## Quantitative Methods

Skew

Skew, or skewness, can be 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 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).

#### Kurtosis

Kurtosis is any measure of the "peakedness" of the probability distribution of a real-valued random variable. In a similar way to the concept of skewness, kurtosis is a descriptor of the shape of a probability distribution and, just as for skewness, there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population. There are various interpretations of kurtosis, and of how particular measures should be interpreted; these are primarily peakedness (width of peak), tail weight, and lack of shoulders (distribution primarily peak and tails, not in between).One common measure of kurtosis, originating with Karl Pearson, is based on a scaled version of the fourth moment of the data or population, but it has been argued that this really measures heavy tails, and not peakedness. For this measure, higher kurtosis means more of the variance is the result of infrequent extreme deviations, as opposed to frequent modestly sized deviations. It is common practice to use an adjusted version of Pearson's kurtosis, the excess kurtosis, to provide a comparison of the shape of a given distribution to that of the normal distribution. Distributions with negative or positive excess kurtosis are called platykurtic distributions or leptokurtic distributions respectively.Alternative measures of kurtosis are: the L-kurtosis, which is a scaled version of the fourth L-moment; measures based on 4 population or sample quantiles.These correspond to the alternative measures of skewness that are not based on ordinary moments.#### 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).

#### Skew And Kurtosis