1. Volatility and standard deviation: measuring risk with numbers

Theory 4: Risk, return and diversification

There is an index they call 'the fear index'. It spikes just when the market has been calm for weeks. And that apparent contradiction is exactly what it measures.

In the first level of this course we already used the word volatility intuitively: how much an asset's price moves. It is time now to give it a precise definition.

Measuring risk with numbers is the indispensable first step before comparing assets, building a portfolio, or evaluating a strategy with something more solid than an impression.

Standard deviation: dispersion in a single number

The most common measure of volatility is the standard deviation of an asset's returns over a given period.

Without going into the exact formula, the idea is straightforward:

• You take the asset's daily returns — or weekly, or monthly — over, say, the past year.
• You work out how far each individual return sits from the average of all of them.
• You aggregate that dispersion into a single number.

The result reads easily. An asset whose daily returns always hover around similar values, with little dispersion, has a low standard deviation. An asset whose daily returns swing wildly from one day to the next, with a lot of dispersion, has a high standard deviation.

Why it is annualized

That figure, computed on daily returns, is usually annualized. It is multiplied by the square root of 252, the approximate number of trading days in a year.

The reason is to be able to compare assets and strategies on a common and more intuitive scale. Talking about '20% annualized volatility' lets you compare stocks, indices or portfolios directly against one another. Comparing isolated daily deviations does not allow that with anything like the same ease.

What has happened versus what is expected

It is worth distinguishing two ways of arriving at this figure.

• Historical volatility (or realized) is computed, as we have just seen, by looking backwards: how much the asset has in fact moved in the recent past.
• Implied volatility is extracted from the current prices of options on that asset. It represents how much volatility the market expects in the future, not how much there has been in the past.

A frequently cited example is the VIX index, calculated from options on the S&P 500 and often nicknamed 'the fear index'.

There is the scene we opened with. The VIX tends to spike when the market anticipates turbulence — just before an important central bank decision, for instance — even if recent historical volatility had been low up to that point.

The gap between the two measures, what has happened versus what is expected to happen, is one of the signals many professional managers use to gauge the market's appetite for risk at any given moment.

What standard deviation cannot see

It is equally important to understand what standard deviation on its own does not capture well.

Being a symmetric measure, it treats a run of sharp gains exactly like a run of sharp losses of the same size. And for most investors those two situations do not feel remotely equally 'risky'.

There is a second problem, a more technical one. Many financial assets, especially during periods of stress, display what is known as fat tails: extreme moves that occur far more often than a perfectly symmetric 'normal' statistical distribution would predict. That distribution is precisely the one the standard calculation implicitly assumes.

This is one of the most serious and persistent criticisms of using volatility as the sole measure of risk. It is why sophisticated managers supplement standard deviation with other measures, rather than trusting a single number:

• The maximum expected loss in a reasonable worst case.
• The maximum drawdown, which we will meet later in this course.

Why it is still the starting point

Even so, and despite its limitations, standard deviation remains the almost universal starting point for talking about risk in finance.

The reason is practical. It is easy to compute, easy to compare across assets, and captures the central intuition reasonably well: how far an asset's actual behavior strays from what would be its 'typical' or expected behavior.

Beta: volatility, but relative

A related but distinct measure is beta. Instead of measuring an asset's absolute volatility, it measures its volatility relative to the broad market, typically an index such as the S&P 500.

• A beta of 1 indicates that the asset tends, on average, to move in line with the market.
• A beta of 1.5 indicates that it amplifies market moves by 50%. If the index rises 2%, the asset tends to rise 3%, and likewise on the way down.
• A beta of 0.5 indicates that it dampens those moves by half.

Beta is particularly useful for two things. The first is understanding how much of an asset's risk is systematic risk. That is the risk it shares with the market as a whole, and which, as we will see in the next module, cannot be eliminated by diversifying. The second is anticipating how a portfolio is likely to behave in a broad market move.

With a warning identical to the one for standard deviation: it is a measure computed on the past, and it guarantees no identical behavior in the future.