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 and equal to zero means it's 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 would mean frequent small negative outcomes, and 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 less than the median (the point where 50% are above/50% below), which is 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 à median à mode. For positive skew, they are separated with a greater than sign, for negative, less than.

Kurtosis
Kurtosis refers to the degree of peak in a distribution. More peak than normal (leptokurtic) means that a distribution also has fatter tails and that there are more chances of extreme outcomes compared to a normal distribution.

The kurtosis formula measures the degree of peak. Kurtosis equals three for a normal distribution; excess kurtosis calculates and expresses kurtosis above or below 3.

In figure 2.5 below, the solid line is the normal distribution; the dashed line is leptokurtic distribution.

Figure 2.5: Kurtosis

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 lesser risk of extreme outcomes).



Basic Probability Concepts

Related Articles
  1. Investing

    What's Skewness?

    Skewness describes how a data distribution leans.
  2. Investing

    What a Normal Distribution Means

    Normal distribution describes a symmetrical data distribution, where most of the results lie near the mean.
  3. Investing

    A Simplified Approach To Calculating Volatility

    Though most investors use standard deviation to determine volatility, there's an easier and more accurate way of doing it.
  4. Investing

    3 Ways To Evaluate the Performance of Alternatives

    Learn about three ways to measure the performance of alternative investments. See how the commonly used Sharpe ratio has drawbacks in measuring volatility.
  5. Investing

    What is Descriptive Statistics?

    Descriptive statistics is the term applied to meaningful data analysis.
  6. Insights

    Be Afraid! Volatility & Fear Measures Back to Brexit Levels

    Safe-haven assets on the rise as volatility increases and geopolitical tensions grow.
  7. Investing

    The Uses And Limits Of Volatility

    Check out how the assumptions of theoretical risk models compare to actual market performance.
  8. Investing

    Understanding Statistics

    Statistics provide the means to analyze data and then summarize it into a numerical form.
  9. Trading

    Ratio Writing: A High-Volatility Options Strategy

    Selling a greater number of options than you buy profits from a decline back to average levels of implied volatility.
Frequently Asked Questions
  1. How does the number of credit card accounts I have affect my credit score?

    Your credit score, which is also referred to as your FICO score, is a measure that creditors use to assess your potential ...
  2. What is the 'three-legged stool'?

    The "three-legged stool" was a retirement terminology from the past that many financial planners used to describe the three ...
  3. If I am looking to get an investment banking job, what education do employers prefer? MBA or CFA?

    If you are looking specifically for an investment banking position, an MBA may be marginally preferable over the CFA. The ...
  4. Is a Simplified Employee Pension (SEP) IRA tax deductible?

    Learn everything you need to know about your SEP IRA, including the benefits to employers and whether or not a SEP IRA is ...
Trading Center