Compound interest: how returns on returns make wealth grow

Basics8 min read · updated 8 September 2026 · 5 sources

With compound interest, it is not only the money you paid in that earns returns, but also the returns you have already earned. That is why wealth does not grow evenly but faster and faster: the curve bends upwards. For long-term investors, this is the single most important lever, more important than the choice of any particular ETF.

Compound interest: the formula

The difference from simple interest lies in the exponent. With simple interest, only the initial capital earns interest each year: final value = initial capital × (1 + r × n). With compound interest, one year's return becomes part of the next year's capital. At first the difference is small; after a few decades it is huge.

The figures are a worked example with an assumed, constant return. On the stock market, annual returns fluctuate a lot; compounding still works, just on the actual, fluctuating performance. How to calculate the average growth rate from it is explained in the article on CAGR.

The rule of 72: when will my money double?

A simple rule of thumb: divide 72 by the return in percent and you get roughly the number of years until your money doubles. At 7%, that is 72 / 7 = 10.3 years; exactly, it is ln(2) / ln(1.07) = 10.24 years. The rule works well between about 4% and 12%. At 3% it takes 24 years, at 10% about 7 years.

The rule also works in reverse: if you want to double your money in 12 years, you need around 72 / 12 = 6% a year. And it shows how much inflation eats away: at 3% inflation, the purchasing power of cash halves in about 24 years.

Compound interest with monthly contributions

Most investors do not pay in once but regularly. Each contribution then compounds for a different length of time: the first for 30 years, the last for just one month. The sum of all contributions with their compound interest gives the final value of a monthly investment plan.

This leads to the most important practical rule: start early. Ten more years with the same contribution often achieve more than a much higher contribution in later years. Whether you are better off investing a lump sum or in installments is covered in the article lump sum vs. dollar-cost averaging; why buying at an “average cost” brings no extra return is explained under dollar-cost averaging.

Working backwards: how much do I need to save?

Often the question is the other way round: how high does the monthly contribution need to be to reach a goal? To answer it, you rearrange the formula for monthly contributions to solve for the contribution. Here too, the power of time shows: if you save for twice as long, you need well under half the contribution, because the early contributions work for longer.

Fluctuating returns

The formulas assume a constant return. If returns fluctuate, the geometric mean is what counts: a year of +50% and one of −50% average 0%, but wealth falls by −25.00%, because 1.5 × 0.5 = 0.75. The geometric average return is then −13.40% a year. The more a portfolio fluctuates, the larger this discount; the article on volatility drag explains why.

Compounding within the year and the effective interest rate

If interest is credited several times a year, capital grows faster than the nominal rate suggests. A high-yield savings account with 5% nominal interest credited monthly effectively yields (1 + 0.05 / 12)^12 − 1 = 5.116% a year. With ETFs the question does not arise in this form: their prices include income continuously. It matters when comparing interest rate offers and when converting monthly into annual returns.

Never simply multiply monthly returns by 12 when it comes to growth: 1% a month is 1.01^12 − 1 = 12.68% a year, not 12%. With fluctuating returns, volatility drag also applies.

What slows compounding down: costs, taxes, inflation

Compounding works in both directions: ongoing costs also “compound” year after year (as lost returns). A difference of just over one percentage point in costs sounds small, but it adds up:

Taxes have a similar effect: whatever is taxed every year is lost to compounding. Because capital gains are usually taxed only when you sell, part of the tax is pushed further into the future and the deferred amount keeps compounding. And inflation eats away purchasing power: a 7% return with 2% inflation becomes (1.07 / 1.02) − 1 = 4.90% a year in real terms. After 30 years, $10,000 then becomes only $42,025 in today's purchasing power instead of $76,123 nominal. More on this under real return.

What the research shows

The distinction between nominal and real interest goes back to Fisher (1930). For long periods, Dimson et al. (2002) compiled returns on equities, bonds and money market instruments over 101 years and many countries; Siegel (2014) documents the development in the US since the early 19th century. Both show that over decades, large differences in wealth arise from small differences in annual return. That is exactly the compounding effect.

Two subtleties from the research are important. First: what counts is the geometric mean, not the arithmetic one. Booth and Fama (1992) show that a broadly diversified portfolio achieves more than the average of its components, because diversification lowers fluctuation and thus reduces the drag on compounding. Second: with individual stocks, compounding is very unevenly distributed. Bessembinder (2018) finds that the majority of US stocks returned less than money market instruments over their lifetime, and a small fraction of companies accounts for almost all of the wealth creation. If you diversify broadly, you are sure to own these few winners too.

The Einstein quote

You often read that Albert Einstein called compound interest the “eighth wonder of the world” or the “most powerful force in the universe”. There is no evidence for this; the quote is a legend. The mathematics does not need the famous name anyway.

Limits and common fallacies

• Constant returns do not exist: calculators with a fixed 7% show a smooth curve. In reality, wealth can fall for years; with withdrawals, the order of returns also matters (see withdrawal rate).
• Past returns are no promise: long series show what was possible, but not what is to come. For ranges instead of point values, a Monte Carlo simulation is the right tool.
• Nominal is not real: a final value in 40 years sounds large; in today's purchasing power it is much smaller.
• Time beats timing: if you wait to catch the “right moment”, you give away years of compounding. Why market timing rarely works is explained under market timing.

How Portfolio Lab calculates it

In the Backtest, Portfolio Lab works with price series in which distributions are reinvested. Compounding is therefore already contained in the performance. The engine deducts ongoing charges pro rata in every period, and regular contributions and withdrawals are booked on the chosen dates, escalated or adjusted for inflation if you wish. The CAGR shown is the geometric growth rate over the whole period; the app calculates real final values using a consumer price index. Details are in the backtest methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.