Encyclopedia of Financial Models, 3 Volume Set

Applications of Order Statistics to Risk Management Problems

RADU TUNARU, PhD

Professor of Quantitative Finance, Business School, University of Kent

Abstract: Value-at-risk (VaR) calculation based on parametric models is in essence an estimation problem. The point estimates should be interpreted accompanied by their confidence intervals. Risk management for complex portfolios may consider simultaneously two or more VaR confidence levels. The quantiles used for VaR estimation at different orders such as 1% and 5% are not independent and therefore should be analyzed jointly. Consequently, it would be useful to establish confidence regions for bivariate VaR estimates that will provide the risk managers with a valuable tool for verifying the accuracy of their estimation process, as requested by external audit. A trade-off between the complexity of probability distribution underlying the model and the degree of robustness achieved is recommended.

While there are many models used for calculations of risk management measures such as value-at-risk (VaR) and expected tail loss (ETL), there are not many tools available to a risk manager to verify whether the models chosen are very good in practice. In this entry, we highlight some practical aspects of VaR and ETL calculus that are underpinned by theoretical results on order statistics. More precisely, we show how to compute VaR and ETL based on quantile sample statistics and how to derive the probability distribution of this estimator. ...