Compound interest

Money can grow in a special way.

When you save money, you earn extra. This extra is called interest.

You can keep that interest to earn even more. The interest earns its own interest! This makes your money grow faster over time. People have known about this for a long time. Long ago, people used clay tablets to write it down. It is a very smart way to grow wealth. Do you want to save money?

Money can grow in a special way called compound interest.

When you save money, you earn interest. This is extra money paid to you. With compound interest, you keep that extra money in your account. Then, that interest earns its own interest! This makes your money grow faster and faster.

How often this happens is called the frequency. Interest can be added every day, every month, or every year. The more often it is added, the more it grows. Many countries use a special rate to help people compare different bank deals. This is called the annual equivalent rate.

People have studied this for a very long time. Ancient people in Babylon wrote about it on clay tablets. In the year 1613, a man named Richard Witt wrote a famous book. His book was all about this math. He used many examples to show how it works. In 1683, Jacob Bernoulli found a special math constant while studying this topic. Even today, math helps us plan for the future.

Compound interest is a special way that money grows over time.

It happens when you earn interest on your original sum of money. You also earn interest on the interest you already made. This is different from simple interest. With simple interest, you only earn money on your first amount. With compound interest, the extra money is added back to the total. This makes the amount grow faster as time goes on. It can be used for saving money or for debts.

How often the interest is added is called the frequency. This can happen every year or every month. It can even happen every day or continuously. If it happens every month, the frequency is 12. Higher frequencies can make the money grow more. Some banks use a special rate to help people compare deals. This is called the annual equivalent rate or AER. It shows the total interest you would get in one year. This rate often includes other things like taxes or fees.

People have studied this math for thousands of years. Ancient people in Babylon used it long ago. A clay tablet from 2000 to 1700 B.C. shows these problems. In the past, some leaders thought it was unfair. Roman law and other laws once condemned it. In 1340, a merchant named Francesco Pegolotti wrote a table for it. His book showed interest on 100 lire for 20 years. Later, Luca Pacioli wrote about the Rule of 72 in 1494. This rule helps you find how long it takes to double money.

Many famous thinkers made big discoveries about this topic. In 1613, Richard Witt published a very important book. His book was called Arithmeticall Questions. It was the first book only about compound interest. He was a math practitioner in London. He used 124 examples to explain his ideas clearly. In 1683, Jacob Bernoulli found a special math constant. He found it while he was studying compound interest. Even Persian merchants used clever math to calculate payments in their heads.

Today, we see compound interest in many parts of life. Many people use it when they have a mortgage. A mortgage is a loan used to buy a home. Some loans use an amortization schedule to pay things off. This helps people know how much to pay each month. You can also see it when you save money regularly. If you add money every month, your total grows even more. Math helps us understand how our money works in the future.

Compound interest is the process of earning interest on both an initial principal sum and the interest that has already accumulated.

This occurs when a person reinvests or retains interest instead of having it paid out. It can also apply to the accumulation of debts for a borrower. Compound interest is different from simple interest. In simple interest, the accumulated interest is not added back to the principal for the next period. Because the interest grows on a larger base each time, the total amount increases more rapidly. This concept is vital for understanding how savings grow and how loans increase over time.

The speed of this growth depends on the compounding frequency. Compounding frequency is the number of times interest is capitalized within a specific unit of time. This happens on a regular basis. The frequency can be yearly, half-yearly, quarterly, monthly, weekly, or even daily. It can also occur continuously. For example, if interest is expressed as an annual rate but compounded monthly, the frequency is 12.

This means the interest is calculated and added to the principal every month. Higher frequencies generally lead to higher total amounts over time.

To help consumers compare different financial products, many countries require specific disclosures. Financial institutions must often show the annual compound interest rate on a comparable basis. This is frequently called the Annual Equivalent Rate (AER) or the effective annual percentage rate (EAPR). Another term used is the annual percentage yield. The effective annual rate represents the total accumulated interest payable by the end of one year, divided by the principal sum. These rates often include the annualized compound interest rate along with other costs like taxes or fees.

History shows that humans have grappled with these math problems for millennia. Traces of mathematicians analyzing compound interest appear in the medieval era. However, a clay tablet from Babylon, dating from 2000 to 1700 B.C., may show the first recorded compound interest problem. In ancient times, the practice was controversial. When lenders charged compound interest, it was often viewed as the worst kind of usury. Because of this, it was severely condemned by Roman law and the common laws of many other nations.

As math progressed, specialized tools for calculation emerged. In about 1340, a Florentine merchant named Francesco Balducci Pegolotti included a compound interest table in his book, *Pratica della mercatura*. His table showed interest on 100 lire at rates from 1% to 8% for up to 20 years. In 1494, Luca Pacioli published *Summa de arithmetica*, which included the Rule of 72. This rule allows one to estimate how many years an investment takes to double by dividing 72 by the interest rate.

Later, in 1613, Richard Witt published *Arithmeticall Questions*. This was a landmark book because it was entirely devoted to the subject, which was then called anatocism. Witt was a London mathematical practitioner who provided 124 worked examples and high accuracy.

Mathematical complexity increases when looking at continuous compounding. This occurs when the number of compounding periods per year increases without limit. In this scenario, the effective annual rate approaches an upper limit.

Mathematicians use the "force of interest" to describe this continuous rate. This concept is a natural consequence of Itô calculus, where financial derivatives are valued at an ever-increasing frequency. As the frequency approaches a limit, the derivative is valued in continuous time. This makes continuous compounding mathematically simpler for valuing complex instruments like derivatives.

Compound interest connects to many different systems in modern life. It is seen in corporate and government bonds, where interest is usually paid twice a year. In these cases, the six-month payment is the disclosed rate divided by two, multiplied by the principal. Canadian mortgage loans often use semi-annual compounding with frequent payments. In contrast, many U.S. mortgages use an amortizing loan system. These use an amortization schedule to apply payments toward the principal and interest. Instead of adding interest to the principal, the interest is paid off monthly through regular payments.