Backtesting Value-at-Risk: A Generalized Markov Framework

29 Pages Posted: 21 Nov 2015

See all articles by Thor Pajhede

Thor Pajhede

University of Copenhagen - Department of Economics

Date Written: November 18, 2015


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

Suggested Citation

Pajhede, Thor, Backtesting Value-at-Risk: A Generalized Markov Framework (November 18, 2015). Univ. of Copenhagen Dept. of Economics Discussion Paper No. 15-18. Available at SSRN: or

Thor Pajhede (Contact Author)

University of Copenhagen - Department of Economics ( email )

Øster Farimagsgade 5
Bygning 26
1353 Copenhagen K.

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