Any distribution has several moments. The moments of a distribution characterize its shape. The moments are the weighted averages of the deviations from the mean, elevated at power 2, 3, 4, etc., using the discrete probabilities of discrete values or the probability density as weights.

Expectations

The first moment is the expectation, or mean, of the function. The second moment is the variance. It characterizes dispersion around the mean. The square root of the variance is the standard deviation. It is identical to the "volatility." The third moment is skewness, which characterizes departure from symmetry. The fourth moment is kurtosis, which characterizes the flatness of distribution.

The expectation of a random variable XhE(X), where E represents the expectation operator. The expectation is the weighted average of all values using probabilities as weights. For discrete distributions, the probability assigned to a value x of random variable X is PfX= x.), or p.

For continuous distributions with density fix), the expectation is:

When using time series, it is common to assign to each observation an identical probability specifically — 1/n when there are n observations:

The mean is a probability weighted average of all possible values.

Variance and Volatility

The variance a2, where a is the volatility, is the sum of the probability weighted squared deviations to the mean. The volatility o is the square root of the variance. The probability of occurrence of x of random variable Xis P(X— x), or p..

The volatility characterizes the dispersion around the mean, named E(X) — [I. It is a convenient measure of risk because it measures the magnitude of possible fluctuations around the mean. With discrete distributions:

For continuous distributions with density fix), the variance is:

The standard deviation o is the square root of variance:

When using time series, it is common to assign to each observation an identical probability p — lln when there are n observations. In general: C2(X) — Ji.p [x - E(X)]2. A useful property facilitating the calculation of the variance is that:

Another property of variance is that it is scaled by a constant, using the square of the constant a2:

This implies that the volatility is also multiplied by the constant a: o(aX) — ac(X). 10.3.3 Skewness and Kurtosis

In general the k central moment of a distribution is the expectation of the deviation from the mean, with power k:

The expectation is the first moment and measures the central tendency. The variance is the second moment and measures the dispersion around the expectation. The third and fourth moments are central moments divided respectively by moment is a3 and a4. The skewness is:

The kurtosis is:

The skewness is 0 and the kurtosis is 3 for the standard normal distribution. The skewness measures the symmetry of the distribution. The kurtosis measures how thick are the tails of the distribution and how narrow is the central section of the distribution. Excess kurtosis is the kurtosis minus 3, and serves for comparing the shape of a distribution to the normal curve. Figure 10.4 illustrates the effect of skewness and excess kurtosis compared to a normal distribution.

FIGURE 10.4 Effect of skewness and excess kurtosis compared to a normal distribution

In finance, kurtosis of periodical returns on a stock, for instance, measured by the relative change of value over a small interval of time t and t + At, or (Vt+At - V)IVt, can arise from "jumps" in the stock prices, or from serial dependence of consecutive returns. Many jumps make extreme values more frequent. Serial dependence means that high returns tend to be followed by large higher returns (in absolute value, either positive or negative), creating higher frequencies of large returns than if returns were independent.

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