Value at Risk : Une nouvelle approche fondée sur les valeurs extrêmes

This paper presents extreme value theory and its application to the computation of the value at risk of a position. This statistical theory allows quantification of the behavior of extreme moveme nts in prices and rates such that a new measure for catastrophe or bankruptcy risk can be defined. Empirically, it is shown that the Fréchet distribution models this type of movement well. Extreme movements are associated with both little tremors like market adjustments or corrections during ordinary periods, and also earthquake-like stock market crashes, bond market collapses or foreign exchange crises observed during extraordinary periods. The approach based on extreme values then covers market conditions ranging from the usual environemnt considered by the existing VaR methods to the financial crises which are the focus of stress testing. The approach based on extreme values is then applied to a position in the French stock market using extreme value theory which characterizes the limit distribution of extreme returns. This method is then compared to different methods of the traditional approach which describe the statistical behavior of all returns (the historic distribution, the normal distribution and conditional processes like the GARCH process or the exponential weighted moving average used in RiskMetrics(tm) used to describe the variance). These empirical results allow to evaluate the French regulation on market risks.