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. Trading

    Trading With Gaussian Models Of Statistics

    The entire study of statistics originated from Gauss and allowed us to understand markets, prices and probabilities, among other applications.
  2. Investing

    What's Skewness?

    Skewness describes how a data distribution leans.
  3. Investing

    Quantitative Analysis Of Hedge Funds

    Hedge fund analysis requires more than just the metrics used to analyze mutual funds.
  4. 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.
  5. Investing

    Find The Right Fit With Probability Distributions

    Discover a few of the most popular probability distributions and how to calculate them.
  6. Investing

    What is Descriptive Statistics?

    Descriptive statistics is the term applied to meaningful data analysis.
  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. Investing

    A Simplified Approach To Calculating Volatility

    Volatility is sometimes greater than anticipated, but the way it’s measured can compound the problems that occur when it’s unexpected.
Frequently Asked Questions
  1. Where else can I save for retirement after I max out my Roth IRA?

    The first option to explore is to determine if you can contribute to a 401(k), 403(b), or 457 plan at work. If your employer ...
  2. How did George Soros "break the Bank of England"?

    In Britain, Black Wednesday (September 16, 1992) is known as the day that speculators broke the pound. They didn't actually ...
  3. What counts as "debts" and "income" when calculating my debt-to-income (DTI) ratio?

    It's important to know your debt-to-income ratio because it's the figure lenders use to measure your ability to repay the ...
  4. Who are Monsanto's main competitors?

    Learn about Monsanto Company's two main operating divisions and its main competitors within each sector, including The Mosaic ...
Trading Center