Statistics of Extreme Risks
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.
KeywordsPareto Distribution Exceedance Probability Generalise Pareto Distribution Extremal Index Gumbel Distribution
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