Standard Deviation

Standard deviation is a measure of how spread out a set of numbers is around their average. A small standard deviation means the numbers cluster tightly near the average; a large one means they scatter widely. In investing it is applied to an asset's returns, where it becomes the standard measure of volatility: the statistic that puts a number on how much a fund or portfolio bounces around from period to period.

It is the same idea reported on statements and in fund fact sheets, almost always annualized and expressed as a percentage. A stock fund might show a standard deviation of 15% a year while a bond fund shows 4%, and the plain meaning is that the stock fund's yearly returns have historically strayed much further from their own average than the bond fund's have. Standard deviation is the statistic; volatility is the property it measures, which is why the two words are often used as if they were one.

Standard deviation is symmetric, and that is its most important limitation. It measures distance from the average in either direction, so a year far above the average adds to it exactly as much as a year equally far below. To the arithmetic, an unexpectedly good year and an unexpectedly bad one are the same kind of event. This is why a careful reader treats standard deviation as a measure of movement, not of harm. Most investors do not lie awake worrying about upside surprises, and measures that count only downside dispersion (such as the Sortino ratio's denominator) exist precisely because standard deviation refuses to distinguish the two.

The bell-curve shorthand is useful and slightly dangerous. When returns follow a roughly normal distribution, about 68% of observations fall within one standard deviation of the average and about 95% within two. So a fund averaging 8% with a 15% standard deviation would, under that assumption, spend roughly two years in three between −7% and +23%. The danger is that real investment returns are not perfectly bell-shaped: extreme losses occur more often than a normal distribution predicts, so the true chance of a very bad year is higher than the tidy 68/95 rule suggests. Standard deviation still describes the ordinary range well; it understates the tails.

Two practical points follow. First, standard deviation is only comparable across the same time frame: a daily figure and an annual figure are different numbers for the same asset, and comparing them is meaningless. Second, it is entirely backward-looking. It summarizes how an investment has behaved, not how it will behave, and an asset can post a low standard deviation for years and then move violently, which is one way a risk-by-past-volatility measure can lull an investor.

Standard deviation is built from how far each observation sits from the average. Take a simple case: an investment returns 10%, then −2%, then 18% over three years. The average is (10 − 2 + 18) ÷ 3 = 26 ÷ 3, or about 8.67%.

Now measure each year's distance from that average and square it: about (1.33)² = 1.78, (−10.67)² = 113.8, and (9.33)² = 87.1. The average of those squared distances is (1.78 + 113.8 + 87.1) ÷ 3 = about 67.6, and the standard deviation is the square root of that, roughly 8.2 percentage points.

So this investment averaged about 8.67% a year and typically strayed around 8 points from that average, which is a wide spread for so few observations. The squaring is what makes the measure penalize large deviations more heavily than small ones, and the final square root is what returns the answer to the same units (percentage points) as the returns themselves. Real calculations use many more observations, but the machinery is identical. All figures are illustrative.

Pros

• It is the standard, widely reported measure of an investment's volatility, so it makes funds and portfolios directly comparable on the same time frame.
• It is a single, intuitive number: bigger means bouncier.
• It feeds directly into other tools, including the Sharpe ratio, which divides excess return by standard deviation.

Cons

• It treats upside and downside swings identically, so it measures movement rather than the losses investors actually fear.
• It assumes a roughly normal distribution and therefore understates the likelihood of extreme losses, which occur more often than the model predicts.
• It is backward-looking; a low past figure is no promise of a calm future.
• It is only comparable across matching time frames, so a daily and an annual figure cannot be set side by side.