Value-at-Risk under Lévy GARCH models: Evidence from global stock markets

Highlights

• The performance of twenty-three VaR models is investigated.

• We implement stringent backtesting for model validation during crisis and post-crisis.

• We find strong support for the use of Lévy distributions during crisis period.

• Normal models are economically efficient during market calm.

Abstract

The aim of this paper is to reconsider the evidence on the forecasting ability of GARCH-type models in estimating the Value-at-Risk (VaR) of global stock market indices with improved return distribution. The performance of twenty-one VaR models that are generated by a combination of three conditional volatility specifications including GARCH, GJR and FIGARCH and seven distributional assumptions for return innovations is investigated. We implement stringent backtesting during crisis and post-crisis periods for developed, emerging and frontier markets. Results show that the skewed-t along with heavy-tailed Lévy distributions considerably improve the forecasts of one-day-ahead VaR for long and short trading positions during crisis period, regardless of the volatility model. However, we find no evidence that a given volatility specification outperforms the others across markets. The relevant models show evidence of long memory in developed markets and conditional asymmetry in frontier markets; whereas the standard GARCH is found to be the best suited specification for estimating VaR forecasts in emerging markets. The inclusion of high volatility period in the estimation sample highlights the predictability of VaR during post-crisis period, where even the normal distribution rivals the more sophisticated ones in terms of statistical accuracy and regulatory capital allocation.

Introduction

The Value-at-Risk (VaR) concept has been established as an industry standard measure of market risk. It provides financial institutions with information on the expected worst loss over a target horizon at a given confidence level. Despite its importance and simplicity, there is no universally accepted method to compute the VaR of a particular portfolio, while different models may lead to significantly different risk measures (see Kuester et al., 2006, McMillan and Kambouroudis, 2009, among others), a main concern in the estimation of market risk with the VaR method is the choice of the appropriate model, e.g., a misspecified model may turn out to be costly for the risk manager, and leads to inaccurate risk estimation. Moreover, the extreme losses experienced by financial institutions during the recent global financial crisis, triggered by the U.S. subprime mortgage debacle of 2007–2008, have raised questions about the reliability of the implemented risk models. These questions have a direct bearing on the debate amongst financial industry, regulators and academicians over probabilistic market models for VaR forecasting, which are capable to properly account for extreme events and increased volatility during financial market downturns.

In obtaining accurate VaR measures, the prediction of future market volatility is of paramount importance, particularly in view of its time-varying nature as well as some prominent stylized facts of stock returns (Cont, 2001). Indeed, there is ample empirical evidence that spells of small amplitude for the price variations alternate with spells of large amplitude; one calls this feature volatility clustering. Numerous econometric models have been suggested to capture the volatility clustering effect, the most widely used one is the GARCH model (Bollerslev, 1986). The normal distribution arising from the Brownian motion assumption as a benchmark process for describing return innovations have dominated former GARCH-based VaR models drawing criticisms that such distributional assumption may not sufficiently capture the frequency of extreme shocks to asset prices, as well as the amplitude of these shocks, and usually leads to risk underestimation. The introduction of more flexible distributions, allowing for skewness and heavy tails into return modeling, exonerates GARCH-type models from such criticisms (Bao et al., 2007, BenSaïda, 2015).

Recently, enhanced conditional volatility models with Lévy distributions have emerged with a view towards improved tail modeling. Examples include the normal inverse Gaussian (Forsberg and Bollerslev, 2002, Venter and de Jongh, 2002, Broda and Paolella, 2009, Wilhelmsson, 2009), the multivariate generalized hyperbolic distribution (McNeil et al., 2005) and used in a GARCH context by Paolella and Polak, 2015a, Paolella and Polak, 2015b, the Meixner distribution (Grigoletto and Provasi, 2008), $\alpha$-stable and tempered stable distributions (Mittnik and Paolella, 2003, Broda et al., 2013, Kim et al., 2008, Kim et al., 2011). The current study pursues the use of Lévy distributions as statistical tools for advanced risk modeling.

More concretely, our objective is to examine the suitability of univariate GARCH-type models in modeling conditional volatility and VaR under different assumptions of error distribution for global stock market indices.¹ Within the class of conditional volatility models, we employ the standard GARCH model, GJR (Glosten et al., 1993) and FIGARCH (Baillie et al., 1996). Four infinitely divisible distributions arising from popular Lévy processes are considered, namely, the Variance Gamma (Madan et al., 1998), the CGMY (Carr et al., 2002), the normal inverse Gaussian (Barndorff-Nielsen, 1997), and the Meixner distribution (Schoutens, 2001).

The forecasting performance of these models is discussed and compared to the benchmark normal GARCH model, fat-tailed time series models using the Student’s t and the skewed-t distribution of Hansen (1994). We also consider two higly competitive models, namely, the mixed normal GARCH and the fast Asymmetric Power ARCH model driven by noncentral t innovations, recently introduced by Krause and Paolella (2014). We empirically test the accuracy of VaR estimates during high volatility period for the MSCI World, MSCI emerging markets and MSCI frontier markets indices by means of an extensive backtesting exercise that includes the frequency-based test (Kupiec, 1995), independence tests (Christoffersen, 1998, Engle and Manganelli, 2004), duration-based tests (Candelon et al., 2011) and a test that jointly accounts for the frequency and the magnitude of VaR exceedances, recently introduced by Colletaz et al. (2013). Such backtesting strategy encompasses almost all risk model validation techniques in the literature, and provides a meaningful framework to evaluate the accuracy of VaR models. Furthermore, we evaluate the economic importance of our results by computing daily capital requirements under the Basel II Accord (Basel Committee on Banking Supervision, 2006).

This paper distinguishes itself from the relevant literature in several ways. (1) It allows for direct comparisons to be made on the performance of a broad set of GARCH-type specifications combined with non-normal Lévy distributions for gauging and managing market risk. Although, the option pricing literature establishes Lévy processes as quite suitable in capturing the behavior of returns innovation; other evaluation metrics and discussions on their applicability as risk management tools are needed. (2) Our empirical assessment of the performance of competing VaR models not only allows to test whether VaR forecasts are misspecified overall, but also identifies how they are misspecified by looking into which individual hypothesis is rejected. The hypotheses induced by the employed backtesting methods have different economic implications. Financial institutions focus on the frequency test since the empirical number of violations (i.e., when the realized losses exceed the VaR values) is used to determine their market risk minimum capital requirements. Even in practice, market participants and regulators do care about the number of VaR violations, they often have a more stringent requirement on the magnitude of their losses beyond VaR, as such quantity is not given by any VaR model and represents the main conceptual deficiency of VaR. The duration independence test considers whether the VaR reflects the time-varying nature of risk. In particular, clustered violations may signal that the VaR model reacts too slowly to changes in market conditions, and hence may induce solvency issues.

Further, to the best of our knowledge, no published article examines all of these practically important issues in the backtesting of VaR estimates. (3) Until recently, a substantial literature of empirical applications narrowly focuses on the computation of the VaR on the left tail of the distribution which corresponds to the long position risk. The short position risk, which is theoretically unlimited, has been neglected and only a few papers have attempted to jointly account for both long and short position risk in equity markets (Giot and Laurent, 2003, So and Yu, 2006, Tang and Shieh, 2006, Diamandis et al., 2011). In our empirical analysis, the performance of VaR models is also examined in terms of the critical issue of model consistency between long and short position risk.

The main contributions of this study are the following. First, our results corroborate the calls for the use of more realistic assumptions in financial modeling. In model estimation, it is shown that allowing for leptokurtic and skewed return distributions significantly improves the fit of conditional volatility models. Second, even though, non-normal Lévy distributions do provide more accurate VaR estimates than those generated by both normal and Student’s t distributions, conditional volatility specification bears on the accuracy of VaR as none of the three GARCH-type models outperform the others across markets. The empirical evidence favors non-normal Lévy-based FIGARCH models in developed markets, whereas the GJR and the standard GARCH models are preferred for the frontier and emerging markets, respectively. Third, this paper offers important implications regarding the recent global financial crisis with respect to the estimation of VaR for developed, emerging and frontier markets. In general, the forecasting performance of the VaR models deteriorates during crises periods, predominantly in the case of developed markets. For emerging and frontier markets, nevertheless, the forecasting performance is less affected. Fourth, the inclusion of the crisis period in the estimation sample, significantly improves the backtesting performance of the competing VaR models during the post-crisis period. Even though, normal time series models rival the more flexible non-normal Lévy models. However, our findings provide compelling evidence for the inadequacy of the Student’s t distribution, as it tends to produce overly conservative risk measures during both high and low volatility periods.

The remainder of the paper is organized as follows. Section 2 outlines the basic concept of VaR and presents the employed time series models. Section 3 presents the alternative non-normal Lévy distributions. Section 4 describes the data and discusses the results of the empirical investigation. Section 5 summarizes the main findings of the paper.