href="#x13_x_13_i62">(4.13) 
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 the two assets move independently of each other, their covariance will be zero. Thus, covariance is a statistical measure of the extent to which two variables move together. The formula for covariance is similar to that for variance, except that instead of squaring the deviations of one variable, such as the returns of fund i, the formula cross multiplies the contemporaneous deviations of two different variables, such as the returns of funds i and j:
where Ri is the return of fund i, μi is the expected value or mean of Ri, Rj is the return of fund j, and μj is the expected value or mean of Rj.
The covariance is the expected value of the product of the deviations of the returns of the two funds. Covariance can be estimated from a sample using Equation 4.16:
where Rit is the return of fund i in time t, and
The estimation of the covariance for a sample of returns from a market index fund and a real estate fund is shown in Exhibit 4.2. Column 8 multiplies the fund's deviation from its mean return by the index's deviation from its mean return. Each of the products of the deviations is then summed and divided by n − 1, where n is the number of observations. The result is the estimated covariance between the returns over the sample period, shown near the bottom right-hand corner of Exhibit 4.2.
Exhibit 4.2 Covariance, Correlation, and Beta
Source: Bloomberg.
Because covariance is based on the products of individual deviations and not squared deviations, its value can be positive, negative, or zero. When the return deviations are in the same direction, meaning they have the same sign, the cross product is positive; when the return deviations are in opposite directions, meaning they have different signs, the cross product is negative. When the cross products are summed, the resulting sum generates an indication of the overall tendency of the returns to move either in tandem or in opposition. Note that the table method illustrated in Exhibit 4.2 simply provides a format for solving the formula, which can be easily solved by software. Covariance is used directly in numerous applications, such as in the classic portfolio theory work of Markowitz.
4.3.2 Correlation Coefficient
A statistic related to covariance is the correlation coefficient. The correlation coefficient (also called the Pearson correlation coefficient) measures the degree of association between two variables,
