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10.1.2 QQ Plots

It is also possible to observe departures from normality using some standard qualitative plots such as those shown in Figure 10.1. The first view in Figure 10.1a is of the histogram of 1-hour returns on the DEM-USD exchange rate. This is transformed to a relative frequency distribution in Figure 10.1b and compared with that of the normal density of the same mean and standard deviation. Now from Table 10.1 the skewness is 0.289 (this shows up in the extra weight in the right-hand tail of the distribution) and the excess kurtosis is 8.34. Fat-tailed densities have higher peaks - that is, more weight around the mean - as well as more weight in the tails. Note that when there is excess kurtosis the mid-range values either side of the mean have less weight than in the normal distribution. So fat-tailed densities are appropriate when market returns are likely to be very small or very large, but returns are less likely to take values in between these two extremes.

The third view of the data in Figure 10.1c is a quantile to quantile plot or QQ plot.5 This is a scatter plot of the empirical quantiles (vertical axis) against the theoretical quantiles (horizontal axis) of a given distribution. To construct a QQ plot that compares an empirical distribution with a normal distribution, apply the standard normal transformation to the empirical returns6 and calculate the quantiles of the transformed empirical distribution. Then the QQ plot is simply a scatter plot of the transformed empirical and the standard normal quantiles.

If returns have excess kurtosis, the probability of large negative or large positive values is greater than under the corresponding normal density function. So the lower quantiles are less than the normal quantiles, and the upper quantiles are

5 An a quantile of a density function is a value x such that Prob(JT< x) = a.

6 That is, transform Xt into Z, = (X, - X)/s, where Xis the sample mean and s is the sample standard deviation. Then the Z, have a sample mean of 0 and a sample standard deviation of 1.

Fat tails show up in QQ plots as deviations below this line at the lower quantiles, and above this line at the upper quantiles

greater. Fat tails show up in QQ plots as deviations below this line at the lower quantiles, and above this line at the upper quantiles. The fat tails of the 1-hour returns are very evident in Figure 10.1c, although the QQ plot in Figure 10.Id shows that daily returns are much closer to being normal.

10.2 Non-normal Distributions

The normal density function is given by

( ) = (2 2)~1/2 exp(-±(x - u)2/o2), - oo < x < oo.

The normal distribution ( ) does not have a nice functional form so we usually work with normal density functions. For non-normal distributions it depends: sometimes it is more convenient to use a density function (§10.2.2 and §10.2.3) and in other cases the distribution function makes the analysis more tractable (§10.2.1).

Normal distributions are completely determined by only two parameters: the mean u, which measures the location of the distribution; and the variance o2, which measures the scale of the distribution. All normal density functions have the same shape and the function ( ) describes the familiar bell-shaped curve. But some families of distributions will have one (or more) shape parameter, in addition to scale and location parameters. For example, hyperbolic densities have two shape parameters and two other parameters that determine the scale and location of the distribution (§10.2.2).

10.2.1 Extreme Value Distributions

Extreme value distributions are a class of distribution that only applies to extreme values. They are commonly used in financial risk management because they focus on extreme values of returns, or exceptional losses. These extreme returns or exceptional losses are extracted from the data, and then an extreme value distribution may be fitted to these values. There are two approaches. One either models the maximal and minimal values in a sample using the generalized extreme value (GEV) distribution, or one models the excesses over a predefined threshold using the generalized Pareto distribution (GPD).

The difference between these approaches is illustrated in Figures 10.2 and 10.3. Figure 10.2a shows how the maximal loss data are obtained: the underlying time series, which may be thought of as a series of daily returns or of P&Ls, is filtered by extracting the maximal loss (or largest negative return) during a week, every non-overlapping week during the sample period. Figure 10.2b shows how the peaks over threshold (POT) data are obtained: the same underlying time series as that of Figure 10.2a is now filtered by setting a high threshold and taking all excesses over this threshold. Histograms of these two types of extreme value data are shown in Figure 10.3 (u - -2 in this example).