Compound interest is interest calculated not just on your original principal, but on the interest that principal has already earned. That one sentence is the entire mechanism, and it sounds so mild that most people underestimate it for years before it becomes obvious.
The reason it is worth understanding properly is not that the formula is hard. It is that the intuition is wrong. Human beings estimate growth linearly, and compounding is not linear, so the gap between what you expect and what actually happens widens the longer you wait to look.
The simplest possible example
Put $1,000 in an account earning 5% a year. After year one, you have $1,050 — the $1,000 plus $50 of interest. After year two, you do not earn another $50. You earn 5% of $1,050, which is $52.50, giving $1,102.50. Year three earns 5% of that, or $55.13.
The interest itself is growing, because the balance it is calculated on keeps getting bigger. With simple interest — 5% of the original $1,000, forever — you would have $1,150 after three years. With compounding you have $1,157.63. A difference of $7.63 that nobody would notice.
Now run it for forty years. Simple interest gives $3,000. Compound interest gives $7,040. Same rate, same principal, same math — the only variable is that the interest was allowed to earn interest.
Why time matters more than the amount
This is the counterintuitive part, and it is worth being concrete about it.
Take two savers, both earning 7% a year. Alice invests $300 a month from age 25 to age 35, then stops completely and never adds another dollar. She contributed $36,000 over ten years. Ben invests nothing until 35, then contributes $300 a month every single month until 65 — thirty years, $108,000 contributed, three times what Alice put in.
At 65, Alice has roughly $358,000. Ben has roughly $340,000. Alice contributed a third as much and finishes ahead, because her money had forty years to compound instead of thirty, and the last decade of compounding on a large balance produces more growth than an entire decade of new contributions on a small one.
The practical lesson is not "Ben should not bother." It is that a dollar invested at 25 is worth several dollars invested at 45, so the cost of waiting is much higher than the cost of contributing a smaller amount now.
The math, if you want it
For a lump sum, the future value is FV = P × (1 + r/n)^(n×t), where P is the principal, r the annual rate as a decimal, n the number of compounding periods per year, and t the number of years.
For regular contributions, each deposit compounds for a different length of time, so the future value of a stream of equal payments is FV = PMT × [((1 + i)^k − 1) / i], where i is the periodic rate and k the number of payments. Total value is the sum of both parts.
You do not need to compute this by hand — but knowing the shape of it tells you where the leverage is. The exponent is time. Everything else in the formula is a multiplier; time is the power the whole thing is raised to.
The Rule of 72
Divide 72 by an annual percentage return and you get roughly the number of years for money to double. At 6%, about 12 years. At 8%, about 9. At 12%, about 6.
The value of this shortcut is that it reframes the question. Instead of asking "is 6% or 8% better," which is obviously answerable but not very informative, you are asking "does my money double every 12 years or every 9?" Over a 36-year horizon, that is the difference between three doublings and four — between 8× and 16× your starting amount.
The same force, pointed at you
Compounding is indifferent to which side of the ledger you are on. A credit card balance at 22% APR compounds against you on roughly the same mechanics, and at a rate no reliable investment matches.
A $6,000 balance at 22%, left alone, is about $7,320 after a year and roughly $10,900 after three. There is no investment strategy with a comparable certainty of a 22% return, which is why paying down high-rate debt is usually the highest-confidence compounding decision available.
What to be careful about in projections
Two honest warnings. First, projections assume a smooth constant rate, and real markets do not deliver one. A 7% average over thirty years can include years down 30% or up 25%; the ending value can differ meaningfully from the smooth-curve estimate depending on when the bad years land, especially once you start withdrawing.
Second, most projections quote nominal returns while your actual goal is purchasing power. A projection showing $1,000,000 in thirty years at 7% is describing dollars that, at 3% inflation, will buy roughly what $412,000 buys today. That is still a good outcome. It is simply not the outcome the headline number implies.
Both of those are reasons to project carefully, not reasons to skip projecting. The direction of the conclusion — start earlier, contribute consistently, let it run — survives any reasonable set of assumptions.