Compound interest and what the number actually means

Compound interest is one formula. Everything difficult about it sits either side of the formula — how often it compounds, when money goes in, and what the resulting number is worth in today's money.

The formula, and the variable that surprises people

A = P × (1 + r/n)^(n×t), where P is the starting amount, r the annual rate as a decimal, n the number of compounding periods per year and t the number of years.

The exponent is what makes the result non-obvious: the balance grows with the number of periods, so time dominates every other input. 1,000 at 7% for 30 years is about 7,610. The same 1,000 at 7% for 20 years is 3,870 — the last decade of a thirty-year run contributes more growth than the first two decades combined.

Nominal, effective, APR, APY

The same rate quoted two ways is the most common source of a mismatch between a calculator and a bank statement.

A nominal rate of 12% compounded monthly is 1% per month, and 1% twelve times is 12.68%, not 12%. That 12.68% is the effective annual rate — the number you can actually compare between products, published as APY on savings in the US, AER in the UK, and often as EAR on borrowing.

APR runs the other way and is defined differently by jurisdiction: it typically includes mandatory fees but ignores intra-year compounding, which is why the APR on a credit card understates what a revolving balance really costs. When comparing two products, convert both to an effective annual rate first, and be suspicious of any comparison that does not say which convention it uses.

Daily compounding is worth roughly a further 0.01–0.02 percentage points over monthly at ordinary savings rates. It is a real difference and a small one; it is almost never worth choosing a worse headline rate for.

Contributions: the timing is not a rounding error

Two conventions, and calculators disagree about which they use:

Ordinary annuity — the contribution lands at the end of each period. This is the usual default.
Annuity due — the contribution lands at the start, so every payment earns one extra period of interest.

Over a long horizon, paying at the start is worth about one extra period of growth on the whole stream — roughly 7% more at a 7% annual rate. If a projection differs meaningfully from your provider's, contribution timing is the first thing to check and compounding frequency is the second.

The deeper point: with regular contributions, the rate matters less than the contribution in the early years. Reaching a target ten years out is almost entirely a savings-rate problem; thirty years out it is a return problem. Which lever to pull depends on where you are on that curve.

Nominal growth is not purchasing power

A balance projected 30 years out in today's currency units is a number you cannot spend. At 2.5% inflation, money loses a little over half its value in 30 years — so 100,000 in 2056 buys roughly what 48,000 buys now.

The clean way to handle this is to work in real terms: take the return net of inflation and project with that. A 7% nominal return with 2.5% inflation is a real return of about 4.4% — 1.07 / 1.025 - 1, not 7 - 2.5, though the subtraction is close enough for a mental estimate. The resulting number is directly comparable to what things cost today, which is the only comparison anyone actually wants to make.

The rule of 72, and where it stops working

Divide 72 by the percentage rate to get the doubling time: 6% doubles in about 12 years. It is accurate to within a few percent for rates between roughly 4% and 12%, and it drifts outside that — at 1% the true doubling time is 69.7 years against the rule's 72, and at 25% the rule says 2.9 years against a true 3.1.

Its real use is as a sanity check. If a projection implies your money doubles in five years, the rule tells you that is about a 14% annual return, and that number is much easier to be sceptical about than a balance is.

Total return versus annualised return

"Up 50%" means nothing without a period attached. A 50% total return over five years is an annualised 8.45% — 1.5^(1/5) - 1 — which is a completely different investment from 50% in one year.

That is the arithmetic behind CAGR, and it is the only fair way to compare holdings of different ages. Simple ROI — gain divided by cost — answers "what did this return in total", which is the right question for a completed one-off project and the wrong one for anything you want to compare against an alternative held for a different length of time.

Two related traps:

Percentage gains and losses are not symmetric. A 50% loss needs a 100% gain to recover. A portfolio that alternates +20% and −20% loses 2% per pair of years, not zero.
Averaging annual returns overstates them. The arithmetic mean of +20% and −20% is 0%; the actual outcome is −4% over the two years. Compounding needs the geometric mean.

What every calculator quietly assumes

Any projection you see, here or anywhere, assumes a constant rate of return, contributions that never miss and never change, no fees, no tax and no withdrawals. Reality differs on all five.

Fees compound too. A 1% annual fee on a 7% return is not 1% of your money — over 30 years it removes roughly a quarter of the final balance, because the fee applies to the growing balance every year.
Tax depends on the wrapper. Interest, dividends and capital gains are usually taxed differently, and tax-sheltered accounts change the answer more than most rate differences do.
Sequence of returns matters once you withdraw. While you are only contributing, the order of good and bad years is irrelevant to the final balance. Once you are drawing an income, a bad first decade is far worse than the same decade later, because you sold units to live on while they were cheap.

The output of a compound interest calculation is a precise consequence of the inputs you gave it, not a prediction. Treat it as a way to compare two plans under identical assumptions — is 15 years at 200 a month better than 10 years at 350 — rather than as a number to plan a retirement date around. And nothing here is financial advice: the arithmetic is general, your situation is not.