What is compound interest? How small differences grow over time

Imagine putting one grain of rice on the first square of a chessboard, two on the second, four on the third, doubling the amount on every square. On the first squares the amount is trivial; but by the time you reach square 64, the rice needed far exceeds the world’s annual production. This old story shows how exponential growth catches human intuition off guard. Compound interest is the most familiar everyday example of the same maths. In this article we treat compound interest not as investment advice but as a mathematical idea: understanding how a small rate turns into large differences over time matters not only for money, but for making sense of many topics, from population growth to computer science to the spread of viruses.

Note: This article is for educational purposes and is not financial advice. Real decisions require a careful look at a product’s terms, taxes and fees, and expert advice where needed.

The difference between simple and compound interest

Suppose you deposit 10,000 TL (Turkish lira) in a bank and earn 10 per cent interest a year.

With simple interest, interest is calculated each year only on the principal. You earn 1,000 TL of interest every year. After 20 years the total is:

With compound interest, each year’s interest is added to the principal, and next year’s interest is calculated on this larger amount. In other words, the interest earns interest too. At the end of the first year you have 11,000 TL. In the second year the interest is not 1,000 but 1,100 TL, and the amount rises to 12,100 TL. By the end of the third year it is 13,310 TL, and the growth speeds up a little more each year.

The compound interest formula

Since the amount is multiplied by $(1 + r)$ each year, the amount after $n$ years is:

Here $P$ is the principal, $r$ is the annual interest rate (as a decimal) and $n$ is the number of years. In our example:

Same rate, same period, same starting money; but 30,000 TL with simple interest and about 67,275 TL with compound interest. The difference is small at first: at the end of the first year both methods give 11,000 TL, and at the end of the second year the difference is only 100 TL. But the gap grows every year, and by year 20 compound interest gives more than twice as much.

Simple interest grows like a straight line: the same amount is added each year. Compound interest grows like an exponential curve: it increases by the same proportion each year. If you plot them, the two lines almost overlap in the first years, then the exponential curve bends upwards and pulls away.

Why do small differences in rate lead to big results?

The most instructive thing about compound growth is how small differences in rate become big differences over long periods. Here is a thought experiment: imagine putting 10,000 TL into two different options for 30 years, one growing at 7 per cent a year and the other at 6 per cent.

The rates differ by only 1 percentage point; but after 30 years the second amount is only about three quarters of the first. That is why, in long-term calculations, an annual fee or deduction of 1 per cent, which looks trivial at first glance, can end up taking close to a quarter of the total growth.

Time is the rate’s most powerful partner

Because $n$ in the formula is an exponent, time affects the result as much as the rate does, or even more. At 7 per cent a year, 10,000 TL reaches about 19,672 TL in 10 years, and about 76,123 TL in 30 years. Tripling the time has multiplied the amount by about 3.9. The gain is even more striking: the 30-year gain (about 66,123 TL) is roughly 7 times the 10-year gain (about 9,672 TL). With exponential growth, the early years call for patience and the later years bring the surprises.

You can test this with a question: if the water lilies on a lake double every day and the lake is fully covered on day 30, on which day was half the lake covered? Our intuition often says day 15; the correct answer is day 29. In exponential processes, most of the growth happens at the very end, in a very short stretch of time. That is why people usually think the end of an exponential process is further away than it really is.

The rule of 72: doubling time in your head

How long does it take for an amount to double? To calculate it exactly we use logarithms:

But there is a practical shortcut: the rule of 72. Divide 72 by the rate written as a percentage; the result is roughly the doubling time.

• For 6 per cent: $72 / 6 = 12$ years. Exact calculation: about 11.9 years.
• For 8 per cent: $72 / 8 = 9$ years. Exact calculation: about 9.0 years.
• For 12 per cent: $72 / 12 = 6$ years. Exact calculation: about 6.1 years.

The rule comes from $\ln 2 \approx 0.693$ and the approximation $\ln(1 + r) \approx r$ for small rates. In theory 69.3 would be more accurate; but because 72 divides exactly by 2, 3, 4, 6, 8, 9 and 12, it makes mental arithmetic much easier, and it gives quite good results across the usual range of rates. The rule drifts further off at very high rates; in those cases you need the exact formula.

The same rule applies to debt. An unpaid debt grows at the same pace as compound interest. Knowing that a high-interest debt can double within a few years is a reminder that compound interest is a double-edged sword.

How often is interest added? The birth of the number e

What happens if interest is added more often than once a year? With an annual rate $r$ and interest added $m$ times a year, the formula becomes:

Let’s look at the one-year growth factors for an annual rate of 12 per cent:

As the frequency increases, the factor grows, but not without limit; it approaches a certain value. This question also caught the interest of the Swiss mathematician Jacob Bernoulli in the late 17th century. Bernoulli studied how many times larger money would be after one year if 100 per cent interest were added more and more often, and showed that the result does not grow forever but approaches a particular number between 2 and 3. Its value is about 2.718:

Today this number is known as e, and it is one of the most important constants in mathematics. When interest is added continuously, the formula takes an elegant form:

For 12 per cent, $e^{0.12} \approx 1.1275$; that is the last row of the table. That an everyday question about money gave birth to a constant that turns up in almost every growth and decay process in nature is one of the lovely surprises of mathematics.

Inflation: nominal and real returns

A point often skipped in compound interest calculations is that the purchasing power of money also changes over time. If prices are generally rising, the things you can buy with your money may not increase even if the number of lira you hold does.

Suppose that in one year your money grows by 50 per cent while prices rise by 45 per cent. The real increase, measured in purchasing power, is not 5 per cent:

So the real increase is about 3.45 per cent. Here $\pi$ is the inflation rate. The rule of thumb “nominal rate minus inflation” is a good approximation at low rates, but it drifts noticeably when rates are high. If inflation is higher than the interest rate, the real return is negative: the amount grows, but its purchasing power falls.

This distinction shows that compound growth involves not only the question “how much?” but also “measured in what?”.

Compound growth beyond money

The same maths turns up in many different fields:

• Population growth: A population growing at 2 per cent a year doubles in about 36 years by the rule of 72; the exact calculation gives about 35 years.
• Bacterial colonies: Under suitable conditions, bacteria that divide at regular intervals reach very large numbers in a short time. This growth does not go on forever, because food and space are limited.
• Radioactive decay: This is a compound process running in reverse; the amount of material halves with every half-life.
• Learning and skills: It is often said that improving a skill a little every day makes a big difference in the long run. The comparison is motivating, but real learning does not follow a smooth exponential curve; spells of rapid progress, plateaus and forgetting are all part of the process.

In the real world no growth stays exponential without limit. But understanding the exponential model is the key to seeing why something that starts “slowly” can suddenly “explode”.

In summary

• With simple interest, interest is calculated only on the principal; with compound interest, it is calculated on the accumulated amount.
• The compound interest formula is $A = P(1 + r)^n$; because time is an exponent, it has a very strong effect on the result.
• Small differences in rate and small fees turn into big differences over long periods.
• The rule of 72 lets you estimate the doubling time in your head.
• When interest is added more and more often, the result approaches a limit set by the number $e$.
• Growth calculations that ignore inflation can be misleading.

If you’d like to look more deeply at exponential growth and percentages, have a look at our maths articles and maths topics. In Kuazara’s interactive lessons you can change the rate and the time yourself, see how simple and compound growth curves pull apart, and understand the formulas without memorising them.