Are range based models good enough? Evidence from seven stock markets

Dockery, Everton; Efentakis, Miltiadis; Al-Faryan, Mamdouh Abdulaziz Saleh Article — Published Version Are range based models good enough? Evidence from seven stock markets Risk Governance and Control: Financial Markets & Institutions Suggested Citation: Dockery, Everton; Efentakis, Miltiadis; Al-Faryan, Mamdouh Abdulaziz Saleh (2018) : Are range based models good enough? Evidence from seven stock markets, Risk Governance and Control: Financial Markets & Institutions, ISSN 2077-429X, Virtus Interpress, Sumy, Vol. 8, Iss. 2, pp. 7-40, https://doi.org/10.22495/rgcv8i2p1 This Version is available at: https://hdl.handle.net/10419/225996 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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EVIDENCE FROM SEVEN STOCK MARKETS Everton Dockery *, Miltiadis Efentakis **, Mamdouh Abdulaziz Saleh Al-Faryan *** * Corresponding author University of Portsmouth Business School, UK Contact details: Economics and Finance, Business School, University of Portsmouth, Richmond Building, Portland Street, Portsmouth P01 3DE, Hampshire, England, UK ** Independent analyst, UK *** University of Portsmouth Business School, UK Abstract How to cite this paper: Dockery, E., Efentakis, M., & Al-Faryan, M. A. S. (2018). Are range based models good enough? Evidence from seven stock markets. Risk Governance and Control: Financial Markets & Institutions, 8(2), 7-40. http://doi.org/10.22495/rgcv8i2p1 Copyright © 2018 The Authors This work is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). http://creativecommons.org/licenses/b y-nc/4.0/ ISSN Online: 2077-4303 ISSN Print: 2077-429X Received: 17.03.2018 Accepted: 02.05.2018 JEL Classification: C53, G15 DOI: 10.22495/rgcv8i2p1 We study the performance of range-based models over varying market conditions and compare their performance against a set of alterative risk measurement models, including the more widely used techniques in practice for measuring the Value-at-Risk (VaR) of seven financial market indices. In particular, we focus on model accuracy in estimated VaRs over quiet and volatile moments utilizing loss functions and likelihood ratio tests for coverage probability. The empirical estimates based on these two criteria find that the range based-model of Yang and Zhang (2000) shows some success in estimated VaR risk measure, especially during quiet periods, than is the case for the other range based models considered. Also, we find that the EWMA and RiskMetrics models have an inconsistent marginal edge over the widely used GARCH and historical simulation specifications and that there is validity in the use of the EWMA and RiskMetrics models over range-based approaches as both capture and thus provide more accurate estimated VaR risk measure of market risk. Keywords: Range Based Models, Value-at-Risk, Market Risk, Financial Markets, Risk Measurement 1. INTRODUCTION Today risk measurement models are universal with financial firms situating value-at-risk (VaR) methods at the fulcrum of their risk management process for the management and reporting of market risk. Essentially, VaR is defined as the maximum expected loss of a portfolio for a given confidence level 𝛼 and a specified time horizon. Its wide use in risk management owes much to its conceptual simplicity for VaR summarizes the market risk associated with any portfolio to just a single number. For example, the VaR at level 𝛼 at the 1-day horizon is the nominal 1-day loss that will not be exceeded. Expressed differently, VaR at level 𝛼 would indicate that over a specified horizon the potential maximum loss for an asset will not exceed VaR at a confidence level of (1−𝛼). Although the literature in this area has grown considerably, partly motivated by the risk-adjusted measures of capital adequacy enforced by the Basil committee which, in turn, spawned the development of increasingly sophisticated risk measurement techniques, such as the equally weighted moving average (EWMA), RiskMetrics, the historical simulation approach to econometric procedures based on autoregressive moving average (ARIMA) models, extensions of the generalized autoregressive conditional heteroscedasticity (GARCH) family of statistical processes, and the application of extreme value theory in an effort to shed light on the forecasting ability of these approaches, the overwhelming evidence have been mixed; see for examples Schlueter and Deuschle (2010), Aloui et al. (2011), Berger (2013), Gerlach et al. (2013), Del Brio et al. (2014), Bams, et al. (2017), and Zhang et al. (2017). Of these studies Schlueter and Deuschle (2010) compare VaR estimates based on ARIMA approaches and report mixed statistical evidence for the predictive ability of wavelet-based forecasts, while Berger (2016) presents a copulabased wavelet approach in order to derive better predictive performance and, in an attempt to improve the forecasting ability for daily S&P 500 returns, Zhang et al. (2017) offers a wavelet-based Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 8 ARIMA approaches. What is apparent from these and other studies is that there remains no universally accepted method that yields accurate estimates of VaR for an asset or a portfolio and or what can be considered as the best risk measurement model; see for a discussion Kuester et al. (2006), Perignon and Smith (2010), Berkowitz et al. (2011), and KochMedina and Munari (2016). Mindful of this, and in light of the frequency of crisis events in financial markets, practitioners have become increasingly fastidious and so want to know when deliberating the choice of risk measurement models which of the available methods are conducive to delivering consistent and accurate estimated VaRs over varying market conditions. For if a particular model is not of a suitable fit it may prove to be costly as a consequence of inaccurate estimates of market risk. This concern about what an inaccurate estimate of market risk might mean for financial firms has been highlighted by the demise of a number of financial firms which incurred large financial losses at the height of the financial crisis that appears to challenge existing approaches to risk measurement and the management of financial risk more specifically. Together, these events are providing the catalyst for a host of regulatory proposals and for calls for risk measurement models to keep pace with the market environment, both of which are directed toward appraising appropriate models for accurate estimates of VaR. There are several reasons to why it is important to evaluate the performance of risk measurement models over varying market conditions. First, financial firms need to know how well risk measurement models perform in quiet as well as at volatile times in relation to each other in order to compare model accuracy, which is tantamount to stress testing. Thus, in principle, the credibility of a risk measurement model will depend on the exactness of the estimates of market risk it delivers. Second, risk measurement models are vehicles for decision making; that is, they are important for risk managers whose primary objective is to maintain the level of risk exposure within defined limits and for regulators whose main task are to ensure the stability of the financial system, and so require accurate measurement and reporting of VaR numbers to be able to take an informed view about the level of risk-taking at a financial firm, as well as to track their market risk exposure over time. Thus, the present study could be of interest to financial firms and their regulatory authorities. The data used throughout this paper is the daily index of seven countries drawn from Europe, North America, Asia, and Latin America, as it offers insights of the risks faced by financial firms in these market over a time period marked by the reactions of market participant to news associated with deteriorating economic conditions, market volatility, the bursting of stock market bubbles, the Asian financial crisis, the Russian and Brazilian crisis, the deflation of the financial bubble centred on the dotcom companies, the catastrophic events of 9/11, and other shocks that impacted the world’s stock markets over the sample period. The empirical analysis is confined to the period January 1992 to December 2002, a period which, as just mentioned, witnessed a number of shocks to global markets. In accordance with the regulatory framework, the accuracy of estimated VaRs is assessed with respect to their one-step-ahead forecasts and 99 percent coverage levels. The methods are then backtested using two evaluation measures − Lopez (1999) loss function approach which is defined to produce higher values when exceptions occur, and Christoffersen’s (1988) likelihood ratio tests for coverage probability, which is independent of the model process producing the estimated VaR and captures whether a particular procedure shows correct conditional coverage. The paper is organized into five sections. Section 2 outlines the range based approaches used to estimate VaR, including a class of alternative approaches, and the evaluation methodology used to assess their statistical accuracy. Section 3 presents the data. Section 4 presents and discusses the empirical results of the model’s evaluation of estimated VaRs. Section 5 contains a summary of our findings and concluding remarks. 2. THE VAR SPECIFICATIONS CONSIDERED As earlier noted, value at risk (VaR) is a widely used statistical framework for estimating the market risk of economic losses in financial markets. By employing VaR in daily risk management, banks and other financial firms can discern the minimum amount, in monetary terms; they might expect to lose with a small probability  over a stated time horizon, usually 1-day or 10-days. Mathematically, we commence by considering the return series {𝑅𝑡}1 𝑇 of a financial asset, such that {𝑅𝑡}1 𝑇 follows a stochastic process: t t t y   (1) Where, 𝐸(𝜀𝑡| 𝛩 𝑡−1)= 0 and 𝐸(ℰ𝑡 2| 𝛩 𝑡−1)=𝜎𝑡2. 1 Let 𝑧𝑡 ≡𝜀𝑡/𝜎𝑡 have the conditional distribution t with zero conditional mean and unit conditional variance defined by 𝑧𝑡Θ𝑡−1 ~Φ𝑡(0,1). Since our approach in this paper is to consider stock indices drawn from a number of capital markets, in preference to the construction of portfolios, we do not consider covariances. Thus the approach we follow is a variance method whereby the Var(a) can be estimated as follows: . ( ) ( ) t t t t VaR        (2) In many applications, researchers assume the expected return 𝜇 equals 0, and we make this assumption. 2 Thus Eq. (2) becomes: . ( ) ( ) t t t t VaR      (3) From Eq. (3) the estimation of VaR entails estimating t() and t. In regard to the models examined in this paper, we assume a parametric distribution (e.g., normal distribution) for t(). The conditional distribution, t(), is assumed to be constant over time or assumed to be Gaussian 1 Specifically, Θ𝑡−1 is the time t-1 information set (-field). 2 This assumption is based on the conjecture that the magnitude of 𝜇 is substantially smaller than the magnitude of the standard deviation 𝜎 and can thus be ignored Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 9 N(0, 1), while conditional variance 𝜎𝑡2 is estimated using different methods of volatility models. The approaches followed for the estimation of VaR (i.e., the estimation and modelling of 𝜎𝑖) includes the widely used parametric and non-parametric approaches and the less representative range-based method which are presented next. 2.1. Range based models Since volatility is recognized as time varying, it is imperative to use the most recent price observations to construct an estimate of volatility. Here, volatility estimates using additional information such as high, low, open prices to achieve better accuracy ─ in addition to the closing price used by the conventional estimators, have been considered in the literature. For example, the theoretical and empirical studies include Parkinson (1980), Garman and Klass (1980), Rogers and Satchell (1991), and Yang and Zhang (2000). For the purpose of this study, we subject these extreme based models to empirical testing to see how well they estimate VaR. Following Garman and Klass (1980), the price in each period of length 𝛵 starts at the closing price of the previous period, with each period divided into two intervals with fractions 𝜃 and 1−𝜃. Since trading is closed during the first interval of length, 𝜃𝛵, the price movement in this interval (before opening) is unobservable. The high and low prices in a data set are those observed from the second interval of length(1−𝜃)𝛵 (i.e., the trading interval) 3 . Parkinson (1980) demonstrates that expectation of the high minus the low squared is proportional to variance and constructs an estimate based on the high minus the low expressed as: 𝜎𝑝=1 4𝑛1n2∑(𝑢𝑖−𝑑𝑖) 2 𝑛 𝑖=1 (4) However, this estimator is only valid when there are no opening jumps and no drift. In contrast, Rogers and Satchell (1991) variance estimator is defined by:     1 1 ˆ n rs i i i i i i i u u c d d c n          (5) The variance estimator, 𝜎𝑟𝑠, of expression (5) is, according to Yang and Zhang (2000) a much better estimator than expression (4) since it is independent of the drift and is equal to zero when the security price makes a one-direction move of either u=c and d=0 for a straight-up move or d=c and u=0 for a straight-down move. 4 Garman and Klass (1980) variance estimator (derived under the assumption of no drift) is calculated as follows 5 : '' 0 ˆ ˆ ˆ 0.383 1.364 0.019 gk C p rs          (6) 3 θΤ, is an effective time period that models the opening jump as an unobservable continuous price movement. The fraction θ, measures the relative size of the opening jump. The case θ=0 means that there is no opening jump, and the case 𝜃 → 1 implies that the price movement in the period is dominated by the opening jump. 4 This is because the price movements in such situations can be explained by the drift term alone (zero variance). 5 This formula was not explicitly given in Garman and Klass (1980). We adopt it from Yang and Zhang (2000). where '2 01 1n i i o n    '2 1 1n ci i c n    The estimators just discussed are only valid under the assumptions of either no drift or no opening jumps. Thus, Yang and Zhang (2000) point out that although the no drift assumption is reasonable for daily financial data, we can often see that “the price of a security goes through a “trendy” phase, in which the drift could be large compared to the volatility”. Since estimators 𝜎𝑝 and 𝜎𝑔𝑘 will underestimate volatility, they further note that the assumption of no opening jumps is not realistic, since opening jumps do occur in reality and, in such cases estimators 𝜎𝑝 and 𝜎𝑟𝑠 will underestimate the volatility. To overcome this, Yang and Zhang (2000) suggests estimating the variance of the underlying security during these periods based on the following:   0 ˆ ˆ ˆ ˆ 1 yz c rs kk         (7) where 𝜎𝑟𝑠 is given in expression (6) and 𝜎0 and 𝜎𝑐 are defined as follows:   2 01 1 ˆ1 n i i oo n     (8)   2 1 1 ˆ1 n ci i cc n     (9) with 1 1n i i oo n  (10) 1 1n i i cc n  (11) In the empirical part of this paper, the constant k was chosen to minimize the variance of the estimator 𝜎𝑦𝑧 and was set equal to: 0.34 1 1.34 1 kn n   (12) The above minimum-variance estimator is an unbiased estimator, which is independent of both the drift and opening jumps of the underlying price movement. For Yang and Zhang (2000), expression (12) is more accurate than the estimator based only on closing prices. Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 10 2.2. The equally weighted moving average model The equally weighted moving average (EWMA) method assumes variances are constant over the forecasting period. For calculating VaR, we estimate the volatility of asset return by a historical moving average variance process. Assuming that returns, 𝑅𝑡, are observable over m days, the equally weighted sample variance is expressed as: 22 1 1 1() 1 m tt j R m     (13) where the choice of window width m is critical. The choice of short windows would suffer from inferior statistical efficiency, though they are likely to do better in capturing the dynamics of short-term volatility. Once 𝜎𝑡2 is estimated, VaR is estimated under the assumption returns are normally distributed. 2.3. RiskMetrics method Perhaps the most practicable volatility models used in the application of risk management has been the RiskMetrics (RM) model of J.P. Morgan (1996), also known as the exponentially weighted moving average, which deals with the insensitivity of the Equally Weighted method to recent innovations and assigns exponentially weights to more distant observations. The RM model expresses the variance as: 2 22 11 (1 ) t t t y        (14) Specifically, 𝜆 𝜖[0,1] is the decay factor reflecting how the impact of past observations decays while forecasting one-day-ahead 𝜎𝑡2. The most recent observations have the largest impact and the impact decays exponentially as the observations move towards the past. With a low value of 𝜆, the weight attached to historical returns decays rapidly as we go further into the past. A high 𝜆 leads to a much lower decay of weights. But once 𝜎𝑡2 is estimated VaR is estimated under the assumption that returns are normally distributed. 2.4. Garch models While the EWMA model captures volatility clustering, Bollerslev’s (1986) Generalized Autoregressive Conditional Hetroscedasticity (GARCH) model allows for both autoregressive and moving average behaviour in variance and covariance. Thus we consider the Exponential GARCH (EGARCH) and the standard GARCH (1, 1) models. The latter is given by: 2 2 2 11t t t         (15) where 𝜎𝑡2 the time-varying conditional variance is modelled as a stochastic process. For a welldefined GARCH(1, 1) the restrictions 𝜑>0, 𝛽 ≥0, |𝛼|<1 and 1−𝛽−𝛼 > 0 are imposed to ensure the conditional variance is positive. 𝛽 measures the extent to which a volatility shock today feeds through into next period’s volatility, while (𝛽+𝛼) measures the rate at which this effect dies out over time. The EGARCH model of Nelson (1991) is used to forecast volatility as given by:   1 21 1 1 11 ln 1 1 qp ii t i j t t t ii a L L z z E z                              (16) where 𝜃 and 𝛾 are the parameters of asymmetry, and 𝐿𝑖 is the lag operator. Eq. (16) is capable of capturing any asymmetric impact of shocks on volatility and allows good and bad news to affect volatility in a different way ─ e.g. small positive shocks will have a greater impact on conditional volatility than small negative shocks, while large negative shocks will have a greater impact on conditional volatility than large positive shocks. 2.5. Historical simulation models The Historical simulation (HS) approach is widely used in the financial industry owing to its flexibility and ease of application. The model constructs the distribution of portfolio value changes, △𝑃, from historical data without imposing distribution assumptions and estimating parameters, and further assumes that trends of past price changes will continue in the future. Thus, the hypothetical future prices for time t+s are obtained by applying historical price movements to current (log) prices as follows: ** , , 1 ,i t s i t s i t s k P P P        (17) Where t is the current time, s=1,2,...,k, k, is the horizon length of going back in time, 𝑃𝑖,𝑡+𝑠 ∗ is the hypothetical (log) price of the i-th asset at time t+s, 𝑃𝑖,𝑡 ∗=𝑃𝑖,𝑡, △ 𝑃𝑖,𝑡+𝑠−𝑘 =𝑃𝑖,𝑡+𝑠−𝑘 −𝑃𝑖,𝑡+𝑠−1−𝑘, 𝑃𝑖,𝑡 is the historical (log) price of the i-th asset at time t. Assuming a time horizon 𝜏=1, the portfolio returns at time 𝑡 + 𝑠 is defined as: ** , , ,p t s p t s p t R P P   (18) where 𝑃𝑝,𝑡 is the current portfolio (log) price. The VaR is obtained from the density function of the computed hypothetical returns. Thus VaR(α) = VaRt,τ is estimated by the negative of the (1−𝛼)𝑡ℎ quantile, 𝑉𝑎𝑅∗; specifically, 𝐹𝑘△𝑃(−𝑉𝑎𝑅)=𝐹𝑘△𝑃(𝑉𝑎𝑅∗)=1−𝛼, where 𝐹𝑘△𝑃(x) is the empirical cumulative distribution function:     * ,, 1 11 , k k P p t s s F x R x x k      Ў 19 A particular advantage of this approach is the no distributional assumptions, meaning that deviation from normality is not an issue. The model is also risk-free in that parameter estimation and correlation effects between various assets are Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 11 modelled implicitly since profits are paired against losses for the entire position (Khindanova and Rachev, 2000; Linsmeier and Pearson, 1999). 2.6. Kernel method As regards the Kernel method, Kernel estimation makes use of non-parametric methods of weighting the historical data when estimating volatility as: 6 2 1 T t i t t wR    (20) The weights are then estimated using a nonparametric kernel method defined by:     1 ˆ ˆ t tT t t fx w fx   (21) where 𝑓󰆹𝑥 is the kernel density estimator, which at point x has the form:   1 1 ˆTi i xx f x K nh h       (22) where K( ) is the kernel function, which may take a variety of functions provided it holds certain regularity properties. A commonly used kernel is the Gaussian kernel defined by:   2 11 exp 2 2 K u u      (23) The unknown factor, h, in Eq. (23) is the window width, otherwise termed the bandwidth. In choosing the bandwidth there is a trade-off between variance and bias. The larger the bandwidth the smaller will be the variance but the greater the bias and vice versa. A number of methods are available for choosing the bandwidth from simple crossvalidation to various “plug-in” methods (Sheather and Jones, 1991). In this study, the bandwidth is defined by: 1/5 ( )1.06h Std X T  (24) since it can be shown to the optimal choice if data are assumed to be normally distributed and the Gaussian Kernel is used (Silverman, 1986). 2.7. Back testing the methods: Model evaluation Back testing is a common procedure in risk management which enables risk managers, at a glance, to validate the statistical accuracy of economic VaR models. In this respect, the present article will consider two alternatives: (i) a test for accuracy of VaR estimates suggested by Lopez 6 A similar method was followed by Pagan & Schwert (1989) and was used by Engel & Gizycki (1999a) to estimate VaR (1999) and (ii) a likelihood ratio tests for coverage probability proposed by Christoffersen (1988). The utility of the testing framework is that it roots in regulatory requirements and thereby allows for an evaluation of the statistical precision of out-ofsample VaR estimates; see Kuester et al. (2006) and Halbleib and Pohlmeier (2012). According to the framework of Lopez (1999), the accuracy of estimated VaRs can be gauged by how well they minimize a loss function representing the concerns of risk managers. Loss functions reflecting such concerns are specified in a negative direction by assigning higher scores when failures occur and the VaR models are then assessed by comparing the expected value of the loss function. A model that minimizes expected losses is preferred to ones that do not reduce losses. The loss function at time t has the form: 11 11 ( , ) if ( , ) if tt t t t t t tt t t t t f R VaR R VaR Lg R VaR R VaR         (25) where 𝑓 ( ) and 𝑔 ( ) are functions that satisfy 𝑓 ( )≥𝑔 ( ) and 𝑅𝑡 the realized return or loss. For the purpose of this exercise we consider three loss functions: a binary loss function which takes account of whether any given days return is greater or smaller than estimated VaR; a quadratic loss function which takes account of the size of the negative returns that exceed estimated VaR, and a firm’s loss function suggested by Sarma et al. (2000) to resolve a possible conflict between the goals of safety and profit resulting from a firm’s use of VaR in internal risk management. The binary loss function treats any negative return smaller than estimated VaR as a violation. Thus we are concerned with the number of violations rather than the magnitude of such violations. Each loss exceeding the VaR is assigned an equal weight of unity, while all other returns have a zero weight, that is. 1 1 1 if 0 if ttt b t ttt R VaR LR VaR         (26) It follows that if the VaR model discussed above provides the correct level of coverage, as defined by its confidence level, then the average binary loss function over the sample will equate 0.05 for the 95 percent confidence level and 0.01 for the 99 percent confidence level. The quadratic loss function reflects the magnitude of the exception, but as well as taking this into account the application of its functional form penalizes large exceptions more severely than would be the case with a linear or binary measure. The quadratic loss function is defined by:   2 11 1 1 if 0 if tt t t t t Q t ttt R VaR R VaR LR VaR          (27) Sarma et al. (2000) term the above the regulatory loss function since it more readily Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 12 reflects the goals of the financial regulator. The implication in this area is that a VaR estimator reporting too high values is likely to persuade a financial firm to hold too much capital, thus imposing the opportunity cost of capital upon them. To solve this Sarma et al. suggest modelling a firm’s loss function in a manner that penalizes failures as well as imposing a penalty reflecting the cost of capital suffered on other days as follows:   11 11 if if tt t t t t F t t t t t R VaR R VaR LaVaR R VaR        (28) where 𝛼 is a measure of the opportunity cost of capital. The first component in Eq. (28) reflects the penalty due to the failure of a model, while the second component signifies the opportunity cost for the risk manager due to risk management. In contrast, test of unconditional coverage consists of examining if the realized (ex-post) coverage rate equilibrates the theoretical coverage rate, which is equivalent to testing if the indicator variable It, which takes a value of 1 if the loss is greater than the estimated VaR (referred to as a violation) and 0 otherwise, follows an independent and identically distributed (i.i.d) process with parameter p; where p equals VaR’s theoretical coverage rate α. The likelihood ratio (LR) test statistic for the unconditional coverage test follows a 𝑥2distribution with one degree of freedom. The test is calculated as follows:         2 1 2ln 1 1 / / TN Nd uc TN N pp LR N T N T               (29) where p is the desired significance level, i.e. one minus the VaR confidence level, N is the number of violations, and T is the number of VaR estimates. 𝐿𝑅𝑢𝑐 is asymptotically distributed chi-squared with one-degree of freedom under the null hypothesis that p is the true probability. Although the test can be used to penalize financial firms, it does not capture asymmetries or leverage effects which will affect the accuracy of any forecasts. An improvement of the unconditional back-testing framework is the test for conditional coverage which requires correct unconditional coverage and simultaneously ensures that the violating series is i.i.d. through a test for independence. Christoffersen’s (1988) LR test for independence thus test the hypothesis that the failure process is independently distributed against the alternative that the process follows a first order Markov process as defined by:             00 10 01 11 00 10 01 11 2 0 0 1 1 1 2ln 1 11 TT TT d ind TT TT LR                 (30) where 𝑇𝑖𝑗 𝑖,𝑗=0,1 is the number of observations with a j following an i in the I𝑡 sequence, and 𝜋0,1 = 𝑇0,1/(𝑇00 +𝑇o,1). Since both the unconditional coverage and the independence properties should be satisfied for an accurate VaR model, Christoffersen’s (1988) proposed the following statistic:   22 d cc uc ind LR LR LR     (30) which is asymptotically a 𝜒2 distribution with two degrees of freedom. For an insight into the statistical back-testing framework over and above the regulatory back-testing approach see Engle and Manganielli (2004) and Ziggel et al. (2014). 3. DATA DESCRIPTION AND SUMMARY STATISTICS The data used in this study consists of daily stock index closing prices, which includes 2870 observations for each series over the period January 1992 to December 2002 for a range of international stock indexes including: the U.S.A, Dow Jones Industrial Index (DJI), Canada, TSX Index (TSX), Germany, DAX30 Index (DAX), Netherlands, AEX Index (AEX), Japan, Nikkei 225 Index (NIKKEI 225), Thailand, Bangkok SET Index (SET), and Brazil, BOVESPA Index (IBOV), for which we calculate the daily VaR. These data were obtained from Data stream. The selection of the time period of 11 years was motivated by the appropriate time span for estimation and testing that this set of daily returns offer as it includes periods marked by events that triggered large price swings in the capital markets considered. 7 Daily market Open, High, Low and Close prices were used which is consistent with normal practice and reflect the view that these four prices have a much higher informational content than other intraday prices. 8 Returns, r, are calculated as the percentage logarithmic differences between the price time t and 𝑡_1, 𝑟𝑡=100∗ (𝑙𝑜𝑔 𝑃𝑡−𝑙𝑜𝑔𝑝𝑡−1). Table 1 displays summary statistics for the data. 7 The period was chosen to cover both volatile periods and periods of relative tranquility in financial markets. For example, it covers the “controlled” devaluation of the Mexican peso in December 1994 that resulted in a spill over of the “peso-crisis” to the rest of Latin America, Asia, and the more developed financial markets. The flotation of the Thai baht and its subsequent 17% decline against the US dollar on the 2nd of July 1997, that resulted in significant global financial market volatility, as well as the devaluation of the Russian rouble; the fiscal deficit problems in Brazil that resulted in global financial market volatility in 1998, the financial distress of Barings Bank and Long Term Capital Management, the stock market crisis that commenced with the terrorist attack in the U.S in 2001, and was intensified in 2002 with the corporate governance scandals of Enron and WorldCom, resulting in significant financial market volatility, especially in the U.S and Europe. 8 Open and Close refer to the price at the opening and closing of the market, while High and Low prices corresponds to the two extremes – the highest and lowest prices recorded from the day’s trading Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 13 Table 1. Preliminary statistics on stock market returns Countries Summary U.S. Canada Germany Netherlands Japan Thailand Brazil Statistics (DJI) (TSX) (DAX) (AEX) (NIKKEI) (SET) (BOVESPA) Mean 0.0337 0.0221 0.0211 0.0328 -0.0343 -0.02241 0.3423 S.D 1.03320 0.93610 1.44420 1.31550 1.47348 1.78500 3.06999 Skewness 0.296418 -0.721448 -0.302624 -0.253861 0.1737 0.443055 0.423815 Kurtosis 8.193869 10.309260 6.638475 7.615715 5.3761 7.564559 9.027373 JB 3267.941 6637.731 1626.910 2578.528 2578.528 2585.4 4430.288 PP-LC -1.49399 -1.44921 -1.46105 -1.72784 -0.67851 -0.70626 -7.73829 PP-R -53.35509 -48.76776 -53.65794 -52.31525 -56.18333 -7.58750 -50.09319 Q(5) 6.10 29.65 7.67 19.04 10.33 48.72 15.84 Q(10) 11.97 33.81 21.00 40.75 15.63 72.73 57.23 Q(20) 25.89 55.09 45.90 64.30 22.97 93.76 105.93 Q2(5) 385.63 259.69 1003.30 1366.00 190.46 440.53 376.69 Q2(10) 636.16 468.22 1908.30 2616.30 315.52 658.86 536.18 Q2(20) 877.21 742.31 2981.40 4392.90 452.29 802.29 715.05 Max 6.1547 4.6835 7.5527 7.4526 7.6605 11.3495 28.8176 Min -7.4549 -8.4652 -8.87447 -7.5310 -7.2340 -10.0280 -17.2292 Note: JB is the Jarque-Bera statistics, which is distributed asymptotically as a chi-square under the null hypothesis of normality. Q(5), Q(10), Q(20) and Q(5)2, Q(10)2, Q(20)2 are the Ljung-Box statistics of up to order 20 of the return and squared return series, respectively. The reported results indicate that the mean daily returns from investing in a fund representative of the selected developed markets’ range between - 0.03 percent (Japan) and 0.03 percent (U.S.), while for a fund representative of emerging market stock indices between -0.02 percent (Thailand) and 0.34 percent (Brazil). Meanwhile, the standard deviations, which may be taken as a direct measure of volatility and the asset’s risk for developed markets range between 0.9361 percent (Canada) and 1.4734 percent (Japan). While for emerging markets this range between 1.7850 percent (Thailand) and 3.0699 percent (Brazil). For the stock markets sampled, the markets of Thailand (11.3) and Brazil (28.8) displayed the largest daily price movement The skewness statistics suggest that all return series are either negatively or positively skewed. The kurtosis statistics suggest departure from normality and all series are highly leptokurtic. From the respective Jarque-Bera statistics, we can reject the normality assumption for return series, while the Ljung-Box statistic values suggest that they exhibit a high degree of autocorrelation in squares, but not in levels for some of the markets. The results of the unit root test, based on Phillips (1987) and PhillipsPerron (1988), (the critical values for rejection of hypothesis of a unit root are –3.4357, -2.8631 and – 2.5676 for 1, 5 and 10 percent respectively) for the logarithmic of close prices (PP-LC) and logarithmic returns (PP-R) indicate that all of the return series are stationary. Figure 1(see Appendix) displays the dynamics of the logarithmic of close prices of the market indexes over the past eleven years. The patterns indicate that the markets experienced significant market falls associated with the South East Asian crisis of 1997, Brazil’s debt problem of 1998, the dot-com bubble in 2000, the 9/11 terrorist attacks in 2001 and the pressure of selling more generally In addition, Figure 2 (see Appendix) shows that the return series are more volatile over specific time periods. We can clearly observe that there are phases with different degrees of volatility. The USA stock market index (DJI) is seen to have a very high degree of volatility especially in 2001 which coincided with bad news. The volatility levels were very similar in other markets, except for the Canadian and Brazilian market, mainly due in part to the bad news coming from the USA which resulted in a succession of large positive and negative returns within a very short time horizon, indicating therefore that stock price risk management is warranted. Volatility clustering is manifestly apparent for all stock index return series indicating the presence of heteroscedasticity. Figure 3 (see Appendix) displays the monthly volatilities of the return series. Interestingly the data shows the standard pattern of volatility across all markets. Although volatility increased in all markets, the industrialized markets seem to exhibit more market jitters than the emerging markets sampled. In particular, there are turbulent months which are then followed by further turbulent months, while relatively calm months tend to bunch together. 4. ESTIMATION AND RESULTS In this section, we apply the parametric and nonparametric models outlined in section 2 to the data for the seven stock markets. The VaR estimates for the financial markets in this study are calculated for a one-day holding period at the 95% and 99% confidence levels in order to evaluate whether model performance changes for different VaR levels. For VaR estimation, we employ estimation window size ranging from 50 to 1250 past observations and a one-day moving average procedure to push the estimation period through time. For the GARCH and EGARCH methods we apply the algorithm advanced by Bernt et al. (1974) to maximise the log-likelihood, while the parameters are estimated using 1250 observations and a one day ahead sample forecasts of volatility is estimated using a rolling window dynamic re-parameterization approach which requires estimating each model, and then using the parameters to create a one-day-ahead forecast of volatility for each market index. We first analyze the performance of the models by using the method advanced by Lopez (1999) to evaluate VaR estimates using regulatory loss functions. The results for the evaluation of model accuracy are presented in Tables 2 to 8 (95% VaR) and (99% VaR), where the Lopez B, Lopez Q, and Lopez F denote the binary, quadratic, and the firm’s loss function respectively. The first column lists the name of the models. The second and third column reports the estimated Lopez B statistic for the Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 14 95%and 99% confidence levels for period 1 (volatile moments). The fourth and fifth column reports the estimated statistics for the Lopez Q test for both confidence levels, while the sixth and seventh column shows the estimated statistics for the Lopez F statistics. The remaining columns presents the results for the second period (quiet times) and the third period (volatile moments). Tables 9 to 15 are assembled in the same manner and report Christoffersen’s likelihood ratio test results for unconditional coverage (or probability failures, 𝐿𝑅𝑢𝑐), serial independence of exceptions (𝐿𝑅𝑖𝑛𝑑), and conditional coverage (𝐿𝑅𝑐𝑐), to determine which model is more accurate. Considering first, period 1, designated ‘volatile’ moments – due to the instability set in motion by the Asian (1997 and 1998), Russian (1998) and Brazilian (1999) crisis. A first observation that emerges from Table 2 to 8 using the Lopez (1999) loss functions, as given by the binary and quadratic loss functions, is that according to their performance in each statistical loss function, no particular model produced sufficiently accurate and consistent estimated VaRs for the 95% and 99% level VaR for developed and emerging markets, although the range based and kernel methods are identified as the least accurate in VaR risk measure. To further illustrate model performance, we examined closely the results for the European markets which are perhaps less prone to market trembles than the emerging markets in our sample. Tables 2 and 3 (see Appendix) show that for the AEX and DAX returns, the Lopez-B statistic ranges from 5.046 (AEX - RM 50.90) to 28.440 (DAX - RS 1250) for the 95% confidence level and from 1.376 (AEX - GK 50) to 22.477 (DAX RS 1250). For these markets the range based and kernel models are again identified as the least recommendable as they provided the least accurate estimated VaRs, though the Historical Simulation (HS) model which, as earlier noted, is one of the most widely used models in practice evidenced poor values, with the Lopez-B statistics ranging from 7.339 (DAX HS 50) to 15.367 (AEX HS 1250) for the 95% confidence level, and (DAX HS 250) 2.064 to 5.275 (AEX HS 1250). More specifically, we see from the values of the firm’s loss function (Lopez F) that during volatile moments all the models, for all returns, become relatively more expensive, and especially in the case of the SET and BOVESPA returns for the estimation period. Comparable levels of performance in the accuracy of estimated VaRs were also observed for period three as a result of the terrorist attack on the Pentagon and the World Trade Centre on September 11th 2001, coupled with the financial shocks linked to the financial collapse of Enron and WorldCom which affected most markets. The estimates in Tables 2 to 8 for the Lopez-B and Lopez-Q statistics indicate that no particular model stood out in terms of delivering sufficiently accurate estimated VaRs for the 95% and 91% level VaR. For the same period the estimates indicate that for the DJI return, the Lopez-B statistic ranges from 4.323 RiskMetrics (RM100.94) to 19.885 Kernel (KS 1250) for the 95% confidence level, and from 1.153 (EGARCH) to 11.239 (KS 1250) for the 99% confidence level. More strikingly, the loss function values fail to identify a dominant VaR model, suggesting that the results are open to interpretation taking into full consideration the loss functions used and the chosen horizon. On this basis, the kernel, range based, and HS models appear to be the least recommendable for their ability in delivering accurate estimated VaRs across all markets and confidence intervals. Tables 9 to 15 (see Appendix) report the LR statistics for the different coverage levels at 95% and 99% level VaR. As can be seen, the accuracy performance of the models in estimated VaRs are broadly similar, though most noticeably the performance of the range-based models are considerably poor in confidence levels for all markets than is the case for the more widely used risk measurement models. Similar conclusions are attained here, comparable with their performance in each statistical loss function. One encouraging result was that in the case of the AEX and DJI returns in period 1 and 3, although there were a large number of violations, the violations did not arrive in clusters as would otherwise have been expected. Interestingly, the poor performance of the models during volatile moments ─ i.e., the Asian financial crisis, the catastrophic events of September 11, 2001, and the fears surrounding the financial distress of Enron is generally in line with results furnished by Danielsson (2002) who investigated similar models over the same time span, albeit with larger window lengths (300, 1000 and 1250 observations), but only for the regulatory 99% level VaR, which produced inferior accuracy results to smaller confidence levels. We find that when we analyzed the models during volatile moments, due to the Asian crisis which followed a period of relative calm in global financial markets, the smaller sample windows were not only able to fully capture the short-term dynamics of the markets but also furnished better model performance in estimated VaRs. For example, the performance of the EWMA procedure for the SET return, displayed in Table 7, where the performance difference as measured by the binary loss function for the largest and smallest sample window size was between 8.486 for the 95% level VaR and 2.293 for the 99% level VaR. Besides, the conditional coverage LR statistic improved by a factor of 48.3 for the 95% level VaR and 19.93 for the 99% level VaR. The improved performance of the VaR models using smaller sampling windows were further confirmed during the catastrophic events of 9/11 2001 (period 3), especially for European and North American markets that were widely affected. Discernibly, model performance in estimated VaR risk measure improved when the largest window size was examined. The results here permit to conclude that the models were able to better capture the extreme returns from the first period, thereby delivering larger estimated VaRs. Thus the performance difference between the two volatile moments, for both the smallest and largest window size, reduced significantly. An examination of results in Table 3 for the DAX return shows that the difference of the binary loss function statistic for the EWMA procedure with 50 and 1250 observations during the first period were 5.275 and 4.358 for the 95% and 99% level VaR respectively, while for the third period this was 4.611 and 4.611. In the second period, quiet times, denoting the absence of any market perturbation or financial distress, the variation in window size had more of a varied influence on model performance in the accuracy of estimated VaR risk measure. We also observe that model accuracy generally improved as the size of sampling windows increased. This was expected since, as earlier remarked, larger size windows were able to fully Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 21 Figure 3. Monthly volatilities (DJI, TSX, DAX, AEX, Nikkei 225, SET, IBOV) Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 22 Table 2. Netherlands (AEX) loss function statistics (part 1) Period 1 Period 2 Period 3 Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% MA Z 50 5,963 1,376 5,964 1,377 0,076 0,112 5,747 1,724 5,748 1,724 0,056 0,082 7,781 3,170 7,784 3,171 0,101 0,147 MA Z 100 5,963 2,294 5,965 2,294 0,074 0,107 5,172 1,580 5,173 1,581 0,059 0,086 8,934 5,476 8,937 5,477 0,098 0,140 MA Z 250 7,569 3,440 7,570 3,441 0,067 0,097 4,023 1,293 4,024 1,293 0,065 0,093 8,934 4,899 8,938 4,901 0,086 0,123 MA Z 500 8,716 4,817 8,718 4,817 0,057 0,081 3,017 1,006 3,018 1,006 0,070 0,100 11,527 7,205 11,533 7,207 0,073 0,103 MA Z 1250 12,615 8,028 12,618 8,029 0,044 0,063 3,736 1,006 3,736 1,006 0,062 0,090 13,545 7,205 13,262 7,207 0,071 0,101 RM 50 .90 5,046 2,294 5,047 2,294 0,075 0,108 7,615 1,580 7,615 1,581 0,053 0,079 7,205 1,153 6,918 1,153 0,099 0,146 RM 100 .90 5,046 2,064 5,047 2,064 0,075 0,108 7,471 1,580 7,472 1,581 0,053 0,079 7,205 1,153 6,918 1,153 0,099 0,147 RM 250 .90 5,046 2,064 5,047 2,064 0,075 0,108 7,471 1,580 7,472 1,581 0,053 0,079 7,205 1,153 6,918 1,153 0,099 0,147 RM 500 .90 5,046 2,064 5,047 2,064 0,075 0,108 7,471 1,580 7,472 1,581 0,053 0,079 7,205 1,153 6,918 1,153 0,099 0,147 RM 1250 .90 5,046 2,064 5,047 2,064 0,075 0,108 7,471 1,580 7,472 1,581 0,053 0,079 7,205 1,153 6,918 1,153 0,099 0,147 RM 50 .94 5,963 2,064 5,964 2,065 0,073 0,107 7,328 1,724 7,328 1,724 0,053 0,078 8,069 1,729 8,071 1,730 0,097 0,144 RM 100 .94 5,505 1,835 5,506 1,835 0,075 0,109 6,322 1,580 6,322 1,581 0,055 0,080 6,916 1,441 6,918 1,442 0,101 0,147 RM 250 .94 5,505 1,835 5,506 1,835 0,075 0,109 6,322 1,580 6,322 1,581 0,055 0,080 6,916 1,441 6,918 1,442 0,101 0,148 RM 500 .94 5,505 1,835 5,506 1,835 0,075 0,109 6,322 1,580 6,322 1,581 0,055 0,080 6,916 1,441 6,918 1,442 0,101 0,148 RM 1250 .94 5,505 1,835 5,506 1,835 0,075 0,109 6,322 1,580 6,322 1,581 0,055 0,080 6,916 1,441 6,918 1,442 0,101 0,148 RM 50 .97 6,881 2,294 6,882 2,294 0,066 0,097 8,764 2,443 8,765 2,443 0,048 0,071 10,086 3,458 10,089 3,459 0,087 0,129 RM 100 .97 6,193 1,376 6,194 1,377 0,073 0,108 5,891 1,437 5,891 1,437 0,055 0,081 7,781 3,170 8,072 3,171 0,098 0,142 RM 250 .97 5,505 1,376 5,506 1,377 0,075 0,110 5,172 1,149 5,173 1,150 0,057 0,084 6,628 2,594 6,631 2,595 0,101 0,145 RM 500 .97 5,505 1,376 5,506 1,377 0,075 0,110 5,172 1,149 5,173 1,150 0,057 0,084 6,628 2,594 6,631 2,595 0,101 0,145 RM 1250 .97 5,505 1,376 5,506 1,377 0,075 0,110 5,172 1,149 5,173 1,150 0,057 0,084 6,628 2,594 6,631 2,595 0,101 0,145 KS 50 10,092 4,817 10,094 4,817 0,054 0,079 11,063 4,598 11,064 4,598 0,040 0,059 12,680 7,493 12,685 7,495 0,077 0,109 KS 100 11,697 6,651 11,700 6,653 0,049 0,071 10,920 5,029 10,921 5,029 0,040 0,059 15,274 9,222 15,569 9,227 0,069 0,097 KS 250 13,991 8,486 13,995 8,488 0,042 0,060 10,920 4,454 10,921 4,455 0,041 0,060 21,037 12,392 21,047 12,397 0,051 0,071 KS 500 19,954 10,780 19,959 10,783 0,033 0,048 9,770 4,023 9,771 4,024 0,042 0,062 25,072 17,003 25,083 16,722 0,044 0,058 KS 1250 21,771 17,661 24,777 17,665 0,026 0,036 14,368 7,471 14,370 7,472 0,033 0,048 24,784 16,427 24,796 24,795 0,044 0,060 GK 50 6,193 1,376 6,194 1,376 0,075 0,111 5,029 1,437 5,029 1,437 0,058 0,084 8,357 3,458 8,360 3,459 0,094 0,136 GK 100 6,193 2,523 6,194 2,523 0,074 0,108 4,598 1,437 4,598 1,437 0,061 0,089 9,510 5,476 9,514 5,477 0,092 0,132 GK 250 7,339 3,211 7,341 3,212 0,068 0,098 3,736 1,293 3,736 1,293 0,067 0,096 9,510 5,187 9,514 5,189 0,083 0,118 GK 500 8,486 4,587 8,488 4,588 0,058 0,084 2,730 1,006 2,730 1,006 0,071 0,102 12,104 7,205 12,109 7,207 0,072 0,102 GK 1250 12,615 8,028 12,618 8,029 0,044 0,063 3,305 1,006 3,305 1,006 0,064 0,092 13,545 7,205 13,262 7,207 0,071 0,102 RS 50 12,156 5,734 12,158 5,735 0,051 0,075 8,477 2,586 8,478 2,587 0,046 0,068 12,392 6,340 12,396 6,342 0,078 0,113 RS 100 12,884 6,422 12,847 6,423 0,050 0,072 7,615 2,730 7,616 2,730 0,048 0,071 12,392 7,781 12,398 7,784 0,079 0,110 RS 250 12,156 7,110 12,159 7,112 0,047 0,067 6,466 2,874 6,466 2,874 0,051 0,074 13,256 7,781 13,262 7,784 0,070 0,100 RS 500 14,679 8,486 14,683 8,488 0,041 0,058 5,747 1,724 5,748 1,725 0,053 0,077 16,715 8,934 16,433 8,937 0,060 0,086 RS 1250 20,413 12,385 20,418 12,389 0,032 0,045 8,477 3,017 8,478 3,018 0,045 0,067 17,579 10,375 17,229 10,379 0,057 0,080 P 50 11,009 5,275 11,011 5,276 0,053 0,078 8,764 2,730 8,765 2,730 0,046 0,069 12,104 5,764 12,396 5,766 0,079 0,116 P 100 11,468 5,275 11,471 5,277 0,052 0,076 8,046 2,874 8,047 2,874 0,049 0,071 12,104 7,205 12,397 7,207 0,080 0,113 P 250 11,468 7,110 11,471 7,112 0,049 0,070 6,322 2,874 6,323 2,874 0,052 0,075 12,968 7,493 12,974 7,496 0,071 0,101 P 500 13,991 8,486 13,994 8,488 0,042 0,060 6,034 1,724 6,035 1,724 0,054 0,078 16,427 8,934 16,433 8,937 0,061 0,087 P 1250 12,385 11,468 19,730 11,471 0,033 0,047 7,902 2,874 7,903 2,874 0,046 0,068 17,579 9,510 17,299 9,514 0,058 0,081 YZ 50 6,193 1,376 6,194 1,376 0,076 0,111 5,029 1,293 5,029 1,293 0,058 0,085 8,069 3,458 8,072 3,459 0,095 0,137 YZ 100 5,963 2,523 5,965 2,523 0,075 0,108 4,598 1,580 4,598 1,581 0,062 0,089 9,510 5,476 9,514 5,477 0,093 0,133 YZ 250 7,110 3,440 7,112 3,441 0,069 0,099 3,736 1,149 3,736 1,150 0,067 0,096 9,510 5,187 9,514 5,189 0,083 0,119 YZ 500 8,257 4,587 8,259 4,588 0,059 0,084 2,730 1,006 2,730 1,006 0,072 0,102 16,427 7,205 12,109 7,207 0,072 0,102 Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 23 Table 2. Netherlands (AEX) loss function statistics (part 2) Period 1 Period 2 Period 3 Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% YZ 1250 12,385 8,028 12,389 8,029 0,045 0,063 3,305 1,006 3,305 1,006 0,064 0,092 13,256 6,916 12,974 6,919 0,072 0,103 GARCH 6,193 2,294 6,194 2,294 0,071 0,103 5,172 1,437 5,173 1,437 0,056 0,082 6,628 1,441 6,341 1,441 0,096 0,141 EGARCH 5,734 2,523 5,735 2,523 0,069 0,100 4,023 1,293 4,024 1,293 0,059 0,085 7,781 1,441 7,782 1,441 0,090 0,135 HS 50 8,028 2,752 8,029 2,753 0,069 0,103 7,184 1,437 7,185 1,437 0,054 0,078 9,222 3,746 9,513 3,747 0,097 0,130 HS 100 7,569 2,294 7,571 2,294 0,070 0,113 5,316 1,724 5,317 1,724 0,058 0,088 8,646 4,611 8,649 4,613 0,095 0,150 HS 250 7,798 2,064 7,800 2,065 0,066 0,113 4,023 1,293 4,024 1,293 0,063 0,104 10,086 2,017 10,090 2,019 0,082 0,158 HS 500 9,404 3,211 9,406 3,212 0,053 0,098 3,305 0,718 3,305 0,718 0,067 0,122 12,104 4,035 12,109 4,036 0,072 0,129 HS 1250 15,367 5,275 15,371 5,276 0,039 0,077 4,454 0,718 4,455 0,718 0,057 0,117 13,883 3,458 13,550 3,459 0,068 0,135 Table 3. Germany (DAX) loss function statistics (part 1) Period 1 Period 2 Period 3 Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% MA Z 50 7,569 2,523 7,571 2,524 0,078 0,113 6,034 1,293 6,035 1,293 0,067 0,098 7,493 2,305 7,495 2,306 0,107 0,157 MA Z 100 7,339 2,064 7,342 2,065 0,075 0,110 5,172 1,149 5,173 1,149 0,070 0,102 8,357 4,035 8,361 4,036 0,102 0,148 MA Z 250 7,798 3,670 7,801 3,671 0,069 0,099 4,741 1,580 4,742 1,580 0,073 0,106 8,934 4,035 8,937 4,036 0,091 0,132 MA Z 500 9,862 5,275 9,866 5,277 0,059 0,084 3,592 0,862 3,592 0,862 0,076 0,110 10,951 6,340 10,955 6,054 0,081 0,116 MA Z 1250 12,844 6,881 12,849 6,883 0,050 0,071 6,609 1,293 6,610 1,293 0,065 0,096 12,104 6,916 12,108 6,630 0,077 0,110 RM 50 .90 5,963 2,294 5,965 2,294 0,077 0,110 7,328 1,580 7,328 1,580 0,064 0,095 7,493 1,153 7,494 0,865 0,105 0,157 RM 100 .90 5,963 2,294 5,965 2,294 0,077 0,111 7,328 1,580 7,328 1,580 0,064 0,095 7,205 1,153 7,206 0,865 0,106 0,157 RM 250 .90 5,963 2,294 5,965 2,294 0,077 0,111 7,328 1,580 7,328 1,580 0,064 0,095 7,205 1,153 7,206 0,865 0,106 0,157 RM 500 .90 5,963 2,294 5,965 2,294 0,077 0,111 7,328 1,580 7,328 1,580 0,064 0,095 7,205 1,153 7,206 0,865 0,106 0,157 RM 1250 .90 5,963 2,294 5,965 2,294 0,077 0,111 7,328 1,580 7,328 1,580 0,064 0,095 7,205 1,153 7,206 0,865 0,106 0,157 RM 50 .94 6,651 2,523 6,653 2,524 0,076 0,109 7,328 1,724 7,328 1,724 0,063 0,093 8,069 1,153 8,071 0,865 0,103 0,154 RM 100 .94 6,193 2,523 6,194 2,523 0,077 0,111 6,897 1,293 6,897 1,293 0,065 0,096 7,781 0,865 7,782 0,865 0,105 0,157 RM 250 .94 5,963 2,523 5,965 2,523 0,078 0,111 6,753 1,293 6,753 1,293 0,065 0,096 7,781 0,865 7,782 0,865 0,105 0,157 RM 500 .94 5,963 2,523 5,965 2,523 0,078 0,111 6,753 1,293 6,753 1,293 0,065 0,096 7,781 0,865 7,782 0,865 0,105 0,157 RM 1250 .94 5,963 2,523 5,965 2,523 0,078 0,111 6,753 1,293 6,753 1,293 0,065 0,096 7,781 0,865 7,782 0,865 0,105 0,157 RM 50 .97 8,716 3,899 8,718 3,900 0,068 0,098 8,477 2,730 8,478 2,730 0,057 0,084 10,663 3,458 10,665 3,459 0,092 0,137 RM 100 .97 7,339 2,523 7,341 2,524 0,075 0,109 6,753 1,580 6,753 1,580 0,065 0,096 8,069 2,017 8,071 2,018 0,102 0,151 RM 250 .97 6,422 2,294 6,424 2,294 0,077 0,111 6,178 1,149 6,178 1,149 0,067 0,099 7,493 2,017 7,495 2,018 0,104 0,154 RM 500 .97 6,422 2,294 6,424 2,294 0,077 0,111 6,178 1,149 6,178 1,149 0,067 0,099 7,205 2,017 7,207 2,018 0,104 0,154 RM 1250 .97 6,422 2,294 6,424 2,294 0,077 0,111 6,178 1,149 6,178 1,149 0,067 0,099 7,205 2,017 7,207 2,018 0,104 0,154 KS 50 10,092 7,110 10,095 7,112 0,058 0,081 13,218 5,316 13,219 5,316 0,046 0,070 12,392 6,052 12,396 6,054 0,082 0,120 KS 100 11,697 7,339 11,702 7,342 0,054 0,075 12,213 5,172 12,214 5,173 0,047 0,070 14,986 8,069 14,992 8,361 0,074 0,106 KS 250 13,991 8,716 13,996 8,719 0,046 0,064 12,356 5,172 12,358 5,173 0,046 0,069 18,156 10,086 18,163 10,090 0,059 0,084 KS 500 18,119 12,156 18,126 12,160 0,037 0,051 12,356 5,747 12,358 5,748 0,046 0,068 21,037 13,833 21,047 13,838 0,053 0,073 KS 1250 21,789 15,826 21,796 15,831 0,032 0,043 17,098 9,914 17,100 9,915 0,035 0,052 23,055 15,562 23,065 15,568 0,050 0,068 GK 50 10,550 4,587 10,553 4,588 0,062 0,090 6,753 2,011 6,753 2,012 0,063 0,093 8,934 3,170 8,936 3,171 0,097 0,142 GK 100 10,321 5,505 10,325 5,506 0,061 0,086 6,034 1,868 6,035 1,868 0,065 0,096 8,934 4,611 8,937 4,613 0,095 0,137 GK 250 10,550 5,963 10,554 5,965 0,055 0,079 5,460 1,437 5,460 1,437 0,068 0,100 9,510 4,611 9,514 4,324 0,087 0,127 GK 500 13,761 8,257 13,766 8,260 0,047 0,066 5,603 1,006 5,604 1,006 0,068 0,100 12,104 6,628 12,109 6,342 0,076 0,109 GK 1250 16,284 9,174 16,290 9,178 0,042 0,059 9,339 2,730 9,340 2,730 0,055 0,082 14,697 8,069 14,703 7,783 0,070 0,101 RS 50 19,266 14,679 19,272 14,682 0,039 0,052 7,615 2,155 7,616 2,155 0,058 0,087 10,663 2,882 10,665 2,883 0,095 0,141 Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 24 Table 3. Germany (DAX) loss function statistics (part 2) Period 1 Period 2 Period 3 Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% RS 100 19,954 14,679 19,960 14,683 0,037 0,049 7,184 2,155 7,184 2,155 0,060 0,089 9,222 4,323 9,226 4,324 0,095 0,137 RS 250 23,624 15,596 23,631 15,602 0,031 0,042 7,759 1,868 7,759 1,868 0,060 0,090 9,798 5,187 9,802 4,901 0,086 0,124 RS 500 27,294 18,807 27,302 18,814 0,027 0,036 8,908 3,161 8,909 3,161 0,056 0,084 12,392 6,916 12,397 6,630 0,075 0,109 RS 1250 28,440 22,477 28,450 22,484 0,026 0,032 14,080 7,902 14,082 7,903 0,041 0,060 16,715 9,510 16,722 9,513 0,063 0,090 P 50 17,890 12,615 17,895 12,618 0,042 0,057 6,897 2,155 6,897 2,155 0,060 0,088 9,510 3,170 9,513 3,171 0,098 0,144 P 100 17,890 13,532 17,896 13,536 0,039 0,053 6,753 2,155 6,753 2,155 0,062 0,091 9,222 4,611 9,226 4,613 0,096 0,138 P 250 20,413 14,679 20,420 14,684 0,034 0,046 7,471 1,724 7,472 1,724 0,062 0,093 9,798 5,187 9,802 4,901 0,086 0,124 P 500 25,459 16,972 25,467 16,978 0,028 0,039 8,477 2,011 8,478 2,012 0,059 0,088 12,392 6,628 12,397 6,342 0,076 0,110 P 1250 27,294 19,954 27,302 19,961 0,027 0,035 12,931 7,184 12,932 7,184 0,043 0,063 16,138 8,934 16,145 8,649 0,064 0,092 YZ 50 10,321 4,817 10,324 4,818 0,062 0,090 6,609 1,868 6,610 1,868 0,064 0,094 8,646 2,882 8,648 2,883 0,098 0,145 YZ 100 10,550 5,734 10,554 5,736 0,061 0,086 5,891 1,724 5,891 1,724 0,066 0,097 8,357 4,323 8,361 4,324 0,097 0,139 YZ 250 11,009 5,963 11,013 5,965 0,055 0,078 5,460 1,580 5,460 1,581 0,069 0,101 9,798 4,899 9,802 4,612 0,087 0,126 YZ 500 13,761 8,257 13,766 8,260 0,047 0,066 5,460 0,862 5,460 0,862 0,069 0,101 12,104 6,628 12,108 6,342 0,077 0,111 YZ 1250 16,284 9,174 16,290 9,178 0,042 0,059 9,195 2,586 9,196 2,586 0,055 0,083 14,121 8,069 14,127 7,783 0,071 0,102 GARCH 6,422 3,899 6,424 3,900 0,071 0,100 6,322 1,293 6,322 1,293 0,066 0,097 8,069 1,153 8,070 0,865 0,100 0,151 EGARCH 7,110 4,587 7,112 4,588 0,068 0,095 5,747 1,149 5,747 1,149 0,066 0,098 8,357 0,865 8,359 0,577 0,094 0,144 HS 50 7,339 3,211 7,341 3,212 0,076 0,113 7,615 3,017 7,616 3,017 0,063 0,086 8,646 4,611 8,648 4,612 0,100 0,130 HS 100 8,028 2,752 8,030 2,753 0,075 0,114 5,891 2,299 5,891 2,299 0,071 0,098 8,357 3,746 8,360 4,036 0,099 0,143 HS 250 8,716 2,064 8,718 2,065 0,066 0,123 4,454 1,149 4,454 1,149 0,075 0,117 8,646 2,017 8,649 2,018 0,090 0,152 HS 500 11,468 2,752 11,472 2,753 0,054 0,110 4,310 0,718 4,311 0,718 0,073 0,131 12,104 4,035 12,108 4,036 0,079 0,124 HS 1250 13,991 4,587 13,996 4,589 0,047 0,087 7,184 0,287 7,184 0,287 0,062 0,119 12,968 3,746 12,973 3,459 0,074 0,133 Table 4. Canada (TSX) loss function statistics (part 1) Period 1 Period 2 Period 3 Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% MA Z 50 5,734 2,523 5,735 2,524 0,049 0,071 4,598 1,437 4,599 1,437 0,061 0,088 4,899 2,305 4,900 2,306 0,048 0,069 MA Z 100 6,193 2,294 6,194 2,295 0,048 0,069 4,310 1,580 4,311 1,581 0,063 0,090 5,476 2,594 5,476 2,594 0,047 0,067 MA Z 250 7,110 2,752 7,112 2,753 0,042 0,060 4,741 1,724 4,743 1,725 0,063 0,091 5,764 2,017 5,764 2,017 0,050 0,073 MA Z 500 8,945 4,817 8,947 4,818 0,036 0,051 6,322 1,724 6,323 1,725 0,058 0,085 3,170 0,576 3,170 0,576 0,063 0,091 MA Z 1250 10,780 6,881 10,782 6,882 0,031 0,044 8,477 4,023 8,479 4,024 0,046 0,066 2,882 0,865 2,882 0,865 0,059 0,084 RM 50 .90 5,505 2,752 5,506 2,753 0,047 0,068 5,316 2,443 5,317 2,443 0,059 0,084 7,205 1,441 7,205 1,441 0,046 0,068 RM 100 .90 5,505 2,752 5,506 2,753 0,047 0,068 5,316 2,443 5,317 2,443 0,059 0,085 6,916 1,441 6,917 1,441 0,046 0,069 RM 250 .90 5,505 2,752 5,506 2,753 0,047 0,068 5,316 2,443 5,317 2,443 0,059 0,085 6,916 1,441 6,917 1,441 0,046 0,069 RM 500 .90 5,505 2,752 5,506 2,753 0,047 0,068 5,316 2,443 5,317 2,443 0,059 0,085 6,916 1,441 6,917 1,441 0,046 0,069 RM 1250 .90 5,505 2,752 5,506 2,753 0,047 0,068 5,316 2,443 5,317 2,443 0,059 0,085 6,916 1,441 6,917 1,441 0,046 0,069 RM 50 .94 5,505 2,523 5,506 2,524 0,047 0,067 5,603 2,155 5,605 2,156 0,058 0,084 6,916 2,017 6,917 2,017 0,045 0,067 RM 100 .94 4,817 2,523 4,818 2,524 0,048 0,069 5,316 1,724 5,317 1,725 0,060 0,086 6,340 2,017 6,341 2,017 0,047 0,068 RM 250 .94 4,817 2,523 4,818 2,524 0,049 0,069 5,172 1,724 5,174 1,725 0,060 0,086 6,340 2,017 6,341 2,017 0,047 0,069 RM 500 .94 4,817 2,523 4,818 2,524 0,049 0,069 5,172 1,724 5,174 1,725 0,060 0,086 6,340 2,017 6,341 2,017 0,047 0,069 RM 1250 .94 4,817 2,523 4,818 2,524 0,049 0,069 5,172 1,724 5,174 1,725 0,060 0,086 6,340 2,017 6,341 2,017 0,047 0,069 RM 50 .97 7,339 3,440 7,341 3,441 0,042 0,061 6,609 2,874 6,611 2,874 0,053 0,076 8,069 3,170 8,358 3,170 0,041 0,060 RM 100 .97 5,734 2,523 5,735 2,524 0,047 0,068 5,029 1,868 5,030 1,868 0,059 0,085 6,340 2,305 6,341 2,306 0,046 0,067 RM 250 .97 5,734 2,523 5,735 2,524 0,048 0,069 4,598 1,868 4,599 1,868 0,061 0,088 5,764 2,017 5,764 2,017 0,047 0,069 RM 500 .97 5,734 2,523 5,735 2,524 0,048 0,069 4,454 1,868 4,455 1,868 0,061 0,088 5,764 2,017 5,764 2,017 0,047 0,069 Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 25 Table 4. Canada (TSX) loss function statistics (part 2) Period 1 Period 2 Period 3 Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% RM 1250 .97 5,734 2,523 5,735 2,524 0,048 0,069 4,454 1,868 4,455 1,868 0,061 0,088 5,764 2,017 5,764 2,017 0,047 0,069 KS 50 11,927 6,422 11,929 6,424 0,032 0,046 11,207 4,741 11,209 4,743 0,041 0,060 11,239 5,187 11,528 5,188 0,033 0,049 KS 100 12,615 8,028 12,618 8,030 0,029 0,041 11,351 5,316 11,353 5,317 0,040 0,058 13,256 6,628 13,834 6,629 0,030 0,045 KS 250 14,908 9,404 14,911 9,406 0,025 0,035 10,201 6,322 10,203 6,323 0,038 0,055 13,256 7,205 13,258 7,205 0,030 0,044 KS 500 17,431 11,697 17,435 11,700 0,021 0,029 12,213 8,190 12,215 8,191 0,034 0,048 10,086 5,476 10,087 5,476 0,035 0,051 KS 1250 19,266 14,450 19,270 14,453 0,018 0,025 22,270 12,931 22,274 12,934 0,023 0,034 11,816 6,052 12,105 6,052 0,031 0,046 GK 50 8,257 4,128 8,259 4,130 0,040 0,057 7,759 2,299 7,760 2,300 0,052 0,076 8,357 3,458 8,934 3,459 0,040 0,058 GK 100 8,716 4,817 8,718 4,818 0,039 0,055 6,753 2,011 6,754 2,012 0,054 0,079 9,222 3,746 9,223 3,747 0,039 0,057 GK 250 9,174 5,505 9,176 5,506 0,034 0,048 7,184 2,730 7,185 2,731 0,054 0,079 7,781 2,882 7,781 2,882 0,045 0,066 GK 500 11,009 6,881 11,012 6,882 0,030 0,042 7,615 3,305 7,617 3,306 0,050 0,072 4,323 0,865 4,323 0,865 0,056 0,082 GK 1250 15,138 9,174 15,141 9,176 0,025 0,035 10,345 6,753 10,347 6,754 0,039 0,055 4,323 1,729 4,323 1,729 0,051 0,073 RS 50 16,055 9,862 16,058 9,865 0,024 0,034 17,098 9,483 17,100 9,484 0,032 0,046 13,833 7,205 14,410 7,782 0,030 0,044 RS 100 15,826 11,009 15,829 11,012 0,024 0,033 15,517 8,621 15,520 8,622 0,033 0,048 14,986 6,916 15,563 6,917 0,028 0,043 RS 250 18,349 11,927 18,352 11,929 0,020 0,029 12,644 7,471 12,646 7,473 0,035 0,050 14,697 7,781 14,699 7,782 0,028 0,041 RS 500 19,725 14,679 19,728 14,682 0,019 0,025 14,511 8,477 14,514 8,479 0,032 0,046 10,375 5,187 10,376 5,188 0,034 0,051 RS 1250 20,413 15,826 20,417 15,829 0,017 0,023 21,408 12,069 21,412 12,072 0,024 0,035 11,816 6,052 12,105 6,052 0,031 0,046 P 50 11,927 7,110 11,929 7,112 0,030 0,043 13,649 6,609 13,651 6,610 0,036 0,054 11,239 5,187 11,817 5,188 0,033 0,049 P 100 12,156 7,339 12,159 7,341 0,030 0,042 12,356 6,034 12,358 6,036 0,038 0,056 11,239 5,187 11,817 5,188 0,032 0,048 P 250 13,761 8,486 13,764 8,488 0,027 0,038 10,345 5,603 10,347 5,605 0,040 0,057 12,392 5,764 12,393 5,764 0,032 0,048 P 500 15,596 9,404 15,599 9,406 0,023 0,033 10,489 7,184 10,491 7,185 0,037 0,053 8,357 4,035 8,358 4,035 0,039 0,057 P 1250 17,890 11,927 17,893 11,929 0,020 0,028 16,523 9,339 16,526 9,341 0,029 0,042 9,510 4,323 9,511 4,323 0,036 0,053 YZ 50 7,569 3,899 7,571 3,900 0,041 0,059 7,471 2,155 7,473 2,156 0,052 0,077 8,357 3,170 8,934 3,170 0,041 0,060 YZ 100 8,028 4,587 8,030 4,589 0,040 0,057 6,322 2,011 6,323 2,012 0,055 0,080 8,934 3,746 8,934 3,747 0,040 0,058 YZ 250 8,945 4,587 8,947 4,589 0,035 0,051 6,466 2,730 6,467 2,731 0,055 0,080 7,781 3,170 7,781 3,170 0,044 0,065 YZ 500 10,321 6,651 10,324 6,653 0,031 0,044 7,328 2,874 7,329 2,875 0,051 0,074 3,746 0,865 3,747 0,865 0,057 0,083 YZ 1250 14,450 8,716 14,452 8,718 0,026 0,036 9,914 6,609 9,916 6,611 0,040 0,056 4,323 1,729 4,323 1,729 0,052 0,075 GARCH 6,881 2,752 6,882 2,753 0,044 0,064 4,598 2,011 4,599 2,012 0,062 0,089 4,323 1,153 4,323 1,153 0,051 0,075 EGARCH 8,486 3,670 8,488 3,671 0,038 0,056 5,747 2,011 5,748 2,012 0,058 0,084 2,305 0,288 2,306 0,288 0,055 0,079 HS 50 6,881 3,211 6,882 3,212 0,046 0,077 6,897 2,155 6,898 2,156 0,053 0,087 7,493 4,035 7,493 4,035 0,047 0,062 HS 100 8,028 2,982 8,029 2,983 0,046 0,074 6,322 1,724 6,323 1,725 0,057 0,089 6,628 2,882 6,629 2,882 0,044 0,069 HS 250 8,945 1,606 8,947 1,606 0,037 0,071 5,747 1,724 5,748 1,725 0,060 0,095 6,340 0,865 6,340 0,865 0,049 0,085 HS 500 11,697 2,294 11,700 2,295 0,030 0,064 6,466 1,149 6,467 1,150 0,058 0,099 2,882 0,576 2,882 0,576 0,065 0,099 HS 1250 13,073 3,899 13,076 3,900 0,027 0,057 9,052 1,580 9,054 1,581 0,043 0,088 2,882 0,288 2,882 0,288 0,059 0,100 Table 5. US (DJI) loss function statistics (part 1) Period 1 Period 2 Period 3 Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% MA Z 50 4,358 2,064 4,359 2,065 0,059 0,083 6,609 1,580 6,610 1,581 0,054 0,079 5,187 1,729 5,477 1,730 0,072 0,104 MA Z 100 4,817 1,835 4,818 1,836 0,058 0,083 5,603 1,580 5,604 1,581 0,056 0,081 5,187 1,729 5,477 1,730 0,069 0,100 MA Z 250 5,734 2,294 5,736 2,295 0,053 0,076 4,885 1,580 4,886 1,581 0,057 0,083 6,340 1,729 6,342 1,730 0,064 0,094 MA Z 500 7,569 3,440 7,571 3,442 0,047 0,067 4,741 1,293 4,742 1,293 0,057 0,083 7,493 2,305 7,494 2,306 0,061 0,090 MA Z 1250 9,633 5,046 9,635 5,048 0,037 0,054 6,466 1,868 6,466 1,868 0,049 0,072 8,069 2,594 8,071 2,595 0,058 0,085 RM 50 .90 5,046 3,211 5,047 3,212 0,056 0,079 6,609 2,011 6,610 2,012 0,052 0,077 4,899 1,729 4,900 1,730 0,070 0,101 RM 100 .90 5,046 3,211 5,047 3,212 0,056 0,079 6,609 2,011 6,610 2,012 0,053 0,077 4,611 1,729 4,612 1,730 0,070 0,101 Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 26 Table 5. US (DJI) loss function statistics (part 2) Period 1 Period 2 Period 3 Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% RM 250 .90 5,046 3,211 5,047 3,212 0,056 0,079 6,609 2,011 6,610 2,012 0,053 0,077 4,611 1,729 4,612 1,730 0,070 0,101 RM 500 .90 5,046 3,211 5,047 3,212 0,056 0,079 6,609 2,011 6,610 2,012 0,053 0,077 4,611 1,729 4,612 1,730 0,070 0,101 RM 1250 .90 5,046 3,211 5,047 3,212 0,056 0,079 6,609 2,011 6,610 2,012 0,053 0,077 4,611 1,729 4,612 1,730 0,070 0,101 RM 50 .94 4,817 3,211 4,818 3,212 0,056 0,079 6,753 1,724 6,753 1,724 0,052 0,076 4,899 1,441 5,189 1,442 0,069 0,100 RM 100 .94 4,358 2,752 4,359 2,753 0,057 0,081 6,322 1,724 6,322 1,724 0,053 0,078 4,323 1,153 4,324 1,154 0,071 0,103 RM 250 .94 4,358 2,752 4,359 2,753 0,057 0,081 6,322 1,724 6,322 1,724 0,053 0,078 4,323 1,153 4,324 1,154 0,071 0,103 RM 500 .94 4,358 2,752 4,359 2,753 0,057 0,081 6,322 1,724 6,322 1,724 0,053 0,078 4,323 1,153 4,324 1,154 0,071 0,103 RM 1250 .94 4,358 2,752 4,359 2,753 0,057 0,081 6,322 1,724 6,322 1,724 0,053 0,078 4,323 1,153 4,324 1,154 0,071 0,103 RM 50 .97 6,193 3,211 6,194 3,212 0,051 0,072 8,190 3,305 8,190 3,305 0,047 0,068 6,340 3,458 6,630 3,459 0,062 0,090 RM 100 .97 4,817 2,294 4,818 2,295 0,057 0,080 6,322 1,580 6,322 1,581 0,053 0,077 5,187 1,729 5,477 1,730 0,069 0,100 RM 250 .97 4,587 2,294 4,589 2,295 0,058 0,082 5,891 1,437 5,891 1,437 0,054 0,080 4,899 1,441 4,900 1,442 0,071 0,102 RM 500 .97 4,587 2,294 4,589 2,295 0,058 0,082 5,891 1,437 5,891 1,437 0,054 0,080 4,899 1,441 4,900 1,442 0,071 0,102 RM 1250 .97 4,587 2,294 4,589 2,295 0,058 0,082 5,891 1,437 5,891 1,437 0,054 0,080 4,899 1,441 4,900 1,442 0,071 0,102 KS 50 11,239 5,734 11,241 5,736 0,037 0,053 12,500 6,466 12,501 6,466 0,037 0,054 9,510 5,187 10,089 5,189 0,050 0,072 KS 100 11,468 6,651 11,471 6,653 0,035 0,050 11,925 6,322 11,927 6,322 0,036 0,052 14,121 6,628 14,700 6,918 0,044 0,065 KS 250 11,697 8,028 11,700 8,030 0,032 0,045 11,925 5,747 11,927 5,748 0,034 0,050 16,715 8,934 17,006 8,936 0,037 0,055 KS 500 14,450 9,174 14,453 9,177 0,028 0,040 13,075 6,034 13,076 6,035 0,033 0,049 17,867 10,375 18,159 10,377 0,035 0,052 KS 1250 19,954 12,615 19,958 12,618 0,022 0,030 16,667 9,770 16,668 9,771 0,027 0,039 19,885 11,239 20,177 11,530 0,033 0,048 GK 50 5,275 2,294 5,277 2,295 0,052 0,075 7,759 2,586 7,759 2,586 0,047 0,069 7,493 2,882 7,783 2,883 0,059 0,086 GK 100 5,505 2,752 5,506 2,753 0,052 0,074 6,753 2,874 6,754 2,874 0,049 0,071 8,934 3,458 9,224 3,459 0,057 0,083 GK 250 6,651 2,752 6,653 2,753 0,049 0,071 5,747 2,299 5,748 2,299 0,050 0,072 8,934 2,882 9,224 2,883 0,054 0,081 GK 500 8,257 3,670 8,259 3,671 0,044 0,064 5,603 1,868 5,604 1,868 0,050 0,073 10,375 3,458 10,377 3,459 0,051 0,076 GK 1250 10,321 5,505 10,324 5,506 0,036 0,052 7,471 3,305 7,472 3,305 0,044 0,065 10,951 4,035 10,953 4,036 0,050 0,074 RS 50 6,193 2,523 6,194 2,524 0,050 0,072 8,046 2,874 8,047 2,874 0,045 0,066 7,781 3,170 8,071 3,171 0,057 0,083 RS 100 6,651 2,982 6,653 2,983 0,049 0,070 6,897 3,017 6,897 3,018 0,047 0,068 9,510 3,746 9,800 3,747 0,055 0,081 RS 250 8,257 3,670 8,259 3,671 0,045 0,065 6,178 2,586 6,179 2,587 0,049 0,071 10,086 3,458 10,377 3,459 0,052 0,077 RS 500 8,486 4,128 8,488 4,130 0,042 0,060 6,466 2,299 6,466 2,299 0,048 0,071 10,951 4,035 11,241 4,036 0,050 0,074 RS 1250 10,550 5,734 10,553 5,736 0,036 0,051 8,046 3,592 8,047 3,592 0,043 0,063 11,239 4,323 11,530 4,324 0,048 0,071 P 50 5,963 2,523 5,965 2,524 0,051 0,074 7,471 2,443 7,472 2,443 0,048 0,071 6,340 2,594 6,630 2,595 0,061 0,089 P 100 5,963 2,752 5,965 2,754 0,051 0,073 6,753 2,443 6,754 2,443 0,050 0,073 7,781 3,458 8,071 3,459 0,059 0,085 P 250 7,339 3,670 7,341 3,671 0,047 0,067 5,603 2,011 5,604 2,012 0,051 0,075 9,222 2,594 9,224 2,595 0,055 0,081 P 500 8,716 4,128 8,718 4,130 0,042 0,061 5,316 1,580 5,317 1,581 0,051 0,075 10,375 3,170 10,377 3,171 0,053 0,078 P 1250 10,550 6,193 10,553 6,194 0,035 0,050 7,471 3,305 7,472 3,305 0,044 0,065 10,663 4,035 10,665 4,036 0,051 0,075 YZ 50 5,046 2,064 5,048 2,065 0,053 0,077 7,471 2,443 7,472 2,443 0,048 0,070 6,628 2,594 6,918 2,595 0,061 0,089 YZ 100 5,275 2,294 5,277 2,295 0,053 0,076 6,753 2,299 6,754 2,299 0,050 0,073 7,781 2,882 8,071 2,883 0,059 0,087 YZ 250 6,651 2,752 6,653 2,753 0,049 0,071 5,603 1,868 5,604 1,868 0,051 0,075 8,646 2,594 8,647 2,595 0,056 0,082 YZ 500 8,257 3,670 8,259 3,671 0,045 0,065 5,316 1,580 5,317 1,581 0,051 0,075 10,375 3,170 10,377 3,171 0,053 0,079 YZ 1250 9,633 4,817 9,635 4,818 0,037 0,054 7,040 2,874 7,041 2,874 0,046 0,067 10,663 3,746 10,665 3,747 0,051 0,076 GARCH 5,734 2,982 5,736 2,983 0,053 0,076 6,178 1,149 6,179 1,150 0,054 0,079 4,899 1,441 4,900 1,442 0,068 0,099 EGARCH 7,110 2,523 7,112 2,524 0,047 0,068 5,747 1,293 5,748 1,293 0,054 0,079 5,187 1,153 5,188 1,153 0,066 0,097 HS 50 6,193 2,752 6,194 2,753 0,053 0,086 7,471 2,443 7,472 2,443 0,050 0,074 6,340 2,882 6,630 2,883 0,062 0,097 HS 100 5,963 2,523 5,965 2,524 0,052 0,083 5,891 2,155 5,891 2,155 0,054 0,078 7,493 2,305 7,783 2,307 0,058 0,102 HS 250 6,881 1,835 6,883 1,836 0,048 0,081 5,029 1,293 5,029 1,293 0,056 0,089 8,357 1,729 8,359 1,730 0,058 0,100 HS 500 8,028 2,294 8,030 2,295 0,043 0,074 5,316 0,862 5,317 0,862 0,054 0,089 7,781 2,017 7,783 2,018 0,059 0,101 HS 1250 11,239 3,440 11,241 3,442 0,034 0,063 6,897 1,293 6,897 1,293 0,046 0,081 10,086 2,017 10,088 2,018 0,055 0,094 Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 27 Table 6. Japan (NIKKEI) loss function statistics (part 1) Period 1 Period 2 Period 3 Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% MA Z 50 6,651 2,752 6,652 2,753 0,077 0,112 5,316 2,299 5,317 2,299 0,066 0,095 5,476 1,153 5,476 1,153 0,082 0,119 MA Z 100 5,734 2,523 5,735 2,523 0,078 0,111 4,454 2,011 4,455 2,012 0,068 0,098 4,323 1,153 4,324 1,153 0,082 0,119 MA Z 250 5,734 1,835 5,735 1,835 0,075 0,109 4,741 1,580 4,742 1,581 0,068 0,099 4,323 1,153 4,324 1,153 0,083 0,120 MA Z 500 7,798 3,211 7,800 3,212 0,066 0,095 4,741 1,437 4,742 1,437 0,071 0,103 4,899 0,865 4,900 0,865 0,078 0,114 MA Z 1250 10,321 3,440 10,323 3,441 0,060 0,089 4,310 1,437 4,311 1,437 0,068 0,099 6,052 1,153 6,053 1,153 0,074 0,109 RM 50 .90 7,339 1,606 7,340 1,606 0,074 0,111 5,891 2,299 5,892 2,299 0,064 0,093 7,781 1,441 7,782 1,441 0,077 0,115 RM 100 .90 7,339 1,606 7,340 1,606 0,075 0,111 5,891 2,299 5,892 2,299 0,064 0,093 7,781 1,441 7,782 1,441 0,077 0,115 RM 250 .90 7,339 1,606 7,340 1,606 0,075 0,111 5,891 2,299 5,892 2,299 0,064 0,093 7,781 1,441 7,782 1,441 0,077 0,115 RM 500 .90 7,339 1,606 7,340 1,606 0,075 0,111 5,891 2,299 5,892 2,299 0,064 0,093 7,781 1,441 7,782 1,441 0,077 0,115 RM 1250 .90 7,339 1,606 7,340 1,606 0,075 0,111 5,891 2,299 5,892 2,299 0,064 0,093 7,781 1,441 7,782 1,441 0,077 0,115 RM 50 .94 6,881 2,064 6,881 2,064 0,074 0,108 5,603 2,155 5,604 2,156 0,063 0,092 7,493 1,153 7,494 1,153 0,076 0,114 RM 100 .94 6,651 2,064 6,652 2,064 0,076 0,111 5,460 2,011 5,461 2,012 0,065 0,095 6,628 1,153 6,629 1,153 0,079 0,116 RM 250 .94 6,651 1,835 6,652 1,835 0,076 0,111 5,460 2,011 5,461 2,012 0,065 0,095 6,628 1,153 6,629 1,153 0,079 0,117 RM 500 .94 6,651 1,835 6,652 1,835 0,076 0,111 5,460 2,011 5,461 2,012 0,065 0,095 6,628 1,153 6,629 1,153 0,079 0,117 RM 1250 .94 6,651 1,835 6,652 1,835 0,076 0,111 5,460 2,011 5,461 2,012 0,065 0,095 6,628 1,153 6,629 1,153 0,079 0,117 RM 50 .97 8,945 3,670 8,946 3,670 0,066 0,097 6,322 3,017 6,323 3,018 0,057 0,083 8,646 1,729 8,647 1,729 0,069 0,103 RM 100 .97 6,193 2,523 6,194 2,523 0,075 0,109 5,460 2,299 5,461 2,299 0,064 0,093 6,340 1,441 6,341 1,441 0,078 0,115 RM 250 .97 5,734 2,523 5,735 2,523 0,077 0,111 5,029 2,155 5,030 2,156 0,066 0,096 5,476 1,153 5,476 1,153 0,080 0,118 RM 500 .97 5,734 2,523 5,735 2,523 0,077 0,111 5,029 2,155 5,030 2,156 0,066 0,096 5,476 1,153 5,476 1,153 0,080 0,118 RM 1250 .97 5,734 2,523 5,735 2,523 0,077 0,111 5,029 2,155 5,030 2,156 0,066 0,096 5,476 1,153 5,476 1,153 0,080 0,118 KS 50 12,615 7,339 12,617 7,341 0,052 0,075 12,644 5,747 12,646 5,748 0,041 0,061 12,104 4,323 12,105 4,323 0,057 0,085 KS 100 14,908 7,339 14,911 7,341 0,048 0,071 14,080 6,609 14,083 6,610 0,040 0,059 13,256 5,187 13,835 5,188 0,054 0,082 KS 250 16,514 8,486 16,517 8,488 0,043 0,063 14,511 6,753 14,514 6,754 0,039 0,058 14,697 5,187 15,276 5,188 0,050 0,076 KS 500 20,413 12,844 20,417 12,847 0,036 0,052 14,080 7,615 14,083 7,616 0,040 0,058 17,291 8,357 17,870 8,647 0,044 0,066 KS 1250 23,394 14,908 23,400 14,911 0,032 0,046 15,805 8,046 15,807 8,047 0,036 0,053 21,037 10,086 21,329 10,376 0,039 0,059 GK 50 9,633 3,899 9,635 3,900 0,061 0,089 7,615 3,305 7,616 3,305 0,055 0,080 10,086 2,882 10,376 2,882 0,063 0,094 GK 100 9,862 3,670 9,864 3,671 0,061 0,090 7,328 2,874 7,329 2,874 0,056 0,082 8,934 2,882 9,223 2,882 0,063 0,093 GK 250 10,092 3,899 10,094 3,900 0,059 0,087 7,040 2,730 7,041 2,730 0,057 0,082 8,357 3,170 8,359 3,171 0,063 0,092 GK 500 12,844 5,275 12,847 5,276 0,052 0,077 7,040 2,586 7,041 2,587 0,058 0,085 9,510 4,035 9,800 4,035 0,060 0,089 GK 1250 13,991 6,422 13,994 6,423 0,048 0,071 7,328 3,017 7,329 3,018 0,055 0,081 10,375 3,746 10,376 3,747 0,059 0,087 RS 50 11,239 4,817 11,241 4,817 0,055 0,081 9,195 3,592 9,197 3,593 0,051 0,074 13,256 5,476 13,835 5,476 0,053 0,080 RS 100 11,468 5,046 11,470 5,047 0,056 0,082 8,333 3,448 8,335 3,449 0,052 0,076 13,833 4,899 14,411 4,900 0,052 0,079 RS 250 11,697 4,817 11,699 4,817 0,054 0,080 8,621 3,448 8,622 3,449 0,052 0,076 13,833 4,323 14,123 4,324 0,053 0,081 RS 500 13,991 5,963 13,994 5,965 0,049 0,072 8,621 3,017 8,622 3,018 0,053 0,078 13,256 4,611 13,835 4,612 0,053 0,080 RS 1250 14,908 7,798 14,911 7,800 0,045 0,066 8,908 3,305 8,909 3,305 0,051 0,075 14,121 4,611 14,411 4,612 0,051 0,079 P 50 9,862 4,128 9,864 4,129 0,061 0,089 8,621 3,879 8,622 3,880 0,053 0,077 12,392 4,899 12,682 4,900 0,056 0,084 P 100 10,092 4,128 10,094 4,129 0,061 0,089 7,902 3,017 7,904 3,018 0,055 0,080 12,680 3,746 13,258 3,747 0,055 0,084 P 250 10,092 3,899 10,094 3,900 0,059 0,088 7,759 2,730 7,760 2,730 0,055 0,081 11,239 3,746 11,529 3,747 0,057 0,086 P 500 12,844 5,275 12,847 5,276 0,052 0,078 7,328 2,730 7,329 2,730 0,057 0,084 12,680 4,035 12,970 4,035 0,055 0,084 P 1250 13,991 5,963 13,994 5,965 0,049 0,073 7,615 3,161 7,616 3,161 0,055 0,080 12,104 3,746 12,394 3,747 0,056 0,085 YZ 50 9,404 3,899 9,405 3,900 0,063 0,092 7,328 3,017 7,329 3,018 0,057 0,083 9,222 2,594 9,512 2,594 0,065 0,096 YZ 100 9,633 3,440 9,635 3,441 0,063 0,092 6,178 2,874 6,179 2,874 0,059 0,085 8,357 2,305 8,359 2,306 0,065 0,096 Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 28 Table 6. Japan (NIKKEI) loss function statistics (part 2) Period 1 Period 2 Period 3 Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% YZ 250 8,945 3,670 8,947 3,670 0,061 0,090 6,609 2,299 6,610 2,299 0,059 0,085 8,069 2,594 8,070 2,594 0,065 0,096 YZ 500 12,844 5,275 12,846 5,276 0,053 0,080 6,322 2,299 6,323 2,299 0,060 0,088 9,222 3,170 9,512 3,171 0,062 0,092 YZ 1250 13,991 5,505 13,994 5,506 0,049 0,074 6,897 2,730 6,898 2,730 0,057 0,083 9,798 3,458 9,800 3,459 0,060 0,090 GARCH 7,110 2,064 7,111 2,064 0,072 0,106 5,603 2,155 5,604 2,155 0,066 0,095 6,340 0,865 6,341 0,865 0,077 0,114 EGARCH 6,193 2,064 6,193 2,064 0,075 0,109 4,741 2,011 4,742 2,012 0,066 0,095 6,052 0,288 6,052 0,288 0,078 0,117 HS 50 6,422 3,899 6,423 3,899 0,074 0,103 6,466 2,586 6,467 2,587 0,063 0,092 6,916 2,594 6,917 2,594 0,076 0,106 HS 100 5,963 2,982 5,965 2,982 0,075 0,113 5,603 2,299 5,605 2,299 0,063 0,101 6,340 2,017 6,341 2,018 0,076 0,113 HS 250 5,505 1,835 5,506 1,835 0,074 0,126 5,891 1,149 5,892 1,150 0,064 0,103 5,764 1,153 5,765 1,153 0,077 0,118 HS 500 8,716 1,835 8,717 1,835 0,065 0,115 5,029 1,149 5,030 1,150 0,069 0,111 5,764 1,153 5,765 1,153 0,073 0,115 HS 1250 10,092 1,835 10,094 1,835 0,060 0,103 4,454 1,006 4,455 1,006 0,068 0,106 6,052 1,153 6,053 1,153 0,072 0,118 Table 7. Thailand (SET) loss function statistics (part 1) Period 1 Period 2 Period 3 Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% MA Z 50 5,505 1,606 5,506 1,606 0,124 0,179 3,161 0,718 3,162 0,719 0,092 0,132 4,035 2,594 4,036 2,595 0,069 0,097 MA Z 100 3,670 1,147 3,671 1,147 0,123 0,176 3,448 0,862 3,449 0,862 0,095 0,137 4,035 2,305 4,037 2,307 0,071 0,100 MA Z 250 5,275 1,376 5,277 1,377 0,114 0,165 2,730 0,431 2,730 0,431 0,102 0,147 3,458 1,441 3,460 1,442 0,073 0,103 MA Z 500 9,174 3,670 9,177 3,671 0,095 0,138 1,868 0,287 1,868 0,287 0,111 0,159 3,170 1,729 3,172 1,730 0,078 0,110 MA Z 1250 13,991 3,899 13,995 3,901 0,075 0,114 3,448 0,718 3,449 0,719 0,101 0,146 1,729 0,865 1,730 0,865 0,104 0,147 RM 50 .90 6,881 2,064 6,882 2,064 0,120 0,175 3,879 1,149 3,880 1,150 0,087 0,125 5,187 2,594 5,189 2,594 0,064 0,092 RM 100 .90 6,881 1,835 6,882 1,835 0,121 0,176 3,736 1,006 3,737 1,006 0,087 0,126 5,187 2,594 5,189 2,594 0,065 0,092 RM 250 .90 6,881 1,835 6,882 1,835 0,121 0,176 3,736 1,006 3,737 1,006 0,087 0,126 5,187 2,594 5,189 2,594 0,065 0,092 RM 500 .90 6,881 1,835 6,882 1,835 0,121 0,176 3,736 1,006 3,737 1,006 0,087 0,126 5,187 2,594 5,189 2,594 0,065 0,092 RM 1250 .90 6,881 1,835 6,882 1,835 0,121 0,176 3,736 1,006 3,737 1,006 0,087 0,126 5,187 2,594 5,189 2,594 0,065 0,092 RM 50 .94 6,422 1,835 6,423 1,835 0,119 0,173 3,592 1,437 3,593 1,437 0,087 0,124 4,899 2,882 4,901 2,882 0,065 0,092 RM 100 .94 5,963 1,606 5,964 1,606 0,122 0,178 3,305 1,006 3,305 1,006 0,089 0,128 4,035 2,882 4,036 2,882 0,067 0,094 RM 250 .94 5,963 1,606 5,964 1,606 0,122 0,178 3,305 1,006 3,305 1,006 0,089 0,128 4,035 2,882 4,036 2,882 0,067 0,094 RM 500 .94 5,963 1,606 5,964 1,606 0,122 0,178 3,305 1,006 3,305 1,006 0,089 0,128 4,035 2,882 4,036 2,882 0,067 0,094 RM 1250 .94 5,963 1,606 5,964 1,606 0,122 0,178 3,305 1,006 3,305 1,006 0,089 0,128 4,035 2,882 4,036 2,882 0,067 0,094 RM 50 .97 8,028 2,982 8,029 2,982 0,107 0,156 5,603 1,868 5,605 1,868 0,078 0,113 5,187 2,882 5,478 2,883 0,060 0,084 RM 100 .97 5,275 1,376 5,277 1,376 0,120 0,174 3,736 1,149 3,736 1,150 0,089 0,128 4,035 2,594 4,036 2,594 0,067 0,095 RM 250 .97 4,817 1,376 4,818 1,376 0,123 0,177 3,448 1,006 3,449 1,006 0,092 0,132 4,035 2,305 4,036 2,306 0,069 0,097 RM 500 .97 4,817 1,376 4,818 1,376 0,123 0,177 3,448 0,862 3,449 0,862 0,092 0,133 4,035 2,305 4,036 2,306 0,069 0,097 RM 1250 .97 4,817 1,376 4,818 1,376 0,123 0,177 3,448 0,862 3,449 0,862 0,092 0,133 4,035 2,305 4,036 2,306 0,069 0,097 KS 50 14,220 6,422 14,224 6,424 0,079 0,116 11,063 4,454 11,065 4,455 0,058 0,086 9,222 3,746 9,513 4,037 0,046 0,066 KS 100 16,055 6,651 16,060 6,654 0,071 0,106 10,920 4,167 10,922 4,168 0,058 0,086 10,663 4,899 10,955 5,190 0,043 0,062 KS 250 18,119 9,174 18,125 9,177 0,064 0,093 9,770 4,023 9,772 4,024 0,060 0,088 10,663 5,187 10,954 5,190 0,042 0,061 KS 500 23,853 16,284 23,862 16,289 0,052 0,072 8,764 4,023 8,766 4,024 0,062 0,091 9,222 4,899 9,513 4,901 0,045 0,065 KS 1250 30,963 22,706 30,975 22,714 0,040 0,053 14,511 6,322 14,515 6,323 0,050 0,074 6,340 3,170 6,343 3,171 0,056 0,079 GK 50 11,009 2,752 11,012 2,754 0,097 0,144 5,460 1,293 5,461 1,294 0,078 0,114 4,611 3,170 4,613 3,171 0,064 0,089 GK 100 10,550 2,752 10,553 2,753 0,092 0,137 4,310 1,580 4,311 1,581 0,085 0,122 4,899 2,882 4,902 2,883 0,066 0,093 GK 250 12,156 3,899 12,160 3,901 0,081 0,121 4,023 1,293 4,024 1,293 0,093 0,135 4,611 1,729 4,613 1,730 0,066 0,094 GK 500 16,743 6,651 16,748 6,653 0,070 0,106 2,874 0,431 2,874 0,431 0,098 0,141 3,458 1,729 3,460 1,730 0,071 0,101 GK 1250 19,495 9,633 19,501 9,636 0,060 0,088 4,598 1,149 4,599 1,150 0,085 0,124 2,017 1,153 2,018 1,153 0,091 0,129 Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 29 Table 7. Thailand (SET) loss function statistics (part 2) Period 1 Period 2 Period 3 Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% RS 50 17,431 7,339 17,436 7,342 0,069 0,103 11,063 4,023 11,065 4,024 0,058 0,086 10,951 4,899 11,242 4,901 0,043 0,063 RS 100 15,826 5,734 15,830 5,736 0,069 0,104 10,632 3,736 10,634 3,737 0,059 0,088 10,375 4,323 10,666 4,325 0,045 0,065 RS 250 16,972 6,651 16,977 6,654 0,066 0,100 9,195 3,448 9,197 3,449 0,063 0,092 8,646 4,035 8,937 4,036 0,048 0,068 RS 500 19,266 11,239 19,272 11,242 0,060 0,086 7,328 3,017 7,329 3,018 0,067 0,098 7,205 3,170 7,496 3,172 0,051 0,073 RS 1250 24,083 14,908 24,090 14,913 0,051 0,072 8,908 4,310 8,910 4,311 0,063 0,091 4,323 1,729 4,325 1,730 0,065 0,093 P 50 12,385 4,587 12,389 4,589 0,084 0,124 7,902 2,730 7,904 2,731 0,067 0,099 8,934 3,746 9,225 3,748 0,049 0,070 P 100 12,385 2,982 12,389 2,983 0,081 0,123 6,753 2,730 6,754 2,731 0,070 0,102 8,069 3,746 8,072 3,748 0,051 0,072 P 250 12,844 3,670 12,848 3,671 0,078 0,118 6,178 2,299 6,179 2,299 0,074 0,108 6,340 3,170 6,343 3,171 0,054 0,077 P 500 17,661 7,110 17,666 7,112 0,067 0,101 5,172 1,580 5,173 1,581 0,079 0,115 6,340 3,170 6,342 3,171 0,058 0,083 P 1250 21,101 11,239 21,108 11,242 0,056 0,082 5,891 3,305 5,892 3,305 0,073 0,105 3,170 1,729 3,172 1,730 0,075 0,107 YZ 50 9,862 2,523 9,865 2,524 0,103 0,152 5,316 1,293 5,317 1,294 0,080 0,117 4,323 3,170 4,325 3,171 0,069 0,096 YZ 100 8,945 2,064 8,947 2,065 0,099 0,147 4,167 1,580 4,168 1,581 0,087 0,125 4,899 2,882 4,901 2,883 0,072 0,101 YZ 250 10,092 3,211 10,095 3,212 0,090 0,132 3,448 1,006 3,449 1,006 0,096 0,138 3,458 1,729 3,460 1,730 0,071 0,100 YZ 500 15,596 5,275 15,600 5,277 0,074 0,112 2,299 0,287 2,299 0,287 0,103 0,148 3,458 1,729 3,460 1,730 0,073 0,104 YZ 1250 18,119 9,174 18,125 9,177 0,063 0,092 4,454 1,006 4,455 1,006 0,090 0,131 1,729 0,865 1,730 0,865 0,096 0,135 GARCH 8,486 2,064 8,488 2,064 0,114 0,168 2,730 0,862 2,731 0,862 0,097 0,138 2,882 0,865 2,883 0,865 0,082 0,118 EGARCH 7,339 1,606 7,341 1,606 0,112 0,164 3,017 1,293 3,018 1,293 0,096 0,138 2,882 1,153 2,883 1,153 0,083 0,118 HS 50 7,110 2,982 7,113 2,983 0,101 0,149 6,322 1,868 6,323 1,868 0,077 0,114 5,476 2,305 5,766 2,306 0,067 0,100 HS 100 6,422 2,523 6,424 2,524 0,102 0,151 4,885 1,580 4,886 1,581 0,084 0,126 6,052 2,017 6,343 2,018 0,061 0,124 HS 250 6,193 2,064 6,195 2,065 0,099 0,153 3,448 0,862 3,449 0,862 0,094 0,139 5,187 1,153 5,190 1,154 0,062 0,137 HS 500 11,239 3,211 11,242 3,212 0,086 0,140 3,017 0,431 3,018 0,431 0,099 0,150 3,746 1,441 3,748 1,441 0,070 0,120 HS 1250 15,138 2,982 15,142 2,983 0,072 0,129 4,023 0,862 4,024 0,862 0,094 0,146 1,729 0,865 1,730 0,865 0,096 0,150 Table 8. Brazil (BOVESPA) loss function statistics (part 1) Period 1 Period 2 Period 3 Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% MA Z 50 6,881 3,670 6,892 3,676 0,156 0,216 4,454 1,293 4,455 1,293 0,106 0,153 6,052 2,305 6,055 2,307 0,103 0,148 MA Z 100 7,798 4,128 7,810 4,135 0,150 0,209 3,592 0,718 3,593 0,719 0,114 0,164 5,476 2,882 5,479 2,884 0,101 0,144 MA Z 250 8,028 5,046 8,042 5,054 0,137 0,188 4,167 1,149 4,168 1,150 0,127 0,183 5,476 1,729 5,478 1,730 0,098 0,142 MA Z 500 10,321 6,193 10,338 6,203 0,119 0,161 2,299 0,718 2,300 0,719 0,143 0,203 5,476 2,017 5,478 2,019 0,097 0,139 MA Z 1250 5,963 2,752 5,974 2,758 0,156 0,218 1,724 0,431 1,725 0,431 0,137 0,196 2,882 0,865 2,883 0,865 0,131 0,188 RM 50 .90 7,110 3,211 7,119 3,216 0,149 0,211 5,891 1,580 5,892 1,581 0,099 0,144 7,205 2,594 7,207 2,595 0,099 0,144 RM 100 .90 7,110 3,211 7,119 3,216 0,150 0,211 5,891 1,580 5,892 1,581 0,099 0,145 7,205 2,594 7,207 2,595 0,099 0,144 RM 250 .90 7,110 3,211 7,119 3,216 0,150 0,211 5,891 1,580 5,892 1,581 0,099 0,145 7,205 2,594 7,207 2,595 0,099 0,144 RM 500 .90 7,110 3,211 7,119 3,216 0,150 0,211 5,891 1,580 5,892 1,581 0,099 0,145 7,205 2,594 7,207 2,595 0,099 0,144 RM 1250 .90 7,110 3,211 7,119 3,216 0,150 0,211 5,891 1,580 5,892 1,581 0,099 0,145 7,205 2,594 7,207 2,595 0,099 0,144 RM 50 .94 7,110 3,440 7,120 3,446 0,149 0,209 5,891 1,868 5,892 1,868 0,099 0,143 7,493 2,594 7,496 2,595 0,098 0,142 RM 100 .94 6,651 3,440 6,661 3,445 0,153 0,214 5,172 1,724 5,173 1,724 0,102 0,148 6,628 2,305 6,631 2,307 0,100 0,145 RM 250 .94 6,651 3,211 6,661 3,216 0,153 0,214 5,172 1,580 5,173 1,581 0,102 0,149 6,628 2,305 6,631 2,307 0,100 0,145 RM 500 .94 6,651 3,211 6,661 3,216 0,153 0,214 5,172 1,580 5,173 1,581 0,102 0,149 6,628 2,305 6,631 2,307 0,100 0,145 RM 1250 .94 6,651 3,211 6,661 3,216 0,153 0,214 5,172 1,580 5,173 1,581 0,102 0,149 6,628 2,305 6,631 2,307 0,100 0,145 RM 50 .97 8,486 4,817 8,499 4,824 0,137 0,190 7,040 2,443 7,042 2,443 0,090 0,131 8,069 3,170 8,073 3,172 0,089 0,129 RM 100 .97 7,569 3,440 7,580 3,446 0,150 0,210 4,741 1,293 4,742 1,293 0,104 0,151 6,628 2,594 6,631 2,595 0,099 0,142 RM 250 .97 7,110 3,440 7,121 3,446 0,153 0,214 4,167 1,293 4,168 1,293 0,108 0,156 6,052 2,594 6,055 2,595 0,101 0,145 Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 30 Table 8. Brazil (BOVESPA) loss function statistics (part 2) Period 1 Period 2 Period 3 Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F Lopez B Lopez Q Lopez F VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% RM 500 .97 7,110 3,440 7,121 3,446 0,153 0,214 4,167 1,293 4,168 1,293 0,108 0,157 6,052 2,594 6,055 2,595 0,101 0,145 RM 1250 .97 7,110 3,440 7,121 3,446 0,153 0,214 4,167 1,293 4,168 1,293 0,108 0,157 6,052 2,594 6,055 2,595 0,101 0,145 KS 50 12,615 8,257 12,635 8,270 0,109 0,145 9,483 4,167 9,485 4,168 0,072 0,104 11,239 6,052 11,244 6,055 0,072 0,103 KS 100 16,514 10,321 16,536 10,336 0,097 0,128 8,908 3,879 8,910 3,880 0,073 0,106 12,968 6,628 12,974 6,631 0,068 0,098 KS 250 19,495 13,303 19,524 13,323 0,085 0,108 9,914 4,454 9,917 4,455 0,074 0,107 13,256 7,205 13,262 7,208 0,063 0,090 KS 500 23,165 16,743 23,198 16,767 0,077 0,092 8,190 4,023 8,193 4,024 0,074 0,107 14,121 7,205 14,127 7,208 0,062 0,090 KS 1250 18,578 12,385 18,604 12,403 0,090 0,115 11,351 5,172 11,354 5,174 0,063 0,092 12,392 5,476 12,397 5,478 0,067 0,097 GK 50 9,862 5,505 9,877 5,513 0,128 0,176 7,328 2,299 7,329 2,299 0,086 0,125 9,222 4,035 9,226 4,037 0,082 0,118 GK 100 10,780 6,193 10,795 6,202 0,125 0,171 6,034 2,155 6,036 2,155 0,092 0,133 10,375 3,746 10,379 3,749 0,081 0,118 GK 250 10,092 6,651 10,109 6,662 0,118 0,159 5,460 1,868 5,461 1,868 0,102 0,147 9,510 4,035 9,514 4,036 0,080 0,117 GK 500 13,991 8,028 14,012 8,041 0,102 0,137 3,879 1,149 3,881 1,150 0,115 0,165 10,086 4,035 10,090 4,036 0,079 0,115 GK 1250 9,633 4,817 9,648 4,825 0,126 0,174 3,448 0,862 3,449 0,862 0,111 0,159 4,323 1,441 4,325 1,442 0,107 0,154 RS 50 11,239 6,881 11,254 6,889 0,122 0,166 7,615 2,586 7,617 2,587 0,082 0,120 10,663 4,611 10,667 4,614 0,077 0,112 RS 100 11,468 6,881 11,485 6,890 0,121 0,164 6,322 2,299 6,323 2,299 0,087 0,127 11,239 4,035 11,244 4,037 0,076 0,112 RS 250 12,385 7,569 12,404 7,581 0,111 0,150 6,034 2,011 6,036 2,012 0,098 0,141 11,239 4,323 11,243 4,325 0,075 0,110 RS 500 15,138 8,257 15,160 8,271 0,098 0,132 4,310 1,293 4,312 1,294 0,108 0,156 11,527 4,323 11,532 4,325 0,075 0,110 RS 1250 11,927 5,734 11,944 5,744 0,117 0,161 4,023 1,006 4,024 1,006 0,104 0,151 4,899 1,441 4,901 1,442 0,101 0,146 P 50 9,862 5,734 9,876 5,742 0,132 0,181 6,609 2,443 6,611 2,443 0,090 0,131 7,781 3,458 7,785 3,460 0,087 0,124 P 100 10,780 5,963 10,795 5,972 0,129 0,177 4,885 2,011 4,886 2,012 0,097 0,139 9,510 2,882 9,514 2,884 0,085 0,124 P 250 9,633 6,651 9,650 6,662 0,120 0,161 5,172 1,868 5,174 1,868 0,108 0,156 9,222 3,746 9,225 3,748 0,083 0,121 P 500 13,532 8,028 13,552 8,041 0,104 0,140 3,305 1,149 3,306 1,150 0,121 0,173 9,222 3,746 9,225 3,748 0,082 0,119 P 1250 8,945 4,587 8,960 4,595 0,131 0,180 2,874 0,575 2,875 0,575 0,116 0,166 4,323 1,153 4,325 1,153 0,111 0,161 YZ 50 9,404 5,505 9,418 5,512 0,132 0,181 6,897 2,299 6,898 2,299 0,089 0,129 8,934 3,746 8,938 3,749 0,084 0,122 YZ 100 9,633 5,963 9,648 5,972 0,129 0,177 5,172 2,011 5,174 2,012 0,095 0,137 10,086 3,458 10,091 3,460 0,083 0,122 YZ 250 10,092 7,110 10,109 7,121 0,119 0,160 5,172 1,724 5,174 1,724 0,106 0,153 9,222 3,746 9,226 3,748 0,082 0,119 YZ 500 13,303 7,798 13,323 7,811 0,105 0,141 3,592 1,149 3,593 1,150 0,118 0,170 9,222 3,746 9,226 3,748 0,081 0,118 YZ 1250 9,404 4,587 9,418 4,595 0,129 0,179 3,017 0,575 3,018 0,575 0,114 0,164 4,323 1,153 4,325 1,153 0,110 0,158 GARCH 6,193 3,211 6,201 3,215 0,155 0,218 4,454 1,149 4,455 1,150 0,107 0,155 4,323 1,441 4,325 1,442 0,109 0,157 EGARCH 6,193 2,982 6,201 2,986 0,152 0,214 4,167 0,575 4,167 0,575 0,106 0,155 5,187 1,153 5,189 1,153 0,109 0,159 HS 50 8,716 3,440 8,727 3,445 0,151 0,227 6,034 2,586 6,036 2,587 0,089 0,129 7,205 2,305 7,208 2,307 0,098 0,150 HS 100 8,028 3,670 8,039 3,674 0,147 0,230 4,741 1,437 4,742 1,437 0,097 0,150 6,340 2,305 6,343 2,307 0,100 0,162 HS 250 8,257 3,440 8,272 3,445 0,137 0,241 4,885 0,862 4,886 0,862 0,110 0,187 5,764 1,441 5,766 1,442 0,098 0,151 HS 500 11,927 3,670 11,945 3,676 0,112 0,212 3,017 0,575 3,018 0,575 0,126 0,233 5,187 1,729 5,190 1,730 0,097 0,146 HS 1250 7,798 2,294 7,812 2,297 0,134 0,251 2,443 0,144 2,443 0,144 0,123 0,246 3,746 0,288 3,748 0,288 0,117 0,229 Table 9. Holland (AEX) Christoffersen test statistics (part 1) Period 1 Period 2 Period 3 LRuc LRind LRcc LRuc LRind LRcc LRuc LRind LRcc VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% MA Z 50 0,80 0,56 0,13 0,17 0,94 0,73 0,78 3,03 0,05 0,42 0,83 3,45 4,87 10,49 6,11 0,88 10,98 11,37 MA Z 100 0,80 5,40 3,19 0,47 4,00 5,87 0,04 2,01 0,01 0,35 0,05 2,37 9,26 34,26 5,89 5,90 15,15 40,16 MA Z 250 5,27 16,05 2,36 0,39 7,63 16,44 1,49 0,55 0,61 0,24 2,10 0,79 9,26 27,51 5,89 3,96 15,15 31,47 MA Z 500 10,47 33,40 6,10 0,85 16,57 34,24 6,67 0,00 2,05 0,14 8,72 0,14 23,12 57,05 2,73 2,42 25,84 59,47 MA Z 1250 38,13 86,75 7,82 3,39 45,96 90,14 2,56 0,00 2,02 0,14 4,58 0,14 37,13 57,05 1,65 7,69 38,77 64,75 RM 50 .90 0,00 5,40 0,01 0,47 0,01 5,87 8,70 2,01 0,25 0,35 8,95 2,37 3,14 0,08 0,35 0,09 3,49 0,17 Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 37 Table 14. Thailand (SET) Christoffersen test statistics (part 1) Period 1 Period 2 Period 3 LRuc LRind LRcc LRuc LRind LRcc LRuc LRind LRcc VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% MA Z 50 0,23 1,36 4,17 0,23 4,40 1,59 5,67 0,62 0,13 5,12 5,79 5,74 0,73 6,18 6,09 5,82 6,82 12,01 MA Z 100 1,78 0,09 2,34 0,12 4,12 0,21 3,94 0,14 3,80 4,35 7,74 4,49 0,73 4,36 6,09 6,82 6,82 11,18 MA Z 250 0,07 0,56 4,72 0,17 4,79 0,73 8,98 2,89 2,68 0,03 11,66 2,92 1,94 0,60 7,95 11,19 9,89 11,79 MA Z 500 12,97 18,64 3,02 0,26 15,99 18,90 18,71 4,97 1,41 0,01 20,12 4,98 2,80 1,53 9,06 9,42 11,86 10,95 MA Z 1250 50,97 21,36 1,73 0,16 52,70 21,52 3,94 0,62 1,30 5,12 5,24 5,74 10,34 0,07 9,42 16,97 19,76 17,04 RM 50 .90 2,92 3,82 6,12 0,38 9,04 4,20 1,99 0,15 0,00 0,19 1,99 0,34 0,03 6,18 3,39 1,51 3,42 7,70 RM 100 .90 2,92 2,46 6,12 0,30 9,04 2,76 2,56 0,00 0,00 0,14 2,56 0,14 0,03 6,18 3,39 1,51 3,42 7,70 RM 250 .90 2,92 2,46 6,12 0,30 9,04 2,76 2,56 0,00 0,00 0,14 2,56 0,14 0,03 6,18 3,39 1,51 3,42 7,70 RM 500 .90 2,92 2,46 6,12 0,30 9,04 2,76 2,56 0,00 0,00 0,14 2,56 0,14 0,03 6,18 3,39 1,51 3,42 7,70 RM 1250 .90 2,92 2,46 6,12 0,30 9,04 2,76 2,56 0,00 0,00 0,14 2,56 0,14 0,03 6,18 3,39 1,51 3,42 7,70 RM 50 .94 1,71 2,46 4,65 0,30 6,36 2,76 3,21 1,18 0,01 0,29 3,22 1,47 0,01 8,23 3,96 4,97 3,97 13,20 RM 100 .94 0,80 1,36 5,85 0,23 6,65 1,59 4,76 0,00 0,07 0,14 4,83 0,14 0,73 8,23 6,09 4,97 6,82 13,20 RM 250 .94 0,80 1,36 5,85 0,23 6,65 1,59 4,76 0,00 0,07 0,14 4,83 0,14 0,73 8,23 6,09 4,97 6,82 13,20 RM 500 .94 0,80 1,36 5,85 0,23 6,65 1,59 4,76 0,00 0,07 0,14 4,83 0,14 0,73 8,23 6,09 4,97 6,82 13,20 RM 1250 .94 0,80 1,36 5,85 0,23 6,65 1,59 4,76 0,00 0,07 0,14 4,83 0,14 0,73 8,23 6,09 4,97 6,82 13,20 RM 50 .97 7,17 11,30 5,57 0,73 12,73 12,03 0,51 4,22 0,31 1,41 0,82 5,62 0,03 8,23 2,88 4,97 2,90 13,20 RM 100 .97 0,07 0,56 4,72 0,17 4,79 0,73 2,56 0,15 0,92 3,17 3,47 3,32 0,73 6,18 6,09 5,82 6,82 12,01 RM 250 .97 0,03 0,56 5,97 0,17 6,00 0,73 3,94 0,00 0,04 3,71 3,98 3,71 0,73 4,36 6,09 6,82 6,82 11,18 RM 500 .97 0,03 0,56 5,97 0,17 6,00 0,73 3,94 0,14 0,04 4,35 3,98 4,49 0,73 4,36 6,09 6,82 6,82 11,18 RM 1250 .97 0,03 0,56 5,97 0,17 6,00 0,73 3,94 0,14 0,04 4,35 3,98 4,49 0,73 4,36 6,09 6,82 6,82 11,18 KS 50 53,24 58,18 2,43 4,65 55,68 62,83 40,66 45,38 3,88 0,27 44,54 45,65 10,54 15,55 1,17 6,09 11,71 21,64 KS 100 72,76 62,05 0,90 4,12 73,66 66,17 38,95 39,41 4,27 2,11 43,22 41,51 17,94 27,51 0,20 3,39 18,13 30,90 KS 250 97,32 109,06 0,71 0,53 98,03 109,59 26,40 36,52 0,94 2,40 27,34 38,92 17,94 30,83 0,20 3,39 18,13 34,22 KS 500 178,11 273,79 2,53 1,36 180,64 275,15 17,13 36,52 0,56 2,40 17,69 38,92 10,54 27,51 1,17 3,96 11,71 31,47 KS 1250 300,13 451,46 2,50 1,44 302,62 452,91 89,69 90,22 14,21 3,31 103,91 93,53 1,21 10,49 1,64 9,06 2,85 19,55 GK 50 25,07 9,15 2,84 0,95 27,90 10,11 0,30 0,55 0,41 2,72 0,71 3,27 0,11 10,49 4,60 9,06 4,71 19,55 GK 100 21,74 9,15 2,21 0,68 23,95 9,84 0,73 2,01 1,84 1,97 2,57 3,99 0,01 8,23 3,96 10,32 3,97 18,55 GK 250 34,18 21,36 3,61 0,16 37,79 21,52 1,49 0,55 2,40 2,72 3,89 3,27 0,11 1,53 4,60 9,42 4,71 10,95 GK 500 80,66 62,05 0,35 4,12 81,01 66,17 7,77 2,89 2,35 0,03 10,12 2,92 1,94 1,53 7,95 9,42 9,89 10,95 GK 1250 115,10 118,38 1,03 0,36 116,13 118,74 0,24 0,15 0,19 3,17 0,43 3,32 8,31 0,08 8,00 13,53 16,31 13,61 RS 50 88,84 74,10 0,79 1,14 89,63 75,24 40,66 36,52 3,88 2,40 44,54 38,92 19,60 27,51 0,69 3,96 20,29 31,47 RS 100 70,20 47,04 1,14 0,17 71,34 47,21 35,63 30,98 5,09 3,05 40,72 34,03 16,33 21,25 0,00 5,31 16,33 26,56 RS 250 83,35 62,05 0,65 0,61 84,00 62,67 20,89 25,76 0,84 3,80 21,73 29,57 8,05 18,32 0,59 6,09 8,63 24,42 RS 500 112,06 152,60 1,30 0,47 113,36 153,07 6,99 18,59 0,02 2,05 7,01 20,64 3,14 10,49 2,01 9,06 5,16 19,55 RS 1250 181,69 238,92 0,90 0,71 182,59 239,63 18,35 42,36 11,87 0,36 30,21 42,72 0,35 1,53 5,31 9,42 5,66 10,95 P 50 36,13 30,22 4,70 1,07 40,83 31,30 10,57 14,29 0,67 0,37 11,24 14,67 9,26 15,55 1,47 6,97 10,73 22,52 P 100 36,13 11,30 4,70 0,80 40,83 12,10 4,08 14,29 0,01 2,68 4,09 16,97 5,85 15,55 1,33 6,97 7,18 22,52 P 250 40,17 18,64 1,31 0,26 41,48 18,90 1,90 8,68 0,05 3,86 1,95 12,54 1,21 10,49 1,64 9,06 2,85 19,55 P 500 91,64 70,01 0,59 3,17 92,23 73,18 0,04 2,01 0,66 1,97 0,70 3,99 1,21 10,49 1,64 9,06 2,85 19,55 P 1250 137,17 152,60 1,64 0,47 138,81 153,07 1,10 23,28 4,45 1,53 5,55 24,81 2,80 1,53 9,06 9,42 11,86 10,95 YZ 50 17,12 7,18 3,41 0,57 20,53 7,75 0,14 0,55 0,53 2,72 0,67 3,27 0,35 10,49 5,31 9,06 5,66 19,55 YZ 100 11,69 3,82 1,85 0,38 13,55 4,20 1,08 2,01 2,11 1,97 3,18 3,99 0,01 8,23 3,96 10,32 3,97 18,55 YZ 250 18,61 13,60 0,62 0,54 19,23 14,14 3,94 0,00 3,80 3,71 7,74 3,71 1,94 1,53 7,95 9,42 9,89 10,95 YZ 500 67,67 40,04 0,71 2,16 68,38 42,19 13,26 4,97 0,79 0,01 14,05 4,98 1,94 1,53 7,95 9,42 9,89 10,95 YZ 1250 97,32 109,06 0,71 0,16 98,03 109,22 0,45 0,00 0,27 3,71 0,72 3,71 10,34 0,07 9,42 16,97 19,76 17,04 Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 38 Table 14. Thailand (SET) Christoffersen test statistics (part 2) Period 1 Period 2 Period 3 LRuc LRind LRcc LRuc LRind LRcc LRuc LRind LRcc VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% GARCH 9,31 3,82 4,44 1,89 13,75 5,71 8,98 0,14 0,37 4,35 9,35 4,49 3,84 0,07 4,97 6,08 8,81 6,14 EGARCH 4,42 1,36 4,91 0,23 9,33 1,59 6,67 0,55 0,19 2,72 6,86 3,27 3,84 0,08 4,97 13,53 8,81 13,61 HS 50 3,63 11,30 5,49 0,80 9,13 12,10 2,37 4,22 0,02 1,41 2,39 5,62 0,16 4,36 2,42 6,82 2,58 11,18 HS 100 1,71 7,18 0,02 0,57 1,73 7,75 0,02 2,01 2,70 6,77 2,72 8,79 0,76 2,80 1,64 2,42 2,40 5,22 HS 250 1,22 3,82 0,07 0,38 1,29 4,20 3,94 0,14 3,80 4,35 7,74 4,49 0,03 0,08 3,39 13,53 3,42 13,61 HS 500 26,80 13,60 0,47 0,54 27,27 14,14 6,67 2,89 2,05 7,46 8,72 10,35 1,25 0,60 6,97 11,19 8,22 11,79 HS 1250 62,72 11,30 0,53 0,73 63,24 12,03 1,49 0,14 0,61 4,35 2,10 4,49 10,34 0,07 9,42 16,97 19,76 17,04 Table 15. Brazil (BOVESPA) Christoffersen test statistics (part 1) Period 1 Period 2 Period 3 LRuc LRind LRcc LRuc LRind LRcc LRuc LRind LRcc VaR 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% 5% 1% MA Z 50 2,92 18,64 3,62 1,22 6,54 19,86 0,45 0,55 1,60 0,24 2,05 0,79 0,76 4,36 0,41 1,92 1,17 6,29 MA Z 100 6,19 24,20 6,19 1,55 12,37 25,75 3,21 0,62 1,10 0,07 4,31 0,69 0,16 8,23 2,88 1,17 3,04 9,40 MA Z 250 7,17 36,67 8,18 8,84 15,34 45,50 1,08 0,15 0,48 3,17 1,55 3,32 0,16 1,53 0,00 3,02 0,16 4,55 MA Z 500 20,15 54,39 6,17 5,23 26,32 59,62 13,26 0,62 3,86 0,07 17,13 0,69 0,16 2,80 0,00 2,42 0,16 5,22 MA Z 1250 0,80 9,15 1,23 0,68 2,04 9,84 20,82 2,89 18,87 0,03 39,70 2,92 3,84 0,07 1,17 0,05 5,01 0,12 RM 50 .90 3,63 13,60 0,02 0,93 3,66 14,53 1,10 2,01 4,45 0,35 5,55 2,37 3,14 6,18 0,79 0,48 3,93 6,67 RM 100 .90 3,63 13,60 0,02 0,93 3,66 14,53 1,10 2,01 4,45 0,35 5,55 2,37 3,14 6,18 0,79 0,48 3,93 6,67 RM 250 .90 3,63 13,60 0,02 0,93 3,66 14,53 1,10 2,01 4,45 0,35 5,55 2,37 3,14 6,18 0,79 0,48 3,93 6,67 RM 500 .90 3,63 13,60 0,02 0,93 3,66 14,53 1,10 2,01 4,45 0,35 5,55 2,37 3,14 6,18 0,79 0,48 3,93 6,67 RM 1250 .90 3,63 13,60 0,02 0,93 3,66 14,53 1,10 2,01 4,45 0,35 5,55 2,37 3,14 6,18 0,79 0,48 3,93 6,67 RM 50 .94 3,63 16,05 0,30 1,07 3,93 17,12 1,10 4,22 2,46 1,41 3,57 5,62 3,96 6,18 0,58 0,48 4,54 6,67 RM 100 .94 2,28 16,05 0,59 1,07 2,87 17,12 0,04 3,03 4,20 0,42 4,25 3,45 1,76 4,36 0,15 0,38 1,92 4,74 RM 250 .94 2,28 13,60 0,59 0,93 2,87 14,53 0,04 2,01 4,20 0,35 4,25 2,37 1,76 4,36 0,15 0,38 1,92 4,74 RM 500 .94 2,28 13,60 0,59 0,93 2,87 14,53 0,04 2,01 4,20 0,35 4,25 2,37 1,76 4,36 0,15 0,38 1,92 4,74 RM 1250 .94 2,28 13,60 0,59 0,93 2,87 14,53 0,04 2,01 4,20 0,35 4,25 2,37 1,76 4,36 0,15 0,38 1,92 4,74 RM 50 .97 9,31 33,40 4,44 0,85 13,75 34,24 5,44 10,43 5,32 0,63 10,76 11,06 5,85 10,49 1,33 0,88 7,18 11,37 RM 100 .97 5,27 16,05 2,36 1,07 7,63 17,12 0,10 0,55 1,17 0,24 1,27 0,79 1,76 6,18 1,32 1,51 3,08 7,70 RM 250 .97 3,63 16,05 3,17 1,07 6,80 17,12 1,08 0,55 2,11 0,24 3,18 0,79 0,76 6,18 0,41 1,51 1,17 7,70 RM 500 .97 3,63 16,05 3,17 1,07 6,80 17,12 1,08 0,55 2,11 0,24 3,18 0,79 0,76 6,18 0,41 1,51 1,17 7,70 RM 1250 .97 3,63 16,05 3,17 1,07 6,80 17,12 1,08 0,55 2,11 0,24 3,18 0,79 0,76 6,18 0,41 1,51 1,17 7,70 KS 50 38,13 91,09 5,87 7,44 44,01 98,54 23,58 39,41 7,16 2,11 30,74 41,51 21,33 41,47 3,20 0,41 24,53 41,87 KS 100 77,99 132,75 8,79 6,17 86,79 138,92 18,35 33,71 3,67 2,71 22,01 36,42 32,86 49,07 3,40 3,36 36,26 52,44 KS 250 115,10 199,89 12,87 12,16 127,97 212,04 27,86 45,38 7,52 6,56 35,38 51,95 34,97 57,05 4,49 0,02 39,46 57,08 KS 500 167,52 285,69 10,67 9,89 178,19 295,58 12,61 36,52 5,73 12,23 18,33 48,75 41,59 57,05 2,89 0,49 44,48 57,54 KS 1250 103,13 178,44 10,72 3,16 113,85 181,61 44,16 61,48 5,99 6,83 50,15 68,31 28,80 34,26 2,85 0,00 31,66 34,27 GK 50 17,12 43,50 5,28 4,17 22,41 47,66 6,99 8,68 6,54 3,86 13,53 12,54 10,54 18,32 1,47 0,30 12,01 18,62 GK 100 23,38 54,39 4,98 8,27 28,36 62,66 1,48 7,05 2,18 0,97 3,66 8,02 16,33 15,55 2,96 0,45 19,29 16,00 GK 250 18,61 62,05 9,23 9,98 27,84 72,03 0,30 4,22 8,61 1,41 8,92 5,62 11,89 18,32 0,26 0,30 12,15 18,62 GK 500 50,97 86,75 9,56 8,18 60,53 94,93 1,99 0,15 17,69 0,19 19,68 0,34 14,79 18,32 0,68 0,30 15,46 18,62 GK 1250 15,69 33,40 5,88 0,85 21,56 34,24 3,94 0,14 11,26 0,10 15,20 0,24 0,35 0,60 0,18 0,15 0,53 0,75 RS 50 26,80 66,00 7,84 3,62 34,63 69,62 8,70 12,30 5,57 3,04 14,27 15,34 17,94 24,31 1,18 0,09 19,12 24,40 Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 39 Table 15. Brazil (BOVESPA) Christoffersen test statistics (part 2) RS 100 28,58 66,00 5,16 9,12 33,73 75,12 2,37 8,68 3,31 0,79 5,68 9,46 21,33 18,32 1,73 0,30 23,06 18,62 RS 250 36,13 78,25 10,87 9,79 47,01 88,04 1,48 5,56 9,08 1,17 10,56 6,73 21,33 21,25 1,73 0,18 23,06 21,43 RS 500 62,72 91,09 11,81 7,44 74,53 98,54 0,73 0,55 14,70 0,24 15,43 0,79 23,12 21,25 4,44 0,18 27,55 21,43 RS 1250 32,27 47,04 4,09 1,51 36,36 48,55 1,49 0,00 12,23 3,71 13,73 3,71 0,01 0,60 0,03 0,15 0,04 0,75 P 50 17,12 47,04 3,41 1,51 20,53 48,55 3,46 10,43 6,78 3,43 10,24 13,86 4,87 12,93 1,65 0,65 6,52 13,58 P 100 23,38 50,68 7,02 5,85 30,40 56,52 0,02 5,56 0,98 1,17 1,00 6,73 11,89 8,23 2,63 1,17 14,52 9,40 P 250 15,69 62,05 10,91 13,64 26,60 75,69 0,04 4,22 9,92 1,41 9,96 5,62 10,54 15,55 0,41 0,45 10,95 16,00 P 500 46,53 86,75 9,02 8,18 55,55 94,93 4,76 0,15 12,10 0,19 16,86 0,34 10,54 15,55 0,41 0,45 10,95 16,00 P 1250 11,69 30,22 3,46 1,07 15,15 31,30 7,77 1,50 14,99 0,05 22,76 1,55 0,35 0,08 0,18 0,09 0,53 0,17 YZ 50 14,30 43,50 6,51 4,17 20,81 47,66 4,74 8,68 5,78 3,86 10,52 12,54 9,26 15,55 1,81 0,45 11,07 16,00 YZ 100 15,69 50,68 5,88 5,85 21,56 56,52 0,04 5,56 0,66 1,17 0,70 6,73 14,79 12,93 1,83 0,65 16,61 13,58 YZ 250 18,61 70,01 9,23 11,60 27,84 81,61 0,04 3,03 9,92 1,67 9,96 4,70 10,54 15,55 0,41 0,45 10,95 16,00 YZ 500 44,37 82,47 9,82 6,19 54,19 88,66 3,21 0,15 14,94 0,19 18,15 0,34 10,54 15,55 0,41 0,45 10,95 16,00 YZ 1250 14,30 30,22 4,38 1,07 18,68 31,30 6,67 1,50 13,96 0,05 20,63 1,55 0,35 0,08 0,18 0,09 0,53 0,17 GARCH 1,22 13,60 5,23 0,54 6,44 14,14 0,45 0,15 3,75 0,19 4,20 0,34 0,35 0,60 0,18 0,15 0,53 0,75 EGARCH 1,22 11,30 0,99 0,73 2,21 12,03 1,08 1,50 2,11 0,05 3,18 1,55 0,03 0,08 1,05 0,09 1,07 0,17 HS 50 10,47 16,05 2,19 1,07 12,66 17,12 1,48 12,30 2,18 3,04 3,66 15,34 3,14 4,36 0,02 1,92 3,17 6,29 HS 100 7,17 18,64 5,57 2,34 12,73 20,98 0,10 1,18 1,17 0,29 1,27 1,47 1,21 4,36 1,64 1,92 2,85 6,29 HS 250 8,21 16,05 7,44 6,50 15,65 22,55 0,02 0,14 2,70 0,10 2,72 0,24 0,41 0,60 0,02 0,15 0,43 0,75 HS 500 32,27 18,64 12,70 2,34 44,96 20,98 6,67 1,50 2,05 0,05 8,72 1,55 0,03 1,53 0,00 3,02 0,03 4,55 HS 1250 6,19 5,40 2,01 0,47 8,20 5,87 11,72 8,09 18,54 0,00 30,26 8,09 1,25 2,47 0,45 0,01 1,71 2,48 Table 16. Model abbreviations (part 1) Abbreviation Description MA Z 50 Equally Weighted Moving Average. Normal Distribution, Sample Window of 50 obs MA Z 100 Equally Weighted Moving Average. Normal Distribution, Sample Window of 100 obs MA Z 250 Equally Weighted Moving Average. Normal Distribution, Sample Window of 250 obs MA Z 500 Equally Weighted Moving Average. Normal Distribution, Sample Window of 500 obs MA Z 1250 Equally Weighted Moving Average. Normal Distribution, Sample Window of 1250 obs RM 0.90 50 RiskMetrics, Lamda=0.90, Sample Window of 50 obs RM 0.90 100 RiskMetrics, Lamda=0.90, Sample Window of 100 obs RM 0.90 250 RiskMetrics, Lamda=0.90, Sample Window of 250 obs RM 0.90 500 RiskMetrics, Lamda=0.90, Sample Window of 500 obs RM 0.90 1250 RiskMetrics, Lamda=0.90, Sample Window of 1250 obs RM 0.94 50 RiskMetrics, Lamda=0.94, Sample Window of 50 obs RM 0.94 100 RiskMetrics, Lamda=0.94, Sample Window of 100 obs RM 0.94 250 RiskMetrics, Lamda=0.94, Sample Window of 250 obs RM 0.94 500 RiskMetrics, Lamda=0.94, Sample Window of 500 obs RM 0.94 1250 RiskMetrics, Lamda=0.94, Sample Window of 1250 obs RM 0.97 50 RiskMetrics, Lamda=0.97, Sample Window of 50 obs RM 0.97 100 RiskMetrics, Lamda=0.97, Sample Window of 100 obs RM 0.97 250 RiskMetrics, Lamda=0.97, Sample Window of 250 obs RM 0.97 500 RiskMetrics, Lamda=0.97, Sample Window of 500 obs RM 0.97 1250 RiskMetrics, Lamda=0.97, Sample Window of 1250 obs KS 50 Kernel Estimation, Sample Window of 50 obs KS 100 Kernel Estimation, Sample Window of 100 obs KS 250 Kernel Estimation, Sample Window of 250 obs KS 500 Kernel Estimation, Sample Window of 500 obs Risk Governance and Control: Financial Markets & Institutions/ Volume 8, Issue 2, Spring 2018 40 Table 16. Model abbreviations (part 2) Abbreviation Description KS 1250 Kernel Estimation, Sample Window of 1250 obs GK 50 Garman and Klass Extreme Value Approach, Sample Window of 50 obs GK 100 Garman and Klass Extreme Value Approach, Sample Window of 100 obs GK 250 Garman and Klass Extreme Value Approach, Sample Window of 250 obs GK 500 Garman and Klass Extreme Value Approach, Sample Window of 500 obs GK 1250 Garman and Klass Extreme Value Approach, Sample Window of 1250 obs RS 50 Rogers and Satchell Extreme Value Approach, Sample Window of 50 obs RS 100 Rogers and Satchell Extreme Value Approach, Sample Window of 100 obs RS 250 Rogers and Satchell Extreme Value Approach, Sample Window of 250 obs RS 500 Rogers and Satchell Extreme Value Approach, Sample Window of 500 obs RS 1250 Rogers and Satchell Extreme Value Approach, Sample Window of 1250 obs P 50 Parkinson Extreme Value Approach, Sample Window of 50 obs P 100 Parkinson Extreme Value Approach, Sample Window of 100 obs P 250 Parkinson Extreme Value Approach, Sample Window of 250 obs P 500 Parkinson Extreme Value Approach, Sample Window of 500 obs P 1250 Parkinson Extreme Value Approach, Sample Window of 1250 obs YZ 50 Yang and Zhang Extreme Value Approach, Sample Window of 50 obs YZ 100 Yang and Zhang Extreme Value Approach, Sample Window of 100 obs YZ 250 Yang and Zhang Extreme Value Approach, Sample Window of 250 obs YZ 500 Yang and Zhang Extreme Value Approach, Sample Window of 500 obs YZ 1250 Yang and Zhang Extreme Value Approach, Sample Window of 1250 obs GARCH GARCH model, Conditional Normal Distribution, Sample Window of 1250 obs EGARCH EGARCH model, Conditional Normal Distribution, Sample Window of 1250 obs HS 50 Historical Simulation, Sample Window of 50 obs HS 100 Historical Simulation, Sample Window of 100 obs HS 250 Historical Simulation, Sample Window of 250 obs HS 500 Historical Simulation, Sample Window of 500 obs HS 1250 Historical Simulation, Sample Window of 1250 obs