A computational study of a quasi-PORT methodology for VaR based on second-order reduced-bias estimation

Abstract

In this paper, we deal with the estimation, under a semi-parametric framework, of the Value-at-Risk (VaR) at a level p, the size of the loss occurred with a small probability p. Under such a context, the classical VaR estimators are the Weissman–Hill estimators, based on any intermediate number k of top-order statistics. But these VaR estimators do not enjoy the adequate linear property of quantiles, contrarily to the PORT VaR estimators, which depend on an extra tuning parameter q, with 0≤q<1. We shall here consider ‘quasi-PORT’ reduced-bias VaR estimators, for which such a linear property is obtained approximately. They are based on a partially shifted version of a minimum-variance reduced-bias (MVRB) estimator of the extreme value index (EVI), the primary parameter in Statistics of Extremes. Due to the stability on k of the MVRB EVI and associated VaR estimates, we propose the use of a heuristic stability criterion for the choice of k and q, providing applications of the methodology to simulated data and to log-returns of financial stocks.

Keywords:

• statistics of extremes
• value-at-risk
• semi-parametric estimation
• heuristics

2000 AMS Subject Classifications :

• 62G32
• 65C05

Acknowledgements

Research partially supported by FCT/OE and PTDC/FEDER. We also would like to thank a referee for the valuable comments on a first version of this paper.