Compound interest: what it is, how it works, its math formula

Compound interest is a fundamental concept in money management and investing. Warren Buffet, one of the greatest investors of the 20th century, achieved his extraordinary success thanks to compound interest, a simple mathematical formula that allows you to increase your earnings exponentially over time. In this guide, we will clearly explain what compound interest is and how it works, how to calculate it correctly, and where to invest to maximize results.

What is compound interest?

Interest is the profit generated by the use of capital and represents the additional amount that is paid on top of the principal over a certain period of time. It therefore defines how much returns an investment.

Through this financial mechanism, the returns generated by an investment are reinvested at the end of each period, increasing the amount of the investment. This means that the interest itself adds to the principal on which the interest is calculated, creating a growth cycle that amplifies the overall return.

In essence, compound interest is a powerful tool for unlocking the full growth potential of your investments over the long term. This strategy allows you to maximize your return overall and achieve substantial gains over time.

This is the formula used by Warren Buffet to grow his wealth exponentially: we can therefore define compound interest as a strategy for reinvesting the profits of an investment.

Simple interest and compound interest: difference

Compound interest offers the opportunity for higher returns over the medium to long term, provided you give up small amounts in the short term. While simple interest generates linear growth, compound interest activates exponential growth, allowing you to maximize returns over time, taking into account the risks associated with each type of investment.

The key difference between simple interest and compound interest lies in the reinvestment effect of interest. In the case of simple interest, the interest generated is calculated only on the initial capital, without being reinvested. Conversely, with compound interest, accrued interest is fed back into the principal, allowing you to generate additional interest that accumulates over time.

An example will make this concept clearer.

Suppose you have invested 100 euros at 5% per annum with simple interest. Each year the interest will be calculated on the capital of 100 euros and will always be equal to 5 euros.

• the initial capital is fixed for all periods

• the accrued interest is always the same amount

• the interest rate is always calculated on the initial capital

Instead, we speak of compound interest when the interest earned at the end of each period accumulates in the initial capital, generating new interest at each maturity.

• the initial capital increases at the end of each period, adding the accrued interest

• interest grows over time

• the interest rate is calculated on a variable capital

The extraordinary effect of compound interest is that it grows exponentially over time and therefore favors those who keep the investment longer over time by reinvesting the returns obtained.

The origin of compound interest

The origin of compound interest is linked to chess. According to legend, a constantly bored Persian emperor announced a ban promising a reward to anyone who knew how to amuse him. After several failed attempts by storytellers, magicians, and artists, a merchant showed up at the king’s court with a game called chess.

The king fell in love with the game right away and spent a whole night playing with the merchant. The next day, enthralled by the game, the king asked the merchant what reward he wanted for distracting him from boredom. Surprisingly, the merchant asked to be rewarded with grain. In particular, he wanted to receive a grain of wheat for the first box, two grains for the second, four for the third, and so on up to the sixty-fourth box.

The emperor agreed, not realizing that a grain of rice doubled 64 times (as many as the squares on the chessboard) would reach an enormous amount that it could never satisfy: 18,446,744,073,709,551,615.

This legend of "grains of wheat" aptly illustrates the exponential and surprising effect of increasing quantities over time.

How compound interest works

Compound interest is the return you receive on previous accumulated interest, generating a multiplier effect on your initial principal. It therefore represents interest on interest.

To better understand how it works, however, it is necessary to analyze in detail the factors that influence compound interest:

• the compounding frequency,
• the interest rate,
• the time period.

The compounding frequency, i.e. the number of times per year that interest is paid out, has a direct impact on compound interest. Usually, interest is calculated and paid annually, semi-annually, quarterly or monthly, but it can also be compounded on a daily basis or continuously. In general, the more frequently you pay interest, the more the future value of your money increases. For example, if interest is calculated and distributed quarterly, you will have the opportunity to reinvest your earnings more quickly, allowing your initial capital to grow more significantly over time.

The interest rate is another crucial element in compound interest. Higher interest rates will contribute more substantially to the overall value of an investment than lower rates. When yields are obtained at a higher interest rate, the annual increase in capital will be greater, since the percentage of interest charged is more significant. Therefore, even a small difference in the interest rate can make a considerable difference in the final result.

Finally, time plays a determining role in compound interest. The longer the investment remains active, the greater the effect of compound interest on its value. Over an extended period, even a small annual gain can add up significantly due to the cumulative effect of compound interest. This is why long-term investments can produce remarkable results compared to short-term ones. The combination of an extended time frame and a high compounding frequency can further magnify the effect of compound interest on your assets.

How to calculate compound interest

Simple Capitalization
From a mathematical point of view, the formula to calculate the amount in the case of simple capitalization, i.e. the sum of interest and final principal, is as follows:

• Final Capital = 𝐶 ∙ (1 + 𝑖 ∙ 𝑡)

where 𝐶 is the starting capital, 𝑖 the simple interest rate is 𝑡 the duration of the investment.

For example, by investing an initial 10,000 euros with an annual interest rate of 10%, after 10 years we would have 20,000 euros calculated with simple interest. After 30 years, the sum of 40,000 euros would be reached.

Compound Capitalization
The formula for calculating the capitalization of interest is different:

• Final Capital = 𝐶 ∙ (1 + 𝑖)^𝑡

where 𝐶 is the starting capital, 𝑖 is the compound interest rate and 𝑡 is the time, which expresses the duration of the investment.

Returning to the previous example, by investing an initial 10,000 euros with an annual interest rate of 10%, after 10 years we would have an amount of 25,938 euros calculated with compound interest. After 30 years, the sum of 174,494 euros would be reached.

This example confirms that the effect of compound interest is greater as the term of the investment increases.

**The shortcut to calculating compound interest

The rule of 72 is a quick and easy formula for estimating the time it takes to double an investment and for evaluating the approximate impact of compound interest. The formula for the rule of 72 is as follows:

72 / r = Y

Where r is the compound annual interest rate and Y is the number of years it takes to double your investment.

For example, considering an interest rate of 5% and using the rule of 72, it would take about 14.4 years for money to double. It’s important to keep in mind that the rule of 72 only provides a rough estimate. In the specific case mentioned, the actual time required to double the principal with annual compound interest would be approximately 14.21 years. Despite a slight discrepancy between these figures, the rule of 72 can be used as an easy way to get a rough estimate.

It is vital to note that the Rule of 72 is an approximation and may not be highly accurate in all situations. It can be especially useful when you want a quick, rough estimate of how long it takes to double an investment or to understand the general effect of compound interest. However, for more precise calculations, it is advisable to use specific financial instruments and formulas.

Where compound interest applies

After ascertaining that compound interest is a strategy that allows you to increase the return generated by an investment by reinvesting profits, let’s now see some examples of how to maximize earnings through compound interest.

The calculation of compound interest applies to many financial investments, especially those with a multi-year duration and to those types of investments or savings that pay interest:

• open a deposit account: although in recent years it has become inconvenient due to low-interest rates, this instrument offers a periodic return against the deposit of a sum.
• invest in bonds that accrue a coupon or interest at periodic intervals reinvesting the profits
• invest in shares over the long term, reinvesting dividends to activate compound capitalisation. Alternatively, you can choose ETFs on the S&P 500 or on the MSCI World
• open a open or negotiated pension fund from an early age: an investment of 5,000 euros a year from the age of 25 and an interest rate of 6% per annum would make it possible to reach a capital of 400,000 in 40 years.
• in general, accumulation investment products are preferable, i.e. products which do not pay a coupon periodically but automatically incorporate it into the invested capital.
• anatocism: a negative example of the application of compound interest. Although it is a practice prohibited by Italian law, it may happen that banks accrue interest on the debit interest already due.

The benefits of compound interest

Compound interest is therefore a powerful tool for investors. All other things being equal, it is in fact preferable to choose investment instruments that discount compound interest rather than simple interest.

By constantly investing a small sum over time, for example, 100 euros a month, it is possible to build an investment strategy that after 20, 30 or 40 years will return a decidedly interesting sum thanks to compound capitalization.
The longer the duration of the investment, the higher the final capital will be.

Original article published on Money.it Italy 2023-07-11 09:44:50. Original title: Interesse composto: cos’è, come funziona, calcolo