Scaling conditional tail probability and quantile estimators

Abstract

In terms of risk measurement, probability and quantile risk estimation have developed enormously in the past decade, from value-at-risk measures to coherent measures such as expected shortfall. These measures allow an investor to determine their risk profile accounting for losses (quantiles) at a given likelihood (probability) and a given time frame (holding period). The authors refer to this framework as conditional extreme value theory (EVT). To begin, they use an AR(1)-Garch(1,1) specification with student-t innovations to model the conditional distribution. They then apply EVT to the conditional filtered series to model the tail returns, allowing them to examine low-probability (out-of-sample) events for single-period horizons. Scaling is important as it allows one to overcome the lack of non-overlapping returns for low-frequency horizons. The approach illustrates the estimation bias that exists when assuming normality is minimized for the conditional EVT approach, both for single-period and multi-period settings (using the respective scaling laws).