Variance and Volatility of Returns

In order to examine the risk of an investment, we measure its variability in a distribution.

Variance and Standard Deviation:

• Measure the spread in returns
• Measure how far actual returns deviate from the mean
• Measure the volatility of asset returns

Variance estimate using realized returns:

$$Var(R)=\frac{1}{T-1}\sum _{t=1}^T(R_t-\bar{R}{)}^2$$

Standard deviation is the square root of variance

$$SD(R)=\sqrt{Var(R)}$$

• Expressed in percentage
• The larger the number, the greater the uncertainty of returns

Example

Using the following data, what is the standard deviation of the S&P 500’s returns for the year’s 2005-2009?

We have:

$$\begin{array}{rl}Var(R)& =\left(\frac{1}{5-1}\right)(3.1+160+5.57+1611+545.7)=581.46\\ SD(R)& =\sqrt{581.46}=24.11\end{array}$$