The Variances of VaR for the Poisson-Gumbel Compound Extreme Value Distribution and for the Poisson-Generalized Pareto Compound Peaks over Threshold Distribution

In this paper, we compared the variances of value at risk (VaR) of loss distribution models: one based on the Poisson-Gumbel compound extreme value distribution and another based on the Poisson-generalized Pareto (GP) compound peaks over threshold distribution. The data used in this study are records of exchange rates between US Dollars and British Pounds from January 2, 1990 to December 29, 2006. By comparison, we found that the variance of VaR for the Poisson-Gumbel compound extreme value distribution is less than the variance of VaR for the Poisson-GP compound peaks over threshold distribution when the variances of other parameter estimates are assumed to be similar. We concluded that if both distribution models can be used to model the loss sample data, then the Poisson-Gumbel compound extreme value distribution is superior than the Poisson-GP compound peaks over threshold distribution.