Backtesting Value-at-Risk: A Generalized Markov Framework

Abstract

Testing the validity of Value-at-Risk (VaR) forecasts, or backtesting, is an integral part of modern market risk management and regulation. This is often done by applying independence and coverage tests developed in Christoffersen (1998) to so-called hit-sequences derived from VaR forecasts and realized losses. However, as pointed out in the literature, see Christoffersen (2004), these aforementioned tests suffer from low rejection frequencies, or (empirical) power, when applied to hit-sequences derived from simulations matching empirical stylized characteristics of return data. One key observation of the studies is that non-Markovian behavior in the hit-sequences may cause the observed lower power performance. To allow for non-Markovian behavior, we propose to generalize the backtest framework for Value-at-Risk forecasts, by extending the original first order dependence of Christoffersen (1998) to allow for a higher, or k’th, order dependence. We provide closed form expressions for the tests as well as asymptotic theory. Not only do the generalized tests have power against k’th order dependence by definition, but also included simulations indicate improved power performance when replicating the aforementioned studies.

Keywords: Value-at-Risk, Backtesting, Risk Management, Markov Chain, Duration-based test, quantile, likelihood ratio, maximum likelihood

JEL Classification: C12, C15, C52, C32