Measuring Volatility

Volatility is most commonly measured using the standard deviation of the returns of the investment over a given period (and this implies normal distribution which is not the case in reality). A higher standard deviation indicates a higher volatility, meaning the price of the asset can change dramatically in a short period in either direction.

Where:

• $\sigma$ - standard deviation of returns
• $T$ number of periods in the time horizon

For example, monthly closing prices for a stock changed from $100 to $109 over 10 months.

1.

Calculate mean value $\mu =(100+101+102+103+104+105+106+107+108+109)÷10=104.5$

2.

Calculate sum of squared deviations ${(100-\mu )}^2+{(101-\mu )}^2\dots {(109-\mu )}^2=82.5$

3.

Calculate variance as a mean of deviations ${\sigma }^2=82.5÷10=8.25$. In Excel/Google Spreadsheets this is VAR.P(). Note, if we would use only subset of samples, we would divide $V\mathit{ariance}=82.5÷9\approx 9.167$, which in Excel/Google Spreadsheets is VAR() or VAR.S().

4.

Standard deviation as square root of variance $\sigma =\sqrt{8.25}\approx 2.87$

If prices are sampled from a normal distribution, approximately 68% of the data values will fall within one standard deviation from the mean. About 95% of the values will be within two standard deviations (2 $\times$ 2.87 in this example), and 99.7% will fall within three standard deviations (3 $\times$ 2.87).

A large standard deviation indicates that the data points can spread far from the mean and a small standard deviation indicates that they are clustered closely around the mean.

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 another measure of volatility combining multiple indicators.

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 are crucial in distinguishing between funds that achieve higher returns through wise investment decisions versus those that take excessive risks.