Value-at-Risk Estimation and the Port Mean-of-Order-P Methodology

Abstract

In finance, insurance and statistical quality control, among many other areas of appli[1]cation, a typical requirement is to estimate the value-at-risk (VaR) at a small level q, i.e. a high quantile of probability 1 −q, a value, high enough, so that the chance of an exceedance of that value is equal to q, small. The semi-parametric estimation of high quantiles depends strongly on the estimation of the extreme value index (EVI), the primary parameter of extreme events. And most semi-parametric VaR-estimators do not enjoy the adequate behaviour, in the sense that they do not suffer the appropriate linear shift in the presence of linear transformations of the data. Recently, and for heavy tails, i.e. for a positive EVI, new VaR-estimators were introduced with such a behaviour, the so-called PORT VaR-estimators, with PORT standing for peaks over a random threshold. Regarding EVI-estimation, new classes of PORT-EVI estimators, based on a powerful generalization of the Hill EVI-estimator related to adequate mean[1]of-order-p (MOp) EVI-estimators, were even more recently introduced. In this article, also for heavy tails, we introduce a new class of PORT-MOp VaR-estimators with the above mentioned behaviour, using the PORT-MOp class of EVI-estimators. Under convenient but soft restrictions on the underlying model, these estimators are consis[1]tent and asymptotically normal. The behaviour of the PORT-MOp VaR-estimators is studied for finite samples through Monte-Carlo simulation experiments.

Keywords

• asymptotic behaviour
• heavy tails
• high quantiles
• mean-of-order-p estimation
• Monte-Carlo simulation
• PORT methodology