Statistics Fundamentals: Symmetry, Skewness, and Kurtosis in Return Distribution

The normal distribution plays a significant role in “mean-variance” models, and it has the following characteristics:

• Mean is equal to the median
• It is completely described by two parameters (mean and variance)
• 68% of observations lie between (+/-) 1 standard deviation, 95% between (+/-) 2 standard deviations, and 99% between (+/-) 3 standard deviations

Skew or skewness refers to the distribution being not symmetrical. The skews could either be:

• Right or positive, where (a) the tail is depicted on the right, (b) mode < median < mean, and (c) it has frequent small losses and infrequent high gains
• Left or negative, where (a) the tail is depicted on the left, (b) the mean<median<mode, and (c) it has frequent large gains and infrequent small losses. Please note investors prefer positive skews because although frequent downside, the mean return falls above the median
• Sample Skewness Formula: {[n/(n-1)*(n-2)] * ∑(Xi – Xbar)3 /s3}; (Cubing preserves the sign of deviation from the mean, while making it a scale free measure). As a friendly reminder, long tail to the right means positive skewness where long tail to the left means negative skewness

Kurtosis measures the sharpness of the peaks (of the distribution). The following are quick charateristics of the degrees of Kurtosis. Higher the kurtosis, higher the deviation.

• Mesokurtic - Defined by a standard normal peak, resembling the bell curve (typically kurtosis = 3) the most from the other two forms
• Leptokurtic - Defined by a more skinny and higher peak, with fatter tails. This is best used for T-Distributions and has a greater kurtosis than "Mesokurtic"
• Platykurtic - Defined by a more wider and smaller peak, with more skinny tails. It has a smaller kurtosis than "Mesokurtic"