5. Variance and standard deviation: the exact arithmetic of volatility

Level 1: Probability: the language of uncertainty

A -5% day does not weigh 5 times more than a -1% day in the volatility calculation. It weighs 25 times more.

That detail is not a quirk of the formula. It is the reason volatility reacts so strongly to a handful of extreme days.

The market theory course introduced volatility intuitively, as how much an asset's prices move. This module gives that idea precise form.

What variance is, exactly

The variance of a random variable is defined as the average of the squared deviations between each possible value and the mean — the expectation — of that variable.

Each squared deviation is weighted by its probability, exactly as was done when computing the expectation in the previous module.

Why the deviations are squared

It is worth understanding why the deviations are squared instead of simply averaged.

If you take the differences between each value and the mean and average them as they are, the result is always zero. By the very definition of the mean, positive and negative deviations cancel each other out exactly.

Squaring does two things at once:

• It removes the sign, avoiding that cancellation.
• It gives proportionally more weight to large deviations than to small ones.

And that second effect is stronger than it looks. A 5% one-day drop contributes 25 times more to the variance than a 1% drop, not merely 5 times more. The reason is that 5 squared is 25 and 1 squared is 1.

Something worth remembering follows from this: variance is especially sensitive to extreme moves, not just to typical day-to-day dispersion.

From variance to standard deviation

Standard deviation is defined as the square root of the variance.

The reason for that extra step is units. Squaring leaves the original units squared as well, percent squared for instance, a quantity with no direct intuitive meaning.

Taking the square root brings the result back into the variable's original units, percentage return for instance.

That is why it is standard deviation, and not variance itself, that is the figure quoted in practice when people talk about volatility.

The calculation, step by step

A numerical example helps to nail down the procedure. Take a stock with these five daily returns: +2%, -1%, 0%, -3%, +2%.

• The mean is (2 - 1 + 0 - 3 + 2) / 5 = 0 / 5 = 0%.
• The deviations from the mean are therefore 2, -1, 0, -3 and 2.
• Squaring them gives 4, 1, 0, 9 and 4, which add up to 18.
• The variance is 18 / 5 = 3.6, in units of percent squared.
• The standard deviation is the square root of 3.6, roughly 1.90%.

Annualizing: why 16 and not 252

There is a very widespread convention for annualizing a daily standard deviation. It is worth understanding exactly why it works the way it does rather than in some more naive way.

The key is in a single sentence: when daily returns are reasonably independent of one another, it is the variances that add up across days, not the standard deviations.

So to go from a daily standard deviation to an annual one, the procedure is:

• First multiply the daily variance by the approximate number of sessions in a year, about 252.
• Then take the square root of the result.

That is mathematically equivalent to multiplying the daily standard deviation directly by the square root of 252, roughly 15.87.

That is exactly why the annualized volatility figures you normally see are on the order of 16 times the daily volatility, not 252 times.

The two assumptions behind the shortcut

This annualization shortcut, however, drags along two assumptions worth keeping in mind at all times:

• That returns from one day to the next are reasonably independent. That is the exact subject of the next module.
• That the level of volatility stays relatively stable across the whole period considered.

Both assumptions are frequently violated in real markets. Volatility clusters in time and shifts regime abruptly.

Hence a practical warning: an annualized volatility figure computed over a period that includes an abrupt regime change can be quite misleading.

Standard deviation and risk

Standard deviation connects directly to the notion of risk. The larger it is, the more spread out the possible outcomes are around the mean.

Two strategies can have exactly the same positive expectancy, the one from the previous module, and still have very different standard deviations.

For equal expectancy, the one with the lower standard deviation is usually preferable. Its results are more predictable, and the risk of going through a damaging losing streak before the favorable average shows up is lower.

In short, standard deviation is literally the formal, computable version of the word volatility used intuitively up to now. Every time the market theory course claimed that one stock is more volatile than another, this module supplies the exact arithmetic behind that claim, number by number.