distribution. Hence no explicit assumption about the return distribution is required here. However, this procedure
implicitly assumes a constant (stable) distribution of the market factors. A general problem arises from the fact that
the empirical distribution function, while being relatively smooth around the mean, shows discrete jumps in the tails
due to the small number of extreme sample values. The higher the desired confidence level is, the more uncertain the
estimation of the corresponding quantile becomes. Accordingly, VaR estimates based on HS react sensitively to
modifications of the data sample. The probability of events that are worse than the sample minimum cannot be
predicted per definitionem. The extreme value theory, described in section 3, offers an opportunity to avoid these
problems.
2.3 Modeling the return distribution
If a parametric approach to VaR estimation is utilized the question arises, which distribution function fits best to the
observed changes of the market factors. As mentioned above it is widely recognized in the literature that empirical
return distributions of financial assets are characterized by fat tails. With respect to modeling the underlying
stochastic process two consequences can be deduced (Jorion 1997, p. 166 f.): either one uses a leptocurtic
distribution, e.g. a t-distribution, or one resorts to a model with stochastic volatility. Of course both approaches can
be combined. The observation of volatility clusters in high frequency (i.e. daily) data rows favours the use of models
with stochastic volatilities. The change of phases of relatively small and relatively high fluctuations of returns can be
captured with GARCH models. Yang and Brorsen (1992) demonstrate that GARCH models are not only relevant for
financial applications, but also appropriate to describe the development of daily spot market prices of agricultural
commodities. Herein a stochastic process of the form
Xt = μ t+σ tε t
(6)
is assumed for the returns. εt are iid rvs. In most applications normal or t-distributions for the disturbance variable εt
are presumed. In a GARCH(1,1) process the variance σ 2 develops according to
σ 2t+1 = ω2 + δX2t + βσ 2
(7)
with ω = γσ2 > 0, δ ≥ 0, β ≥ 0, δ + β < 1
σ2 is a long-term average value of the variance, from which the current variance can deviate in accordance with
(7). Obviously the use of models with stochastic volatility implies a permanent updating of the variances and thus
the VaR forecasts.
While conditional models are superior for short term forecasts, their value vanishes with increasing time horizon.
Christoffersen and Diebold (2000) argue that the recent history of data series has little to tell about the probability of
events occurring far in the future. This applies especially to the prediction of rare events like disasters, which are