, but the Romans continued to use it widely in trade and banking. Today, compound interest remains a fundamental concept in banking, investment, and economics, ensuring that money grows over time.

Compound interest is calculated on both the initial principal and the accumulated interest from previous periods, unlike simple interest, which is calculated only on the principal. This means that with investments, compound interest causes money to grow faster, while with debt, it can result in a much larger amount to repay. We can easily calculate the compound interest using a compound interest calculator. Compared to simple interest, compound interest leads to quicker growth of the total amount.

The formula we use for compound interest is: 

\(A = P (1 + \frac{r}{n})^{nt} - P\)

Where, 
A = Final amount after interest
P = Principal (starting/initial money)
r = Annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = time in years

Interest can be calculated in compound interest on different frequencies of time like daily, monthly, quarterly, and annually. The higher the number of compounding periods, the larger the effect of the interest. In other words, it can be defined as interest on interest.

Compound Interest and simple interest are very closely related to each other. Therefore, it is very important for us to understand the concepts of simple and compound interest to solve problems easily. Let us try to understand their differences with the help of a simple interest vs compound interest table, as shown below:
 

A student must keep in mind the key properties to understand the concept of compound interest. Below are some of the key properties that students must know.
 

• Growth of interest: Compound interest grows exponentially as the interest is calculated on both the principal and the accumulated interest. 
   
• Time period: The longer the time period, the greater the growth as interest continues to accumulate on previous interests.
   
• Interest rate: A higher interest rate results in quicker accumulation of interest and a faster growth. This could impact on any kind of savings or investment.

Compound interest grows faster than simple interest because the interest earned in each period itself earns more interest. When the compounding frequency is high, we get a greater yield. We can calculate compound interest in first finding the final amount and then subtracting the principal from it. The value we get after we subtract the principal amount from the final amount is the interest. Let us learn how to find compound interest for different time periods next.

Compound interest formula for different time periods.

The formula for compound interest varies with the number of compounding periods per year. The formulas for different periods are given as;

1. Annual compounding: Here, the number of compounding periods per year is one.

\(A = P\left(1 + \frac{R}{100}\right)^{T} \)

2. Semi-annual compounding: Here, we compound twice per year. Hence, the number of compounding periods per year is two.

\(A = P\left(1 + \frac{R}{200}\right)^{2T} \)

3. Quarterly compounding: Here, we compound four times per year. Hence, the number of compounding periods per year is four.

\(A = P\left(1 + \frac{R}{400}\right)^{4T} \)

4. Monthly compounding: Here, we compound twelve times per year. Hence, the number of compounding periods per year is twelve.

\(A = P\left(1 + \frac{R}{1200}\right)^{12T} \)

5. Weekly compounding: Here, we compound fifty-two times per year. Hence, the number of compounding periods per year is fifty-two.

\(A = P\left(1 + \frac{R}{5200}\right)^{52T} \)

6. Daily compounding: Here, we compound for all the days of the year. Hence, the number of compounding periods per year is 365.

\(A = P\left(1 + \frac{R}{36500}\right)^{365T} \)

7. Continuous compounding: We use continuous compounding when the interest is added every moment. The continuous compound interest formula is given as;

\(A = Pe^{\frac{RT}{100}} \)

For compound interest, period of time is the most important one. For how long are you investing or taking the loan, you have to decide accordingly. Frequency of time is the only determining factor for the interest amount needs to pay in compound interest. 

The formula for compound interest is given as,

\(CI = P\left(1 + \frac{R}{100}\right)^{T} -P\)

Derivation of compound interest formula

Let us now try to derive the compound interest formula. By understanding the derivation, we can learn how to find compound interest.

Let,
P = principal
R = annual interest rate (%)
n = number of compounding periods per year
T = time in years

Step 1: First, let us convert the rate to per-period interest. If interest is compounded n times per year, the interest rate for each compounding is given as:

Rate per period \((r) = \frac{R}{n \cdot 100} \)

Step 2: Now, let us find the value of amount after one period.

\(P_{1} = P(1 + r) \)

Step 3: The amount after two compounding periods, where P1 becomes the new principal, is written as;

\(P_{2} = P_{1}(1 + r) = P(1 + r)(1 + r) = P(1 + r)^{2} \)

Step 4: Similarly, the amount after k compounding periods is given as;

\(P_{k} = P(1 + r)^{k} \)

Step 5: Since we compound the interest for n times per year for T years,

\(k = nT \)

Step 6: Let us now substitute into pk

\(A = P(1 + r)^{nT} \)

Now substitute \(r=\frac{R}{n\cdot100}\)​:

\(A = P\left(1 + \frac{R}{n \cdot 100}\right)^{nT} \)

This is the formula for the final amount in compound interest.

The final formulas are given as:

Amount:
\(A = P\left(1 + \frac{R}{n \cdot 100}\right)^{nT} \)

Compound interest:
\(CI = A - P\)

As a financial concept, compound interest helps in growing the money over a period of time. So here are a few reasons why compound interest is significant.
 

• Loans: If a loan with compound interest is not managed properly, the debt can grow rapidly. Understanding how interest accumulates can help prevent this and ensure proper management. 
   
• Savings: If you understand how compound interest works, it would help to maximize your savings and investments by choosing any option with higher compounding frequencies.
   
• Retirement: Savings can grow at an increased rate through compound interest, which will be helpful during retirement.
   
• Credit card: Credit card bills interest rates are compounded, so it sometimes helps in avoiding compulsive shopping.

Compound interest can help you make smart financial decisions. So here are some tips and tricks to master the concept: 
 

• Understand the compounding frequency: Due to compound interest, the money can grow exponentially depending on the frequency of time period
   
• Use online calculators for precise calculations: To make sure you are not making mistakes in calculations, use online compound interest calculators to quickly and accurately find how much money will grow.
   
• Keep in mind the various synonyms for per year: There are different synonyms used for per year. Remember that per year, annual, or per annum all mean the same.
   
• Make sure to memorize the formulas: It is very important to remember and memorize the standard formula for compound interest. This would make it easier to learn the other formulas, as they are variations of the standard formula.
   
• Rule of 72: It is a calculation for compound interest, where one can find out how an investment can be doubled by multiplying 72 by annual interest rate. 
   
• While teaching the concept of compound interest, teachers should start by introducing the idea of interest on interest. Before explaining the formulas, clarify the core idea to the children. In compound interest, we earn interest on the interest we have already earned. 
   
• Parents should help their children by incorporating hands-on activities that make the concept of compound interest concrete. For example, put $100 in a box labeled bank and add 10% each time, representing a month or a year. Finally, show them how the amount grows faster each round. 
   
• Teachers should ensure students learn the formula step by step. Break down the formula and teach them its derivation. 
   
• Parents should allow the students to explore in their own way and assist them in connecting with technology. Let them work on a banking app that displays interest earned and an online compound interest calculator.

Compound interest is a powerful financial tool used in various aspects of our lives. Here are some real-world applications of compound interest:

Used in savings accounts: All banks use compound interest to calculate how much our savings will grow over time.

Repayment for loans: When you borrow money from the bank on compound interest and agree to repay by a certain date, it's significant to pay back the loan before the interest accumulates. Otherwise, debt would grow, and you would end up having to pay even more money than the initial amount.

Education funds and savings: When saving money for your college, like a college fund. The bank uses compound interest, so the money grows over time. 

Credit cards: If you don’t pay your credit card bill in full, compound interest is added to the remaining balance, making the debt grow quickly.

Investment in stocks and mutual funds: The returns are often reinvested, and with compound interest, the invested amount grows exponentially over long periods.