A New Estimator of Value at Risk under Peaks over Threshold Framework using Optimal Loss Function

Abstract

Value at Risk (VaR) is one of the most popular measures of risk associated with financial instruments. The generalized Pareto distribution (GPD) has been widely used to fit observations exceeding the tail threshold in the peaks over threshold (POT) framework. In this paper we propose a new estimator of GPD parameters and hence VaR& Expected Shortfall (ES) under POT framework. The procedure minimizes the differences between the empirical distribution function and the theoretical distribution function of GPD using an optimal loss function. A simulation study is carried out in presence of outliers to compare the performance of proposed estimator of VaR and ES with some of the existing estimators with respect to bias and mean square error. The study observed that the proposed estimator performs on par with some of the existing robust methods considered in the study in terms of mean square error for certain values of shape parameter(k<0) and moderately large sample size. The efficiency of proposed estimator is more than existing robust estimators considered in this study. In addition, the study includes comparison of these estimators using real dataset.