4.2.3 Skewness
The third central moment is the expected value of a variable's cubed deviations:
(4.10)
A problem with the third central moment is that it is generally affected by the scale. Thus, a distribution's third central moment for a variable measured in daily returns differs dramatically if the daily returns are expressed as annualized returns. To provide this measure with a more intuitive scale, investment analysts typically use the standardized third moment (the relative skewness or simply the skewness). The skewness is equal to the third central moment divided by the standard deviation of the variable cubed and serves as a measure of asymmetry:
Skewness is dimensionless, since changes in the scale of the returns affect the numerator and denominator proportionately, leaving the fraction unchanged. By cubing the deviations, the sign of each deviation is retained because a negative value cubed remains negative. Further, cubing the deviations provides a measure of the direction in which the largest deviations occur, since the cubing causes large deviations to be much more influential than the smaller deviations. The result is that the measure of skewness in Equation 4.11 provides a numerical measure of the extent to which a distribution flares out in one direction or the other. A positive value indicates that the right tail is larger (the mass of the distribution is concentrated on the left side), and a negative value indicates that the left tail is larger (the mass of the distribution is concentrated on the right side). A skewness of zero can result from a symmetrical distribution, such as the normal distribution, or from any other distribution in which the tails otherwise balance out within the equation. The top illustration of Exhibit 4.1 depicts negatively skewed, symmetric, and positively skewed distributions.
Exhibit 4.1 Skewness and Kurtosis
4.2.4 Excess Kurtosis
The fourth central moment is the expected value of a variable's deviations raised to the fourth power:
(4.12)
As with the third central moment, a problem with the fourth central moment is that it is difficult to interpret its magnitude. To provide this measure with a more intuitive scale, investment analysts do two things. First, they divide the moment by the standard deviation of the variable raised to the fourth power (to make it dimensionless):
The resulting measure, known as kurtosis, is shown in Equation 4.13 and serves as an indicator of the peaks and tails of a distribution. In the case of a normally distributed variable, the estimated kurtosis has a value that approaches 3.0 (as the sample size is increased). The second adjustment that analysts often perform to create a more intuitive measure of kurtosis is to subtract 3.0 from the result to derive a measure, known as excess kurtosis. Excess kurtosis provides a more intuitive measure of kurtosis relative to the normal distribution because it has a value of zero in the case of the normal distribution:
(4.14)
Since 3.0 is the kurtosis of a normally distributed variable, after subtracting 3.0 from the kurtosis, a positive excess kurtosis signals a level of kurtosis that is higher than observed in a normally distributed variable, an excess kurtosis of 0.0 indicates a level of kurtosis similar to that of a normally distributed variable, and a negative excess kurtosis signals a level of kurtosis that is lower than that observed in a normally distributed variable.
Kurtosis is typically viewed as capturing the fatness of the tails of a distribution, with high values of kurtosis (or positive values of excess kurtosis) indicating fatter tails (i.e., higher probabilities of extreme outcomes) than are found in the case of a normally distributed variable. Kurtosis can also be viewed as indicating the peakedness of a distribution, with a sharp, narrow peak in the center being associated with high values of kurtosis (or positive values of excess kurtosis).
In summary, the mean, variance, skewness, and kurtosis of a return distribution indicate the location and shape of a distribution, and are often a key part of measuring and communicating the risks and rewards of various investments. Familiarity with each can be a critical component of a high-level understanding of the analysis of alternative investments.
4.2.5 Platykurtosis, Mesokurtosis, and Leptokurtosis
The level of kurtosis is sufficiently important in analyzing alternative investment returns that the statistical descriptions of the degree of kurtosis and the related terminology have become industry standards. If a return distribution has no excess kurtosis, meaning it has the same kurtosis as the normal distribution, it is said to be mesokurtic, mesokurtotic, or normal tailed, and to exhibit mesokurtosis. The tails of the distribution and the peakedness of the distribution would have the same magnitude as the normal distribution.
The middle illustration in Exhibit 4.1 depicts that kurtosis can be viewed by the fatness of the tails of a distribution. If a return distribution has negative excess kurtosis, meaning less kurtosis than the normal distribution, it is said to be platykurtic, platykurtotic, or thin tailed, and to exhibit platykurtosis. If a return distribution has positive excess kurtosis, meaning it has more kurtosis than the normal distribution, it is said to be leptokurtic, leptokurtotic, or fat tailed, and to exhibit leptokurtosis.
The bottom illustration in Exhibit 4.1 depicts leptokurtic, mesokurtic, and platykurtic distributions. A leptokurtic distribution (positive excess kurtosis) with fat tails and a peaked center is illustrated on the left. A platykurtic distribution (negative excess kurtosis) with thin tails and a rounded center is illustrated on the right. In the middle is a normal mesokurtic distribution (no excess kurtosis). The key to recognizing excess kurtosis visually is comparing the thickness of the tails of both sides of the distribution relative to the tails of a normal distribution.
4.3 Covariance, Correlation, Beta, and Autocorrelation
An important aspect of a return is the way that it correlates with other returns. This is because correlation affects diversification, and diversification drives the risk of a portfolio of assets relative to the risks of the portfolio's constituent assets. This section begins with an examination of covariance, then details the correlation coefficient. Much as standard deviation provides a more easily interpreted alternative to variance, the correlation coefficient provides a scaled and intuitive alternative to covariance. Finally, the section discusses the concepts of beta and autocorrelation.
4.3.1 Covariance
The covariance of the return of two assets is a measure of the degree or tendency of two variables to move in relationship with each other. If two assets tend to move in the same direction, they are said to covary positively, and they will have a positive covariance. If the two assets tend to move in opposite directions, they are said to covary negatively, and they will have a negative covariance. Finally, if