18. Statistics of Extreme Risks

When we model returns using a GARCH process with normally distributed innovations, we have already taken into account the second stylized fact (see Chapter 12). The distribution of the random returns automatically has a leptokurtosis and larger losses occurring more frequently than under the assumption that the returns are normally distributed. If one is interested in the 95%-VaR of liquid assets, this approach produces the most useful results. For the extreme risk quantiles such as the 99%-VaR and for riskier types of investments the risk is often underestimated when the innovations are assumed to be normally distributed, since a higher probability of particularly extreme losses than a GARCH process $ \varepsilon_t$ with normally distributed $ Z_t$ can produce.

Thus procedures have been developed which assume that the tails of the innovation's distribution are heavier. The probability of extreme values largely depends on how slowly the probability density function $ f_Z(x)$ of the innovations goes to 0 as $ \vert x\vert \to
\infty$. The rate at which it diminishes must be estimated from the data. Since extreme observations are rare, this produces a difficult estimation problem. Even large data sets contain only limited information on the true probability of an extreme loss (profit). In such a situation methods from extreme value statistics produce a more realistic estimate of the risk. In this chapter a short overview of the basic ideas and several of the latest applications are given.