A comparison of extreme value theory approaches for determining value at risk

Abstract

This paper compares a number of different extreme value models for determining the value at risk (VaR) of three LIFFE futures contracts. A semi-nonparametric approach is also proposed, where the tail events are modeled using the generalised Pareto distribution, and normal market conditions are captured by the empirical distribution function. The value at risk estimates from this approach are compared with those of standard nonparametric extreme value tail estimation approaches, with a small sample bias-corrected extreme value approach, and with those calculated from bootstrapping the unconditional density and bootstrapping from a GARCH(1,1) model. The results indicate that, for a holdout sample, the proposed semi-nonparametric extreme value approach yields superior results to other methods, but the small sample tail index technique is also accurate. D

Selecting copulas for risk management

“…The marginal distributions constructed in the first step are based on the semi-parametric 11 Brooks et al (2005) conclude that this estimator outperforms other methods for tail index estimation when applied in Value-at-Risk calculations.…”

Market capitalization and Value-at-Risk

“…Reference papers comparing the performance of different VaR models are Bao et al (2003Bao et al ( , 2006, Brooks et al (2005), Kuester et al (2006) and Pritsker (1997). Many of the studies on the computation of VaR compare and propose different methods using data on large capitalization firms, major indices or highly traded currencies.…”

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“…Moreover, their findings, where there is overlap, agree with ours. 3 Other comparison-type studies include Pritsker (1997), although in that article, none of the methods considered herein, namely generalized autoregressive conditional heteroskedasticity (GARCH), mixed normal-GARCH (MN-GARCH), extreme value theory (EVT), and conditional autoregressive VaR (CAViaR), are used; and Brooks et al (2005), who also use a variety of models (but none of which is identical to those in our study), including variants of the unconditional EVT approach and the regular GARCH model coupled with resampling strategies, and also a variety of nonparametric tail estimators. 4 To the extent that a commercial bank and the regulator are interested in the aggregate VaR across different trading activities, the question arises whether first to aggregate profit and loss data and proceed with a univariate forecast model for the aggregate, or to start with disaggregate data.…”