a = h(1 - β )2 + 2h(h -1)
(1 - δ - β)2(1 - 2δβ - β2)
( κ -1)(1 - ( δ + β )2)
+ 4(h -1 - h( δ + β ) + ( δ + β )h)( δ - δβ ( δ + β ))
1 - ( δ + β )2
b = ( δ - δβ ( δ + β ))
1 - ( δ + β )2h
1 - ( δ + β )2
к denotes the kurtosis of the return distribution.
Comparing (9) with (8) reveals systematic differences, which become larger with increasing h. If h goes to infinity,
δ and β in (9) converge to zero and hence the stochastic terms vanish, whereas the first deterministic term increases.
That means that the average levels of the h-period-volatility coincide in both cases, but the square-root-rule
magnifies the fluctuations of the volatility, while they actually become smaller with increasing time horizon.
Diebold et al. (1997) illustrate the magnitude of the difference of both methods of volatility forecasting by means of
simulation experiments. Similar calculations in the context of our application are presented in section 4.
3 Extreme-Value-Theory
From the discussion in section 2.2 some pitfalls of traditional methods of VaR estimation became obvious, in
particular if the prediction of very rare events is desired and leptocurtic distributions are involved. Now we turn to
the Extreme-Value-Theory (EVT) in order to improve the estimation of extreme quantiles1. EVT provides statistical
tools to estimate the tails of probability distributions. Some basic concepts are briefly addressed below. A much
more comprehensive treatment can be found in Embrechts et al. (1997).
3.1 Basic concepts
A main objective of the EVT is to make inferences about sample extrema (maxima or minima)2. In this context the
so called Generalized Extreme Value distribution (GEV) plays a central role. Using the Fisher-Tipplet theorem it
can be shown that for a broad class of distributions the normalized sample maxima converge towards the
Generalized Extreme Value distribution with increasing sample size. If X1, X2, ...,Xn are iid random variables from
an unknown distribution F, and let an und bn be appropriate normalization coefficients, then for the sample maxima
Mn = max(X1,X2,∙∙∙,Xn) holds:
1 We emphasize that EVT is not the only method to cope with extreme events and fat tailedness. For example, Li (1999) uses a
semiparametric approach to VaR estimation, which takes into account skewness and kurtosis of the return distribution in
addition to the variance. Moreover, stress testing is a rather widespread technique that may be used as a complement to
traditional VaR methods. It gauges the vulnerability of a portfolio under extreme hypothetical scenarios
2 Embrechts et al. (1997, p. 364) express the objective of EVT vividly as „mission improbable: how to predict the
unpredictable“.