# What is compound interest: the formula and how to do it

July 12, 2023

6 min

Beginner

What is compound interest and what’s it formula? Let’s just say that Einstein called it ‘the eighth wonder of the world’, and that combined with a great flair for investing, it is responsible for Warren Buffett’s wealth. So let’s find out what is behind this investment ‘miracle’.

## What is compound interest?

Before we begin to explain the formula, it is worth examining the concept behind what compound interest is: that of the time value of money.

In essence, it says that it is better to have €100 today than €100 tomorrow. This is because money of ‘today’ can be invested, grown and used to generate value ‘tomorrow’.

In fact, according to this principle, the value of money changes depending on when it is acquired.

So how does one generate value over time? Compound interest is the easy answer!

It is calculated not only on the initial capital of an investment, but also on the interest amounts accumulated over time. In other words, it is based on the principle of reinvesting the interest earned to generate further interest.

For example, if you invest 1,000 euro at an annual rate of 5% with compound interest, at the end of the first year you will have earned 50 euro, bringing the total to 1050 euro. In the second year, the interest calculation will be on 1050 euro instead of 1000 euro, generating 52.50 euro. At the end of the second year, the total will be 1102.50 euros.

In a nutshell, while direct interest provides linear growth, compound interest allows exponential growth.

What happens, however, if the time value of money is not exploited?

By receiving €100 tomorrow instead of today, you would have lost time that you could have spent accumulating interest. Time thus becomes an indirect cost, called ‘opportunity cost‘, i.e. the loss of profit opportunities, without actually making a loss.

If you do not invest €100 in an account with an annual interest rate of 10%, you would miss the opportunity to generate €10 per year of direct interest and an increasing amount of compound interest.

The rate at which compound interest grows depends on how frequent the accumulation is, i.e. how often the basic interest is accrued, whether annually, semi-annually or monthly. The more frequent the accumulation, the steeper the growth curve will be.

## Compound interest: the formula

Now that we understand what it is, it is worth dwelling on the calculation of compound interest.

There are various online calculators that can simplify the process, but let us briefly look at the formula for simulating the outcome of an investment with compounding.

A = P × (1 + r/n)^(nt)

where:

• A is the final amount (i.e. the value of the investment at the end of the time period);
• P is the initial capital (i.e. the amount of money invested at the beginning);
• r is the annual interest rate, expressed in decimal form (e.g. a rate of 5% becomes 0.05);
• n is the frequency with which interest is calculated and capitalised in a year (e.g. if it is capitalised quarterly, n equals 4);
• t is the number of years the investment is maintained.

Here is an example of how to apply the formula for calculating compound interest:

Suppose you invest €1,000 at an annual rate of 5% with compound interest, and you hold the investment for 3 years, with the interest calculated and capitalised annually. The formula becomes:

A = 1000 × (1 + 0.05/1)^(1×3) = €1157.63

Thus, at the end of 3 years, the value of the investment will be €1157.63, with a compound interest gain of €157.63.

## How do you invest with compound interest?

Now that we know what compound interest is from a mathematical point of view as well, let’s get down to practice. How do you achieve it and with which assets?

You can potentially compound any type of financial instrument. Usually it is enough to keep your capital invested in the asset you think will grow. In fact, selling generating gains could trigger the payment of taxes, making it impossible to reinvest the total profit.

Therefore, there is no particular compounding feature, or operations to be done actively: it is generally enough to maintain the investment over the long term.

There are several instruments, however, that generate liquid fixed annuities such as bonds that give dividends, deposit accounts or real estate. In that case you need to actively reinvest the profit you receive in order to generate compound interest.

Sounds too good to be true? They don’t call it the ‘miracle of compound interest’ for nothing, but there are some conditions to consider.

First of all, the volatility of the market, which means that you cannot always count on stable or even positive interests. This of course depends on the risk rate of the chosen financial instrument.

Moreover, compound interest can also work against you if you have debts or loans with high interest, as the accumulated interest can quickly increase your debt.

## Alternative forms of compounding

In order to apply the principle behind compound interest, the return on investment does not have to be ‘interest’, but can come in the form of dividends or capital gain.

The mechanism of compounding can therefore also be applied to trading, although it is not as direct and involves more risk. However, in theory, if you close a position in profit, and then reinvest all that you have gained, this second position will be larger and can therefore yield more profit if closed in the positive. So the effect is very similar, and always requires reinvestment.

Once you have applied the formula and chosen the form of compounding you prefer, nurture your capital with patience and dedication. For what is compound interest if not the seed of a great tree?