What is compound interest: the formula, examples and how Italy taxes it

What "interest on interest" really means

Compound interest is interest calculated not only on the money you put in, but also on the interest that money has already earned.

In one sentence the difference looks small; over thirty years it is enormous. With simple interest, €1,000 at 5% earns €50 every year, always the same: the base never changes. With compound interest, the first year's €50 is added to the capital, so in the second year the 5% is calculated on €1,050, in the third on €1,102.50, and so on.

That is the whole mechanism. There is nothing more sophisticated behind it, and yet it is the reason the term weighs so heavily in any savings plan. To see the two curves side by side, go to simple versus compound interest.

Why growth accelerates

The practical consequence is that growth is not linear. In the early years almost all of the balance comes from what you have contributed. In the later years most of it comes from interest.

With €1,000 to start, €100 a month and 5% a year:

• after 10 years you would have about €17,175, of which €13,000 is your own contributions
• after 20 years, about €43,816, with €25,000 contributed
• after 30 years, about €87,694, with €37,000 contributed

Look at the last stretch: more than half of the final value is interest, not your money. And the ten years from year twenty to year thirty add more than the first twenty combined.

There is a precise date when that happens, and it comes late: with these numbers the accumulated interest does not overtake the accumulated contributions until month 294, around year 24 and a half. For almost a quarter of a century the balance is mostly your own money. That is why many people give up before: the effect they were waiting for arrives almost entirely at the end.

The formula, and what each part means

With regular contributions and monthly compounding:

FV = C·(1+i)ⁿ + P·((1+i)ⁿ − 1)/i

• FV is the future value
• C is the starting capital
• P is each month's contribution
• i is the annual rate divided by 12
• n is the number of months

The first term is what your starting capital does on its own. The second is what your contributions do, each compounding for the time it has left. The first month's contribution works for thirty years; the last month's, not at all. It is the same idea seen from another angle.

Compounding more often helps, but far less than it seems

"The more frequent the compounding, the better" is repeated everywhere and it is true, but it is usually said without the number next to it. With a nominal rate of 5% a year, this is what each frequency adds:

Going from annual to monthly adds 0.1162 points. Going from monthly to daily adds a hundredth of a point, and from there to the mathematical limit practically nothing remains. Frequency matters at first and runs out quickly; the term and the rate never run out. When a product boasts "daily compounding", you now know what it is worth.

How Italy taxes it: 26% and 12.5%

This is the part almost no article on compound interest tells you, and it is the one you need to read a real offer.

In Italy, interest, dividends and capital gains on financial products carry a 26% substitute tax (article 3 of Decree-Law 66/2014, in force since 1 July 2014). Italian government bonds and those of white-list states, postal savings bonds and equivalent securities are the exception, taxed at 12.5%. On top of that comes the 0.2% annual stamp duty on the value of financial products.

For compound interest what counts is not only the rate but when it is applied:

• on a deposit account the bank withholds 26% at every interest payment. A 3% gross rate becomes 2.22% net, and it is the 2.22% that compounds. What is withheld earns nothing further.
• in an accumulating fund or ETF the income is reinvested without tax along the way: the tax is paid on the gain when you sell. The capital compounds gross until the end.

An example with real numbers

€10,000, a 3% gross return, twenty years, no fees and ignoring stamp duty, to isolate the effect of taxation.

Case A, a deposit account paying interest yearly. Every year the interest is credited and taxed at 26%. The capital compounds at 2.22%. After twenty years: about €15,512.

Case B, everything deferred to the end. Nothing is taxed along the way, so the capital compounds at the full 3% and reaches €18,061. The gain is €8,061, taxed at 26% in one go: €2,096. About €15,965 remains.

About €450 of difference in favour of deferral, at the same rate and the same return: for twenty years the capital compounded on a larger balance. None of this says which product suits you, which depends on fees, risk and horizon; it only explains why two wrappers with the same gross return can leave you different amounts. You can reproduce both cases in the compound interest calculator, with the "taxing every year" or "taxing only on exit" option.

The rule of 72, and where it stops working

To estimate how many years money takes to double, divide 72 by the annual rate in per cent: at 6% about 12 years, at 4% about 18, at 8% about 9. It is a good approximation between 4% and 10%; below and above it drifts from the exact value, and it ignores both tax and extra contributions.

Inflation is removed by dividing, not subtracting

A nominal 5% with 2% inflation does not leave 3% real but 2.94%: (1.05 ÷ 1.02) − 1. Over a year the difference is negligible; over thirty years it is not. The inflation calculator shows how much purchasing power an amount keeps after a given number of years.

And against you

The same mechanism works in debt. Unpaid interest on a revolving card or an overdraft is added to what you owe and earns further interest: the same curve, with the opposite sign. In Italy interest on interest is limited by article 1283 of the Civil Code, but high rates do the rest even without compounding.