Figure 10.4 Generalized extreme value densities and the tail index (a) Weibull and Gumbel; (b) Frechet and Gumbel.

Hill estimator; Alexander McNeil of ETHZ, Switzerland, has some free extreme value software (EVIS) for S-Plus users at www.math.ethz.ch/-mcneil.

Now let us turn to the case that extreme values are based on the POT model. Here all excess losses (or returns) over a high and predefined threshold are recorded, as in Figure 10.2b. It is possible to state results only in terms of the excesses over a positive threshold, because results for a negative threshold follow by symmetry. The distribution function Gu of these excess losses, X - , has a simple relation to the distribution F(x) of X, the underlying loss (or returns) series. In fact

Gu(y) = Prob(Z- < y\X > u) = [F(y + u) - F(u)]/[l - )]. (10.5)

For many choices of underlying distribution F(x) the distribution Gu(y) will belong to the class of generalized Pareto distributions given by:

G«-\i-(i + /P)- if/o. (106)

The parameters P and i; will depend on the type of underlying distribution F(x) and on the choice of threshold u. Some generalized Pareto densities for different values of P and i; are shown in Figure 10.5. As for the GEV distributions, the effect of the tail index £, is to increase the weight in the tails.

Generalized Pareto distributions also have useful applications to the measurement of portfolio risk. Recall that, when discussing the advantages and limitations of VaR as a risk measure in §9.2.3, it was noted that VaR is not a good risk measure because it is not coherent. However, conditional VaR, which is the average of all losses that exceed VaR, is a coherent risk measure. Now consider the mean excess loss over a threshold u:

e(u) = E(X - u\X > u). (10.7)

Depending on the distribution of X, e{u) can take a more or less simple functional form.10 As mentioned above, the excess over threshold normally has a generalized Pareto distribution, and then the mean excess loss has a simple functional form. In fact under (10.6)

( ) = [ + /[1- . ( .8)

Conditional VaR = + e(u), where - VaR, so to calculate the conditional VaR from historical data on portfolio P&L, we can take the losses in excess of the VaR level, estimate the parameters P and of a generalized Pareto distribution, and compute the quantity e(u) with - VaR in (10.8).

The classical (MLE) approach to fitting the parameters of the GPD model is only applicable when sample sizes are large. However peaks over a threshold are, by definition, exceptional events that occur with very low frequency. Therefore special estimation procedures for the POT model have been developed (Pickands, 1975; Bernardo and Smith, 1994). The combination of a GPD distribution for loss impact and a Poisson process for frequency is called the peaks over threshold (POT) model (Leadbetter et al., 1983; Smith, 1987). Recently it has found useful application in the measurement of operational risks. Alexander (2001c) gives a short review and King (2001) describes a particular model.

Peaks over a threshold are, by definition, exceptional events that occur with very low frequency. Therefore special estimation procedures for the POT model have been developed

l0If A" is exponentially distributed with density function f(x) = Xexp(-Xx) for x > 0, then e(u) is a constant 1 . If X has a distribution with fatter (thinner) tails than the exponential the mean excess increases (decreases) as

increases.

10.2.2 Hyperbolic Distributions

Hyperbolic distributions are a class of distributions that have great potential for modelling financial asset returns and have recently attracted considerable attention (Barndorff-Nielsen, 1977; Eberlein and Keller, 1995; Eberlein, 2001). Value-at-risk estimates based on hyperbolic distributions therefore have much in common with VaR estimates based on extreme value distributions. But whereas extreme value theory concentrates only on the tails of the distribution, the hyperbolic density fits the whole range, the centre as well as the extremes.

A hyperbolic distribution is characterized by a log density function in the shape of a hyperbola. The class of hyperbolic distributions contains the normal distribution, whose log density is a parabola, as a limiting case. Hyperbolic A hyperbolic distribution distributions can arise from quite natural assumptions about the behaviour of is characterized by a log prices and volatility. For example, a stochastic volatility model might assume density function in the that returns are normally distributed with mean 0 and variance o2, where o2 shape of a hyperbola nas a generalized inverse Gaussian distribution. In this case the density function that describes returns is a two-parameter hyperbolic density