VALUE AT RISK ESTIMATION AND VALIDATION IN THE SERBIAN CAPITAL MARKET IN THE PERIOD 2005-2015

Globalna finansijska kriza ukazala je na značajne slabosti postojećih modela merenja rizika na finansijskim tržištima. Posebno je izražena potreba za unapređivanjem i daljim razvojem modela finansijskog rizika na tržištima kapitala u zemljama u tranziciji. U ovom radu bavimo se mogućnošću predviđanja i ponašanjem raznih klasa VaR modela usredsređujući se na tržište kapitala u Srbiji od 2005. do kraja 2015. godine. U postupku procene tržišnog rizika implementirali smo različite VaR modele, a za validaciju VaR modela, koristili smo dvofazni postupak „backtestinga”. Rezultati ukazuju da se VaR model, baziran na simetričnom GARCH modelu sa GED raspodelom inovacija, ponaša opravdano dobro u periodu testiranja izvan uzroka. Najbolje rezultate daje filtrirana istorijska simulacija za interval poverenja od 99%. Ovi rezultati istraživanja ukazuju da standardni VaR modeli koji se najčešće koriste u finansijskim institucijama potcenjuju predviđanja rizika na tržištu kapitala u Srbiji u svim tržišnim okolnostima. Autori stoga sugerišu regulatornim telima i investitorima da za tržište kapitala u Srbiji uvedu robusnije i kompleksnije mere rizika.


Introduction
The last global financial crisis indicated the need for fundamental changes in risk management.The growing magnitude and complexity of trading securities, accompanied with the increased market volatility over the last decade, have pushed financial institutions and regulators to adopt a wide array of risk measurement models (Milojević and Terzić, 2014).There is an ongoing discussion among the financial markets practitioners, as well as academics, about the return distribution and risk in financial markets, with a view to improving their modeling to maximize the utility and precision of these models.
Value at Risk (herein after VaR) is one of the most popular models of market risk analysis in the risk industry, used both by institutions for internal risk management, and by regulators to dictate the regulatory market risk capital requirements.Its popularity increased after the Bank for International Settlements and the Securities and Exchange Commission applied VaR as a measure to quantify risk.The Basel Committee on Banking Supervision imposed on financial institutions the use of VaR (see BIS (2011) by the Basel Committee on Banking Supervision for an extended analysis of the application of VaR for risk measurement in the regulatory context).VaR can be estimated by a number of methods, including variance-covariance, historical simulation and Monte Carlo simulation methods (Beder, 1995;Hendricks, 1996;Mahoney, 1996;and Alexander and Leigh, 1997).
In financial risk management, VaR certainly represents a significant step forward in relation to the more traditional measures, mostly based on the sensitivity to market variables (Živković, 2007).Notwithstanding a number of wellknow drawbacks, VaR remains by far the most prominent and commonly-used quantitative method for risk measurement.Moreover, Danielsson et al. (2005) argue that some of the VaR drawbacks may not be that important in practice.
The model aggregates the market risk exposures of a portfolio into one numbersignifying the loss in the portfolio's value.In terms of market risk, the aim is to assess the magnitude of large potential losses of a portfolio due to adverse price fluctuations.VaR is used to provide a number that states, with a certain level of confidence, the maximum loss that can occur within a defined time horizon, usually anything from one day up to a few months.The loss is calculated as the drop of market prices of the entire portfolio or one of its portions.VaR has arisen as a powerful tool to capture market risk exposure of a portfolio, sector, asset class or security over some specified period of time.Regulators and the financial industry advisory committees recommend VaR as a way of measuring risk (Milojević and Terzić, 2014).
The VaR model assumes that positions can be liquidated over a specified period.This liquidity assumption has proved invalid over the course of recent market events in both developed and emerging equity markets.The fact that the recent financial market crisis originated in the US sub-prime mortgage market had important implications on the sudden drop in developed equity markets liquidity.The situation becomes even more complex when we consider frontier stock markets, where stocks became practically illiquid.
The accuracy of VaR, when applied to relatively illiquid financial instruments and frontier equity markets, is the subject of our empirical research.For VaR outputs to be reliable, we should implement the rigorous processes for model validation and calibration.Those processes are a critical feature of financial risk management to ensure that VaR models are producing reliable and consistent VaR figures based on which we can make competent decisions.There are frontier equity markets which have inherent valuation risk that could be material and which would not be picked up at the 95/99 percent VaR levels, highlighting the need to determine whether VaR is appropriate for particular asset classes in those markets.
The equity market in Serbia is a frontier market.Frontier markets, or "pre-emerging" markets, have recently been gaining attention from investors worldwide.While these markets are still in their early stages of development, some investors consider them an attractive opportunity for long-term economic growth with strong return potential.This high growth potential is, however, accompanied by greater risk and inefficiencies which are inherent in mladim tržištima u nastajanju.Ova tražišta se često karakterišu kao vrlo volatilna, rizična i neefikasna, ali su nedavna pobošljanja i uslovi olakšali ulaganje na ovim ranije zanemarenim tržištima.Ipak, mnogi investitori tvrde da rizici i nelikvidnost ovih investicija mogu nadvladati bilo koji potencijalni prinos, te su se stoga mnogi povukli sa ovih tržišta izazvavši veliki pad cena akcija i nelikvidnost (slučaj Srbije).
Iako postojeći modeli za izračun VaR-a koriste različite metodologije, mogu se generalno klasifikovati u dve široke kategorije: indirektni pristup i direktni pristup (Fabozzi et al, 2008 the young economies of these frontier markets. Frontier markets are often characterized as being highly volatile, risky, and inefficient, but recent improvements and conditions have facilitated investment in these previously overlooked markets.However, many investors argue that the risks and illiquidity of these investments may outweigh any potential returns, and therefore many withdrew from those markets causing a large drop in stock prices and illiquidity (case of Serbia).
There are multiple papers testing the performance of various VaR models in developed and developing markets, but research papers dealing with VaR calculation and verification in the Serbian financial market are very rare.Although there are studies that calculate the VaR, to the best of our knowledge, this is a unique study which compares the performance of VaR models in the Serbian market economy using the data from a ten-year period, which covers the volatile market conditions, and incorporates volatility modeling models into the VaR calculations.Several authors have tested certain standard VaR models in the Serbian equity market, yet the analyses were primarily based on short time series and simple backtesting procedures (Jeremić and Terzić, 2008;Djorić and Djorić, 2011;Radivojević et al, 2010).One of the most detailed studies on the use of VaR methodology adapted to the domestic equity market was empirically researched by Terzić (2014).Milojević and Terzić (2014) tested the performance of different VaR models in Serbia in the period 2005-2013.Thus, the aim of this paper is to test and validate different VaR models in the Serbian equity market in the period from 2005 until the end of 2015 by introducing a two-stage backtesting procedure of VaR models and by testing the stability and reliability of tested VaR models by Milojević and Terzić (2014) over different horizons.The paper, thus, aims to extend the very scarce empirical research on VaR estimation in this financial market and provide valuable information to regulators, financial institutions and investors for setting the market risk capital requirements and evaluating the market risk exposure of their trading portfolios based on VaR methodology.
The paper is organized in the following way: after the introductory section, section 2 presents the methodology of research with descriptions of tested VaR models and backtesting techniques.Section 3 gives the description of the analyzed data and statistical characteristics of the Serbian stock market, along with the backtesting findings and results.Section 4 concludes by summarizing the main results of the empirical research.

Research Methodology
VaR is usually defined as the maximum potential loss that a portfolio can suffer within a fixed confidence level during a holding period (Alexander, 2008).More specifically, VaR (1 -α) is defined as the threshold that is exceeded 100 * α times out of 100 trials on average. 1 -α is the confidence level where α ∈ (0, 1) is a real number.The cases in which ex post portfolio returns are lower than VaR estimates are called violations.One of the main inputs to the VaR calculation is the confidence level.Once the confidence level is set, VaR must be calculated in such a way that the violations should be equal to 100 * α (Köksal et al, 2013).
Although the existing models for calculating VaR employ different methodologies, they can be generally classified into two broad categories: indirect-VaR approach and direct-VaR approach (Fabozzi et al, 2008).In the empirical part of the paper, we include the indirect-VaR approach which consists of the classical parametric approach, nonparametric approach, and semiparametric approach.According to Fabozzi et al. (2008), all those types of VaR models follow the common structure which can be summarized in three steps: 1) mark to market, 2) estimate the distribution of the asset return, and 3) compute VaR by inverting the distribution function.The difficulty of this category of approaches lies in the second step because financial returns usually exhibit volatility clustering (high autocorrelation), significant kurtosis (peaked and fat tailed), marginal skewness (timevarying nature) and, in the case of indexes, autocorrelation of returns.
The second input that is necessary to calculate VaR is the standard deviation or volatility of returns.The key object of interest is the conditional standard deviation.It is wellknown that conditional volatility is not constant over time and that it is highly persistent.Modeling and forecasting volatility is crucial for investors who are interested in the forecast of the variance of a time-varying portfolio return over the holding period.
Volatility is an important factor in risk estimation because it affects all quantiles of the return distribution including the very extreme ones.Therefore, any probabilistic model for risk estimation should include a GARCH-type component (Stoyanov et al, 2011).The most recent and advanced VaR methods make use of GARCH models to calculate the conditional standard deviation (Köksal et al, 2013).
After adoption of different VaR calculations for the analysis of the stock index exposure to market risk, the next step is to give an answer on the question whether VaR, calculated ** For the sake of simplicity we will henceforth assume a zero conditional mean, μ t = 0.It is a common assumption in risk management when short (e.g., daily or weekly) return horizons are considered.It is readily justified by the fact that the magnitude of the daily volatility (conditional standard deviation) σ t easily dominates that of μ t for most portfolios of practical interest.This is also indirectly evident in the fact that, in practice, accurate estimation of the mean is typically much more difficult than accurate estimation of volatility.Still, conditional mean dynamics could easily be incorporated into any of the GARCH models by considering additional returns r t -μ t in place of r t .*** According to the GARCH model VaR is estimated under alternative distributional assumptions: normal, t-student, and generalized error.Final selection of GARCH-type VaR is based on distribution that best fits the data according to the maximum value of LLF.
The drawback of the Kupiec test is that it does not take into account the sequence of violations.To fix this shortcoming, Christoffersen (1998) designed a test that places emphasis to the predecessor of a violation.As stated by Christoffersen (1998), testing solely for the purpose of the correct uncoditional coverage of a VaR-model neglects the possibility that violations might cluster over time.Thus, the second test aims to check whether the process of VaR violations is serially independent.Christoffersen (1998) also developed the conditional coverage test, which represents an incorporated test of the hypothesis of unconditional coverage and independence.
The second stage of the validation process is based on ranking VaR models using two loss functions.Loss function allows for the sizes of tail losses to influence the rankings of VaR models.Models that generate higher tail losses would generate higher values under this sizeadjusted loss function than models that generate lower tail losses.In order to select superior VaR models, each model will be graded by two loss functions: quadratic loss function and absolute loss function. (1) (2) The QL and AL functions do not penalize the model when exceptions do not occur.A well performing model yields a lower score.

Data and Empirical Results
The main objectives in this section are the time-series analysis of the Serbian stock index returns and the development of appropriate risk estimation models.Before trying to model and forecast a given time series of returns, it is desirable to have a preliminary look at the data to get a 'feel' for them and to understand their main properties.This will be invaluable later on in the modeling process.For this purpose, we first describe the data using summary statistics and graphical methods.Secondly, we introduce the modeling of volatility to find a suitable statistical model to describe the data generating process.Thirdly, we employ VaR models as described in Section 3 to forecast the future values of the market risk.Finally, we verify our risk measures through backtesting techniques.
In data analysis, we start with the descriptive statistical properties of the sample data.The daily historical data on the Serbian blue chip stock index (BELEX 15) from 4 th October 2005 until 5 th October 2015 are taken from the Belgrade Stock Exchange.Using the same data set, we apply the five models to estimate the 1% and 5% 1-day VaR.We thus use the data from the period both before and after the 2008 financial crisis.This series of prices is converted to log returns series using the equation ( 3): (3) Data sets for VaR estimation start 500 days later and cover the period from 8 th October 2007 to 5 th October 2015, for the purpose of initializing the estimation.The rest of the data series are used for backtesting purposes.Through the simulation methodology, we attempt to determine how each VaR approach would have performed over a realistic range of indices over the sample period.We investigate out-ofsample (forecasting) performance based on the unconditional coverage test and Christoffersen's 2018 procenjenih VaR brojeva za analizirani indeks akcija predstavljena je u tabeli 8.
Before analyzing the predictive performance of VaR models, some stylized facts of Belex 15 return series are examined and tested in Tables 2, 3, 4 and 5. Graphic representation of levels and daily index changes in the analyzed period is given in Figure 1.
In graph 1, the daily returns are represented as a time series.As it can be seen, the period from 2005 to the first half of 2007 is characterized by the significant increase in index value with the entry of foreign investors into the domestic capital market (primary axis).From the figure (secondary axis) we also observe that there are some periods in which returns are more volatile than in others, which is in favor of a model that accounts for time-varying volatility.Volatility clustering is clearly observable in the graph.
Graph 2 presents the daily returns in a Q-Q plot.QQ plot shows that the fat-tailed distributions are asymmetric against the normal distribution because positive tails are unequal to their counterparts.The fat-tailed and asymmetric properties of return distribution motivate the use of non-normality distribution innovations in this study.
Table 2 shows the descriptive statistics of the daily index returns.The average return is close to zero and it is not possible to statistically reject a zero mean return (p-value of 24% is much higher than 5% significance level).The standard deviation of returns completely dominates the mean of returns at short horizons such as the daily ones.The unconditional standard deviation is 1.35%.Belex 15 exhibits kurtosis above 3 (14.36),which indicates that the daily return distribution has fat tails.Fat tails imply a higher probability of large losses (and gains) than the normal distribution would suggest.Capturing these fat tails appropriately is crucial in risk management.Another feature of the return series is the presence of skewness.In fact, the index is positively skewed with the magnitude of 0.15.These indicate that the return series is non-normal.Takođe Jarque-Bera test za nivo značajnosti od 5%, pokazuje da raspodela prinosa nema normalnu distribuciju, kao što je prikazano na tabeli 3. Rezultati dobijeni testom vode do odbijanja nulte hipoteze o normalnoj raspodeli.
Nakon preliminarne deskriptivne i statističke analize, zaključujemo da je dnevni prinosi BELEX 15 indeksa pokazuju uobičajeno kretanje koje je podrobno dokumentovano u finansijskoj literaturi (Mandelbrot, 1963;Fama, 1965 We also confirmed that this non-normality via the Jarque-Bera test performed at the 5% level, as presented in Table 3.The results obtained by the test lead to the rejection of the null hypothesis of normality. The time series of stock index prices is not a stationary process as it is easy to show that the variance increases over time.However, the first differentials within the series, namely (P t − P t−1 ), do form a stationary series.To test whether a given time series of Belex 15 log returns is stationary or not, we apply an indirect test for the existence of a unit root (Augmented Dickey-Fuller Test).We also conduct a stationary test using several scenarios, as shown in Table 4.When it comes to stationarity, we are mainly concerned with the stability of variances and covariances throughout the sample.The test statistics value is compared to the relevant critical value for the Augmented Dickey-Fuller Test.If the test statistic is less than the critical value, we reject the null hypothesis and conclude that no unit-root is present.According to that, the time series data of Belex 15 index is stationary.
Finally, Table 5 shows the Ljung-Box Q test statistics (white noise test) for autocorrelation in returns.The test performed at the 5% significance level and with up to 25 lags, leads to the conclusion that autocorrelation is present in the considered return series.Engle's ARCH test performed at the 5% level with up to 25 lags leads to the conclusion that ARCH effects are present in the data i.e. the squared returns are serially correlated and we have a conditional heteroskedasticity in the log returns.The above features motivate the need for heteroskedasticity models in calculating VaR.We can conclude that the log returns and squared returns exhibit a strong interdependency, and as a result, an ARMA-ARCH/GARCH model may be in order here.
After the preliminary descriptive and statistical analysis, we can conclude that the daily log returns of BELEX 15 index exhibit common patterns that are well-documented in financial literature (Mandelbrot, 1963;Fama, 1965    u beskonačnost, autoregresivno ponašanje, zakošenost i zadebljani repovi.
The second model is Risk Metrics methodology, where daily variances are computed using an exponentially weighted moving average with the decay factor of 0.94.Unlike the simple movingaverage volatility estimate, an exponentially weighted moving average allows the most recent observations to be more influential in the calculation than observations further in the past.This has the advantage of capturing shocks in the market better than the simple moving average and thus we excluded the equal weighted moving average alternative in our VaR calculations.
Another approach for estimating the variance input to VaR calculations involves the use of modified Risk Metrics model with the optimized decay factor for the Serbian stock market.We obtained the optimal decay factor using forecast evaluation statistics such as root mean square error (RMSE).In order to find the optimal value, we set up an optimization problem for minimizing the mean square error between the EWMA estimate and realized volatility, by varying the lambda value.The optimal obtained lambda value for BELEX 15 data set is 0.83, as shown in Graph 3.
Empirical research has firmly established the fact that volatility clustering is a stylized fact regarding asset returns.Different econometric models have been suggested to explain the time varying volatility and the most widely used ones are the GARCH-type models (Stoyanov et al, 2011).In our study we fitted time series models GARCH(1,1) and EGARCH(1,1) to clean the clustering of volatility effect assuming the normal, classical Student-t model and generalized error distribution on the innovations.In order to obtain conditional volatility based on models from GARCH family and later include it in the formula for VaR calculation, model parameters are calculated using the maximum likelihood estimation method.Maximum likelihood estimates of the parameters are obtained by numerical maximization of the loglikelihood function using NUMXL time series forecasting software.The estimates of optimal parameters,from different specifications of conditional heteroscedasticity models (GARCH and EGARCH) assuming normal and non-normal distributed innovations, are presented in Table 6.Odabir modela je izvršen u skladu sa LLF kriterijumom.U tabeli 7 su sažeto prikazane LLF vrednosti svakog modela i naši rezultati pokazuju da je GARCH model sa GED raspodelom inovacijama razuman model koji dobro odgovara podacima jer ima najveću vrednost verodostojnosti a pretpostavke modela su uglavnom zadovoljene.Shodno tome, koristimo GARCH model sa GED raspodelom za predviđanje volatilnosti u linearnoj VaR formuli (videti tabelu 1).
Rezultati backtestinga VaR modela za Belex 15 su prikazani u tabeli 8. Uzeli smo u razmatranje rezultate statističkih testova (racio test verodostojnosti) u svrhu backtestinga za bezuslovni pokrivenost, i osobinu nezavisnosti i uslovnu pokrivenost.Izvor: Autori Model selection was done according to the log likelihood function criteria.We summarize the LLF value of each model in Table 7 and our results indicate that GARCH model with GED innovations is a reasonable model that fits the data well, because it has the highest loglikelihood value and the model assumptions are largely satisfied.Consequently, we use GARCH model with generalized error distributed innovations for volatility forecasting in the linear VaR formula (see Table 1).
The next modeling stage is called model verification.Verification is the general process of checking whether the model is adequate.This section focuses on backtesting techniques for verifying the accuracy of VaR models.In order to validate the risk model, we formally test whether the sequences of exceptions satisfy the standard backtesting conditions.Formally, we need to test the null hypothesis that the probability of having an exception is α.A typical way to examine a VaR model is to count the number of VaR violations when portfolio losses exceed the VaR estimates.An accurate VaR approach produces a number of VaR exceedances as close as possible to the number of VaR exceedances specified by the confidence level.If the number of violations is higher than the selected confidence level would indicate, then the model underestimates the risk.On the other hand, if the number of violations is smaller, then the model overestimates the risk.The test is conducted as T 1 /T, where T is the total number of observations, and T 1 is the number of violations for the specific confidence level.
Comparing the number of exceedances by visual inspection (see Graph 4), the results indicate that EWMA and RM VaR models are too optimistic, particularly with their daily 99% VaR forecasts being too low.In contrast, it seems that all other models produce exceptions that are consistent with the confidence level, i.e. unconditional coverage.
Za validaciju VaR modela izabrali smo kriterijum koji predstavlja rezultate dva testa i bazirali naše nalaze na činjenici da VaR modeli ispunjavaju oba svojstva istovremeno: bezuslovnu pokrivenosti i nezavisnost tj.uslovnu pokrivenost.Prema tome, u slučaju prva tri VaR modela nijedan nije ispunio u potpunosti oba kriterijuma za backtesting za VaR modele.Možemo zaključiti da istorijska simulacija, RM i optimalni EWMA 1% i 5% VaR model dovode do pogrešnog prestavljanja stvarne izloženosti riziku u posmatranom vremenskom okviru.Izostanak uredno naznačene dinamike u modelu istorijske simulacije može dovesti do ignorisanja dobro poznate stilizovane činjenice o zavisnosti prinosa, i ono što je najznačanije, do grupisanja varijansi u klastere.Ovo obično uzrokuje da istorijska simulacija reaguje previše sporo na promene tržišnog rizika.As evident from Table 8, the satisfactory performance with regards to Christoffersen'sjo intconditional coverage test is recorded for the parametric model (GARCH-GED VaR) at 99% confidence level and semi-parametric model (FHS VaR) at both confidence levels.Very weak performance is recorded for HS (500), RM and EWMA (0.83) 1% and 5% VaR.Although HS approach satisfied the unconditional coverage condition at both confidence levels, the clustering of exceptions is evident.RM and EWMA with the optimized decay parameter were the worst performers, but EWMA 1% VaR model unlike RM 1% VaR was able to generate the independent VaR failures.The results of backtests for HS and RM VaR in the Serbian stock market are consistent with the earlier findings of Terzić et al. (2013), which leads us to conclude that HS and RM VaR models are not suitable for measuring market risk in this financial market.Thus, the failure of these models is confirmed in various time series samples and risk horizons.
As thevalidationcriterion for VaR models we chose the results of two tests and based our findings on the fact that VaR models must simultaneously satisfy both unconditional coverage and independence properties, i.e. conditional coverage property.Accordingly, none of the first three VaR models fully satisfied the both backtesting criteria for VaR models.We can conclude that HS, RM and optimal EWMA 1% and 5% VaR lead to misrepresentation of actual risk exposures within the observed time horizon.The lack of properly specified dynamics in the HS methodology causes it to ignore wellestablished stylized facts on return dependence, most importantly variance clustering.This typically causes the HS VaR to react too slowly to changes in the market risk environment.
On the other hand, the use of more sophisticated models based on GARCH volatility forecasting, especially the combination of historical simulation with GARCH estimates of volatility (FHS), significantly improves  Ovi rezultati potvrđuju da regulatorna tela i investitori na finansijskim tržištima u zemljama u tranziciji treba da promene tradicionalno viđenje i razumevanje modelovanja rizika i prinosa.Potrebno je da uvedu robusnije i kompleksnije mere rizika, dok VaR modeli razvijeni za mirna i stabilna razvijena tržišta nisu pogodni za tržište kapitala u Srbiji.Zaključujemo da VaR modeli bazirani na pretpostavci o normalnoj raspodeli nisu pogodni za merenje finansijskih rizika na tržištu kapitala u Srbiji i daju nove dokaze da su kompleksnije i komplikovanije verzije VaR modela prikladnije za ovo tržište.Ovi rezultati mogu imati veliku vrednost za domaće i strane investitore.According to the employed two-stage backtesting procedure, the best performing VaR model must primarily satisfy both unconditional coverage and Christoffersen's independence test and then provide superior tail loss forecasts, in the sense of minimizing statistical errors.The optimal VaR model should minimize the loss functions.The choice of a model relies on the analysis of loss function scores, which are presented in Table 9.For this purpose, we applied the QL and AL functions, as shown and described in equations 1 and 2.

Zaključak
In terms of QL loss function at 99% confidence level, FHS istheabsolute "winner".RM and EWMA rank second and third, respectively.The GARCH-GED VaR model, not rejected by the statistical tests, ranks fourth.HS model is the worst performer.With respect to AL loss function, the FHS performs well again.It produces the lowest AL, while GARCH-GED ranks fourth.The RM remainsin the same position, while HS and EWMA switch ranks.The high QL and AL function scores of the GARCH model may potentially indicate that this model tends to overestimate VaR.
Out of these five models, at 95% confidence level, in terms of QL and AL loss functions, FHS VaR performs well again.It produces the lowest QL, and AL value and ranks as the second.This model seems to stand out, although the differences in scores are relatively small.The 95% GARCH-GED VaR also seems to be an adequate choice for modeling VaR for the Belgrade Stock Exchange index.The considered RM and EWMA 95% VaR models do not perform very well and are not well suited for modeling VaR on this market.Their daily 95% VaR figures seem too optimistic, with forecasts being too low.
These results confirm that regulators and investors in transitional financial markets should change their traditional perception and understanding of risk and returns modeling.They need to introduce more robust and complex risk measures, while VaR models developed for tranquil and well behaved developed markets are not adequate for the Serbian capital market.We conclude that VaR models based on normal distribution assumption are not suitable for financial risk measurement on the Serbian stock market, and provide new evidence that more complex and complicated extensions of VaR models are well suited for this market.These results might be valuable for domestic and foreign investors.
cl -korespondirajući kvintil standardne normalne varijable * μ t -uslovna srednja vrednost ** GARCH VaR*** (normalna, GED i studentov-t raspodela inovacija) α cl -korespondirajući kvintil raspodele koji najbolje odgovara recent developments in financial econometrics and are likely to produce more accurate risk assessments.In this study, the following types of each of the aforementioned VaR approaches are applied: • Historical simulation, non-parametric model, which uses a large quantity of historical data to estimate VaR but makes minimal assumptions about the risk factor return distribution.• Parametric VaR based on GARCH model family and EWMA models.• Filtered historical simulation, semiparametric model, which computes VaR by combining historical simulation with GARCH estimates of volatility.Here, filtering refers to using the estimated GARCH volatility to create standardized innovations from which we compute empirical quantile for VaR.

*
Jeremić Z., Terzić I.,Milojević  M. Value at risk estimation and validation in the Serbian capital market in the period 2005-2015 Table 1.Selected VaR models VAR model VAR formulation Description Historical simulation Risk Metrics α cl -corresponding quantileof standard normal variable* -conditional mean** GARCH VaR*** (normal, generalized error and t-student distribution of innovation) α cl -corresponding quantile of distribution that best fits In the RM VaR we have assumed a standard normal distribution for the return innovation.
, i.e. the unconditional coverage test and Christoffersen's (1998) independence and conditional coverage tests.Unconditional coverage and independence are fundamental properties of an adequate VaR model.The risk model has the correct forecasting power and effect only if both properties are satisfied.
Jeremić Z., Terzić I., Milojević M. Value at risk estimation and validation in the Serbian capital market in the period 2005-2015

15 GARCH
Jeremić Z.,Terzić I., Milojević  M. Value at risk estimation and validation in the Serbian capital market in the period 2005-2015 Table 6.Estimated optimal GARCH and EGARCH parameters for Normal, Generalized error and Student-t innovation distribution of BELEX

Table 2 .
Descriptive statistics for the Belex 15 return series in the period 2005-2015