How smoothing on observed volatility, distort the estimated volatility
For simplicity, consider an investment that experiences the following six months of returns: –3%, –2%, –1%, +1%, +2%, and +3%. Since this series has a sample mean of 0%, the sample variance of the series is calculated as:
The sample volatility (or standard deviation) of the monthly return series is 2.37% (rounded).
Now consider the measured volatility if the returns of the best and worst months are changed to +2% and –2% from +3% and –3%. The sample variance of this new series is calculated as:
and the sample volatility is 1.90% (rounded).
If the highest and lowest returns are smoothed, the observed volatility can be substantially reduced. In this example, the observed volatility of the smoothed series is approximately 80% of the size of the unsmoothed series.
Smoothing also affects the measured correlation between returns on different assets. Continuing with the previous example, suppose that a second asset has corresponding actual monthly returns of –5%, –3%, –1%, 1%, 3%, and 5%.
Using un-smoothed returns, the estimated correlation between the two assets is 99.4%. However, if the highest and lowest returns of the first asset are smoothed as described previously, then the measured asset correlation is just 95.8%.