Investing Basics · definition
Compound Interest
Compound interest is interest calculated on both the original amount and the interest already earned, so a balance grows faster over time than with simple interest.
Compound interest is interest earned on interest. With simple interest, a return accrues only on the original amount (the principal). With compound interest, each period's interest joins the balance, and the next period's interest is calculated on that larger sum. The result is growth that accelerates: slowly at first, then unreasonably.
The mechanism is ancient. Clay tablets from Babylon already distinguish simple from compound interest, and a famous school exercise from around 1700 BC asks how long a sum takes to double at 20% compounded, the Rule of 72's great-great-grandparent.
Key takeaways
- Compound interest pays interest on the principal and on previously earned interest.
- Time is the dominant variable: the curve bends upward late, which is why long horizons matter more than impressive rates.
- Compounding frequency (yearly, monthly, daily) nudges the effective rate upward for the same stated rate.
- The same mathematics works against you in compounding debts and in recurring fees.
- Real returns vary year to year and can be negative; the formula describes arithmetic, not a guarantee.
The formula, and a feel for it
Final amount = principal × (1 + rate) ^ periods
A $1,000 balance growing 5% a year becomes $1,050 after year one, $1,102.50 after year two (5% of $1,050), and about $4,322 after thirty years. The striking part is the composition: of that $4,322, only $1,000 is deposit and roughly $1,500 would have come from simple interest; the remaining ~$1,800 is interest on interest. The longer the run, the more the snowball outweighs the snow.
Try it below: the solid line compounds, the dashed line grows by the same percentage without compounding.
Interactive · see the maths
Compound growth visualiser
An illustration of the compounding formula, not a forecast or advice. Pick any starting amount, growth rate and period.
Formula: final = start × (1 + rate)years. Real-world investment returns vary from year to year and can be negative; this tool only illustrates the arithmetic of compounding at a constant rate.
The Rule of 72
Divide 72 by the annual growth rate to estimate the years a balance needs to double: at 6%, about 12 years; at 3%, about 24. The shortcut works because of the mathematics of logarithms and is accurate enough for single-digit rates. It also runs in reverse for purchasing power: at 3% inflation, money halves in value roughly every 24 years, the same arithmetic with the sign flipped.
Frequency: how often counts
| Compounding at 4% nominal | Effective annual rate |
|---|---|
| Yearly | 4.00% |
| Quarterly | 4.06% |
| Monthly | 4.07% |
| Daily | 4.08% |
More frequent compounding lifts the effective rate, with rapidly diminishing returns toward a mathematical ceiling (continuous compounding, e^r - 1). The practical lesson is regulatory: because frequency manipulates the apparent rate, disclosure rules force lenders and banks to quote standardised annual rates (APR/APY, in the EU the effective JKP) so products can be compared honestly, as discussed under interest rate.
Two hundred years of proof: Franklin's gift
The most charming documented case of long-horizon compounding is Benjamin Franklin's 1790 bequest: £1,000 each to Boston and Philadelphia, locked up to compound in loans for two hundred years. Despite imperfect management, wars and inflation, the Boston trust alone had grown to roughly $5 million when it paid out in 1990, and Philadelphia's to about $2.3 million. Franklin designed the experiment deliberately, calling compound interest the mechanism by which money "begets money": a primary-source counterweight to the misattributed Einstein quote (no credible evidence Einstein ever called compounding the eighth wonder; quote investigators class it as apocryphal).
The dark side of the same curve
Everything above runs equally hard in reverse. Credit-card debt at 20% doubles in under four years untouched. And recurring costs compound invisibly: an investment portfolio earning 6% gross loses roughly a quarter of its 40-year end value to an annual 1% fee, which is the quiet arithmetic behind the cost obsession described under index fund and John Bogle. Negative returns compound too: lose 50% and a 50% gain only brings you back to 75% of the start, the asymmetry that makes deep drawdowns so expensive in bear markets.
Compounding beyond bank interest
The concept extends past savings accounts: reinvested dividends compound share counts, reinvested bond coupons compound income, and retained corporate earnings compound inside businesses, the engine Warren Buffett has spent six decades describing. Wherever growth feeds the base that generates more growth, the curve appears, and the practice of feeding it steadily is described under dollar-cost averaging.
Frequently asked questions
Is compound interest guaranteed to grow an investment?
No. Compounding is arithmetic that applies to whatever returns occur, positive or negative. Real investment returns vary and can compound losses.
Why does starting early matter so much?
Because the steep part of the curve comes last. Ten early years contribute more end value than ten later years at the same rate, purely through the exponent.
Did Einstein really praise compound interest?
There is no credible source. The quote first appears decades after his death and is classified as misattributed.
Sources
This entry is for education only. Investing Value describes how financial concepts work; it does not provide investment, tax or legal advice, and nothing here is a recommendation to buy or sell any asset.