Things appear completely different for the 12-week-VaRs. HS and the VCM overestimate the medium-term VaRs
relative to EVT. For example, the 95-percent quantile for the farrows (hogs and margin), derived from the extreme
value distribution amounts to 0.207 Euro (0.162 and 9.567) while the VCM and HS display values of 0.362 Euro
(0.282 and 19.422) and 0.361 Euro (0.266 and 18.562), respectively. The short term underestimation of the VaRs by
HS and VCM is overcompensated by a too conservative time scaling via the square-root-rule.7 This bias becomes
larger with increasing time horizon.
Table 2 further reports asymptotic standard errors of the estimated quantiles. The VCM seemingly shows the
smallest estimation error. Some caution is necessary when interpreting the figures. The standard error of the VCM is
calculated according to:
SE( Xp1 ) = σ (2n) ~ 12cp (27)
X n denotes the estimated p-quantil, c is the p-quantil of the standard normal distribution and n denotes the
pp
numbers of observations. However, using (27) is only correct in case of normally distributed rvs. Since the normality
assumption was rejected by the data, the displayed standard errors are incorrect as well. Calculation of the standard
errors of the HS and the EVT is based on the expressions given in Jorion (1998 p. 99) and Danielsson and de Vries
(1997). The figures in table 2 highlight the aforementioned pitfall of HS when it comes to an estimation of extreme
quantiles. The standard errors of HS are relatively large for the given sample size of 405 observations. In this respect
EVT offers a better alternative.
Usually some kind of validation is conducted subsequent to the VaR estimation. This is usually done by an out-of-
sample-prediction (backtesting). For that purpose the sample period is divided into an estimation period and a
forecast period. Comparing theoretically expected and actually observed VaR overshoots occurring within the
forecast period allows to validate competing models statistically. However, such a validation is not possible in this
application due to the relatively short observation period of the price series. An overshoot of a 99%-VaR would
occur only once during 100 periods. In our application such an event is expected to happen once within 100 ∙12
weeks, i.e. once within 23 years. We conjecture that validation of VaR models will in general be difficult, if the
usual short term horizon in financial applications is extended largely. These difficulties are intensified by the fact
that an EVT estimation requires excessive data.
v,,,τ.. . .. ______ .... ι ι . .... . . 1 a .. . . , , . , .,
7 Mc Neil and Frey (2000) criticize the forecast applied here along h and favor a two-stage procedure, which considers
conditional heteroscedasticity in a first stage via GARCH estimation and applies EVT to the residuals of the conditional
estimation model in a second stage.
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