-Daily P&L .......Maximum loss over 5 days

-10 -15

Daily P&L Excess over threshold ( =-2)

Figure 10.2 (a) Filtering data for maximum loss; (b) peaks over threshold.

Maximal loss and excess over threshold are different types of random variables, but each may be defined in terms of the underlying random variable in a straightforward way. The distributions of these two types of extreme values can be built from first principles without too much difficulty. Indeed, it is possible to understand the fundamental concepts for extreme value distributions with a basic knowledge of statistics.

First consider the case where an extreme value is defined as the maximum (or minimum) of a sample {x,, . . ., xn] of size n. The sample size depends on the non-overlapping periods specified in the model. For example, one might wish to model the maximum daily loss experienced during a week, as in Figure 10.2a. In this case the model will take a sample size = 5. We start with the usual construction that each value x, in a random sample size n can be viewed as an observation on a random variable Xh where Xx, . . ., X„ are independent

Figure 10.3 Relative frequency of: (a) maximum loss; (b) peaks over threshold.

and identically distributed random variables.7 Since min(Z,, . . ., X„) = - max(-A,, . . ., - X„), we can without loss of generality derive results for the maximum in a sample (the results can easily be converted to results for the minimum). Let M„ = max(X,, . . ., X„). This has the distribution function F"(x),s so the limiting distribution of M„ as - is degenerate. For this reason M„ is transformed into standardized extreme values by subtracting its location parameter a„ and dividing by its scale parameter P„. With this transformation the standardized extreme values

Y„ = (M„ - a„)/p„

7The X, are independent, otherwise the sample would not be truly random, and they have the same distribution function F(x)=PTob(Xi<x). otherwise the sample would not be drawn from the same population.

"Because Prob(Af„ < ) = Probl.V, < v and X2 < v and ... and X„ < x) = Prob(X] < x)Prob(Z2 < x)... Prob(Z, < x).

will have a non-degenerate distribution. In fact there are only three possible limiting distributions for the standardized extremes Y„ and all three may be expressed in the single formula,

F(y) = Prob(Y„<y)

exp{-exp(-j/)} exp{-(l + l\yy

if £ = 0, ifO,

(l + b) > 0;

(10.4)

hence the term generalized extreme value (GEV) distributions for this type of distribution. GEVs depend on just one parameter , which is called the tail index; £, defines the shape in the tail of the GEV distribution. It is the reciprocal of the shape parameter of the distribution.

The tail index \ defines the shape in the tail of the distribution

The value £, = 0 corresponds to the Gumbel distribution. The corresponding density function has a mode at 0, positive skew and declines exponentially in the tails as n increases. It is the extreme value distribution corresponding to normal or lognormal underlying returns.

When £, < 0 the formula (10.4) defines the Weibull distribution, whose density converges to a mass at zero as £, -» -oo. The lower tail remains finite in the Weibull density, and this is appropriate if the original returns were uniformly distributed.

When £, > 0 we have the Frechet distribution, whose density also converges to a mass at zero as oo, but more slowly than the Weibull. The tail in the Frechet declines by a power as increases. Since this is the distribution of sample maxima if returns are generated by GARCH processes, Student t-distributions or stable Pareto distributions, the Frechet distribution is commonly used in financial risk management to model the maximum loss of a portfolio over a given holding period.9

The Frechet is the distribution of sample maxima if returns are generated by GARCH processes, Student t-distributions or stable Pareto distributions

Figure 10.4 shows different Weibull and Frechet density functions, depending on the tail index . Of course, when i; is small both Weibull and Frechet densities will be approximately the same as the Gumbel density.

The theory of extreme value distributions has obvious applications to measuring portfolio risk when extreme events occur in financial markets. For example, Embrechts et al. (1998, 1999b) advocate the use of GEVs to estimate market VaR during market crashes. It also has applications to insurance and reinsurance (McNeil, 1997). GEV distributions may be fitted to empirical data by maximum likelihood estimation (MLE) as described in §A.6.4. However, although MLE provides consistent estimators, the size of the data set used for the optimization is generally rather small because it will only contain the extreme losses. Therefore Embrechts et al. (1997) suggest using the

9 Note that although the Weibull has weight in the lower tail and the Frechet has weight in the upper tail, that does not imply that the Weibull should be applied to sample minima and the Frechet should be applied to sample maxima. Each can be applied to either sample maxima or sample minima, simply by changing the sign of the data.