Statistics of Extreme Risks

  • Jürgen FrankeEmail author
  • Wolfgang Karl Härdle
  • Christian Matthias 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 stylised fact (see Chapter 13.). 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 there is a higher probability of particularly extreme losses than a GARCH process "t with normally distributed Zt can produce.


Pareto Distribution Exceedance Probability Generalise Pareto Distribution Extremal Index Gumbel Distribution 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Jürgen Franke
    • 1
    Email author
  • Wolfgang Karl Härdle
    • 2
    • 3
  • Christian Matthias Hafner
    • 4
  1. 1.FB MathematikTU KaiserslauternKaiserslauternGermany
  2. 2.Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Centre for Applied Statistics and Economics School of Business and EconomicsHumboldt-Universität zu BerlinBerlinGermany
  3. 3.Graduate Institute of StatisticsNational Central UniversityJhongliTaiwan
  4. 4.Inst. StatistiqueUniversité Catholique de LouvainLeuven-la-NeuveBelgium

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