Title: (F)CLT for functionals of quantile and dispersion estimators: Applications in risk management
Authors: Marcel Brautigam - ESSEC Business School & Sorbonne University (France) [presenting]
Marie Kratz - ESSEC Business School, CREAR (France)
Abstract: CLTs and FCLTs for Functionals of Quantile and Dispersion Estimators are provided. Those will allow us to quantify and explain the statistical pro-cyclicality of standard risk measures. First, we derive the joint bivariate asymptotic distributions of functions of quantile estimators (the non-parametric sample quantile and the parametric location-scale quantile estimator) with functions of measure of dispersion estimators (the sample variance, sample mean absolute deviation or any higher order absolute central sample moment) - assuming an underlying identically and independently distributed sample. In a second step, we extend the joint asymptotics between the sample quantile and any absolute centred sample moment for a broad class of augmented GARCH processes. These results support the empirical findings about the pro-cyclicality of traditional risk measurements. In particular, it proves that part of the procyclicality is intrinsically caused by the way of the historical risk estimation. Further, the exact degree of pro-cyclicality depends on the choice of the risk measure as well as the measure of dispersion considered but does not vanish in any case.