An Optimal Threshod Selection Approach for the Value at Risk of the Extreme Events

The extreme value theory (EVT) can be used to predict rare events and is a popularmeans for estimating the value at risk (VaR) by extrapolating the tails of a distribution. The focusof the present study is on one of the EVT methods, peak-over-threshold (POT). In this approachthe thresholds of a generalized Pareto distribution (GPD) are assigned based on a Hill plot withshape parameter (xi) for various exceedance levels, where(xi)is estimated by applying Hill's estimatorbased on the optimized k of the order statistics. A new approach for selectingkvia Hill's estimatorobtained by using a type 8 quantile function is presented herein. The threshold obtained is then used to compute the VaR and the expected shortfall (ES) for an event. An illustration of the efficacy of the new approach based on real data obtained from Danishfire loss insurance claims is also included