Compound interest: what it is and how it works

A widely repeated line says Einstein called compound interest the eighth wonder of the world. There's no record of him saying it: the oldest known version comes from an ad published in an American newspaper in 1925 (Quote Investigator, 2019). The math behind the line, though, is real.

With compound interest, each period's return is added to the amount invested and starts earning too. That's why it's called interest on interest. Over a few years, the difference from simple interest is small. Over a few decades, it becomes most of the result.

This article shows that math with fictional numbers and what research says about it. It doesn't say where to invest or promise any return: the rates used here are examples.

Simple and compound interest

With simple interest, the return is always calculated on the initial amount. With compound interest, it's calculated on the value at the end of the previous period, which already includes the interest earned so far.

With $10,000 earning 6% a year, the difference looks like this:

In the first year, both come out the same. After 30 years, compound interest reaches more than double simple interest. Each year's math is just the previous year's value times 1.06.

When you invest every month, time matters more

Someone who invests a little every month runs this math on each new investment, and money that goes in early has more years to grow. In the example below, a person invests $500 a month, or $6,000 a year, and the money earns 6% a year. To keep it simple, each year's money goes in together at the end of the year.

After 10 years, almost a quarter of the value came from interest. After 30, more than 60%. And someone who invests for 30 years ends up with more than double what someone investing for 20 years does: the 10 extra years add $253,635, more than all the money invested over the 30 years.

Why we underestimate this math

Intuition tends to picture money growing in a straight line. Researchers call this mistake exponential growth bias, and it shows up in several studies:

• In an experiment with 99 university students in the US, one question asked how much someone saving $400 a month at 10% a year would have. For 40 years, the median answer was less than 10% of the right value. Across all the questions, 90% of participants on average came in below the right value (McKenzie and Liersch, 2011).
• In a representative sample of US adults, about a third ignored compounding entirely, and 96% underestimated growth, even though they could use a calculator. Going from ignoring compounding to getting the math right is associated with 55% to 90% more accumulated wealth, other things equal (Levy and Tasoff, 2016).
• In Sweden, the question was how much 100 kronor is worth after 30 years at 7% a year. The right answer is 761, and the median answer was 410 (Almenberg and Gerdes, 2012).
• Households that underestimate exponential growth borrow more and save less (Stango and Zinman, 2009).
• In the US, 67.1% of adults aged 50 and over got a basic compound interest question right (Lusardi and Mitchell, 2014). Worldwide, in a survey of more than 140 economies, only 33% of adults were financially literate (Klapper, Lusardi and van Oudheusden, 2015).

Showing the math can help, but not always. In the 2011 study, when the exponential growth of savings was shown clearly, students and employees of a large company became more motivated to save. In Levy and Tasoff's study, though, a chart of the growth didn't change the answers.

The rule of 72

To estimate how long an amount takes to double, divide 72 by the annual rate. At 6% a year, that's 12 years, and the exact math gives 11.9 (Weisstein, MathWorld). The rule works for a lump sum at a constant rate. For monthly investments there's no shortcut that simple (McKenzie and Liersch, 2011), which is why it's worth doing the math.

Average returns mislead

Returns change from year to year, and a simple average promises more than a portfolio delivers. A 50% rise followed by a 50% fall averages zero, but $100 turns into $150 and then $75.

What a portfolio actually accumulates follows the compound rate, which falls below the simple average when years vary. In 16 rich countries, from 1870 to 2015, stocks returned an average of 6.89% a year above inflation, but the compound rate was 4.64% a year (Jordà and others, 2019). These are index numbers, before costs and taxes, and the past doesn't guarantee the future.

Costs and inflation compound too

A fee charged every year compounds the same way, only against you. The SEC, the US securities regulator, gives the example of $100,000 invested for 20 years, earning 4% a year. With a 1% annual fee, the portfolio ends up nearly $30,000 smaller than with a 0.25% fee (SEC, 2014).

In the example of $10,000 over 30 years, earning 5% a year instead of 6% because of fees ends at $43,219 instead of $57,435: almost a quarter less.

Inflation weighs too, because what increases what your money can buy is the return above it, the real return. If an investment earns 10% in a year when prices rise 4%, the real gain is 5.8%, not 6%, because this math also compounds: 1.10 divided by 1.04 gives 1.058.

When compounding works against you

On a debt, the same math runs the other way. In the third quarter of 2026, US banks charged an average of 22.36% a year on credit card accounts that were charged interest, in preliminary data (Federal Reserve). Compounded monthly, that rate doubles a balance in a little over three years.

What this changes over an investing career

In the math in this article, three things weigh more than they seem: how long the money stays invested, how steadily you invest, and what leaks out along the way in fees, inflation and interest on debt. The first two work in favor of compounding. The third compounds the same way, in the opposite direction.

Sources

When a journal article is paywalled, the link goes to a free version from the authors when there is one, or to the page with the abstract.

• Almenberg, J. and Gerdes, C. (2012). Exponential growth bias and financial literacy. Applied Economics Letters, 19(17), 1693–1696. DOI: 10.1080/13504851.2011.652772.
• Federal Reserve (2026). Consumer credit, G.19. Commercial bank interest rate on credit card plans, accounts assessed interest, third quarter of 2026, preliminary.
• Jordà, Ò., Knoll, K., Kuvshinov, D., Schularick, M. and Taylor, A. M. (2019). The rate of return on everything, 1870–2015. The Quarterly Journal of Economics, 134(3), 1225–1298. DOI: 10.1093/qje/qjz012. The figures cited come from the 2017 working paper version.
• Klapper, L., Lusardi, A. and van Oudheusden, P. (2015). Financial literacy around the world: insights from the Standard & Poor's Ratings Services Global Financial Literacy Survey.
• Levy, M. and Tasoff, J. (2016). Exponential-growth bias and lifecycle consumption. Journal of the European Economic Association, 14(3), 545–583. DOI: 10.1111/jeea.12149.
• Lusardi, A. and Mitchell, O. S. (2014). The economic importance of financial literacy: theory and evidence. Journal of Economic Literature, 52(1), 5–44. DOI: 10.1257/jel.52.1.5. The figure cited comes from the 2013 working paper version.
• McKenzie, C. R. M. and Liersch, M. J. (2011). Misunderstanding savings growth: implications for retirement savings behavior. Journal of Marketing Research, 48(SPL), S1–S13. DOI: 10.1509/jmkr.48.SPL.S1.
• Quote Investigator (2019). The eighth wonder of the world is compound interest.
• SEC (2014). How fees and expenses affect your investment portfolio. Investor Bulletin.
• Stango, V. and Zinman, J. (2009). Exponential growth bias and household finance. The Journal of Finance, 64(6), 2807–2849. DOI: 10.1111/j.1540-6261.2009.01518.x.
• Weisstein, E. W. Rule of 72. Wolfram MathWorld.