Non-normality of the return distribution

Introduction to Portfolio Analysis in R

Kris Boudt

Professor, Free University Brussels & Amsterdam

Volatility describes "normal" risk

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Standard Deviation of portfolio returns:
- Take the full sample of returns
  
  $$SD = \sqrt{\frac{(R_1-\mu)^2 + (R_2-\mu)^2 + ... + (R_T-\mu)^2}{T-1}}$$
Semi-Deviation of portfolio returns:
- Take the subset of returns below the mean
  
  $$ SemiDev = \sqrt{\frac{(Z_1-\mu)^2 + (Z_2-\mu)^2 + ... + (Z_n-\mu)^2}{n}}$$

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Is it symmetric?
- Check the skewness
Are the tails fatter than those of the normal distribution?
- Check the excess kurtosis

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Zero Skewness
- Distribution is symmetric
Negative Skewness
- Large negative returns occur more often than large positive returns

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Zero Skewness
- Distribution is symmetric
Negative Skewness
- Large negative returns occur more often than large positive returns
Positive Skewness
- Large positive returns occur more often than large negative returns

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Introduction to Portfolio Analysis in R