of the variable. With n = 1, Equation 4.1 is the formula for expected value:
(4.2)
The expected value of a variable is the probability weighted average of its outcomes:
(4.3)
where probi is the probability of Ri.
Equation 4.3 expresses the first raw moment in terms of probabilities and outcomes. Using historical data, for a sample distribution of n observations, the mean is typically equally weighted and is estimated by the following:
(4.4)
Thus, Equation 4.4 is a formula for estimating Equation 4.2 using historical observations. The historical mean is often used as an estimate of the expected value when observations from the past are assumed to be representative of the future. Other raw moments can be generated by inserting a higher integer value for n in Equation 4.1. But the raw moments for n > 1 are less useful for our purposes than the highly related central moments.
4.2.2 The Formulas of Central Moments
Central moments differ from raw moments because they focus on deviations of the variable from its mean (whereas raw moments are measured relative to zero). Deviations are defined as the value of a variable minus its mean, or expected value. If an observation exceeds its expected value, the deviation is positive by the distance by which it exceeds the expected value. If the observation is less than its expected value, the deviation is a negative number. Each central moment applies the following equation to the deviations:
where μ = the expected value of R.
The term inside the parentheses is the deviation of R from its mean, or expected value. The first central moment is equal to zero by definition, because the expected value of the deviation from the mean is zero. When analysts discuss statistical moments, it is usually understood that the first moment is a raw moment, meaning the mean, or expected value. But the second through fourth moments are usually automatically expressed as central moments because in most applications the moments are more useful when expressed in terms of deviations.
The variance is the second central moment and is the expected value of the deviations squared, providing an indication of the dispersion of a variable around its mean:
(4.6)
The variance is the probability weighted average of the deviations squared. By squaring the deviations, any negative signs are removed (i.e., any negative deviation squared is positive), so the variance [V(R)] becomes a measure of dispersion. In the case of probability weighted outcomes, this can be written as:
The variance shown in Equation 4.7 is often estimated with a sample of historical data. For a sample distribution, the variance with equally weighted observations is estimated as:
The mean in Equation 4.8,
(4.9)
In investment terminology, volatility is a popular term that is used synonymously with the standard deviation of returns. Other central moments can be generated by inserting a higher integer value for n in Equation 4.5. But the central moments for n = 3 (skewness) and n = 4 (kurtosis) are typically less intuitive and less well-known than their scaled versions. In other words, rather than using the third and fourth central moments, slightly modified formulas are used to generate scaled measures of skewness and kurtosis. These two scaled measures are detailed in the next two sections.
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):
