Standard deviation of returns

The calculation takes each period's return, measures its distance from the average, squares those distances, averages them, then takes the square root. When the returns in hand are a sample rather than the whole history, that averaging step divides by one less than the count, which is what spreadsheet and portfolio software reports by default. Squaring stops the gaps above and below the mean cancelling each other out, and it makes one large miss count for more than several small ones. The square root at the end puts the answer back into the same unit as the returns, which is why it is quoted in percent rather than in the percent-squared unit of variance.

It is the standard input wherever risk needs a number: the denominator of the Sharpe ratio, the per-asset term that combines with correlation to build the covariance matrix portfolio maths runs on, and the figure people mean when they say volatility. Periods convert by the square root of time, so a monthly figure is annualised by multiplying by $\sqrt{12}$. If returns followed a normal distribution, roughly two thirds of them would land within one standard deviation of the average and about 95 percent within two.

The mistake is taking it as a full description of risk. It leans on two assumptions markets do not honour. The first is that returns are normally distributed, when real returns have fatter tails, so the move the model calls a once-in-a-lifetime event turns up far more often than that. The second is that an upward deviation deserves the same weight as a downward one, which is not how anyone experiences their own portfolio. It also says nothing about the order returns arrive in, and that order is exactly what drawdown captures.

This is not a finance-specific measure. It is the ordinary statistical standard deviation applied to a series of returns, calculated by the same steps taught in a statistics course: standard deviation.