Measuring value-at-risk and expected shortfall of newer cryptocurrencies: new insights

Mappadang, Agoestina; Nugroho, Bayu Adi; Lestari, Setyani Dwi; Elizabeth; Lestari, Titi Kanti Article Measuring value-at-risk and expected shortfall of newer cryptocurrencies: new insights Cogent Business & Management Provided in Cooperation with: Taylor & Francis Group Suggested Citation: Mappadang, Agoestina; Nugroho, Bayu Adi; Lestari, Setyani Dwi; Elizabeth; Lestari, Titi Kanti (2024) : Measuring value-at-risk and expected shortfall of newer cryptocurrencies: new insights, Cogent Business & Management, ISSN 2331-1975, Taylor & Francis, Abingdon, Vol. 11, Iss. 1, pp. 1-29, https://doi.org/10.1080/23311975.2024.2416096 This Version is available at: https://hdl.handle.net/10419/326626 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Cogent Business & Management ISSN: 2331-1975 (Online) Journal homepage: www.tandfonline.com/journals/oabm20 Measuring value-at-risk and expected shortfall of newer cryptocurrencies: new insights Agoestina Mappadang, Bayu Adi Nugroho, Setyani Dwi Lestari, Elizabeth & Titi Kanti Lestari To cite this article: Agoestina Mappadang, Bayu Adi Nugroho, Setyani Dwi Lestari, Elizabeth & Titi Kanti Lestari (2024) Measuring value-at-risk and expected shortfall of newer cryptocurrencies: new insights, Cogent Business & Management, 11:1, 2416096, DOI: 10.1080/23311975.2024.2416096 To link to this article: https://doi.org/10.1080/23311975.2024.2416096 © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group Published online: 14 Nov 2024. Submit your article to this journal Article views: 588 View related articles View Crossmark data Citing articles: 1 View citing articles Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=oabm20 Banking & Finance | ReseaRch aRticle Cogent Business & ManageMent 2024, VoL. 11, no. 1, 2416096 Measuring value-at-risk and expected shortfall of newer cryptocurrencies: new insights agoestina Mappadanga , Bayu adi nugrohob , setyani Dwi lestaria, elizabetha and titi kanti lestaric aFaculty of economics and Business, universitas Budi Luhur, south Jakarta, indonesia; bYKPn school of Business, Yogyakarta, indonesia; cPostgraduate Program Magister of applied economics, universitas atmajaya, south Jakarta, indonesia ABSTRACT a significant amount of historical returns is needed for the generalized autoregressive conditional heteroscedasticity (gaRch) models to be calibrated. newer cryptocurrencies, such as non-fungible tokens (nFts), have relatively limited data to create robust parameter estimates. this study uses a newly developed method, the exponentially weighted moving average (eWMa) model, that takes into account the fat-tailed distributions of returns and volatility response to forecast Value-at-Risk (VaR) and expected shortfall (es). We employ thorough back tests of daily VaR and es forecasts, which are widely utilized for regulatory approval and are considered to be industry standards. We also use loss function ratios to select the best model. Our results indicate that simpler models are just as good as the complicated ones, provided the simpler models capture fat-tailed distributions of returns. the primary findings hold up through several tests. 1. Introduction non-fungible tokens (nFts) are essentially Blockchain-based virtual asset rights with distinctive identities and data (e.g. collectibles, video, audio, art, in-game items) (ante, 2022). nFts are different from traditional cryptocurrencies in that each one has a unique identity and collection of data, making them non-interchangeable. their ability to provide proof of ownership and authenticity for digital goods is what has caused them to become increasingly popular, providing new opportunities for musicians, artists, and other makers to make money off of their work (Belguith et al., 2024). even though the sector is still in its infancy, transaction volumes for nFt assets have increased significantly in recent years (karim et al., 2022). thus, it is particularly important to understand the risk characteristics of nFts. the modeling of quantile risk levels for cryptocurrencies is a significant research topic. this branch of study has become increasingly intricate, assessing numerous versions of the gaRch (generalized autoregressive conditional heteroscedasticity) model series that (Bollerslev, 1986) first introduced. however, the complexity of modeling options for cryptocurrency risk modeling in the academic literature contrasts sharply with the current industry practice. Value-at-risk (VaR) and expected shortfall (es) are used by several online sites that analyze, utilize, offer volatility forecasts, and use an equally weighted methodology. For instance, the cryptocurrency exchange Okex provides VaR prediction for Bitcoin on its blog. the assumption used to define this forecast is that Bitcoin returns follow a normal distribution. the accuracy of gaRch models requires a significant number of historical returns (chu et al., 2017; ardia et al., 2019; guesmi et al., 2019; sosa et al., 2019; tiwari et al., 2019; alexander and Dakos, 2020; hattori, 2020; segnon and Bekiros, 2020; Maciel, 2021; aggarwal, 2022; nugroho, 2022). While some cryptocurrencies, such as Bitcoin and ethereum, have been in circulation for quite some time, the constant arrival of new cryptocurrencies that attract investor interest indicates that there is often inadequate data © 2024 the author(s). Published by informa uK Limited, trading as taylor & Francis group CONTACT agoestina Mappadang [email protected] universitas Budi Luhur, south Jakarta, indonesia https://doi.org/10.1080/23311975.2024.2416096 this is an open access article distributed under the terms of the Creative Commons attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. the terms on which this article has been published allow the posting of the accepted Manuscript in a repository by the author(s) or with their consent. ARTICLE HISTORY Received 13 February 2024 Revised 21 august 2024 accepted 8 October 2024 KEYWORDS expected shortfall; exponentially-weighted moving average; eVt; gaRch; value-at-risk JEL CLASSIFICATION c46; c58; F31 SUBJECTS Finance; business, management and accounting; financial accounting 2 a. MaPPaDang el al. to generate viable estimations for newer cryptocurrencies. alexander and Dakos (2023) were the first to introduce an asymmetric eWMa tailored for a volatility response to address the problem encountered when employing equally weighted and gaRch models. however, no research has been conducted to assess the performance of the asymmetric eWMa model for downside risk measures of newer cryptocurrencies such as non-fungible tokens (nFt). similarly, a significant limitation of the existing literature is the lack of discussion on simpler models despite practitioners most popularly using such models. in addition, there are various gaps in the literature on cryptocurrency risk metrics. For example, the literature on traffic lights for VaR and es backtesting is scarce, which is a common industry practice. similarly, few studies have investigated the VaR and es of short positions on cryptocurrencies, although they are traded as conveniently as long positions. additionally, a few other studies have investigated the prediction accuracy of multivariate models. this study has applied the multivariate model as a robustness check. the main objective of this research is to fill all of these gaps in the literature. the sections that ensue comprise the literature review and discussions of the computation of VaR and es based on univariate and multivariate settings, the empirical results, the discussion, and the conclusion, respectively. 2. Literature review the RiskMetricstM eWMa method is a prominent model owing to its clarity and convenience (longerstaey and spencer, 1996). some researchers have concentrated on evaluating its forecasting accuracy using conventional assets and cryptocurrencies. there is some evidence in cryptocurrency research that applies an integrated gaRch (igaRch). note that the eWMa model fits the integrated volatility model. For example, köchling et al. (2020) discovered that igaRch best fits Bitcoin and other cryptocurrencies. similarly, Baur et al. (2018) find that crypto volatility is integrated. it also turns out that while very complex volatility models can obtain precise out-of-sample downside risk estimates, simpler models may generate equally reliable methods. For example, Bonello and suda (2018) contrasted VaR estimates for Bitcoin by employing both singleand dual-regime gaRch, assuming gaussian and student-t distributions. they discovered these features could achieve precise VaR forecasts at the 95% confidence level. Backtesting daily 1% VaR estimates for Bitcoin, troster et al. (2019) discovered that a standard gaRch model is comparable to the more highly complicated gaRch methods used in their research. trucíos (2019) reviewed VaR estimated for Bitcoin around 2011 and 2017 using six different models and discovered that only the bootstrap VaR method generates reliable predictions at a 99% confidence level. VaR predictions based on simpler volatility models, such as the standard gaRch, as used by trucíos and taylor (2023), can be considered reliable for Bitcoin and ethereum. acereda etal. (2020) discovered that more sophisticated configurations for Bitcoin VaR do not outclass simplified ones, provided that a fat-tailed distribution is used instead of a normal distribution. silahli etal. (2021) discovered that a simple benchmark model works well in numerous VaR backtests for a wide range of crypto assets. even when only eWMa models are considered, contradictory research is evident. For instance, silahli et al. (2021) assert that a standard eWMa method generates precise VaR estimates for all cryptocurrencies. in contrast, liu et al. (2020) find that a similar approach rejects VaR backtests. (nekhili and sultan (2020) observed the out-of-sample results of a RiskMetricstM eWMa method and found that it provides precise VaR forecasts only at a 95% confidence level; however, es forecasts of nearly all cryptocurrencies examined, an eWMa provides reliable predictions based on the eR test. although previous studies have investigated some volatility methods, the variety of investigated models has been limited. liu et al. (2020) concentrated on eWMa-based methodologies and skipped over other, more complicated systems. their findings did not provide any definitive evidence for the general reliability of eWMa-type frameworks in predicting cryptocurrency volatility when compared with other approaches that were either more complicated or simpler. By contrast, catania and grassi (2022) engaged in highly complex gas models and another model that used an assumption of a fat-tailed distribution of return. they then tested those models with an already sophisticated benchmark, the β-skew-t-egaRch framework. they found that those models often produced similar results regarding their ability to predict risks. cOgent Business & ManageMent 3 Moreover, the growing finance literature concerning newer cryptocurrencies, such as non-fungible tokens (nFts), includes portfolio management, price bubbles, hedging properties, investors’ attention, business models, and wash trading. Menvouta etal. (2023), who developed a machine-learning trading technique, discovered that having nFts in a portfolio of traditional and cryptocurrency assets increases the sharpe Ratio. similarly, the gerber statistic demonstrated that nFts have a low correlation with typical asset classes, which may lead to increased portfolio diversification (ko etal., 2022). the results of the squared wavelet coherence method demonstrate the existence of favorable diversification properties (umar etal., 2022). Further, serneels (2023) recently proposed some tactics that can be used to identify potentially fraudulent wash trading behavior in the nFt markets. in addition, the findings from the nonlinear autoregressive distributed lag demonstrated that nFts have the potential to serve as safe havens for the united states dollar during the cOViD-19 timeframe (Zhang et al., 2022). Furthermore, geopolitical risks strongly predict cryptopunks and Decentraland returns based on the quantile regression methodology (urom et al., 2022). according to the Q-joint spillover model, nFts should not be regarded as a separate asset class under adverse market conditions (Xia et al., 2022). additionally, an experimental technique was used by Zarifis and cheng (2022), who discovered four nFt commerce strategies: nFt creators, fan tokens, nFt marketplaces, and computer games. Further, the attention index for nFts introduced by Wang, (2022) adequately explains nFt returns. similarly, google search volume for “nFt” positively correlates with significant cryptocurrency returns (Pinto-gutiérrez et al., 2022). Based on the information above, the study concerning risk management of newer cryptocurrencies applying a simple but powerful model is still very limited. thus, the main goal of this research is to fill this gap in the literature. 3. Methodology 3.1 Value-at-risk (VaR) VaR is a loss we are reasonably confident we cannot surpass if the current portfolio is held for a certain period. VaR has two fundamental specifications: the significance level, α, and the timeframe, typically determined on trading days, for which VaR is calculated. For example, a 5% daily VaR, which relates to a 95% confidence level, is a loss scale we expect to encounter with a 5% frequency when the existing portfolio is maintained for a day. Put another way, it is VR F F aleft tail long position right tail t t t , , , { α α α =−− () − − − () − 1 1 1sshort position () (1) F is a one-day ahead forecast created at the time t and it is based on the return distribution assumption. Following alexander and Dakos (2023), the benchmark model returns, applying an equally weighted average model, is assumed to be normal distribution. thus, XN ∼ () µσ ,2 (2) using the standard normal transformation, we have PX X P XPZ tt tt t tt t tt t < () =−<−     =< −     = , σ µ σ µ σ µ σ α xx (3) if z is a normal value, it generates: X tt t , σ µ σ θσ −= () −1 (4) 4 a. MaPPaDang el al. if θσ θ σ −− () = − 11 1( ) , then the 100α% 𝒽-day normal VaR is VR a t tt , σ θ σσ µ =− () − −11 (5) if a random variable T has a student-t distribution with ν degrees of freedom, its density function is f V V VVV V tt () = ()     +     + () −− −−+      πτ τ 12 22 2 1 2 1 / 1 1 1 (6) the α quintile of the standard Student-t is − () =− () −− tt VV 11 αα 1 (7) the student-t VaR is S VR V V VV tudent t−a α ασ µ ,=− () − () − −−11 21t (8) 3.2 Expected shortfall (ES) es represents the potential losses if the loss surpasses VaR. the es informs us of how much we can expect to lose if the VaR is exceeded: E a x dx S VR tt αα α () = () ∫ 1 0 (9) let 𝒳 be the discounted h-day return. the returns of the benchmark model were assumed to mimic a normal distribution: X =+ ∼ () Z ZN σ µ tt, ,0 1 (10) if z is a normal value, it produces EE SX S t tt t , αα αµ () = () − (11) if 𝒳 has a student-t distribution with mean µ ℏ , standard deviation σ t , and ν degrees of freedom, then (12) xv α () denotes the α quantile of the standardized 𝒮tudent-t distribution with ν degrees of freedom. is density. the standardized student-t density is (13) the es in a standardized 𝒮tudent-t distribution with ν degrees of freedom is (14) cOgent Business & ManageMent 5 3.3 EWMA and EGARCH the eWMa for the variance estimate at time t of returns is (15) λ is the smoothing constant. introduced an asymmetric volatility response (η) to the original eWMa model (aeWMa): (16) Further, the variance estimate in the 𝒮tudent-t egaRch (1,1) model: (17) ϕε ε γ ε ε tt t t () = + −   () θE (18) 3.4 Multivariate setting in these settings, let be the vector of nFt returns at time t. the benchmark model is based on the (19) Σt denotes the covariance matrix. the covariance matrix in the eWMa model: (20) the covariance matrix of the taeWMa (λ, η, ν) model is given by: (21) For the multivariate gaRch model, the covariance matrix is based on: Σt tt t = DCD (22) C QQ Q t tt t = () () −− diag diag 12 12// (23)  t is the conditional correlation, and  t is the variances (diagonal matrix) of the univariate egaRch. t is based on the aDcc model (cappiello et al., 2006): (24) 3.5 Measuring the accuracy and model selection this study employs the Basel committee’s industry-standard traffic light test. additionally, this research utilizes backtesting for es, initiated by costanzino and curran (2018). Following alexander and Dakos (2023), this research advances the test to include leftand right-tail VaR. the violation parameter XVR t a   is (25) 6 a. MaPPaDang el al. the cumulative value of VaR violations X N VaR α () over the entire forecasting period t = 1, …, N is computed as XX N VR N VR aa αα () = () = ∑ t t 1 (26) if the VaR model is precisely fitted, the total number of VaR violations is modeled as a binomial distribution: (27) if X VaR is the total number of violations during the forecasting period and z is the standard normal transform, then the likelihood of obtaining X VaR is θ (z), where z is the standard normal distribution function. green region if θ (z) < 95%, yellow if 95% ⩽ θ (z) < 99.99%, and red if θ (z) ≥ 99.99%. the three-zone technique is introduced to reconcile the two error types: type i refers to the potential for an accurate model to be labeled as inaccurate based on the outcomes of its backtesting; type ii refers to the potential for an erroneous model not to be labeled as such. the backtesting findings are thought to be compatible with an accurate model in the green zone, where there is little chance of mistakenly accepting an inaccurate model. the backtesting results are unlikely to come from an accurate model in the red zone. Backtesting findings could be consistent in the yellow zone. similarly, this research advances the test to include the leftand right-tail es: (28) the variables express the intensity of each VaR exceeding and . greater magnitude returns that surpass both the VaR and es dominate XS t E () α , while returns that surpass the VaR but not the es are given comparatively less weight. the cumulative es violation is XX N S N S EE αα () = () = ∑ t t 1 (29) as stated by costanzino and curran (2018), X N SE α () is N µσ EESS N,2 () (30) µ E S is 0.5 (1 – α ) N and σ E S 2 is (1 – α )(4 – 3(1 – α ))/12. the probability of receiving X es or less provided the total actual es exceedances during the forecast period of X es, is θ (z), where z is the standard normal distribution function for X es. green region if θ (z) < 95%, yellow if 95% ⩽ θ (z) < 99.99%, and red if θ (z) ≥ 99.99%. Furthermore, this study also provides the conditional coverage (cc) test, introduced by christoffersen (1998). LRCC           expexp nn nnnn 10 01 00 11 10 1 11 01 01 11 11 (31) ψ 01 is the proportion of violations, provided that the last return is not a violation, and ψ 11 is the proportion of violations, given that the last return is a violation. Moreover, Mcneil and Frey (2000) introduced a methodology for backtesting es. it is based on a time series of standardized exceedance residuals (eR), defined as cOgent Business & ManageMent 7 (32) the eR test statistic is (33)  µ is based on 1000 bootstrap simulations. if numerous models obtain the correct unconditional/conditional coverage, the practitioner encounters the dilemma of being unable to choose between different options. in this case, comparative techniques were applied to select the model with the best performance. Fissler and Ziegel (2016) demonstrate that VaR and es are jointly elicitable: FZL S VR VR S S T α α αα α α α =− () + +− () − 1 1 E dr E E t tt t t t t a alog (34) the following packages from R statistical software were used: rmgarch (ghalanos, 2022a), rugarch (ghalanos, 2022b), customized gas (ardia et al., 2019, 2022), customized ufRisk (Feng et al., 2022) and other customized R codes. 4. Empirical results 4.1 Data the data include 379 daily log returns of aPe, icP, and sanD from March 18, 2022, to March 31, 2023. this study selected aPe, icP, and sanD because they were newer and among the ten largest nFts when writing this study. the nFt market is open 24 hours a day, seven days a week; thus, when calculating returns, this study used the closing price at midnight (utc). these data were obtained from coinMarketcap and restricted by the availability of the aligned return series. One hundred twenty-nine observations were applied to obtain the one-day-ahead VaR and es. the remaining data (n = 250) were used to backtest the models. the Basel committee proposes a straightforward backtest for determining the statutory risk-management capital requirement that covers 250 days. thus, the rolling window method was utilized to calculate VaR and es. losses are represented as negative daily log returns. Figure 1 illustrates the existence of volatility clustering. in addition, table 1 reveals that the kurtosis values were greater than 3, indicating a leptokurtic distribution. the findings of the JarqueB era, D’agostino, and A nscombeG lynn tests indicate that the null hypothesis of no excess kurtosis, skewness, and normality could not be supported. the assumption of a zero mean applies to all nFts, with a mean close to zero. selecting the appropriate distributions that better capture the fat-tailed and skewed nature of cryptocurrency returns is essential (Yang and Xu, 2021). instead of manually selecting the appropriate distributions, we use a dynamic version of the optimal univariate gaRch selection procedure (antonakakis etal., 2021). in other words, we use the custom R code to select the distributions automatically. the codes are in the R package of connectednessapproach (David gabauer, 2022). hence, we use egaRch with the student-t distribution. it is important to understand that the main aim of this research is to use the eWMa method with ad hoc parameters, and the egaRch model is used as a comparison. Further, the student-t egaRch parameters, shown in table 2, show that the response parameter θ is small and insignificant. Moreover, the asymmetry parameter γ is significant for all nFts. the extremely significant γ supports the assumption that returns follow a fat-tailed distribution. additionally, conditional volatility’s response to market shocks is gauged by the gaRch error parameter α. if α is quite large (e.g. above 0.1), then volatility is very sensitive to market events. the persistence of conditional volatility, 14 a. MaPPaDang el al. Figure 4. VaR (dot-dash), losses (long dash), and es (solid) estimates for aPe (long position), the trading days from 25 July 2022 to 31 March 2023. Notes: the red dots are the total violations/exceedances. cOgent Business & ManageMent 15 Table 8. Backtesting outcomes for one-day-ahead 1% VaR and es (Long Position), 25 February 2022—31 March 2023, multivariate settings. Panel a: 1 % VaR long position avalanche Polygon solana X N V aR θ (z) CC X N V aR θ (z) CC X N V aR θ (z) CC Benchmark 14 0.9999 0.000 12 0.9996 0.000 18 0.9999 0.000 teWMa (94%) 3 0.3942 0.807 3 0.3942 0.807 4 0.5896 0.076 teWMa (92.5%) 2 0.2087 0.481 1 0.0769 0.168 2 0.2087 0.481 taeWMa (94%, 1%) 3 0.3942 0.807 2 0.2087 0.481 3 0.3942 0.032 taeWMa (94%, 2%) 2 0.2087 0.481 1 0.0769 0.168 2 0.2087 0.481 taeWMa (94%, 3%) 1 0.0769 0.168 1 0.0769 0.168 2 0.2087 0.481 taeWMa (92.5%, 1%) 2 0.2087 0.481 1 0.0769 0.168 2 0.2087 0.481 taeWMa (92.5%, 2%) 1 0.0769 0.168 1 0.0146 0.014 1 0.0769 0.168 taeWMa (92.5%, 3%) 1 0.0769 0.168 0 0.0146 0.014 1 0.0769 0.168 taDCC 3 0.3942 0.807 4 0.5896 0.076 3 0.3942 0.032 Panel B: 1 % es long position avalanche Polygon solana X N S E θ (z) eR X N SE θ (z) eR X N S E θ (z) eR Benchmark 6 0.9999 0.554 4 0.9998 0.068 7 0.9999 0.616 teWMa (94%) 0 0.1123 0.317 0 0.1123 0.817 1 0.2499 0.317 teWMa (92.5%) 0 0.0397 0.317 0 0.0109 1.000 0 0.0397 0.317 taeWMa (94%, 1%) 0 0.1123 1.000 0 0.0397 0.784 1 0.1123 0.317 taeWMa (94%, 2%) 0 0.0397 1.000 0 0.0109 0.869 1 0.0397 0.317 taeWMa (94%, 3%) 0 0.0109 1.000 0 0.0109 0.755 1 0.0397 0.317 taeWMa (92.5%, 1%) 0 0.0397 0.317 0 0.0109 0.848 0 0.0397 0.317 taeWMa (92.5%, 2%) 0 0.0109 0.317 0 0.0023 0.755 0 0.0109 0.317 taeWMa (92.5%, 3%) 0 0.0109 0.317 0 0.0023 1.000 0 0.0109 0.317 taDCC 0 0.1123 0.179 0 0.2499 0.755 1 0.1123 1.000 Notes: see table 3. Table 9. Backtesting outcomes for one-day-ahead 2.5% VaR and es (Long Position), 25 February 2022—31 March 2023, multivariate settings. Panel a: 2.5 % VaR long position avalanche Polygon solana X N V aR θ (z) CC X N V aR θ (z) CC X N V aR θ (z) CC Benchmark 21 0.9999 0.000 18 0.9999 0.000 23 0.9999 0.000 teWMa (94%) 11 0.9987 0.011 9 0.9892 0.049 10 0.9961 0.026 teWMa (92.5%) 10 0.9961 0.026 7 0.9370 0.114 12 0.9996 0.004 taeWMa (94%, 1%) 9 0.9892 0.049 4 0.5896 0.076 7 0.9370 0.114 taeWMa (94%, 2%) 5 0.7538 0.117 4 0.5896 0.076 7 0.9370 0.114 taeWMa (94%, 3%) 4 0.5896 0.076 4 0.5896 0.076 5 0.7538 0.117 taeWMa (92.5%, 1%) 7 0.9370 0.114 4 0.5896 0.076 8 0.9727 0.082 taeWMa (92.5%, 2%) 5 0.7538 0.117 4 0.5896 0.076 7 0.9370 0.114 taeWMa (92.5%, 3%) 3 0.4962 0.807 3 0.3942 0.032 4 0.5896 0.076 taDCC 9 0.9892 0.049 7 0.9370 0.114 11 0.9987 0.000 Panel B: 2.5 % es long position avalanche Polkadot solana X N S E θ (z) eR X N S E θ (z) eR X N S E θ (z) eR Benchmark 9 1.0000 0.038 7 0.9999 0.038 10 1.0000 0.506 teWMa (94%) 3 0.9990 0.989 1 0.9784 0.786 3 0.9948 0.732 teWMa (92.5%) 2 0.9948 0.709 1 0.8274 0.898 2 0.9998 0.848 taeWMa (94%, 1%) 2 0.9784 0.695 1 0.2499 0.774 3 0.8274 0.651 taeWMa (94%, 2%) 2 0.4463 0.722 1 0.2499 0.779 2 0.8274 0.683 taeWMa (94%, 3%) 1 0.2499 0.391 0 0.2499 0.784 1 0.4463 0.743 taeWMa (92.5%, 1%) 2 0.8274 0.658 0 0.2499 0.867 2 0.9310 0.711 taeWMa (92.5%, 2%) 1 0.4463 0.684 0 0.2499 0.844 1 0.8274 0.647 taeWMa (92.5%, 3%) 0 0.1123 0.646 0 0.1123 0.843 1 0.2499 0.679 taDCC 2 0.9784 0.678 2 0.8274 0.750 3 0.9990 0.729 Notes: see table 4. 16 a. MaPPaDang el al. Table 10. Backtesting outcomes for one-day-ahead 1% VaR and es (short Position), 25 February 2022 – 31 March 2023, multivariate settings. Panel a: 1 % VaR short position avalanche Polygon solana X N V aR θ (z) CC X N V aR θ (z) CC X N V aR θ (z) CC Benchmark 6 0.8685 0.015 9 0.9892 0.049 5 0.7538 0.082 teWMa (94%) 1 0.0769 0.649 4 0.5896 0.095 1 0.0769 0.095 teWMa (92.5%) 1 0.0769 0.649 3 0.3942 0.807 1 0.0769 0.095 taeWMa (94%, – 1%) 2 0.2087 0.404 4 0.5896 0.095 1 0.0769 0.095 taeWMa (94%, – 2%) 2 0.2087 0.404 4 0.5896 0.095 1 0.0769 0.095 taeWMa (94%, – 3%) 2 0.2087 0.404 3 0.3942 0.807 1 0.0769 0.095 taeWMa (92.5%, – 1%) 1 0.0769 0.649 3 0.3942 0.807 1 0.0769 0.095 taeWMa (92.5%, – 2%) 2 0.2087 0.404 3 0.3942 0.807 1 0.0769 0.095 taeWMa (92.5%, – 3%) 1 0.0769 0.649 2 0.2087 0.481 1 0.0769 0.095 taDCC 2 0.2087 0.404 3 0.3942 0.807 2 0.2087 0.875 Panel B: 1 % es short position avalanche Polygon solana X N SE θ (z) eR X N S E θ (z) eR X N S E θ (z) eR Benchmark 3 0.6571 0.401 6 0.9784 0.019 3 0.4464 0.378 teWMa (94%) 1 0.0109 0.317 1 0.2499 0.817 0 0.0109 0.317 teWMa (92.5%) 1 0.0109 0.317 0 0.1123 1.000 0 0.0109 0.317 taeWMa (94%, – 1%) 1 0.0397 1.000 1 0.2499 0.784 0 0.0109 0.317 taeWMa (94%, – 2%) 1 0.0397 1.000 1 0.2499 0.869 0 0.0109 0.317 taeWMa (94%, – 3%) 1 0.0397 1.000 1 0.1123 0.755 0 0.0109 0.317 taeWMa (92.5%, – 1%) 1 0.0109 0.317 0 0.1123 0.848 0 0.0109 0.317 taeWMa (92.5%, – 2%) 1 0.0397 1.000 0 0.1123 0.755 0 0.0109 0.317 taeWMa (92.5%, – 3%) 1 0.0109 0.317 0 0.0397 1.000 0 0.0109 0.317 taDCC 1 0.0397 0.179 1 0.1123 0.755 1 0.0397 1.000 Notes: see table 5. Table 11. Backtesting outcomes for one-day-ahead 2.5% VaR and es (short Position), 25 February 2022 – 31 March 2023, multivariate settings. Panel a: 2.5 % VaR short position avalanche Polygon solana X N V aR θ (z) CC X N V aR θ (z) CC X N V aR θ (z) CC Benchmark 11 0.9987 0.000 13 0.9998 0.001 9 0.9892 0.004 teWMa (94%) 7 0.9370 0.005 10 0.9961 0.026 8 0.9727 0.001 teWMa (92.5%) 7 0.9370 0.005 11 0.9987 0.011 8 0.9727 0.001 taeWMa (94%, – 1%) 7 0.9370 0.005 10 0.9961 0.026 7 0.9370 0.026 taeWMa (94%, – 2%) 6 0.8685 0.015 9 0.9892 0.049 5 0.7538 0.216 taeWMa (94%, – 3%) 3 0.3942 0.216 7 0.9370 0.404 3 0.3942 0.649 taeWMa (92.5%, – 1%) 6 0.8685 0.015 11 0.9987 0.011 8 0.9727 0.011 taeWMa (92.5%, – 2%) 5 0.7538 0.042 10 0.9961 0.026 7 0.9370 0.042 taeWMa (92.5%, – 3%) 3 0.3942 0.216 7 0.9370 0.404 4 0.5896 0.404 taDCC 5 0.7538 0.042 9 0.9892 0.049 11 0.9987 0.000 Panel B: 2.5 % es short position avalanche Polygon solana X N S E θ (z) eR X N S E θ (z) eR X N S E θ (z) eR Benchmark 7 0.9990 0.153 9 0.9999 0.063 6 0.9785 0.125 teWMa (94%) 1 0.8274 0.989 4 0.9948 0.786 1 0.9311 0.732 teWMa (92.5%) 1 0.8274 0.709 1 0.9990 0.898 1 0.9311 0.848 taeWMa (94%, – 1%) 1 0.8274 0.695 4 0.9948 0.774 1 0.8275 0.651 taeWMa (94%, – 2%) 1 0.6571 0.722 3 0.9784 0.779 1 0.4463 0.683 taeWMa (94%, – 3%) 1 0.1123 0.391 2 0.8274 0.784 1 0.1123 0.743 taeWMa (92.5%, – 1%) 1 0.6571 0.658 3 0.9990 0.867 1 0.9311 0.711 taeWMa (92.5%, – 2%) 1 0.4463 0.684 2 0.9948 0.844 1 0.8275 0.647 taeWMa (92.5%, – 3%) 1 0.1123 0.646 2 0.8274 0.843 1 0.2500 0.679 taDCC 2 0.4463 0.678 3 0.9784 0.704 1 0.9990 0.729 Notes see table 6. cOgent Business & ManageMent 17 4.5 ADCC-GARCH Roll parameters table 13 presents the aDcc-gaRch rolling parameters. the α parameter measures conditional volatility’s response to market shocks. if α is large (e.g. above 0.1), then volatility is highly sensitive to market events. the β parameter computes the persistence of conditional volatility. Volatility takes time to settle down when β is large (e.g., above 0.9). the µ parameter shows the level of the long-term average volatility. in addition, the asymmetric parameter ( ϕ γ 1 ), is positive, indicating that negative residuals do not increase the conditional volatility significantly more than positive shocks. 5. GARCH – vine copula simulation in this section, we provide the findings of Monte carlo simulation-based tests. the sample size makes this test important. nonetheless, no VaR model prescribes the sample size or data frequency—these are personal preferences. in addition, the selection of data frequency and sample size are closely related. For example, if we use 10-day returns, we need a 20-year sample; if we use weekly returns, we need a 10-year sample; if we use daily returns, we need a 2-year sample; and so on. so far, we have computed VaR and es for daily returns. this section also deals with 10-day VaR and es. there are two approaches to computing VaR and es for the long horizon (i.e. h = 10): analytic, such as the square root of time, and numerical simulation, such as Monte carlo. the analytic approach is based on the assumption that daily returns are normal and i.i.d. however, risk factor returns at the daily or weekly frequency rarely have normal distributions. hence, we use the Monte carlo simulation approach. We mainly use copulas. copulas, multivariate distributions with uniform marginals, can create a wide array of risk factor return distributions. sklar (1959) shows that a random vector. XX X d = …( , ., ˙ 1) with joint distribution F and marginals FF d   1,.,…, it has a copula function ˘ C that gives Fx xCFx Fx ddd   111 ,, ,., ˘        (35) Table 12. Joint VaR and es loss function ratios. Panel a α = 1% long position α = 2 5. % long position avalanche Polygon solana avalanche Polygon solana Benchmark 1.000 1.000 1.000 1.000 1.000 1.000 teWMa (94%) 0.919 0.916 0.907 0.956 0.955 0.946 teWMa (92.5%) 0.927 0.928 0.918 0.964 0.966 0.956 taeWMa (94%, 1%) 0.903 0.900 0.892 0.941 0.939 0.932 taeWMa (94%, 2%) 0.874 0.869 0.865 0.914 0.911 0.907 taeWMa (94%, 3%) 0.836 0.830 0.830 0.879 0.874 0.875 taeWMa (92.5%, 1%) 0.911 0.910 0.902 0.949 0.949 0.941 taeWMa (92.5%, 2%) 0.880 0.878 0.874 0.920 0.919 0.915 taeWMa (92.5%, 3%) 0.841 0.836 0.837 0.884 0.881 0.882 taDCC 0.837 0.796 0.831 0.912 0.889 0.906 Panel B α =1% short position α = 2 5. % short position avalanche Polygon solana avalanche Polygon solana Benchmark 1.000 1.000 1.000 1.000 1.000 1.000 teWMa (94%) 0.946 0.924 0.942 0.986 0.963 0.984 teWMa (92.5%) 0.955 0.936 0.953 0.994 0.974 0.994 taeWMa (94%, – 1%) 0.930 0.907 0.926 0.971 0.948 0.970 taeWMa (94%, – 2%) 0.900 0.876 0.898 0.943 0.919 0.944 taeWMa (94%, – 3%) 0.860 0.836 0.862 0.906 0.882 0.910 taeWMa (92.5%, – 1%) 0.938 0.918 0.936 0.978 0.958 0.979 taeWMa (92.5%, – 2%) 0.906 0.885 0.907 0.949 0.927 0.952 taeWMa (92.5%, – 3%) 0.866 0.843 0.869 0.911 0.888 0.917 taDCC 0.862 0.806 0.862 0.941 0.884 0.942 Notes: Values greater than one indicate outperformance of the benchmark model and vice versa. 18 a. MaPPaDang el al. Cu uFFu Fu dd d         11 1 1 1 ,, ,.,        (36) where uFx iii    . Multiple pair-copula constructions exist for more than or equal to three assets. Bedford and cooke (2002) proposed a graphical model to help arrange several pair copulas, such as cand D-vine copulas. there are R-vine copulas for more than or equal to five assets (aas et al., 2009). see czado et al. (2012) for further information on pair copula. in addition, we also added another model to compute VaR and es: the extreme value theory (eVt). Because we focus on the tail of the return distribution, extreme value theory is an essential instrument for modeling the tail distribution without making any assumptions about the distribution center. in eVt, the cumulative distribution function (CDF) is (37) β  and ξ are the scale and the shape parameters, respectively. if ξ > 0, it implies heavy tail distribution. then, VaR and es forecasts under eVt are given by (38) (39) β  and ξ  are estimated using maximum likelihood. Table 13. aDCC-gaRCH roll. optimal parameters across rolls (First 2, Last 2) Roll-1 Roll-2 Roll-13 Roll-14 µ Ava 0.001 0.001 −0.001 −0.001 ω Ava 0.000 0.004 0.000 0.000 α Ava 0.133 0.138 0.149 0.110 β Ava 0.813 0.812 0.819 0.866 Shape 4.839 4.761 5.203 5.711 µ Poly 0.000 0.000 −0.003 0.000 ω Poly 0.000 0.000 0.000 0.000 α Poly 0.079 0.061 0.159 0.166 β Poly 0.889 0.904 0.780 0.771 Shape 3.237 3.366 4.219 4.833 µ Sola 0.002 0.003 −0.002 −0.002 ω Sola 0.001 0.001 0.001 0.001 α Sola 0.141 0.132 0.212 0.177 β Sola 0.712 0.732 0.388 0.476 Shape 4.868 5.007 5.071 4.991 aDCC ϕα 1 0.035 0.039 0.030 0.012 ϕβ 1 0.904 0.913 0.938 0.936 ϕ γ 1 0.092 0.089 0.039 0.088 µ shape 4.209 4.234 4.297 4.628 Notes: the values are taken from the rmgarch package, fixed-rolling window approach, and refit every 30 days. cOgent Business & ManageMent 19 similar to previous studies (ghorbel and trabelsi, 2014; Mejdoub and ghorbel, 2018; nugroho, 2023), we do the following steps for gaRch – Vine copula simulation using R statistical packages (alexios ghalanos, 2022; Brechmann and schepsmeier, 2013; nagler et al., 2021): Step 1: We obtain one-step-ahead volatility based on a heavy-tail distribution ˘ (,., )  tt 11 , and compute one-step-ahead standardized residuals, . We do this step recursively. We use gaRch with the student-t distribution due to autocorrelated returns. Step 2: We compute the pseudo-observations from the standardized residuals computed in Step 1. Step 3: Fit a Vine copula to the data estimated in Step 2. We have found that C-Vine is the best fit. We also selected the best copula family to fit our data. gumbel copula is the best choice. Step 4: We simulate 10000 pseudo-observations based on parameters selected in step 3. Step 5: We transform the pseudo-observations into returns. Step 6: We compute 10-day VaR and es recursively using an equally weighted average model (the benchmark), eWMa models, eWMa models adjusted for fat-tailed distribution, gaRch, and the extreme Value theory (eVt) recursively. We split the data 70 – 30. hence, we have around 300 out-of-sample results. table 14 shows the autocorrelation tests and copulas selection. Panel A indicates that the standardized residuals are free from autocorrelation issues. Panel B shows that the C-Vine copula and Gumbel family are selected for further simulation process. Further, Figure 5 shows the density obtained from the simulation. Moreover, the backtesting results for 10-day 1% VaR and es (long position) are presented in table 15. the VaR estimates reveal that exceptions generated by all models except the benchmark (aPe) occurred in the green regions. in addition, the cc test was performed. it was determined that most models were accurate for estimating VaR. similarly, the es estimates show that the exceptions or exceedances generated by all models except the benchmark (aPe) occurred in the green regions. Furthermore, table 16 presents the findings from another significance level (α = 2.5%). the benchmark model was not consistently accurate. interestingly, the eVt model was the only model that predicted VaR accurately for aPe. it is possible that tAEWMA models need more than three percentage adjustments to be accurate. as expected, the benchmark model could not consistently provide accurate forecasts. in addition, the backtesting results for ten-day VaR and es (short position) are presented in tables 17 and 18. the results show that models other than the benchmark provided good forecasts. in addition, table 19 shows the joint VaR and es loss function ratios used to rank the accuracy of each model for forecasting the ten-day es and VaR. Values greater than one indicate the outperformance of the benchmark model. as expected, the gaRch and eVt models, which were more complicated, consistently outperformed the benchmark. interestingly, the simpler models always topped the benchmark, provided that the models had the appropriate parameter values. Table 14. the goodness of fit tests. Panel a aPe iCP sanD Ljung-Box (5) 0.507 (0.956) 3.092 (0.390) 2.507 (0.504) Ljung-Box (10) 0.233 (0.905) 5.056 (0.497) 3.884 (0.683) Sign Bias 0.359 (0.719) 0.587 (0.390) 1.209 (0.227) Panel B ℓog-ℓikelihood aiC BiC C-Vine 244.680 −481.360 −466.830 D-Vine 235.180 −464.360 −453.470 Gaussian 204.808 −403.617 −392.724 Student-t222.882 −433.764 −411.976 Clayton 225.280 −444.561 −433.667 Gumbel 236.488 −466.977 −456.083 Frank 206.756 −407.512 −396.618 Joe 219.765 −433.531 −422.637 Notes: Panel a shows statistical tests confirming that autocorrelation does not exist. the numbers in parentheses are the p-values. Panel B shows that C-Vine and Gumbel Copula are preferred over others. 20 a. MaPPaDang el al. Further, Figures 6 and 7 show the exceedances or violations. to save space, only an analysis of the aPe is presented. Figure 6 indicates that the benchmark model has the worst breaches of VaR and es for the long position of aPe. similarly, Figure 7 shows that the benchmark model also has the worst violations of VaR and es for the short position of aPe. 6. Discussion When the density function of a distribution has a greater peak value and more mass in the tails compared to the usual density function with the same variance, the distribution is leptokurtic. leptokurtosis is one of the key "stylized facts" that come to light when the empirical distributions on financial asset returns are examined. the skewness of return densities is also noticeable, as they frequently have a bigger lower tail and a strong negative skew. When risk factor return distributions have leptokurtosis and negative skewness, the VaR based on normal distribution will likely underestimate the VaR at high confidence levels (Figure 8). academically speaking, there is still a dearth of nFt literature on measuring nFts’ Value-at-Risk and expected shortfall. understanding the risk profiles and finding accurate and simpler models are essential due to the market dynamics of nFts. the market for nFts saw strong expansion in 2021–2022 before experiencing a sharp downturn. late in 2021 and early 2022, the market exploded, with nFts traded for staggering prices. nevertheless, the market saw a correction by april 2022, and nFt prices reverted to more fair levels. to the best of our knowledge, the current nFt literature has been focusing on herding behavior (De silva et al., 2024), the diversification abilities of nFts (aharon and Demir, 2022; ko et al., 2022; Zhang et al., 2022), bubbles (Maouchi et al., 2022; Wang et al., 2022), etc. the issue encountered when employing an equally weighted model (the practitioners’ approach) is not caused by frequent small jumps in asset prices but by infrequent large jumps (alexander, 2009). When a lengthy period of averaging is applied, the significance of a single extraordinary event is averaged across a large sample of returns. consequently, an estimate of the market’s volatility based on a long-moving average will not react to a sudden, short-term shock. additionally, the model attempts to Table 15. Backtesting outcomes for 1% 10-day VaR and es (Long Position), gaRCH – vine copula simulation. Panel a: 1 % VaR long position aPe iCP sanD X N V aR θ (z) CC X N V aR θ (z) CC X N V aR θ (z) CC Benchmark 8 0.996 0.047 5 0.913 0.538 1 0.191 0.391 teWMa (94%) 3 0.638 0.970 0 0.047 0.057 1 0.191 0.391 teWMa (92.5%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391 taeWMa (94%, 1%) 3 0.638 0.970 0 0.047 0.057 1 0.191 0.391 taeWMa (94%, 2%) 2 0.413 0.804 0 0.047 0.057 1 0.191 0.391 taeWMa (94%, 3%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391 taeWMa (92.5%, 1%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391 taeWMa (92.5%, 2%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391 taeWMa (92.5%, 3%) 0 0.047 0.057 0 0.047 0.057 1 0.191 0.391 gaRCH 3 0.638 0.970 0 0.047 0.057 2 0.413 0.970 eVt 0 0.047 0.057 0 0.047 0.057 1 0.191 0.391 Panel B: 1 % es long position aPe iCP sanD X N S E θ (z) eR X N SE θ (z) eR X N S E θ (z) eR Benchmark 5 0.996 0.333 5 0.775 0.247 1 0.037 0.397 teWMa (94%) 0 0.305 0.551 0 0.007 0.256 0 0.037 0.513 teWMa (92.5%) 0 0.037 0.723 0 0.007 0.862 0 0.037 0.526 taeWMa (94%, 1%) 0 0.305 0.552 0 0.007 0.463 0 0.037 0.512 taeWMa (94%, 2%) 0 0.126 0.555 0 0.007 0.170 0 0.037 0.513 taeWMa (94%, 3%) 0 0.037 0.559 0 0.007 0.797 0 0.037 0.517 taeWMa (92.5%, 1%) 0 0.037 0.727 0 0.007 0.231 0 0.037 0.524 taeWMa (92.5%, 2%) 0 0.037 0.745 0 0.007 0.431 0 0.037 0.523 taeWMa (92.5%, 3%) 0 0.007 0.785 0 0.007 0.869 0 0.037 0.522 gaRCH 0 0.305 0.488 0 0.007 0.496 0 0.126 0.497 eVt 0 0.007 0.946 0 0.007 0.196 1 0.037 0.500 Notes: see table 3. cOgent Business & ManageMent 21 transform a forecast of constant volatility into an estimate of time-varying volatility. Moreover, because it is assumed that returns would have constant volatility, practitioners widely employ the benchmark model, which is computed using an equally weighted moving average. extreme market occurrences can significantly impact the VaR estimate, a significant issue with the equally-weighted VaR method. the VaR estimates won’t accurately reflect the state of the current market. the eWMa approach was designed to overcome the limitations of the benchmark model. this approach provides more recent observations, which are more important. On average, extreme events lose significance when the data window moves. alexander and Dakos (2023) further enhance the eWMa model. specifically, this model captures the volatility response. this is an essential feature because the Figure 5. Density obtained from gaRCH – vine copula simulation. Notes: the density is obtained from the 10000 gaRCH-Vine Copula simulation. 22 a. MaPPaDang el al. Table 16. Backtesting outcomes for 2.5% 10-day VaR and es (Long Position), gaRCH – vine copula simulation. Panel a: 2.5 % VaR long position aPe iCP sanD X N V aR θ (z) CC X N V aR θ (z) CC X N V aR θ (z) CC Benchmark 18 0.999 0.000 10 0.999 0.004 5 0.913 0.538 teWMa (94%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538 teWMa (92.5%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538 taeWMa (94%, 1%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538 taeWMa (94%, 2%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538 taeWMa (94%, 3%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538 taeWMa (92.5%, 1%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538 taeWMa (92.5%, 2%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538 taeWMa (92.5%, 3%) 8 0.996 0.047 4 0.809 0.824 5 0.913 0.538 gaRCH 8 0.996 0.047 9 0.998 0.004 5 0.913 0.538 eVt 1 0.191 0.391 0 0.047 0.057 2 0.413 0.804 Panel B: 2.5 % es long position aPe iCP sanD X N S E θ (z) eR X N S E θ (z) eR X N S E θ (z) eR Benchmark 6 1.000 0.084 3 0.999 0.183 1 0.776 0.176 teWMa (94%) 2 0.996 0.174 0 0.550 0.354 1 0.776 0.187 teWMa (92.5%) 0 0.996 0.206 0 0.550 0.405 1 0.776 0.209 taeWMa (94%, 1%) 1 0.996 0.159 0 0.550 0.350 1 0.776 0.177 taeWMa (94%, 2%) 1 0.996 0.146 0 0.550 0.346 1 0.776 0.170 taeWMa (94%, 3%) 0 0.996 0.132 0 0.550 0.342 1 0.776 0.166 taeWMa (92.5%, 1%) 0 0.996 0.193 0 0.550 0.391 1 0.776 0.192 taeWMa (92.5%, 2%) 0 0.996 0.170 0 0.550 0.379 0 0.776 0.180 taeWMa (92.5%, 3%) 0 0.996 0.152 0 0.550 0.370 0 0.776 0.172 gaRCH 3 0.995 0.094 0 0.998 0.228 2 0.776 0.198 eVt 0 0.037 0.524 0 0.007 0.499 1 0.126 0.488 Notes: see table 4. Table 17. Backtesting outcomes for 1% 10-day VaR and es (short Position), gaRCH – vine copula simulation. Panel a: 1 % VaR short position aPe iCP sanD X N V aR θ (z) CC X N V aR θ (z) CC X N V aR θ (z) CC Benchmark 6 0.965 0.095 4 0.809 0.804 4 0.809 0.538 teWMa (94%) 3 0.638 0.391 0 0.047 0.057 1 0.191 0.391 teWMa (92.5%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391 taeWMa (94%, – 1%) 3 0.638 0.391 0 0.047 0.057 1 0.191 0.391 taeWMa (94%, – 2%) 2 0.413 0.391 0 0.047 0.057 1 0.191 0.391 taeWMa (94%, – 3%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391 taeWMa (92.5%, – 1%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391 taeWMa (92.5%, – 2%) 1 0.191 0.391 0 0.047 0.057 1 0.191 0.391 taeWMa (92.5%, – 3%) 0 0.047 0.391 0 0.047 0.057 1 0.191 0.391 gaRCH 3 0.638 0.970 2 0.413 0.804 2 0.413 0.391 eVt 3 0.638 0.804 1 0.191 0.391 3 0.638 0.804 Panel B: 1 % es short position aPe iCP sanD X N S E θ (z) eR X N S E θ (z) eR X N S E θ (z) eR Benchmark 1 0.918 0.295 0 0.550 0.347 1 0.550 0.163 teWMa (94%) 0 0.305 1.000 0 0.007 0.440 0 0.037 0.933 teWMa (92.5%) 0 0.037 1.000 0 0.007 0.685 0 0.037 1.000 taeWMa (94%, – 1%) 0 0.638 1.000 0 0.007 0.875 0 0.037 0.922 taeWMa (94%, – 2%) 0 0.126 1.000 0 0.007 0.745 0 0.037 0.915 taeWMa (94%, – 3%) 0 0.037 1.000 0 0.007 0.062 0 0.037 0.912 taeWMa (92.5%, – 1%) 0 0.037 1.000 0 0.007 0.811 0 0.037 1.000 taeWMa (92.5%, – 2%) 0 0.037 1.000 0 0.007 0.501 0 0.037 1.000 taeWMa (92.5%, – 3%) 0 0.007 1.000 0 0.007 0.240 0 0.037 1.000 gaRCH 0 0.305 0.559 0 0.126 1.000 1 0.126 0.828 eVt 3 0.305 0.505 0 0.037 0.515 3 0.305 0.423 Notes: see table 5. cOgent Business & ManageMent 23 Table 18. Backtesting outcomes for 2.5% ten-day VaR and es (short Position), gaRCH – vine copula simulation. Panel a: 2.5 % VaR short position aPe iCP sanD X N V aR θ (z) CC X N V aR θ (z) CC X N V aR θ (z) CC Benchmark 10 0.999 0.000 9 0.998 0.005 11 0.999 0.000 teWMa (94%) 8 0.996 0.824 4 0.809 0.804 5 0.913 0.285 teWMa (92.5%) 8 0.996 0.970 4 0.809 0.804 5 0.913 0.285 taeWMa (94%, – 1%) 8 0.996 0.824 4 0.809 0.804 5 0.913 0.285 taeWMa (94%, – 2%) 8 0.996 0.970 4 0.809 0.804 5 0.913 0.285 taeWMa (94%, – 3%) 8 0.996 0.538 4 0.809 0.804 5 0.913 0.285 taeWMa (92.5%, – 1%) 8 0.996 0.970 4 0.809 0.804 5 0.913 0.285 taeWMa (92.5%, – 2%) 8 0.996 0.970 4 0.809 0.804 5 0.913 0.285 taeWMa (92.5%, – 3%) 8 0.996 0.970 4 0.809 0.804 5 0.913 0.285 gaRCH 7 0.987 0.824 5 0.913 0.970 6 0.965 0.057 eVt 7 0.987 0.538 5 0.913 0.804 3 0.638 0.538 Panel B: 2.5 % es short position aPe iCP sanD X N S E θ (z) eR X N S E θ (z) eR X N S E θ (z) eR Benchmark 6 0.999 0.005 4 0.999 0.009 1 0.999 0.008 teWMa (94%) 2 0.996 0.239 0 0.550 0.614 1 0.776 0,256 teWMa (92.5%) 0 0.996 0.294 0 0.550 0.865 1 0.776 0.291 taeWMa (94%, – 1%) 1 0.996 0.219 0 0.550 0.590 1 0.776 0.242 taeWMa (94%, – 2%) 1 0.996 0.203 0 0.550 0.570 1 0.776 0.233 taeWMa (94%, – 3%) 0 0.996 0.185 0 0.550 0.554 1 0.776 0.228 taeWMa (92.5%, – 1%) 0 0.996 0.376 0 0.550 0.785 1 0.776 0.679 taeWMa (92.5%, – 2%) 0 0.996 0.350 0 0.550 0.727 0 0.776 0.252 taeWMa (92.5%, – 3%) 0 0.996 0.330 0 0.550 0.683 0 0.776 0.241 gaRCH 1 0.978 0.159 0 0.776 0.434 2 0.916 0.096 eVt 6 0.978 0.134 3 0.776 0.310 3 0.305 0.257 Notes see table 6. Table 19. Joint VaR and es loss function ratios, gaRCH – vine copula simulation. Panel a α =1% long position α = 2 5. % long position aPe iCP sanD aPe iCP sanD Benchmark 1.000 1.000 1.000 1.000 1.000 1.000 teWMa (94%) 0.653 0.697 0.718 0.802 0.988 0.911 teWMa (92.5%) 0.663 0.707 0.739 0.810 0.999 0.929 taeWMa (94%, 1%) 0.649 0.685 0.714 0.799 0.988 0.908 taeWMa (94%, 2%) 0.643 0.693 0.708 0.793 0.985 0.903 taeWMa (94%, 3%) 0.635 0.697 0.699 0.785 0.978 0.895 taeWMa (92.5%, 1%) 0.659 0.705 0.735 0.806 0.999 0.926 taeWMa (92.5%, 2%) 0.652 0.704 0.729 0.800 0.996 0.920 taeWMa (92.5%,3%) 0.643 0.695 0.720 0.793 0.989 0.912 gaRCH 0.519 0.606 0.468 0.678 0.892 0.774 eVt 0.243 0.440 0.422 0.295 0.555 0.453 Panel B α =1% short position α = 2 5. % short position aPe iCP sanD aPe iCP sanD Benchmark 1.000 1.000 1.000 1.000 1.000 1.000 teWMa (94%) 0.752 0.758 0.561 0.899 0.877 0.738 teWMa (92.5%) 0.818 0.749 0.578 0.908 0.887 0.752 taeWMa (94%, – 1%) 0.813 0.737 0.559 0.895 0.877 0.736 taeWMa (94%, – 2%) 0.805 0.734 0.554 0.889 0.874 0.731 taeWMa (94%, – 3%) 0.795 0.727 0.547 0.880 0.868 0.725 taeWMa (92.5%, – 1%) 0.824 0.748 0.575 0.825 0.887 0.750 taeWMa (92.5%, – 2%) 0.816 0.745 0.571 0.819 0.884 0.745 taeWMa (92.5%, – 3%) 0.805 0.738 0.564 0.810 0.875 0.739 gaRCH 0.657 0.669 0.467 0.692 0.665 0.649 eVt 0.752 0.804 0.799 0.898 0.865 0.689 Notes: Values greater than one indicate outperformance of the benchmark model and vice versa.