Compound Interest is often referred to as the "magic" of finance because it has the power to turn small investments into substantial fortunes over time. Whether you're saving for retirement, investing in the stock market, or simply stashing away money in a savings account, understanding the mathematics behind compound interest is essential.

Compound Interest is an intriguing mathematical effect that puzzled many scientists and businessmen over time. Baron Rothschild once said:

"I don't know what the Seven Wonders of the World are,

but I know the eight, compound interest".

Albert Einstein said that Compound Interest is "the greatest mathematical discovery of all time".

"Compound interest is the greatest mathematical discovery of all time."

The Basics of Compound Interest

Before we delve into the mathematics, let's refresh our understanding of compound interest. Unlike simple interest, which calculates interest solely on the initial principal amount, compound interest factors in the accumulated interest from previous periods. Essentially, you earn interest not just on your initial investment but also on the interest that has already been added to your account.

I am sure you used compound interest before. Every bank deposit has the option to compound the paid interest.

Let's say you deposit $1,000 for three months with a 10% annual interest rate, compounded and renewed automatically. This means that after three months, you will receive 2.5% interest ($25), which will be added to your initial $1,000 deposit. The total deposit then becomes $1,025. In the next three months, your interest will be 2.5% of $1,025, which is $25.625, and this amount will be added to your principal deposit. At this point, you will have $1,050.625 in your principal deposit, and the interest will be calculated based on this sum for the following three months.

That is the compound interest. Off-course, 10% is a lot for any bank deposit nowadays, but we chose 10% for the sake of example and to make the calculation easier.

The Fundamentals of Corporate Science, a basic textbook for Chartered Financial Analyst says the following:

“The effect of Compound Interest is not great over short time periods, but it really starts to add up as the horizon grows. To take an extreme case, suppose one of your frugal ancestors had invested $5 for you at [only] 6% interest 200 years ago. How much would you have today? The future value factor is a substantial (1.06)200 = 115,125.91, so you would have $5 x 115,125.91 = $575,629.53 today. Notice that the simple interest is just $5 x .06 = $.30 per year. After 200 years this amounts to $60. The rest is from reinvesting. Such is the power of compound interest.”

Compound Interest Formula

The formula for compound interest is as follows:

A = P (1 + r/n)nt

Where:

  • A represents the future value of the investment or loan, inclusive of interest.
  • P stands for the principal amount (the initial sum of money).
  • r denotes the annual interest rate (expressed as a decimal).
  • n signifies the number of times the interest is compounded per year.
  • t indicates the number of years the money is invested or borrowed for.

Understanding the Components

  • Principal Amount (P): This is your starting point—a sum of money you either invest or borrow.
  • Annual Interest Rate (r): The annual interest rate is a percentage that reflects the interest added to your investment or loan each year. It's crucial to use the decimal form of this rate in the formula, so if the interest rate is 5%, you'd use 0.05.
  • Compounding Frequency (n): Compounding frequency refers to how often interest is added to the principal. The more frequent the compounding, the faster your money grows. Common values for n include annually (1), semi-annually (2), quarterly (4), and monthly (12).
  • Time (t): Time represents the number of years your money is invested or borrowed. The longer the time, the greater the impact of compound interest.

The Power of Exponential Growth

The most captivating aspect of compound interest is its ability to generate exponential growth. Over time, the interest you earn becomes part of the principal, leading to interest on your interest. This compounding effect can yield astounding results.

Consider a hypothetical example: depositing $1,000 at an annual interest rate of 5%, compounded annually. After 10 years, your investment would grow to approximately $1,628.89. However, if you were to keep the money invested for 30 years, it would soar to a staggering $4,321.94—over four times your initial investment!

If we put this into an excel table, this is how it looks:

# A B
1 Principal 1000
2 Interest Rate % (expressed as decimal fraction) 0.05
3 Compoundings per year 1
4 Years 10
5 A 1628.89

The excel formula to put in B5 cell:

=B1*((1+((B2/100)/B3))^(B3*B4))

Here you can find and Excel sheet where you can calculate the expected return of an investement after a certain number of years at a certain interest rate.

Download Compound Interest Excel Sheet

The Rule of 72

A useful shortcut for estimating how long it takes for your money to double with compound interest is the Rule of 72. Simply divide 72 by the annual interest rate, and you'll get an approximation of the number of years for your investment to double.

For example, with an annual interest rate of 6%, your money would roughly double in about 12 years (72 ÷ 6 = 12).

The Rice Grain Parable

A Chess Table with 64 squares

Legend has it that someone once presented a chessboard as a gift to a king. In response, the king asked the person what they desired in return. The individual's request was intriguing: they asked the king to provide, over the next 64 days, an increasing number of rice grains for each square of the chessboard. It began with just one grain of rice on the first day, two grains on the second day, four on the third day, and so forth, continuing until all 64 squares were filled. The king agreed to this seemingly simple request.

Initially, everything seemed manageable. However, as they reached the 32nd day, the king realized that fulfilling this request had become a daunting challenge. By that point, he had to provide 2,147,483,648 grains of rice, which equated to approximately 50,000 kilograms. The total grains given up to that moment had reached 4,294,967,295, equivalent to about 100,000 kilograms. As they approached the 64th square, an astronomical number with 18 zeros needed to be provided, surpassing the total amount of rice available on the entire planet!

This story vividly demonstrates the power of exponential growth, and it highlights how the compound interest effect harnesses this remarkable phenomenon.

The Power Of Repeated Investment

We saw earlyer what happens with an initial deposit of $1,000 invested with 5% per year, compounded yearly for 10 years. What if you add each year the same ammount to that investment? What happens with the ammount in 10 years and how can you calculat that?

What happens if each year you add the same $1,000 to the initial investment?

To calculate what happens if you add to the initial investment regurarly, we use the annuity formula, or often called the future value of an annuity formula:

A = P * ((1 + r)t - 1) / r * (1 + r)

Here's a breakdown of the components of the formula:

  • A: This represents the future value of the investment or annuity. It's the amount of money you will have at the end of the investment period.
  • P: This is the periodic payment or cash flow amount, often referred to as the "annuity payment." It represents the regular contributions or deposits made at the end of each compounding period (e.g., monthly, quarterly, annually).
  • r: This is the interest rate per compounding period, expressed as a decimal. It represents the rate at which your investment grows or the cost of borrowing money if you're dealing with loans. It's important to note that for consistent calculations, the interest rate should be expressed on an annual basis and adjusted for the compounding frequency. For example, if the annual interest rate is 6% and compounding occurs quarterly, you would use 0.06/4 = 0.015 for 'r'.
  • t: This is the number of compounding periods or the length of time the investment will be held or the annuity will be paid. It can be measured in years, months, quarters, or any other consistent time unit as long as it matches the compounding frequency and the interest rate.

The formula calculates the future value (A) of an investment or annuity by taking into account the periodic payments (P), the interest rate (r), and the number of compounding periods (t). It works by compounding each payment over time, and the result is the total value of the investment at the end of the specified time period.

Here's a practical example of how to use the formula:

Suppose you want to calculate the future value of an investment where you deposit $1,000 into an account that earns 5% interest compounded annually for 10 years. Using the annuity formula:

P (annuity payment) = $1,000
r (annual interest rate) = 0.05 (5% expressed as a decimal)
t (number of years) = 10
A = $1,000 * ((1 + 0.05)^10 - 1) / 0.05 * (1 + 0.05)
A = $1,000 * ((1.05^10) - 1) / 0.05 * (1 + 0.05)
A = $1,000 * (1.628894627 - 1) / 0.05 * (1 + 0.05)
A = $1,000 * (0.628894627 / 0.05) * (1 + 0.05)
A = $1,000 * 12.57789254 * 1.05
A = $13,206.79

# A B
1 Annual Investment 1000
2 Interest Rate % (expressed as decimal fraction) 0.05
3 Years 10
4 A 13,206.79

The excel formula to put in B4 cell:

=B1*((1+B2)^B3-1)/B2*(1+B2)

So, the future value of your $1,000 investment, earning 5% interest compounded annually for 10 years, will be approximately $12,577.89. This formula is a valuable tool for making financial projections and decisions in various investing and loan scenarios.

Download Annuity Formula Excel Sheet

Harnessing Compound Interest for Financial Freedom

The principles of compound interest and exponential growth are potent tools on your journey to financial freedom. Whether you invest in stocks, real estate, or other assets, putting your money to work for you can lead to significant wealth accumulation over time. While there's no one-size-fits-all strategy, the key is to take action, apply these principles, and adapt them to your unique financial goals.

Remember, today is the day to start. Compound interest is a real force, and your financial future can be shaped by how you leverage it. So, take the knowledge you've gained here, explore strategies that work for you, and embark on your path to financial success.

Annual Investment Calculator

Use the Anual Investment Calculator above to calculate your future results. Insert the amount you want to invest each year, the average expected interest rate and the number of years you want to invest and compound the interest. You can add a principal if you like, as an initial investment or calculate only based on the principal, by leaving annual investment zero.

$0.00

Principal:
Annual Investment:
Annual Interest Rate (%):
Investment Period (Years):