## Skew

Skew, or skewness, can be mathematically defined as the averaged cubed deviation from the mean divided by the standard deviation cubed. If the result of the computation is greater than zero, the distribution is positively skewed. If it's less than zero, it's negatively skewed. If equal to zero, it's roughly symmetric. For interpretation and analysis, focus on downside risk. Negatively skewed distributions have what statisticians call a long left tail (refer to graphs on previous page), which for investors can mean a greater chance of extremely negative outcomes. Positive skew implies a long right tail, which for investors can mean a greater chance of extremely positive outcomes. With a positively skewed return distribution, extremely bad scenarios are not as likely.

A nonsymmetrical or skewed distribution occurs when one side of the distribution does not mirror the other. Applied to investment returns, nonsymmetrical distributions are generally described as being either positively skewed (meaning frequent small losses and a few extreme gains) or negatively skewed (meaning frequent small gains and a few extreme losses).

 Positive Skew Negative Skew

 Figure 2.4

For positively skewed distributions, the mode (point at the top of the curve) is typically less than the median (the point where 50% are above/50% below), which is also typically less than the arithmetic mean (sum of observations/number of observations). The opposite rules apply to negatively skewed distribution: mode is greater than median, which is greater than arithmetic mean.

Positive: Mean > Median > Mode Negative: Mean < Median < Mode

Notice that by alphabetical listing, it's mean > median > mode. For positive skew, they are separated with a “greater than” sign, for negative, “less than.”

## Kurtosis

While skewness refers to the asymmetry of the tails, Kurtosis refers to the tails in general. Leptokurtic distributions have one or both tails fatter than normal distribution, impying a higher likelihood of extreme returns (e.g., returns three of more standard deviations on one or both sides of the mean) than the normal distribution predicts.

Kurtosis equals three for a normal distribution; excess kurtosis calculates and expresses kurtosis above or below three.

In figure 2.5 below, the green line is the density estimate from normally distributed data, and the blue line is the density estimate from data that arise from a leptokurtic distribution. Leptokurtosis, or heavy tailedness, is only apparent when you zoom in on the tail (right panel).

Figure 2.5: Kurtosis (Left panel: Mesokurtic and leptokurtic distributions having the same standard deviation. Right panel: Same as the left panel, but zoomed in on values three and more standard deviations from the mean,showing a heavier tail for the leptokurtic distribution.)

## Sample Skew and Kurtosis

For a calculated skew number (average cubed deviations divided by the cubed standard deviation), look at the sign to evaluate whether a return is positively skewed (skew > 0), negatively skewed (skew < 0) or symmetric (skew = 0). A kurtosis number (average deviations to the fourth power divided by the standard deviation to the fourth power) is evaluated in relation to the normal distribution, on which kurtosis = 3. Since excess kurtosis = kurtosis - 3, any positive number for excess kurtosis would mean the distribution is leptokurtic (meaning fatter tails and greater risk of occasional extreme outcomes than the normal distribution would predict).

Basic Probability Concepts

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