Estimating and backtesting risk under heavy tails

TL;DR

This paper addresses bias in risk estimation under heavy tails, proposing a new bias correction algorithm that improves accuracy and efficiency in small samples, especially with heavy-tailed or heteroscedastic data.

Contribution

It introduces a novel bias correction algorithm for risk estimation, applicable to generalized Pareto distributions and GARCH models, enhancing accuracy under heavy tails.

Findings

Bias correction improves risk estimation accuracy

Algorithm enhances efficiency with heavy tails and heteroscedasticity

Application to GARCH models demonstrates practical benefits

Abstract

While the {estimation} of risk is an important question in the daily business of banking and insurance, many existing plug-in estimation procedures suffer from an unnecessary bias. This often leads to the underestimation of risk and negatively impacts backtesting results, especially in small sample cases. In this article we show that the link between estimation bias and backtesting can be traced back to the dual relationship between risk measures and the corresponding performance measures, and discuss this in reference to value-at-risk, expected shortfall and expectile value-at-risk. Motivated by the consistent underestimation of risk by plug-in procedures, we propose a new algorithm for bias correction and show how to apply it for generalized Pareto distributions to the i.i.d. setting and to a GARCH(1,1) time series. In particular, we show that the application of our algorithm leads to gain in efficiency when heavy tails or heteroscedasticity exists in the data.