Units and Dimensional Analysis

Every quantity in a finance problem carries units — dollars, shares, dollars per share, percent per year, years — and a formula that mixes them incorrectly will produce a nonsense answer even if the arithmetic inside it is done perfectly. Dimensional analysis is simply the habit of tracking those units through a calculation and confirming that what comes out the other end has the units the question actually asked for.

A concrete example: estimating annualized volatility from daily returns requires multiplying daily standard deviation by $\sqrt{252}$ (the number of trading days in a year), because variance scales linearly with time under the usual random-walk assumption, and standard deviation is the square root of variance, so it scales with $\sqrt{\text{time}}$. If a candidate instead multiplies daily volatility by $252$ directly, checking units catches it immediately: daily volatility has units of "percent per $\sqrt{\text{day}}$," and multiplying by a plain count of days (units of "days") does not cancel correctly to leave "percent per $\sqrt{\text{year}}$" — only multiplying by $\sqrt{252}$ does.

The same discipline catches errors in position sizing (dollars of risk should come out in dollars, not shares or percent), in Greeks (delta is dimensionless-ish, a hedge ratio, while gamma has units of delta per dollar of underlying move), and in any formula assembled under interview time pressure, where the fastest way to know something is wrong is that the units on the left and right of an equals sign don't match, well before working out whether the number itself is plausible.