Compound interest means your money earns money, and then that new money also starts earning money, making your savings grow much faster over time.

Key idea: The more frequently interest is compounded, the quicker your money grows because it is added to the principal more often.

The 'compounding frequency' tells you how often your earned interest is added to your principal. This can happen yearly, every six months, every three months (quarterly), monthly, weekly, or even daily. The more often your interest is compounded, the faster your money grows.

For example, if you have an annual interest rate, but it is compounded monthly, it means the interest is calculated and added to your balance 12 times a year. Each time, your balance gets a little bigger, and then the next month's interest is calculated on that new, slightly larger amount.

“The more often your interest is compounded, the faster your money grows.”

Key idea: The annual equivalent rate (AER) helps you compare different financial products by showing the true yearly interest, considering how often it is compounded.

When you are comparing different savings accounts or loans, it can be tricky because they might use different compounding frequencies. To make it easier for you to compare, many countries require financial institutions to show an 'annual equivalent rate' (AER).

This AER tells you the true annual rate of return, taking into account how often the interest is compounded. It helps you see the real cost of a loan or the real earnings from a savings account over a year, no matter how often the interest is actually calculated. Other names for AER include effective annual rate or annual percentage yield.

Key idea: Compound interest has a long history, understood by ancient cultures and later formalized by mathematicians and merchants.

Compound interest has been around for a very long time. Ancient civilizations, like the Babylonians around 2000 to 1700 B.C., already had some understanding of it. However, it was not always seen as a good thing. In Roman law and other legal systems, charging compound interest was often considered a bad practice, similar to charging too much interest (usury).

Over time, mathematicians and merchants started to study it more closely. For example, an Italian merchant named Francesco Balducci Pegolotti created tables for compound interest in the 1300s. Later, in 1494, Luca Pacioli introduced the 'Rule of 72', a quick way to estimate how long it takes for an investment to double with compound interest. You just divide 72 by the interest rate. Richard Witt's book in 1613 was entirely about compound interest, showing how important it had become.

“The 'Rule of 72' is a quick way to estimate how long it takes for an investment to double with compound interest: just divide 72 by the interest rate.”

Key idea: A formula helps calculate the total amount after compound interest, considering the principal, rate, frequency, and time.

To figure out how much money you will have with compound interest, you can use a specific formula. This formula takes into account your starting amount, the interest rate, how often the interest is added, and for how long.

Let us say you put $1,000 into a savings account with a 5% annual interest rate, compounded yearly. After one year, you would have $1,000 plus 5% of $1,000, which is $50. So, $1,050. In the second year, you would earn 5% interest on the full $1,050, not just the original $1,000. This means you would earn $52.50 in the second year, making your total $1,102.50. This extra $2.50 is the interest earned on your interest from the first year.

Key idea: The compound interest formula helps calculate the future value of an investment or loan, considering how often interest is added.

The main formula for compound interest helps you find the total amount you will have (A) after a certain time. It looks a bit complex, but each part has a simple meaning.

Let us use an example: You invest $1,000 (P) at an annual interest rate of 4% (r), compounded monthly (n = 12), for 5 years (t).

First, convert the annual rate to a monthly rate by dividing by 12: 0.04 / 12 = 0.00333. Then, figure out the total number of compounding periods: 5 years × 12 months/year = 60 periods. Plug these numbers into the formula.

Key idea: Continuous compounding is a theoretical idea where interest is added constantly, representing the fastest possible growth rate.

Sometimes, interest is compounded so frequently that it is almost constant. This is called 'continuous compounding'. Instead of adding interest every month or day, it is like interest is being added every tiny fraction of a second. This is often used in advanced financial calculations, especially for things like valuing complex financial products.

While it sounds complicated, the idea is that the more often interest is compounded, the faster your money grows, up to a certain limit. Continuous compounding represents that theoretical maximum growth. It uses a special mathematical number called 'e' (approximately 2.71828) in its formula.

“Continuous compounding is like interest being added every tiny fraction of a second.”