Skewness risk in forecasting models utilized in the financial field is the risk that results when observations are not spread symmetrically around an

average In colloquial, ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean the sum of the numbers divided by ...

value, but instead have a

skewed distribution In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real number, real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For ...

. As a result, the

mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...

and the

median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...

can be different.

Skewness In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimodal ...

risk can arise in any quantitative model that assumes a

symmetric distribution In statistics, a symmetric probability distribution is a probability distribution—an assignment of probabilities to possible occurrences—which is unchanged when its probability density function (for continuous probability distribution) or pro ...

(such as the

normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...

) but is applied to skewed data. Ignoring skewness risk, by assuming that variables are symmetrically distributed when they are not, will cause any model to understate the risk of variables with high skewness. Skewness risk plays an important role in

hypothesis testing A statistical hypothesis test is a method of statistical inference used to decide whether the data provide sufficient evidence to reject a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic. T ...

. The

analysis of variance Analysis of variance (ANOVA) is a family of statistical methods used to compare the Mean, means of two or more groups by analyzing variance. Specifically, ANOVA compares the amount of variation ''between'' the group means to the amount of variati ...

, one of the most common tests used in hypothesis testing, assumes that the data is normally distributed. If the variables tested are not normally distributed because they are too skewed, the test cannot be used. Instead, nonparametric tests can be used, such as the Mann–Whitney test for an unpaired situation or the

sign test The sign test is a statistical test for consistent differences between pairs of observations, such as the weight of subjects before and after treatment. Given pairs of observations (such as weight pre- and post-treatment) for each subject, the sign ...

for a paired situation. Skewness risk and

kurtosis risk In statistics and decision theory, kurtosis risk is the risk that results when a statistical model assumes the normal distribution, but is applied to observations that have a tendency to occasionally be much farther (in terms of number of standar ...

also have technical implications in calculation of

value at risk Value at risk (VaR) is a measure of the risk of loss of investment/capital. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically us ...

. If either are ignored, the value at risk calculations will be flawed.

Benoît Mandelbrot Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of #Fractals and the ...

, a French mathematician, extensively researched this issue. He feels that the extensive reliance on the normal distribution for much of the body of modern finance and

investment theory Investment is traditionally defined as the "commitment of resources into something expected to gain value over time". If an investment involves money, then it can be defined as a "commitment of money to receive more money later". From a broade ...

is a serious flaw of any related models (including the Black–Scholes model and CAPM). He explained his views and alternative finance theory in a book: ''The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin and Reward''. In options markets, the difference in implied volatility at different strike prices represents the market's view of skew, and is called volatility skew. (In pure Black–Scholes, implied volatility is constant with respect to strike and time to maturity.)

Skewness for bonds

Bonds have a skewed return. A bond will either pay the full amount on time (very likely to much less likely depending on quality), or less than that. A normal bond does not ever pay ''more'' than the "good" case.

References

* Mandelbrot, Benoit B., and Hudson, Richard L., ''The (mis)behaviour of markets : a fractal view of risk, ruin and reward'', London : Profile, 2004, {{ISBN, 1-86197-765-4 * Johansson, A. (2005
"Pricing Skewness and Kurtosis Risk on the Swedish Stock Market"
Masters Thesis, Department of Economics, Lund University, Sweden *Premaratne, G., Bera, A. K. (2000). Modeling Asymmetry and Excess Kurtosis in Stock Return Data. Office of Research Working Paper Number 00-0123, University of Illinois Statistical deviation and dispersion Investment Risk analysis Mathematical finance Applied probability

Skewness for bonds

See also

References