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
with normally
distributed
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 of the innovations goes to 0 as
. 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.