Quantitative Methods - Skew And Kurtosis

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 lesser 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. Career Education & Resources

    How Hard are the CFA Exams?

    Learn about the difficulty of the CFA exams with a description of the tests, some statistics on pass rates and suggestions that can help you pass the exams.
  2. Professionals

    What it Takes to be a Financial Analyst

    A financial analyst researches companies and economic conditions to make business, sector and industry recommendations.
  3. Career Education & Resources

    Financial Analyst: Career Path & Qualifications

    Read about what it takes to become a financial analyst in a corporation or securities firm, and learn how far you can rise in the profession.
  4. Career Education & Resources

    Financial Planner: Career Path & Qualifications

    Learn what education and certifications you need to become a financial planner, as well as the future prospects and earnings potential for financial planners.
  5. Career Education & Resources

    Where to Find Non-Profit Finance Jobs

    The non-profit sector offers a stable selection of jobs for those who seek other types of fulfillment from their jobs than just purely financial.
  6. Career Education & Resources

    Portfolio Manager: Career Path & Qualifications

    Learn about the basic requirements for getting hired as a portfolio manager, and discover how most professionals in the field rise into the position.
  7. Your Practice

    4 Professional Associations Advisors Should Join

    These four professional organizations are among the most respected and well known in the industry.
  8. Professionals

    Equity Research: Career Path and Qualifications

    Find out what equity research analysts do on a day-to-day basis, and learn more about the typical career progression for these securities professionals.
  9. Professionals

    What's on the CFA Level II Exam?

    The Chartered Financial Analyst Level II exam is the second of three tests that CFA candidates must pass.
  10. Professionals

    Financial Data Analyst: Career Path & Qualifications

    Learn more about the career options available to financial data analysts, and determine whether the profession is a good match for you.
RELATED TERMS
  1. Personal Financial Advisor

    Professionals who help individuals manage their finances by providing ...
  2. CFA Institute

    Formerly known as the Association for Investment Management and ...
  3. Chartered Financial Analyst - CFA

    A professional designation given by the CFA Institute (formerly ...
  4. Security Analyst

    A financial professional who studies various industries and companies, ...
RELATED FAQS
  1. What are the differences between a Chartered Financial Analyst (CFA) and a Certified ...

    The differences between a Chartered Financial Analyst (CFA) and a Certified Financial Planner (CFP) are many, but comes down ... Read Full Answer >>
  2. How do I become a Chartered Financial Analyst (CFA)?

    According to the CFA Institute, a person who holds a CFA charter is not a chartered financial analyst. The CFA Institute ... Read Full Answer >>
  3. What types of positions might a Chartered Financial Analyst (CFA) hold?

    The types of positions that a Chartered Financial Analyst (CFA) is likely to hold include any position that deals with large ... Read Full Answer >>
  4. Who benefits the most from prepaid expenses?

    Prepaid expenses benefit both businesses and individuals. Prepaid expenses are the types of expenses that are bought or paid ... Read Full Answer >>
  5. If I am looking to get an Investment Banking job. What education do employers prefer? ...

    If you are looking specifically for an investment banking position, an MBA may be marginally preferable over the CFA. The ... Read Full Answer >>
  6. Can I still pass the CFA Level I if I do poorly in the ethics section?

    You may still pass the Chartered Financial Analysis (CFA) Level I even if you fare poorly in the ethics section, but don't ... Read Full Answer >>
Hot Definitions
  1. Harry Potter Stock Index

    A collection of stocks from companies related to the "Harry Potter" series franchise. Created by StockPickr, this index seeks ...
  2. Liquidation Margin

    Liquidation margin refers to the value of all of the equity positions in a margin account. If an investor or trader holds ...
  3. Black Swan

    An event or occurrence that deviates beyond what is normally expected of a situation and that would be extremely difficult ...
  4. Inverted Yield Curve

    An interest rate environment in which long-term debt instruments have a lower yield than short-term debt instruments of the ...
  5. Socially Responsible Investment - SRI

    An investment that is considered socially responsible because of the nature of the business the company conducts. Common ...
Trading Center