produce positive returns as it is to produce negative returns (the median is zero), but there are more big negative returns than big positive returns (the distribution is skewed), so the mean is less than zero. As a risk manager, understanding the impact of skew on the mean relative to the median and mode can be useful. Be careful, though, as this rule of thumb does not always work. Many practitioners mistakenly believe that this rule of thumb is in fact always true. It is not, and it is very easy to produce a distribution that violates the rule.
Kurtosis
The fourth central moment is similar to the second central moment, in that it tells us how spread-out a random variable is, but it puts more weight on extreme points. As with skewness, rather than working with the central moment directly, we typically work with a standardized statistic. This standardized fourth central moment is known as the kurtosis. For a random variable X, we can define the kurtosis as K, where:
(3.66)
where σ is the standard deviation of X, and μ is its mean.
By standardizing the central moment, it is much easier to compare two random variables. As with skewness, multiplying a random variable by a constant will not change the kurtosis.
The following two populations have the same mean, variance, and skewness. The second population has a higher kurtosis.
Notice, to balance out the variance, when we moved the outer two points out six units, we had to move the inner two points in 10 units. Because the random variable with higher kurtosis has points further from the mean, we often refer to distribution with high kurtosis as fat-tailed. Figures 3.5 and 3.6 show examples of continuous distributions with high and low kurtosis.
FIGURE 3.5 High Kurtosis
FIGURE 3.6 Low Kurtosis
Like skewness, kurtosis is an important concept in risk management. Many financial assets exhibit high levels of kurtosis. If the distribution of returns of two assets have the same mean, variance, and skewness, but different kurtosis, then the distribution with the higher kurtosis will tend to have more extreme points, and be considered more risky.
As with variance and skewness, the equation for kurtosis differs depending on whether we are calculating the population kurtosis or the sample kurtosis. For the population statistic, the kurtosis of a random variable X can be calculated as:
(3.67)
where μ is the population mean and σ is the population standard deviation. Similar to our calculation of sample variance, if we are calculating the sample kurtosis, there is going to be an overlap with the calculation of the sample mean and sample standard deviation. We need to correct for that. The sample kurtosis can be calculated as:
Later we will study the normal distribution, which has a kurtosis of 3. Because normal distributions are so common, many people refer to “excess kurtosis,” which is simply the kurtosis minus 3.
(3.69)
In this way, the normal distribution has an excess kurtosis of 0. Distributions with positive excess kurtosis are termed leptokurtotic. Distributions with negative excess kurtosis are termed platykurtotic. Be careful; by default, many applications calculate excess kurtosis.
When we are also estimating the mean and variance, calculating the sample excess kurtosis is somewhat more complicated than just subtracting 3. The correct formula is:
(3.70)
where
Part IV Distributions
Normal Distribution
The normal distribution is probably the most widely used distribution in statistics, and is extremely popular in finance. The normal distribution occurs in a large number of settings, and is extremely easy to work with.
In popular literature, the normal distribution is often referred to as the bell curve because of the shape of its probability density function (see Figure 3.7).
FIGURE 3.7 Normal Distribution Probability Density Function
The probability density function of the normal distribution is symmetrical, with the mean and median coinciding with the highest point of the PDF. Because it is symmetrical, the skew of a normal distribution is always zero. The kurtosis of a normal distribution is always 3. By definition, the excess kurtosis of a normal distribution is zero.
In some fields it is more common to refer to the normal distribution as the Gaussian distribution, after the famous German mathematician Johann Gauss, who is credited with some of the earliest work with the distribution. It is not the case that one name is more precise than the other as with mean and average. Both normal distribution and Gaussian distribution are acceptable terms.
Normal distributions are used throughout finance and risk management. Previously, we suggested that log returns are extremely useful in financial modeling. One attribute that makes log returns particularly attractive is that they can be modeled using normal distributions. Normal distributions can generate numbers from negative infinity to positive infinity. For a particular normal distribution, the most extreme values might be extremely unlikely, but they can occur. This poses a problem for standard returns, which typically cannot be less than –100 percent. For log returns, though, there is no such constraint. Log returns also can range from negative to positive infinity.
Normally distributed log returns are widely used in financial simulations, and form the basis of a number of financial models, including the Black-Scholes option pricing model. As we will see, while this normal assumption is often a convenient starting point, much of risk management is focused on addressing departures from this normality assumption.
Because the normal distribution is so widely used, most practitioners are expected to have at least a rough idea of how much of the distribution falls within one, two, or three standard deviations. In risk management it is also useful to know how many standard deviations are needed to encompass 95 percent or 99 percent of outcomes. Table 3.2 lists some common values. Notice that for each row in the table, there is a “one-tailed” and “two-tailed” column. If we want to know how far we have to go to encompass 95 percent of the mass in the density function, the one-tailed value tells us that 95 percent of the values are less than 1.64 standard deviations above the mean. Because the normal distribution is symmetrical, it follows that 5 percent of the values are less than 1.64 standard deviations below the mean. The two-tailed value, in turn, tells us that 95 percent of the mass is within +/–1.96 standard deviations of the mean. It follows that 2.5 percent of the outcomes are less than –1.96 standard deviations from the mean, and 2.5 percent are greater than +1.96 standard deviations from the mean. Rather than one-tailed and two-tailed, some authors refer to “one-sided” and “two-sided” values.
TABLE 3.2 Normal Distribution Confidence Intervals
Application: