See what time does—and what it cannot promise.

Each period, interest is applied to the balance you already have — including the interest you earned in previous periods — and then your contribution is added. That "interest on interest" is what makes the curve bend upward. This hub uses the same end-of-period convention as standard annuity math.

Divide 72 by the annual percentage return to get the rough number of years for money to double. At 7% that is about 10.3 years. It is a mental shortcut, accurate to within a few percent for rates between roughly 4% and 12%; the mathematically exact constant for continuous compounding is 69.3.

Less than people expect. At 7% over 20 years, moving from annual to monthly compounding on a $10,000 lump sum adds a few hundred dollars — real but small. The rate, the time horizon and the fees all matter far more than the cadence.

Not simply return minus inflation, though that is close enough at low numbers. The exact real return is (1 + nominal) ÷ (1 + inflation) − 1. At 7% nominal and 3% inflation that is 3.88%, not 4%. Over 30 years the difference in ending balance is substantial.

Roughly a fifth of your ending balance over 30 years. A 1% annual expense ratio does not take 1% of your final balance — it takes 1% of the whole balance every single year, so the compounding you lose compounds too. It is the single most controllable variable in the whole projection.

Three months of essential expenses if your income is stable and you have no dependants, six months for most people, and twelve if your income is variable or you are self-employed. Size it on essential expenses — rent, food, utilities, insurance and minimum debt payments — not on income.

Usually a small buffer first, then the debt, then finish the fund. One month of expenses stops the next surprise from going straight onto the card; after that, high-interest debt at 20%+ beats a savings account at any realistic yield. Coming back to finish the fund afterwards costs the least overall.

A total US stock market fund, a total international stock fund and a total bond fund, held in fixed proportions and rebalanced. It is popular because it is cheap, diversified across thousands of holdings, and simple enough to keep going through a bad year — which is the part most strategies fail.

Decide the percentage of capital you are prepared to lose on the trade, then divide that dollar amount by the distance to your stop. Risking 1% of a $100,000 account with an 8% stop gives a $12,500 position. The position size falls out of the risk, rather than the other way round.

Return above the risk-free rate, divided by volatility — how much return you got per unit of risk taken. It is what makes two strategies with different volatilities comparable. Above 1 is generally considered good, but it is extremely sensitive to the window you compute it over, so treat any short-sample Sharpe with suspicion.

The approximate percentage change in a bond’s price for a one-percentage-point change in yield. A duration of 8 means a 1% rise in rates costs roughly 8% of the price. It is a first-order estimate: for large rate moves it understates the gain and overstates the loss, because it ignores convexity.

Price divided by earnings per share — how many dollars you pay for a dollar of annual earnings. Inverted, it is the earnings yield. A P/E of 20 is a 5% earnings yield. It is only meaningful compared against the same company's history and its sector; comparing a utility to a software company on P/E tells you nothing.