Received: December 2016 Revised: August 2017 Accepted: 16 November 2017 DOI: 10.1002/for.2503 SPECIAL ISSUE ARTICLE Value-at-risk under market shifts through highly flexible models Ahmed BenSaïda1 Sabri Boubaker2,3 Duc Khuong Nguyen4 Skander Slim1 HEC Sousse—LaREMFiQ Laboratory, University of Sousse, Tunisia South Champagne Business School, Troyes, France International School, Vietnam National University, Hanoi, Vietnam IPAG Business School, Paris, France Correspondence Ahmed BenSaïda, IHEC Sousse, BP 40, Route de la ceinture, Sahloul 3, 4054, Tunisia Email: ahmedbensaida@yahoo.com Abstract Managing market risk under unknown future shocks is a critical issue for policymakers, investors, and professional risk managers Despite important developments in market risk modeling and forecasting over recent years, market participants are still skeptical about the ability of existing econometric designs to accurately predict potential losses, particularly in the presence of hidden structural changes In this paper, we introduce Markov-switching APARCH models under the skewed generalized t and the generalized hyperbolic distributions to fully capture the fuzzy dynamics and stylized features of financial market returns and to generate value-at-risk (VaR) forecasts Our empirical analysis of six major stock market indexes shows the superiority of the proposed models in detecting and forecasting unobservable shocks on market volatility, and in calculating daily capital charges based on VaR forecasts K E Y WO R D S APARCH, flexible distribution, volatility, value-at-risk, regime-switching I N T RO DU CT ION Stock markets around the world have, over the last two decades, experienced large fluctuations and swings, which considerably increased the risk of investment in stock portfolios The subprime crisis of 2007 in the USA and the subsequent global financial crisis of 2008–2009 made investors lose confidence in stock markets and fear cross-market transmission of contagious shocks Market fluctuations in times of increasing economic and financial uncertainty also contributed to amplify this tendency (Caldara, Fuentes-Albero, Gilchrist, & Zakrajs˘ek, 2016; Popp & Zhang, 2016) These facts naturally call for a better understanding of financial market dynamics in the context of financial globalization and an improvement of financial models to accurately estimate and forecast market risk Failure to so could lead to huge portfolio losses and possible financial disasters In this paper, we address the issue of market risk forecasting through the development of highly flexible models Journal of Forecasting 2018;1–15 based on a Markov-switching asymmetric power autoregressive conditional heteroskedasticity (APARCH) process (msAPARCH, henceforth) under the skewed generalized t (SGT) and generalized hyperbolic (GH) distributions We typically show how this type of model can be used to accurately forecast the market risk of a diversified portfolio of stocks, represented by its value-at-risk (VaR), in the presence of unobservable shocks that affect market volatility The main advantage of the proposed models is the possibility to capture not only the hidden structural changes in the conditional volatility processes of portfolio returns, but also the fuzzy dynamics and stylized facts of financial market returns Our motivation is threefold First, the VaR has been recommended by the Basel Committee on Banking Supervision (2006), (2009) as a meaningful tool to measure and monitor market risk, either on the downside or upside of market movements The VaR technique can be easily implemented It provides a straight way for investors to estimate the potential loss over a certain time period wileyonlinelibrary.com/journal/for Copyright © 2018 John Wiley & Sons, Ltd at a given confidence interval, and helps financial institutions determine their minimum capital requirements under the Basel regulatory framework Second, the identification of an appropriate model for VaR forecast is challenging While the Basel market risk framework recommends the use of VaR, the choice of methodologies for VaR calculations is left for the financial institutions Recent research makes it clear that standard VaR generates unreliable results due to the existence of stylized facts of asset and portfolio returns including, among others, return time variations, asymmetric volatility, heavy-tailed distributions, and nonlinearity in both mean and volatility dynamics (see, e.g., Aloui, Aïssa, & Nguyen, 2011) Thus financial models, which not incorporate these irregular patterns, may produce biased VaR estimations leading to inappropriate investment and financial policies Finally, structural changes under complex economic processes, abrupt policy adjustments, and sudden supply/demand shocks to the economies are important characteristics of portfolio returns that require careful modeling For instance, Sarno and Valente (2005) document that a vector equilibrium correction model of stock returns, which allows for both regime-switching behavior and international spillovers across stock market indexes, outperforms alternative models in terms of market timing ability and density forecasting performance A large literature has been devoted to the VaR estimation and forecasting using various methodologies (e.g., Billio & Pelizzon, 2000; Giot & Laurent, 2003; Nieto & Ruiz, 2016; Rodríguez & Ruiz, 2012; Salhi, Deaconu, Lejay, Champagnat, & Navet, 2016) to the extent that VaR estimates are central to ass et allocation and capital adequacy decisions of banking and financial institutions For example, Giot and Laurent (2003) use several univariate and multivariate GARCH-based models with the skewed-t distribution to model the VaR for traders that have both short and long trading positions Their empirical evidence shows that the skewed-t APARCH model performs rather well for all portfolios under consideration While considering both downside and upside risks (short and long positions, respectively), the majority of previous studies assume, however, GARCH-based approaches with normal or Student's t-distributions and not take regime shifts into account Our study thus goes one step further by simultaneously incorporating regime shifts and flexible distributions to model the conditional return and volatility before the VaR is estimated and predicted Using daily data of six international stock market indexes, we show that our proposed msAPARCH models under the skewed generalized t and the generalized hyperbolic distributions outperform the GARCH-t benchmark model in data fitting and forecasting the future volatility Moreover, the new msAPARCH BENSAÏDA ET AL models perform at least as well as the benchmark model in terms of VaR forecasting performance Indeed, the empirical results of the backtesting procedures for the 5% and the 1% VaR vary across evaluation tests, portfolios, and trading positions When VaR forecasts are used to compute daily capital charges, our results indicate that the msAPARCH models provide the most efficient allocation of regulatory capital for both long and short trading positions Therefore, financial institutions can achieve substantial savings in daily capital charges when flexible Markov-switching models are used to calculate VaR on sell-side positions The remainder of this paper is organized as follows Section develops the methodology Section discusses the data and presents the empirical estimation Section exposes the forecasting results and provides insights into the financial and economic implications of the proposed models in terms of daily capital charges Section concludes the paper 2.1 THEO RETICAL D EVELOPMENT Regime-switching APARCH model The asymmetric power ARCH (APARCH) model developed by Ding, Granger, and Engle (1993) is considered as highly flexible, since it can nest a large variety of other GARCH-type models Some authors (see Diamandis, Drakos, Kouretas, & Zarangas, 2011; Giot & Laurent, 2003; Nieto & Ruiz, 2016; Rodríguez & Ruiz, 2012) suggest measuring the VaR using asymmetric models, such as APARCH to cope with different responses of the volatility to negative and positive shocks The fractionally integrated version of this model (FIAPARCH) is not recommended in this context, because incorporating a long-memory effect in the volatility while computing the VaR is against the Basel accords, which requires short-run forecasts (Nieto & Ruiz, 2016) These fractionally integrated models provide, however, a good fit compared to their normal versions (Gencer & Demiralay, 2016; Slim, Koubaa, & BenSaïda, 2017) To the best of our knowledge, incorporating a regime-switching version in the APARCH model has never been conducted in the literature, owing to an aggravated problem of tractable likelihood function when the dynamics of the conditional volatility are excessively complex Indeed, Sajjad, Coakley, and Nankervis (2008) compared the performance of VaR forecasts of a regime-switching GARCH (MS-GARCH) with a single-regime APARCH model and found substantial improvements in the MS-GARCH Dendramis, Spungin, and Tzavalis (2014) found that the performance of a BENSAÏDA ET AL regime-switching model is considerably improved if combined with the EGARCH effect Forecasting becomes more complicated when the conditional distribution differs from one regime to another—in the case that it has one or more regime-dependent parameters—because each regime is characterized by a separate quantile depending on the distribution Other studies (Billio & Pelizzon, 2000; Salhi et al., 2016) forecasted the VaR with regime-switching distributions under the assumption of homoskedastic volatility Furthermore, as pointed out by Hamilton and Susmel (1994), the tractability problem prevents the numerical estimation of conditional volatility regime-switching models from properly reaching convergence, because the volatility at any given time t depends on all paths engendered by the unobservable different regimes, and the likelihood function becomes intractable This problem is worsened when the number of estimated parameters in the conditional variance equation increases A solution was recently proposed by BenSaïda (2015) and proved to overcome the path dependency problem encountered in the numerical estimation of regime-switching conditional volatility models It consists of dissociating between the diverse volatility states; hence each conditional volatility process depends only on its lagged values and the residuals within the same generating regime Finally, the likelihood function (not in logarithm form) is computed as a weighted average between the different likelihoods under each regime multiplied by the respective filtered probabilities Let {st } be a state variable indicating a first-order Markov chain, that is, for K regimes st = {1, 2, … , K} For a given time series rt , for example, the returns of a given financial asset, the regime-switching APARCH(P, Q) model (msAPARCH) is expressed in Equation 1: ⎧ rt = ct,st + 𝜀t,st √ ⎪ i.i.d 𝜀 ⎪ t,st = zt,st ht,st , zt,st ∼ (0, 1) P Q ⎨ ∑ ∑ ( )𝛿 𝛿k ∕2 ⎪ h𝛿k ∕2 = 𝜅k + 𝛽 h + 𝛼𝑗,k ||𝜀t−𝑗,st || − 𝛾𝑗,k 𝜀t−𝑗,st k i,k t,s t−i,s t t ⎪ 𝑗=1 i=1 ⎩ (1) All the coefficients 𝜅 k , 𝛽 i,k , 𝛼 j,k , 𝛾 j,k , 𝛿 k are held constant within the same unknown regime k ∈ st , yet different from one regime to another To ensure the positivity of the conditional volatility ht , the coefficients must be 𝜅 > 0, 𝛽i ⩾ 0, 𝛼𝑗 ⩾ 0, 𝛿 ⩾ 0, and − < 𝛾𝑗 < The parameters 𝛾 j represent the leverage effect, and the power term 𝛿 denotes the Box–Cox transformation of the conditional standard deviation One major drawback of the APARCH model is the derivation of the stationarity condition in Equation 2, which depends on the density of zt,st :1 Q P ∑ ( ) 𝛿k ∑ | | 𝛼𝑗,k E |zt,st | − 𝛾𝑗,k zt,st + 𝛽i,k < 𝑗=1 (2) i=1 To construct a tractable likelihood (BenSaïda, 2015), the conditional variance of a regime-switching msAPARCH model at time t in Equation can be written as 𝛿 ∕2 ⎛ ht,11 ⎞ ⎜ ⋮ ⎟= ⎜ 𝛿K ∕2 ⎟ ⎝ ht,K ⎠ ( 𝛿1 ∕2 ) ⎞ ⎛ ht−i,1 𝛽i,1 ⋮ + ⊙⎜ ⋮ ⎟ ⎜ 𝛿K ∕2 ⎟ 𝛽i,K i=1 ⎝ ht−i,K ⎠ ( ) [ | ( 𝛾𝑗,1 ) | Q 𝛼𝑗,1 ∑ | rt − ct,1 | |− | ⋮ ⋮ ⋮ + ⊙ | | | − c 𝛼 r 𝛾𝑗,K 𝑗=1 𝑗,K t,K || | t 𝜅1 ⋮ 𝜅K ( ⊙ ) P ∑ rt − ct,1 ⋮ rt − ct,K ( ⎛ 𝛿1 ⎞ )]◦⎜⎜ ⋮ ⎟⎟ ⎜𝛿 ⎟ ⎝ K ⎠, (3) where ⊙ represents the Hadamard product, and A◦B is the Hadamard or element-wise matrix power The state variable {st } evolves according to a first-order Markov chain with a probability transition matrix P, such that each column sums to one Each element of P denotes the probability of being in regime j at time t, knowing that at time t − the regime was i It is expressed in Equation 4: 𝑝i,𝑗 = Prob (st = 𝑗|st−1 = i) (4) In the case of two regimes, the transition matrix becomes ( ) 𝑝 1−q P= 1−𝑝 q The ergodic (unconditional probability) 𝝅 is the eigenvector of P corresponding to the unit eigenvalue normalized by its sum In other words, P𝝅 = 𝝅, and 1′K 𝝅 = 1, where 1K is a (K × 1) column vector with all elements equal to one (Hamilton, 1994, p 681) In the special case of two regimes, the unconditional probabilities are expressed as follows: ⎧ 1−q ⎪ 𝜋1 = 2−𝑝−q ⎨ 1−𝑝 ⎪ 𝜋2 = 2−𝑝−q ⎩ The maximum likelihood (ML) is a preferable estimation method for Markov-switching conditionally heteroskedastic models (Hamilton & Susmel, 1994) A key ingredient to compute the likelihood function over all specified regimes is the ex ante (filtered) probability 𝑝𝑗,t = Prob (st = 𝑗|It−1 ), or the probability of being in regime j at Ding et al (1993) derive the stationarity condition for Gaussian errors, Karanasos and Kim (2006) for Student's t, GED, double exponential, and Giot and Laurent (2003) for the skewed Student's t-distribution 4 BENSAÏDA ET AL time t given the information available at t−1, which evolves according to the filtering equation defined in Equation 5: K ∑ 𝑝i,𝑗 𝑓 (rt |st−1 = i) 𝑝i,t−1 𝑝𝑗,t = i=1 K ∑ , (5) 𝑓 (rt |st−1 = k) 𝑝k,t−1 k=1 where pi,j are the transition probabilities defined in Equation 4, and f(·) is the conditional density function.2 Therefore, the log-likelihood function to maximize is expressed in Equation 6: [K ] T ∑ ∑ LLF = ln 𝑝𝑗,t 𝑓 (rt |st = 𝑗) (6) t=1 𝑗=1 Finally, the regime-independent conditional volatility at time t is the weighted average of the different volatilities in each sate; that is, ht = K ∑ 𝑝𝑗,t ht, 𝑗 (7) 𝑗=1 2.2 and a location parameter to impose zero mean: ( ) B 𝜂2 , 𝜓−1 𝜂 𝜇 = −2𝜃𝜆 ( ) , 𝜓 B 𝜂, 𝜂 where sgn(·) is the sign function, and B (·, ·) is the beta function The shape parameters 𝜂 > and 𝜓 > 2, and the skewness parameter |𝜆| < BenSaïda (2015, Theorem 2) has developed a closed-form quantile function needed in VaR forecasting:3 ) ] [ ( + 𝜆 𝜃 sgn x − 1−𝜆 F −1 (x) = 𝜇 + [ (9) ]1 , 𝜂 )−1 ( 𝜓 , −1 I −1 2x−(1−𝜆) 𝜂 𝜂 1−𝜆 𝜆+sgn (x− ) where x ∈ [0, 1] and Iw−1 (a, b) is the inverse regularized incomplete beta function Flexible conditional distributions The conditional distribution of zt in Equation can be of any type An essential condition, however, is that the employed distribution must be defined over the whole real interval R, and must have zero mean and unit variance (BenSaïda, 2015; Slim et al., 2017) Bali and Theodossiou (2007) and Cheng and Hung (2011) suggest using the skewed generalized t (SGT) distribution for VaR forecasting owing to its attractiveness in nesting a variety of other distributions Dendramis et al (2014) point out that a skewed distribution is preferred in VaR forecasting In our framework, we employ two highly flexible distributions, namely the SGT and the generalized hyperbolic (GH) that allow skewness Together, they can nest nearly 30 other distributions, including Hansen (1994) skewed t, generalized error distribution (GED), variance gamma, generalized inverse Gaussian, and more (see BenSaïda & Slim, 2016), for a detailed description 2.2.1 with a scaling parameter to impose unit variance: ) ( B 𝜂1 , 𝜓𝜂 , 𝜃=√ )2 ( ( ) ( 𝜓 ) ( 𝜓−2 ) 𝜓−1 + 3𝜆2 B 𝜂 , 𝜂 B 𝜂 , 𝜂 − 4𝜆2 B 𝜂 , 𝜂 2.2.2 The literature has proposed several parametrizations of the GH functional form, with a common drawback of ambiguous meaning of the parameters (Scott, Würtz, Dong, & Tran, 2011), and where it is difficult to separate the fluctuations in the mean and variance from the fluctuations in the shape of the conditional density Consequently, we adopt the parametrization of BenSaïda and Slim (2016), where the GH density is re-expressed in terms of the mean, variance, and skewness The standardized GH density function (with zero mean and unit variance) is thus defined in Equation 10: √ ( )𝜈−1 [ ]𝜈−1 𝜁 − 𝜚2 ( zt − 𝜇 )2 +1 𝑓GH (zt ; 𝜈, 𝜁, 𝜚) = √ 𝜗 2𝜋 𝜗K𝜈 (𝜁 ) ( ) √ ( z − 𝜇 )2 𝜁 t × K𝜈− √ +1 𝜗 − 𝜚2 The skewed generalized t The standardized SGT density (with zero mean and unit variance) is defined in Equation as 𝜂 𝑓SGT (zt ; 𝜂, 𝜓, 𝜆) = )[ ] 𝜓+1 , ( 𝜂 |zt −𝜇|𝜂 𝜓 𝜃 B 𝜂 , 𝜂 + 1+sgn z −𝜇 𝜆 𝜂 𝜃𝜂 ( (t )) (8) e The filtered probability has a first-order recursive structure To jump-start the estimation, we set 𝑝𝑗,1 = K1 , ∀𝑗 = 1, … , K √𝜁 𝜚 1−𝜚2 ( zt −𝜇 𝜗 ) , (10) with a scaling parameter to impose unit variance: [ ( )]− K𝜈+1 (𝜁 ) 𝜚2 K𝜈+2 (𝜁) K𝜈+1 (𝜁)2 𝜗= , + − 𝜁K𝜈 (𝜁 ) K𝜈 (𝜁) − 𝜚2 K𝜈 (𝜁)2 The generalized hyperbolic Bali and Theodossiou (2007) and Cheng and Hung (2011) used extensive numerical integrations to compute the quantile function of the SGT, which is heavily expensive in computer time BENSAÏDA ET AL and a location parameter to impose zero mean: K𝜈+1 (𝜁 ) 𝜚 , 𝜇 = −𝜗 √ − 𝜚 K𝜈 (𝜁 ) 3.1 where K𝜈 (·) is the modified Bessel function of the third kind The shape parameters 𝜁 > and 𝜈 ∈ R, and the skewness parameter |𝜚| < Unfortunately, there is still no closed-form quantile function of the GH As a consequence, and for the purpose of VaR forecasting, we compute the quantiles using numerical integrations 2.3 Forecasting volatility and VaR under regime switching The regime-independent m-step-ahead optimal volatility forecast evaluated at time t, ĥ t+m , is defined in Equation 11: ĥ t+m = K ∑ 𝑝̂𝑗,t+m ĥ t+m, 𝑗 (11) 𝑗=1 In other words, we compute the m-step-ahead volatility forecasts under each regime j, j = 1, … , K, denoted ĥ t+m, 𝑗 using the APARCH-like variance Equation Next, we filter the probability at m periods( ahead 𝑝̂𝑗,t+m as) in Equation 5; the conditional density 𝑓 r̂t+m |s(t+m−1) = 𝑗 is computed using the regime-dependent conditional forecasted volatility ĥ t+m, 𝑗 Similarly, the VaR forecasts for an investor who has taken either a long or short position (VaRL,t and VaRS,t , respectively) at time t depend on (i) the forecasted conditional volatility and on (ii) the quantiles inferred from the employed distribution In a regime-switching context, however, the conditional distribution f, which can be either SGT or GH, differs from one regime to another due to regime-dependent parameters Consequently, the VaR at 𝛼% significance level can be expressed as follows: ( K √ ) ⎧ ∑ ⎪ VaRL,t = 𝑝̂𝑗,t ĉ t,𝑗 + q𝑗 (𝛼) ĥ t,𝑗 ⎪ 𝑗=1 (12) ( ⎨ K √ ) ∑ ⎪ VaR = ̂ 𝑝̂𝑗,t ĉ t,𝑗 + q𝑗 (1 − 𝛼) ht,𝑗 , S,t ⎪ 𝑗=1 ⎩ where qj (𝛼) and q𝑗 (1 − 𝛼) are the left and right quantiles under regime j at the significance level 𝛼 and − 𝛼, respectively Furthermore, ĉ t,𝑗 and ĥ t,𝑗 are the estimated (or forecasted) conditional mean and variance during regime j, respectively The forecasted conditional variance is deduced using Equation 11 𝑝̂𝑗,t represents the estimated/forecasted filtered probability as computed from Equation 5.4 We develop all necessary codes under the MATLABⓇ programming language Codes for estimations are based on the msgarch toolbox of BenSaïda (2015) EMPIRICAL RESULTS Data and summary statistics The data are composed of six Morgan Stanley Capital International (MSCI) daily closing price indexes collected from Datastream These are World, Emerging Markets (EM), BRIC countries, US, UK, and Germany from January 1, 2000 to December 31, 2015 Prices are plotted in Figure 1, and the returns (logarithmic difference of the prices) are plotted in Figure In our study, stock market indexes are viewed as synthetic assets that are made of multiple tradable assets One may make these investments through holding the stocks issued by exchange traded funds (ETFs) to benefit from diversification effects with tax advantage and lower management fees Moreover, it is also an industry standard for risk management to consider market indexes to assess the consistency of VaR models, to the extent that financial institutions hold various trading lines in domestic and foreign equities The approach we undertake is general enough and can be applied to a set of known tradable assets.5 Descriptive statistics of the returns are presented in Table The Jarque–Bera test overwhelmingly rejects normality for all samples The means of the returns are statistically not different from zero; hence we not include drift terms ct in the mean equations (Equation 1) 3.2 GARCH model under Student's t As a benchmark, we select the classical single-regime GARCH(1, 1) model under Student t-distribution (GARCH-t) Indeed, numerous studies (e.g., Hansen & Lunde, 2005; Slim et al., 2017) argue that sophisticated models rarely beat the GARCH specification when evaluating forecasting performance Our choice for Student's t is based on several studies showing that this distribution yields impressive fitting and forecasting performance when combined with the GARCH(1, 1) (e.g., Salhi et al., 2016) Estimation results are presented in Table 3.3 Regime-switching APARCH models Financial markets are usually volatility states, high and low, market conditions, respectively BenSaïda, 2015) Furthermore, subject to two different reflecting bear and bull (Dendramis et al., 2014; practitioners usually set MSCI indexes are tracked by ETFs, which are marketable securities, and traded publicly like common stocks on stock exchanges ETF investors profit from price gains and dividends of the index constituents One index may have several ETFs associated with it For instance, the MSCI World is currently tracked by 11 ETFs 6 BENSAÏDA ET AL EM 1800 1600 1400 1200 1000 800 Price Price World 00 02 04 06 08 10 12 1400 1200 1000 800 600 400 14 00 02 04 06 08 Year Year BRIC US 10 12 14 10 12 14 10 12 14 10 12 14 10 12 14 12 14 500 2000 Price Price 400 300 200 1500 1000 100 00 02 04 06 08 10 12 14 00 04 06 08 Year UK Germany 1000 Price 2000 Price 02 Year 1500 1000 00 800 600 400 02 04 06 08 10 12 14 00 02 04 06 Year 08 Year FIGURE Index prices [Colour figure can be viewed at wileyonlinelibrary.com] World EM 0.1 0.1 Return Return 0.05 0 -0.05 00 02 04 06 08 10 12 -0.1 00 14 06 US 0.1 -0.1 02 04 06 08 10 12 -0.1 00 14 02 04 06 Year UK Germany 0.1 Return Return 08 Year 0.1 -0.1 00 08 BRIC Return Return 04 Year 0.1 00 02 Year 02 04 06 08 10 12 14 -0.1 00 02 Year 04 06 08 10 Year FIGURE Index returns [Colour figure can be viewed at wileyonlinelibrary.com] the recursive lags of the conditional volatility equation to one Thus we perform our analysis by assuming that the market evolves according to two regimes (K = 2), high and low volatility states The orders of the APARCH model are set to P = and Q = in Equation Tables and present the estimation results of the msAPARCH(1, 1) models BENSAÏDA ET AL TABLE Summary statistics for MSCI returns Country World EM 3.8E−5a Mean BRIC 0.0001a USA 0.0002a UK 7.2E−5a Germany −2.0E−5a 2.7E−6a Minimum −0.0733 −0.0999 −0.1191 −0.0951 −0.0916 Maximum 0.0910 0.1007 0.1351 0.1104 0.0926 0.1113 SD 0.0104 0.0123 0.0149 0.0125 0.0120 0.0151 −0.3357 −0.4882 −0.3386 −0.2023 −0.1725 −0.0185 Skewness −0.0867 Kurtosis 10.582 10.965 11.948 11.401 Number of obs 4,174 4,174 4,174 4,174 4,174 9.3751 4,174 J-B statistic 10, 049b 11, 170b 13, 968b 12, 269b 7, 069b 3, 228b a Mean is statistically not different from zero at 5% confidence level confidence level b Normality 7.3147 is rejected at 5% TABLE GARCH model results Coefficients 𝜅 𝛼1 𝛽1 d.f Maximum log-likelihood Bayesian criterion BIC per observation World EM BRIC USA UK Germany 8.8E−7*** (3.929) 0.0830*** 2.1E−6*** (4.440) 0.0828*** 3.4E−6*** (4.686) 0.0876*** 1.2E−6*** (4.094) 0.0882*** 1.3E−6*** (4.225) 0.1008*** 1.8E−6*** (3.912) 0.0892*** (9.523) (9.234) (9.385) (9.450) (10.21) (9.842) 0.9091*** (101.3) 8.6197*** 0.9019*** (87.29) 9.0591*** 0.8951*** (84.40) 8.8295*** 0.9053*** (98.70) 6.7302*** 0.8921*** (91.81) 9.0147*** 0.9050*** (98.75) 9.2868*** (7.305) (7.763) (7.447) (8.770) (6.917) (7.153) 14,038.6 13,199.1 12,421.0 13,394.8 13,440.8 12,352.4 −28,043.9 −26,364.8 −24,808.7 −26,756.2 −26,848.2 −24,671.4 −6.7187 −6.3164 −5.9436 −6.4102 −6.4322 −5.9107 Note This table reports the estimation output of the GARCH model under Student's t-distribution Numbers in parentheses are the asymptotic t-statistics of the estimated coefficients d.f stands for Student's t degrees of freedom Asterisks indicate significance at a confidence level of: ***1% under the SGT and GH, respectively The stationarity conditions in Equation are verified at the obtained maximum log-likelihoods Estimation performance is measured with the Bayesian information criterion (BIC) per observation Markov switching models are capable of detecting periods of crisis and periods of tranquility The transition between these periods is governed by an estimated probability matrix Therefore, to enhance the validity of the results—both estimation and forecasting—the in-sample period should cover different market conditions, such as pre- and postfinancial crisis, in order to detect multiple sets of estimated coefficients, where each set corresponds to one regime Estimation results show that the msAPARCH models under SGT and GH are remarkably good fits for financial returns due to their high flexibility The estimated coefficients have different values depending on the regime Both models are preferred to the classical GARCH model according to the BIC Moreover, the msAPARCH-SGT slightly outperforms the msAPARCH-GH for all markets FO RECASTING RESULTS In this section, we compare (i) the performance of each model in forecasting the one-step-ahead volatility, and (ii) the 1-day-ahead VaR The forecast procedures are implemented using a rolling window technique More precisely, the parameter estimates are updated every month starting from the beginning of the out-of-sample period, while keeping the sample size used for estimation fixed; that is, the window length contains 4,174 observations The 1-day-ahead volatility and VaR are hence forecasted for the following month using the last coefficient updates.6 4.1 One-step-ahead volatility forecasts To statistically compare the results of the 1-day-ahead forecasting accuracy of the volatility models, we utilize the model confidence set (MCS) of Hansen, Lunde, and Nason Ardia and Hoogerheide (2014) study several estimation frequencies—daily, weekly, monthly, and quarterly—and find that the impact of the updating frequency (and hence of the parameter estimates) when performing rolling window estimation on the quality of VaR forecasts is remarkably insignificant Therefore, we consider that the monthly update of the coefficients is sufficient 8 BENSAÏDA ET AL TABLE Regime-switching APARCH model under SGT distribution results Coefficients Regime 𝜅 𝛼1 𝛾1 𝛽1 EM BRIC USA UK Germany 0.0002 (1.050) 0.0903 9.3E−5 (0.718) 0.1177*** 0.0002 (0.697) 0.2122*** 8.7E−6 (0.784) 0.0555 5.5E−5 (0.245) 0.3040 0.0002 (0.141) 0.2711** (1.421) (4.197) (4.540) (0.967) (1.332) (2.113) 1.0000 (0.859) 0.8835*** 1.0000*** (436.9) 0.8189*** 0.7818*** (5.926) 0.7400*** 0.9513 (0.797) 0.9325*** 0.8590*** (3.809) 0.5901*** 0.8089*** (3.473) 0.7915*** (247.5) (26.82) (17.82) (85.76) (4.282) (10.70) 1.0778*** (2.833) 1.6151*** 1.3844*** (4.992) 1.9723*** 1.2537*** (4.368) 2.1176*** 1.5898*** (6.171) 2.3896*** 1.3453* (1.912) 0.2770*** 0.6852** (2.147) 0.3643*** (5.476) (8.501) (6.893) (10.86) (5.766) (5.220) 21.527 (585.1) −0.1824*** 42.295*** (7.6E4) −0.2665*** 25.869*** (1.3E4) −0.1491** 24.536*** (735.4) −0.2710*** 27.947*** (2.4E5) −6.6E−9 36.710*** (1.0E6) −4.3E−5 (−0.119) (−3.953) (−1.962) (−5.082) (−0.013) (−0.102) 2.3166 1.3136 1.3026 1.6824 2.3485 1.2385 1.6E−5 2.2E−5 1.9E−6 7.4E−5 4.6E−5 5.9E−5 (0.561) (0.403) (0.437) (0.397) (1.314) (1.348) 0.0228** (2.279) 0.9950 0.0365*** (4.198) 0.1378 0.0359 (3.566) 0.1213 0.1293 (0.327) 0.9993 0.0636*** (3.647) 0.9669*** 0.0687*** (2.744) 0.9771** (1.439) (0.874) (1.228) (0.237) (2.857) (2.142) 𝛿 0.9828*** (51.92) 1.0163*** 0.9635*** (114.7) 1.2611** 0.9535*** (101.83) 2.0002*** 0.8094*** (13.02) 1.3786*** 0.9270*** (128.0) 1.2765*** 0.9166*** (105.9) 1.3590*** 𝜂 (5.747) 3.0811*** (2.390) 1.8602*** (3.782) 2.2144*** (2.715) 0.6694*** (8.002) 2.2479*** (8.478) 2.3147*** 𝜓 (8.719) 47.275*** (11.68) 29.652*** (11.60) 13.292*** (10.55) 30.945*** (21.62) 28.880*** (16.31) 33.919*** 𝛿 𝜂 𝜓 𝜆 Regime World Expected duration 𝜅 𝛼1 𝛾1 𝛽1 𝜆 Expected (1.035) (3057) (1404) (1391) (28449) (40970) −0.0114 (−6.000) 1.0000 −0.0856* (−1.770) 1.7798 −0.1464*** (−3.575) 2.2073 −1.9E−6 (−0.002) 1.2384 −0.1319*** (−4.797) 15.141 −0.1426*** (−4.898) 7.5000 0.5683*** (6.060) 0.0000 0.2388 (1.380) 0.4381*** 0.2323* (1.661) 0.5470*** 0.4056*** (4.053) 0.1925** 0.5742*** (7.063) 0.9340*** 0.1926*** (3.175) 0.8667*** (7.2E−6) 0.6985 (3.030) 0.4247 (4.614) 0.3711 (2.357) 0.5760 (54.16) 0.1343 (28.39) 0.1417 14,170.3 13,281.8 12,499.0 13,576.4 13,839.7 12,591.5 −28,190.5 −6.7538 −26,413.4 −6.3281 −24,848.0 −5.9530 −27,002.7 −6.4693 −27,529.2 −6.5954 −25,033.0 −5.9974 duration Transition matrix P1,1 = p P2,2 = q Unconditional probability 𝜋 of regime (𝜋 = − 𝜋 ) Maximum log-likelihood Bayesian criterion BIC per observation Note This table reports the estimation output of the Markov-switching APARCH model (two regimes) under the skewed generalized t-distribution Regime corresponds to the high volatility state (on average) Numbers in parentheses are the asymptotic t-statistics of the estimated coefficients computed using White (1982) method Asterisks indicate significance at a confidence level of: *10%; **5%; ***1% (2011) The MCS enables the comparison of a selection of competitive models under a specific loss function Contrary to the superior predictive ability (SPA) test of Hansen (2005), which enables the comparison of the performance of a benchmark forecasting model simultaneously to that of a whole set of competitors under a specific loss func- BENSAÏDA ET AL TABLE Regime-switching APARCH model under GH distribution results Coefficients Regime 𝜅 𝛼1 𝛾1 𝛽1 𝛿 𝜈 𝜁 𝜚 Regime Expected duration 𝜅 𝛼1 𝛾1 𝛽1 𝛿 𝜈 𝜁 𝜚 Expected World EM BRIC USA UK Germany 5.7E−5 (1.291) 0.0553 1.5E−5 (0.576) 0.0682*** 0.0003 (0.775) 0.1833*** 0.0002 (0.598) 0.1190** 0.0001 (1.371) 0.1120*** 0.0003 (0.486) 0.1548*** (0.078) (3.484) (4.702) (2.430) (10.69) (8.922) 1.0000 (0.047) 0.9329*** 1.0000*** (448.5) 0.8181*** 0.8508*** (5.589) 0.7546*** 0.9876*** (2.720) 0.7770*** 1.0000*** (137.3) 0.8789*** 0.9999*** (510.8) 0.8853*** (120.3) (22.65) (21.48) (15.53) (68.92) (56.75) 1.2023*** (7.718) 3.6827*** 1.8378*** (4.980) 4.1313** 1.2414*** (4.618) −4.8603*** 1.1414*** (3.688) −1.6268 1.0894*** (7.996) −3.6657*** 0.5689*** (4.967) −2.7595*** (5.235) (2.462) (−8.569) (−0.664) (−1912) (−1947) 2.0E−7 (0.001) −0.1359*** 0.7386 (0.092) −0.2369*** 28.351*** (22.68) −0.6808*** 1.1010* (1.925) −0.4862 12.510*** (1.7E4) −0.4318*** 37.043*** (2.3E4) −0.3902 (−4.952) (−3.541) (−3.749) (−0.949) (−4.353) (−1.223) 270.95 1.5421 1.4543 1.0775 2.2445 1.2540 4.5E−5 0.0001 1.6E−5 5.1E−5 2.0E−7 8.9E−6 (0.265) (0.357) (0.522) (1.073) (1.606) (0.600) 0.0000 (0.000) 0.2332*** 0.0403*** (3.948) 0.2466 0.0471*** (4.751) −0.0120*** 0.0689*** (2.819) 0.9681** 0.0075*** (3.316) 1.0000*** 0.0277*** (3.090) 0.9977*** (9E12) (1.384) (−1.0E8) (2.185) (5.3E4) (731.1) 0.0000 (0.000) 2.1699*** 0.9667*** (125.6) 0.5428 0.9517*** (95.29) 1.4981*** 0.9322*** (93.56) 1.2675*** 0.9803*** (192.8) 2.0946*** 0.9510*** (136.4) 1.6985*** (1706) (1.380) (3.305) (6.774) (350.0) (4.720) −0.9537 (−0.165) 1.9287 −14.238 (−0.948) 0.3476 −8.6448* (−1.758) 13.273 0.3130 (0.020) 26.341 22.023*** (1.1E4) 7.9757*** 36.721*** (2.8E4) 41.320*** (0.065) (0.723) (1.125) (0.812) (8518) (3.1E4) −0.3415 (−0.121) 1.7673 −0.9936*** (−79.90) 1.1802 −0.4044*** (−3.076) 2.0348 −0.5715** (−2.325) 1.9805 −0.4343** (−1.973) 1.0000 −0.5862** (−2.534) 2.3968 0.9963*** (32.76) 0.4342 0.3516** (2.085) 0.1527 0.3124** (2.533) 0.5085*** 0.0720 (0.791) 0.4951*** 0.5545*** (5.326) 0.0000 0.2025** (2.539) 0.5828*** (0.167) 0.9935 (0.933) 0.5665 (4.822) 0.4168 (4.537) 0.3524 (0.671) 0.6918 (8.824) 0.3435 duration Transition matrix P1,1 = p P2,2 = q Unconditional probability 𝜋 of regime (𝜋 = 1-𝜋 ) Maximum log-likelihood Bayesian criterion BIC per observation 14,140.3 13,280.8 12,496.7 13,564.4 13,580.4 12,506.4 −28,130.5 −6.7395 −26,411.5 −6.3276 −24,843.4 −5.9519 −26,978.8 −6.4635 −27,010.6 −6.4712 −24,862.7 −5.9566 Note This table reports the estimation output of the Markov-switching APARCH model (two regimes) under the generalized hyperbolic distribution Regime corresponds to the high volatility state (on average) Numbers in parentheses are the asymptotic t-statistics of the estimated coefficients computed using White (1982) method Asterisks indicate significance at a confidence level of: *10%; **5%; ***1% tion, the MCS does not assume that a particular model is the true model The MCS is performed by using the mean squared error (MSE) and the quasi-likelihood (QLIKE) loss functions These loss functions are invariant to noise in the proxy for the true unobserved volatility, as proved by Patton (2011), Proposition The loss functions are 10 BENSAÏDA ET AL TABLE Volatility forecasting performance Model MSE GARCH-t msAPARCH-SGT msAPARCH-GH QLIKE GARCH-t msAPARCH-SGT msAPARCH-GH World EM BRIC USA UK Germany 0.2231 0.2735 0.3951 0.1317 0.2472 0.8647 (0.3307) (0.0284) (0.0594) (0.3798) (0.5633) (0.4780) 0.2177 (1.0000) 0.2192 0.2628 (0.2868) 0.2613 0.3785 (1.0000) 0.3805 0.1264 (0.4238) 0.1259 0.2365 (1.0000) 0.2369 0.8666 (0.8760) 0.8608 (0.3307) (1.0000) (0.1602) (1.0000) (0.8605) (1.0000) 1.5791 1.7500 1.4611 1.5853 1.8279 1.5052 (0.3359) (0.2351) (0.6848) (0.8760) (0.7855) (0.6382) 1.7076 (1.0000) 1.7133 1.4421 (0.3101) 1.4354 1.5812 (0.6848) 1.5775 1.8927 (0.0775) 1.8182 1.4847 (0.8114) 1.4828 1.6145 (0.0181) 1.5564 (0.6822) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) Note This table reports the mean losses for the different volatility models over the out-of-sample period (January 1, 2016–June 26, 2017) with respect to two evaluation criteria (MSE ×107 and QLIKE) Consistent p-values of the model confidence set (MCS) of Hansen et al (2011) are reported (in parentheses) under the selected loss functions The model with the highest p-value is given the best rank expressed in Equations 13 and 14, respectively T𝑓 ( ) ∑ ̂ MSE = r̂t − ht T𝑓 t=1 ] T𝑓 [ r̂t2 ∑ r̂t QLIKE = − ln − T𝑓 t=1 ĥ t ĥ t 4.2 (13) (14) where Tf is the number of forecasting data points, r̂t2 and ĥ t refer to the squared-return and the variance forecast from a particular model, respectively We conduct out-of-sample volatility forecasting on an extended period starting from January 1, 2016 until June 26, 2017, with a total of 387 observations The results are reported in Table 5, √ and the one-step-ahead conditional standard deviations h∗t+1 are plotted in Figure Over- all, we observe significant improvements in the effectiveness of the volatility forecasting models after incorporating both regime dynamics and highly flexible distributions For the MSE loss function, the model confidence set shows that the msAPARCH-SGT model achieves the best performance for three markets (World, BRIC, and UK), because this model has the smallest value of the loss function among other forecasting models with p-values that are equal to one For EM, USA, and Germany, however, the msAPARCH-GH is the best performing model Regarding the QLIKE loss function, the msAPARCH-GH clearly outperforms the competing models for five markets (EM, BRIC, USA, UK, and Germany), and the msAPARCH-SGT provides the most accurate forecasts for World data VaR performance To measure the performance of the investigated VaR models, the predicted 1-day-ahead VaR for both long and short positions are compared to the realized profit and loss using a new set of improved tests, as introduced by Ziggel, Berens, Weiß, and Wied (2014), which are the unconditional coverage (UC), independence (IND), and conditional coverage (CC) tests The new set of tests analyze UC and IND assumptions by using simulations with i.i.d Bernoulli random variables One advantage of the new tests is that they perform better compared to existing backtesting procedures, mainly because of their finite-sample size and power properties The specification of the three tests is given in Appendix A The Basel committee further requires daily forecasts of the VaR for returns over a holding period of 10 days (Basel Committee on Banking Supervision, 2006) However, most researchers often predict the 1-day-ahead VaR that corresponds to returns over a holding period of day, since it provides the most reliable measure for model comparison (see, e.g., Ardia & Hoogerheide, 2014; Berger & Missong, 2014; Cheng & Hung, 2011; Dendramis et al., 2014; Nieto & Ruiz, 2016; Takahashi, Watanabe, & Omori, 2016), and because the Basel accords allow the 10-day-ahead VaR to be obtained from shorter period forecasts by using the square-root-of-time rule (Nieto & Ruiz, 2016).7 The square-root-of-time rule, however, can lead to overestimating the VaR under a “realistic” data-generating process for financial returns featuring serial dependence, mean reversion, and heavy tails (Daníelsson & Zigrand, 2006) A thorough analysis of the suitability of alternative timescale conversion rules that are designed to mitigate potential distortions in the multi-period VaR calculation is beyond the scope of the current paper and would be an interesting avenue for further research BENSAÏDA ET AL 11 World EM GARCH-t msAPARCH-SGT msAPARCH-GH Realized 0.04 GARCH-t msAPARCH-SGT msAPARCH-GH Realized 0.03 0.02 0.02 0.01 Jan 16 Jun 16 Jan 17 Jun 17 Jan 16 BRIC US GARCH-t msAPARCH-SGT msAPARCH-GH Realized 0.02 0.01 0.01 Jun 16 Jan 17 Jun 17 Jan 16 Jun 16 Jan 17 Time (days) Time (days) UK Germany 0.03 GARCH-t msAPARCH-SGT msAPARCH-GH Realized 0.02 Jun 17 GARCH-t msAPARCH-SGT msAPARCH-GH Realized 0.03 0.02 0.06 Jun 17 GARCH-t msAPARCH-SGT msAPARCH-GH Realized 0.04 0.02 0.01 Jan 16 Jan 17 Time (days) 0.03 Jan 16 Jun 16 Time (days) Jun 16 Jan 17 Jun 17 Jan 16 Time (days) Jun 16 Jan 17 Jun 17 Time (days) FIGURE One-step-ahead conditional standard deviations [Colour figure can be viewed at wileyonlinelibrary.com] We compute the 1-day-ahead VaR from January 1, 2016 until June 26, 2017, for a total of 387 observations Tables and present the backtesting results for the 5% and 1% VaR, respectively We report the two-tailed p-values for both long and short trading positions Although the models appear to be remarkably accurate, their performance varies substantially across markets and tests For comparison purposes, we refer to the GARCH-t as the benchmark model In the UC test, the best-performing model is the GARCH-t with the highest p-values for seven out of 12 cases for the 5% VaR, and eight out of 12 cases for the 1% VaR Regarding the IND test, the msAPARCH models outperform the benchmark in 10 cases for the 5% VaR; however, the benchmark performs as well as the regime-switching models for the 1% VaR (six cases for the benchmark against six cases for the msAPARCH) According to the CC test, both the msAPARCH-GH and the benchmark model are preferred for the 1% VaR, since they exhibit the highest p-values for all of the trading positions across markets In the relatively less extreme case of the 5% VaR, the most accurate forecasts are given by the regime-switching models in nine cases out of 12 4.3 Daily capital charges Finally, in order to provide further insights into the economic implications of the proposed VaR models, we compute the daily capital charges under the Basel II Accord (Basel Committee on Banking Supervision, 2006) The benefit of using daily capital charges (DCC) is twofold First, the DCC can be used as a supplemental measure for model selection, since it provides meaningful information about savings, in terms of capital requirements, incurred by VaR models with equal accuracy Second, the DCC is straightforwardly related to the cost of risk management, and hence it can help internal model builders to decide on the tradeoff between a correctly specified VaR and smooth capital requirements Formally, the risk capital charges on a daily basis are defined in Equation 15: } { 60 1∑ VaRt−i ; VaRt−1 , (15) DCCt = max (3 + k) 60 i=1 where DCCt denotes the daily capital charges at day t, VaRt−1 designates the 1% VaR for day t − and ⩽ k ⩽ is a penalty factor depending on the number 12 BENSAÏDA ET AL TABLE Backtesting results of the VaR at the 5% level Model Metric World Long Short EM Long Short BRIC Long Short US Long Short UK Long Short Germany Long Short GARCH-t PF UC IND CC PF 3.10% 0.070 0.050 0.010 3.62% 4.39% 0.496 0.714 0.738 6.20% 4.91% 0.992 0.454 0.524 5.17% 5.43% 0.702 0.434 0.398 5.94% 4.65% 0.816 0.506 0.734 5.69% 4.91% 0.870 0.964 0.094 4.65% 1.81% 0.000 0.856 0.084 5.17% 4.65% 0.852 0.240 0.284 8.27% 3.62% 0.234 0.392 0.332 4.91% 4.39% 0.572 0.132 0.598 6.98% 3.10% 0.074 0.144 0.058 8.79% 5.43% 0.676 0.086 0.476 8.01% msAPARCH-SGT msAPARCH-GH UC 0.164 0.376 0.908 0.424 0.482 0.700 0.828 0.004 0.972 0.086 0.004 0.012 IND 0.190 0.802 0.914 0.330 0.578 0.572 0.860 0.282 0.242 0.150 0.252 0.460 CC 0.176 0.726 0.142 0.934 0.996 0.356 0.170 0.074 0.876 0.382 0.044 0.070 PF UC IND CC 2.84% 0.028 0.196 0.074 5.17% 0.860 0.272 0.966 5.17% 0.906 0.906 0.162 5.69% 0.552 0.374 0.680 5.43% 0.636 0.522 0.720 4.393 0.534 0.762 0.640 4.39% 0.606 0.740 0.538 8.53% 0.004 0.232 0.062 5.43% 0.672 0.976 0.414 6.46% 0.242 0.062 0.656 6.72% 0.132 0.736 0.464 6.98% 0.090 0.304 0.368 Note This table reports the p-values of the new UC, IND, and CC backtests of Ziggel et al (2014) PF denotes the percentage of violations of VaR at 5% Numbers in bold correspond to the model with the best performance TABLE Backtesting results of the VaR at the 1% level Model Metric World Long Short EM Long Short BRIC Long Short US Long Short UK Long Short Germany Long Short GARCH-t PF UC IND CC PF 0.52% 0.444 0.550 0.758 1.29% 1.55% 0.364 0.374 0.484 1.55% 1.29% 0.680 0.716 0.842 1.81% 0.52% 0.498 0.748 0.832 0.78% 0.78% 0.602 0.006 0.052 2.07% 0.52% 0.388 0.554 0.700 1.03% 0.78% 0.756 0.736 0.408 1.81% 0.52% 0.496 0.606 0.872 2.58% 1.03% 0.976 0.822 0.060 1.81% 1.29% 0.454 0.068 0.764 2.33% 0.52% 0.328 0.522 0.714 4.39% 1.29% 0.614 0.178 0.688 2.84% UC 0.418 0.306 0.194 0.832 0.066 0.878 0.132 0.000 0.174 0.014 0.000 0.002 IND 0.444 0.692 0.372 0.372 0.584 0.306 0.426 0.044 0.384 0.008 0.442 0.168 CC 0.828 0.302 0.138 0.792 0.064 0.124 0.174 0.014 0.154 0.044 0.000 0.008 PF UC IND CC 1.29% 0.514 0.430 0.854 1.03% 0.988 0.462 0.332 1.55% 0.370 0.644 0.326 0.78% 0.870 0.348 0.904 2.07% 0.062 0.598 0.082 1.03% 0.852 0.348 0.108 1.29% 0.502 1.000 0.742 2.58% 0.004 0.034 0.012 1.55% 0.330 0.508 0.356 2.33% 0.022 0.098 0.046 3.62% 0.000 0.698 0.000 2.84% 0.004 0.966 0.014 msAPARCH-SGT msAPARCH-GH Note This table reports the p-values of the new UC, IND, and CC backtests of Ziggel et al (2014) PF denotes the percentage of violations of VaR at 1% Numbers in bold correspond to the model with the best performance TABLE Average daily capital charges under the Basel II Accord Model World EM BRIC USA UK Germany GARCH-t 0.0122 0.0180 0.0212 0.0141 0.0152 0.0186 msAPARCH-SGT 0.0112 0.0176 0.0196 0.0110 0.0146 0.0123 0.0120 0.0185 0.0201 0.0126 0.0136 0.0132 0.0580 Long positiona msAPARCH-GH Short position GARCH-t 0.0376 0.0581 0.0653 0.0440 0.0494 msAPARCH-SGT 0.0300 0.0484 0.0524 0.0350 0.0463 0.0477 msAPARCH-GH 0.0326 0.0500 0.0526 0.0353 0.0446 0.0481 Note This table reports the average daily capital charges (ADCC) incurred by VaR models (at 1%) Bold numbers indicate the most economically efficient models in terms of minimum ADCC a Since the VaRs of long positions are negative, we report the absolute value of the computed ADCC considered to be a charge of VaR violations in the previous 250 trading days as given in Appendix B (Basel Committee on Banking Supervision, 2006) Table reports the average daily capital charges (ADCC) over the forecasting period, using our 1% VaR forecasts for both long and short positions derived from the BENSAÏDA ET AL 13 different models Our forecasting period contains 387 observations, but the daily capital charges are computed for only 137 rolling windows as they require the estimation of the number of violations over the previous 250 trading days The smaller the ADCC value, the more economically efficient the model The results in Table show that the msAPARCH-SGT model is the less demanding, in terms of capital requirements for long position risk, in all markets except for the UK, where the msAPARCH-GH beats the other models Moreover, substantial savings in daily capital charges are incurred by Markov-switching models, when it comes to provision against short position risk, for all markets under consideration The best-performing model is the msAPARCH-SGT as it provides efficient allocation of regulatory capital in five out of six cases, followed by the msAPARCH-GH model In both positions, the Markov-switching model under the GH distribution is the second-best model, as it yields the next lowest capital charges, except for EM under long position, where the benchmark GARCH-t model is the second less demanding ORCID Ahmed BenSaïda http://orcid.org/ 0000-0001-5591-4393 CO N C LU S I O N The development of appropriate models for estimating and forecasting market risk is of paramount importance for optimal designs of portfolios and the determination of minimum capital requirements, particularly in times of frequent fluctuations and increased uncertainty Among various techniques that have been used in previous studies (e.g., sensitivity scenario analysis, stress-testing procedures, multivariate GARCH-based model, and copulas), the VaR has become a widely used approach to market risk Our paper introduces Markov-switching APARCH (msAPARCH) models based on two highly flexible distributions—the skewed generalized t and the generalized hyperbolic, to forecast VaRs of diversified portfolios for both short and long trading positions The main findings are as follows: (i) the proposed msAPARCH with flexible distributions are suitable to capture the stylized facts of stock returns including the regime changes They outperform the standard GARCH with Student's t-distribution, which is used in our study as a benchmark model in terms of volatility forecasting and data fitting; (ii) the msAPARCH also perform as well as the benchmark model when we forecast the 1-day-ahead 1% VaR and 5% VaR; and finally (iii) the use of msAPARCH-based VaR forecasts leads to the lowest daily capital charges, which indicates a better capital allocation efficiency The fact that the proposed models not perform better than the standard GARCH with Student's t-distribution in forecasting the VaR could potentially be explained by the ignorance of interdependence between stock markets in times of financial globalization, which typically requires a multivariate setting Future research can extend our flexible msAPARCH framework to include the multivariate dependence among financial markets (or multiple assets of a portfolio) In addition, the effect of liquidity risks on VaR also needs to be accounted for, because of the illiquid scenarios under market turmoil and crises Several works have developed liquidity-adjusted VaR models (e.g., Al Janabi, Hernandez, Berger, & Nguyen, 2017; Berkowitz, 2000; Weiß & Supper, 2013) Al Janabi et al (2017), for instance, propose a model 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D., Berens, T., Weiß, G N., & Wied, D (2014) A new set of improved value-at-risk backtests Journal of Banking and Finance, 48, 29–41 Ahmed BenSaïda is an Associate Professor of Finance, a member of the Scientific council, and a former Head of the department of Finance at HEC Sousse, Tunisia Sabri Boubaker is a Professor of Finance at South Champagne Business School, Troyes, France This paper was written while Sabri Boubaker was a visiting Professor at the International School, Vietnam National University, Hanoi, Vietnam Duc Khuong Nguyen is a Professor of Finance, Deputy director for research, and head of the department of Finance, Auditing and Accounting at IPAG Business School, Paris, France Skander Slim is an Associate Professor of Finance, a member of the Scientific council, and a former Head of the department of Economics and Statistics at HEC Sousse, Tunisia BENSAÏDA ET AL 15 How to cite this article: BenSaïda A, Boubaker S, Nguyen DK, Slim S Value-at-risk under market shifts through highly flexible models Journal of Forecasting 2018;1–15 https://doi.org/10.1002/for.2503 hypothesis of i.i.d violations, they focus solely on testing the independence of VaR violations Extending the current state-of-the-art, Ziggel et al (2014) suggest a new test for both the property of independent as well as the property of identically distributed VaR violations Specifically, the i.i.d property is simply formulated i.i.d asIt (𝛼) ∼ Bern (𝛼) , ∀t, and the test statistic is given by Ωi.i.d.,m = APPENDIX A: Va R BACKTESTING Unconditional coverage test The UC test is an industry standard, mostly due to the fact that it is implicitly incorporated in the “traffic light” system proposed by the Basel Committee on Banking Supervision (2006), (2009), which remains the reference backtest methodology for banking regulators The test consists in examining if the realized coverage rate equals the theoretical coverage rate (𝛼) of the VaR for a backtesting sample of Tf nonoverlapping observations Under the assumption of possibly nonstationary VaR violations sequence, Ziggel et al (2014) define the UC property as T𝑓 (A1) where It (𝛼) is the hit variable on day t, which takes values of if the loss exceeds the reported VaR measure, and otherwise Under the null hypothesis of correct UC, the test statistic is given by Ωuc = T𝑓 ∑ It (𝛼) + 𝜀, + (T𝑓 − tm ) + m ∑ (ti − ti−1 )2 + 𝜖 (A3) i=2 Backtesting is a formal statistical framework that consists in verifying if actual trading losses are in line with model-generated VaR forecasts, and relies on testing over VaR violations A violation (also called a hit) is said to occur when the realized trading loss exceeds the VaR forecast We briefly present the three tests used in our empirical assessment of VaR models ∑ E[ It (𝛼)] = 𝛼, T𝑓 t=1 t12 where {t1 , … , tm } = {t|It = 1} denotes the set of points in time associated with the sequence of VaR violations, T𝑓 ∑ It (𝛼) is the sum of observed VaR violations and m = t=1 Based on squared durations between consecutive violations, the test explicitly allows us to detect clusters in the VaR violation process This feature is economically meaningful as the test would reject VaR models that yield inaccurate forecasts of large losses that occur in rapid succession during financial turmoil Conditional coverage test The CC hypothesis can be tested using the following weighted test statistic: ) ( (A4) Ωcc,m = a 𝑓 (Ωuc ) + (1 − a) g Ωi.i.d.,m where ⩽ a ⩽ is the weight of the UC test, 𝑓 (Ωuc ) = ] | ( ) [ | Ωuc |, g Ωi.i.d.,m = Ωi.i.d.,m − I | − | | 𝛼T𝑓 {Ωi.i.d.,m ⩾̂r} , and r̂ is r̂ | | an estimator of the expected value of the test statistic Ωi.i.d.,m under the null hypothesis of i.i.d VaR violations In the empirical application, we set a = 0.5 For all three tests (i.e., UC, IND, and CC), we use Monte Carlo simulations to approximate the distribution of the test statistic under the corresponding null hypothesis and obtain the associatedp-values of the test (see Ziggel et al., 2014, for further details) APPENDIX B: PENALTY FACTOR (A2) t=1 where 𝜀 is a normally distributed random variable Independence test In the financial econometrics literature, an enhancement of the unconditional backtesting framework is achieved by additionally testing for the independence property of violations yielding a combined test of conditional coverage (CC) Examples include the test of Christoffersen (1998) against an explicit first-order Markov alternative, the regression-based test for higher-order dependence in the violation series (Engle & Manganelli, 2004) and duration-based tests of Christoffersen and Pelletier (2004) and Candelon, Colletaz, Hurlin, and Tokpavi (2011) While these standard backtests embed the null TABLE B1 Three-zone approach of the Basel II Accord for penalty structure Zone Number of violations Green 0–4 Yellow Red Penalty factor k 0.00 0.40 0.50 0.65 0.75 0.85 10+ 1.00 Note The number of violations is given for a forecasting period of 250 trading days Source: Basel Committee on Banking Supervision (2006) ... Nguyen DK, Slim S Value- at- risk under market shifts through highly flexible models Journal of Forecasting 2018;1–15 https://doi.org/10.1002/for.2503 hypothesis of i.i.d violations, they focus... Slim, S., Koubaa, Y., & BenSaïda, A (2017) Value- at- risk under Lévy GARCH models: Evidence from global stock markets Journal of International Financial Markets, Institutions and Money, 46, 30–53... 2017) argue that sophisticated models rarely beat the GARCH specification when evaluating forecasting performance Our choice for Student's t is based on several studies showing that this distribution