4.1 Skewness

Skewness, [latex]S[/latex], measures the degree of asymmetry of a probability distribution. Analytically, it is given by

Skewness can take on both positive and negative values because we raise deviations from the mean to power 3. The skew for a normal distribution is 0 because of the symmetry of the bell curve and the normal distribution is commonly used as a benchmark for comparison. Negative skewness indicates a long left tail while positive skewness indicates a long right tail.

The three stocks have the same mean and standard deviation, implying the same risk-return characteristics. If the standard deviation is an adequate measure of risk in this context, you should be indifferent about which stock you buy. In all likelihood, however, you are not indifferent. If you invest in the Negatively Skewed stock depicted in green in the top panel of Figure 7, you are more likely to observe extreme negative returns compared to the Normal stock and less likely to observe extreme positive returns. The standard deviation understates downside risk when returns are negatively skewed. The opposite would be true for the Normal and Positively Skewed stocks depicted in the bottom panel of Figure 7. Whenever a random variable is not normally distributed, the standard deviation is not an adequate measure of risk, and you would like to consider measures such as skewness and kurtosis in making decisions.