Skewness risk in

financial modeling Financial modeling is the task of building an abstract representation (a model) of a real world financial situation. This is a mathematical model designed to represent (a simplified version of) the performance of a financial asset or portfolio o ...

is the risk that results when observations are not spread symmetrically around an

average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...

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-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimodal d ...

. As a result, the

mean There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithme ...

and the

median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...

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 d ...

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) ...

(such as the

normal distribution In 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 e^ The parameter \mu ...

) 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 at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...

. The

analysis of variance Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statisticia ...

, 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 unpaired situation or the

sign test The sign test is a statistical method to 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 ...

for 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 for investments. 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 used by ...

. 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 roughness" of phy ...

, 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 the dedication of money to purchase of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort. In finance, the purpose of investing is ...

is a serious flaw of any related models (including the Black–Scholes model and

CAPM CAPM may refer to: * Capital asset pricing model, a fundamental model in finance * Certified Associate in Project Management, an entry-level credential for project managers {{Disambig ...

). 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 In financial mathematics, the implied volatility (IV) of an option contract is that value of the volatility of the underlying instrument which, when input in an option pricing model (such as Black–Scholes), will return a theoretical value equa ...

at different strike prices represents the market's view of skew, and is called

volatility skew Volatility smiles are implied volatility patterns that arise in pricing financial option (finance), options. It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices. In particular ...

. (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