Modeling Extreme Values in the Financial Market and Estimating Loss Functions

We estimate the extreme value distribution of the losses on the monthly S&P 500 indexreturns during the period from January, 1871 to January 2013 to quantify the tail probability andthe extreme quantile of the loss distribution. In estimating the generalized extreme valuedistribution (GEV), we use the Fisher-Tippett theorem in specifying the limiting distribution forcentered and normalized maxima. We use the estimated extreme value distribution to calculatethe probability of observing an unprecedented annual maximum loss on the stock market indexover the next period. We also compute the return level which is exceeded by the annualmaximum negative return in any particular year with a given level of probability. As an alternativeto the GEV distribution estimation with the block maxima data extracted from the sample of thedisjoint time period, we estimate the limiting distribution of scaled ex cesses over a highthreshold to improve the efficiency of the GEV estimation. We estimate the generalized Paretodistribution (GPD) and compute the Value-at-Risk (VaR) and the expected shortfall (ES) andconfirm that the risk measures under the normal distribution underestimate the extreme quantileestimation.