Measuring Volatility

Volatility is the standard deviation of an asset’s returns — never of its prices. That distinction is the whole of it, and getting it backwards is the most common error in amateur risk analysis. A price series that climbs steadily from $100 to $109 has a large dispersion of price levels around their mean and essentially zero volatility, because every period’s return is identical. Compute the standard deviation of the prices and you will report healthy volatility for the least volatile series imaginable.

So always convert to returns first. For a price series $P_t$, the period return is either simple, $R_t=P_t∕P_{t-1}-1$, or logarithmic, $r_t=\text{ln}<!--\; FUNCTION\; APPLICATION\; -->(P_t∕P_{t-1})$. Log returns are the convention for volatility work because they add across periods, which is what makes the square-root scaling below valid.

Worked example. Take twelve monthly closing prices:

1.

Convert to log returns. $r_1=\text{ln}<!--\; FUNCTION\; APPLICATION\; -->(104∕100)=3.92\%$, $r_2=\text{ln}<!--\; FUNCTION\; APPLICATION\; -->(99∕104)=-4.93\%$, and so on — eleven returns from twelve prices. You always lose one observation.

2.

Take the mean of the returns: $\overline{r}=1.50\%$ per month.

3.

Sum the squared deviations from that mean: $\sum <!--\; FUNCTION\; APPLICATION\; -->{(r_t-\overline{r})}^2=0.02795$.

4.

Divide by $n-1=10$ to get the sample variance: $s^2=0.002795$. Use $n-1$, not $n$, — you are estimating the volatility of a process from a sample instead of describing a fixed population. In Excel or Google Sheets this is VAR.S(); VAR.P() divides by $n$ and will bias your estimate low.

5.

Take the square root: $s=5.29\%$ monthly.

Scaling to another horizon. Volatility quoted for one period converts to another by the square root of time, because variance — not standard deviation — adds across independent periods:

where $\sigma$ is the per-period standard deviation of returns and $T$ is the number of those periods in the target horizon. Annualizing the example above: $5.29\%\times \sqrt{12}=18.3\%$. For daily data the multiplier is $\sqrt{252}$, the approximate count of trading days in a year.

Contrast that 18.3% with what the wrong method gives. The standard deviation of those twelve prices is 5.79 — a number in dollars, not percent, that scales with the share price and answers no question anyone asked.

The rule depends on returns being serially uncorrelated. They are not exactly — volatility clusters, and trends and mean reversion both break the assumption — so treat $\sqrt{T}$ scaling as a convention that is close enough for comparison and wrong in the tails, which is where you needed it to hold.

Reading the number. If returns were normally distributed, roughly 68% of monthly returns would fall within one standard deviation of the mean, 95% within two, and 99.7% within three. Returns are not normally distributed — the fat tails discussed throughout this book mean the three-sigma events arrive far more often than 0.3% of the time — so use those bands as a rough scale for what is ordinary, and never as a probability of what is extreme.

These are tools used to track volatility in particular markets. For example, the Cboe Volatility Index (VIX) measures the stock market’s expectation of volatility implied by S&P 500 index options. CNN’s Fear & Greed Index is a sentiment gauge that counts volatility (via the VIX) as one of its seven inputs — a mood ring, not a measurement.

During periods of economic uncertainty, such as the financial crisis of 2008, volatility tends to increase.

Volatility is a key component of various ratios used to evaluate performance and risk of individual assets and portfolios, and they are what separates funds that earn higher returns through skill from those that simply took more risk.