Statistics of Extreme Risks

  • Jürgen Franke
  • Wolfgang Härdle
  • Christian M. Hafner
Part of the Universitext book series (UTX)


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 ε t with normally distributed Z t can produce.


Pareto Distribution Exceedance Probability Generalize Pareto Distribution Empirical Distribution Function Extremal Index 
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Recommended Literature

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jürgen Franke
    • 1
  • Wolfgang Härdle
    • 2
  • Christian M. Hafner
    • 3
  1. 1.University of KaiserslauternKaiserslauternGermany
  2. 2.CASE-Center for Applied Statistics and EconomicsHumboldt-Universität zu BerlinBerlinGermany
  3. 3.Econometric Institute, Faculty of EconomicsErasmus University RotterdamRotterdamThe Netherlands

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