Abstract: In this paper, simulation techniques are used to estimate value-at-risk of the CARBS equity indices and a global minimum variance portfolio. The empirical analysis in this paper is divided into two parts, the first part deals with simulating normally distributed returns in order to estimate VaR. In the second part calibrated univariate GARCH models are used to simulate returns series that are consistent with the stylised facts of financial time series. When a normal distribution is assumed, the GARCH model forecast of the returns produces the most reliable result. Finally, when garch processes are simulated, the EGARCH model is superior

Labuschagne, Coenraad C.A.

Oberholzer, Niel

Venter, Pierre J.

English

University of Johannesburg Institutional Repository

Simulation Techniques and Value-At-Risk of theCARBS IndicesCoenraad CA Labuschagne, Niel Oberholzer, and Pierre J VenterAbstract In this paper, simulation techniques are used to estimate value-at-risk ofthe CARBS equity indices and a global minimum variance portfolio. The empiricalanalysis in this paper is divided into two parts, the first part deals with simulatingnormally distributed returns in order to estimate VaR. In the second part calibratedunivariate GARCH models are used to simulate returns series that are consistentwith the stylised facts of financial time series. When a normal distribution is as-sumed, the GARCH model forecast of the returns produces the most reliable result.Finally, when garch processes are simulated, the EGARCH model is superior.1 IntroductionMonte Carlo simulation has been a valuable computational tool in finance, partic-ularly for the valuation of financial instruments and risk measurement (Boyle et al.(1997) [2]). According to Danielsson (2011) [5] Monte Carlo simulation can be de-fined as the process of replicating market outcomes by generating computer basedrandom numbers. Furthermore, due of advances in computing power and shortcom-ings of other VaR methods, Monte Carlo simulation is the preferred method when itcomes to financial risk measurement.Coenraad CA LabuschagneUniversity of Johannesburg, Department of Finance and Investment Management, Auckland Park,Johannesburg, 2006, South Africa, e-mail: ccalabuschagne@uj.ac.zaNiel OberholzerUniversity of Johannesburg, Department of Finance and Investment Management, Auckland Park,Johannesburg, 2006, South Africa, e-mail: nielo@uj.ac.zaPierre J VenterUniversity of Johannesburg, Department of Finance and Investment Management, Auckland Park,Johannesburg, 2006, South Africa, e-mail: pventer@uj.ac.za12 Coenraad CA Labuschagne, Niel Oberholzer, and Pierre J VenterFinancial returns series are usually not normally distributed. The results obtainedshow that the return series considered in this study show signs of leptokurtosis,which is consistent with the stylised facts of financial time series (Embrecths et al.(2005) [7]). Therefore, assuming the normal distribution when simulating returnscould lead to unrealistic results. Furthermore, numerous financial models assume aconstant volatility, in reality volatility is very volatile.The problems mentioned above are investigated in this paper. The empirical anal-ysis in this paper is divided into two parts, the first part deals with simulating nor-mally distributed returns in order to estimate VaR. In the second part univariateGARCH models will be used to simulate returns that consist of volatility cluster-ing and are not normally distributed (Alexander (2008) [1]). The accuracy of thedifferent methods is compared using a back-testing procedure. This provides an op-portunity to compare the reliability of all the models used in this thesis.The GARCH simulation component of the empirical analysis requires the useof the rugarch R statistical programming package (Ghalanos (2014) [6]). Theremainder of this paper is structured as follows: in the next section the relevantliterature on GARCH simulation is discussed, thereafter the methods implementedin this study are discussed in detail, this is followed by the empirical results andinterpretation.2 Literature reviewIn this section, the different studies on GARCH simulation and financial risk man-agement are briefly discussed. Oberholzer et al. (2015b) [9] simulated symmetricGARCH processes using historical data of the JSE returns, which were then com-pared to the actual returns. They finally conclude that the symmetric GARCH modelis not a good fit. Oberholzer et al. (2015b) [9] mention that areas for future researchincludes the simulation of other GARCH type models.Labuschagne et al. (2015) [8] simulated GJR-GARCH and EGARCH processesvia Monte Carlo to estimate volatility skews for the BRICS equity indices, thevolatility skews were then compared to the volatility skews generated via the riskneutral historic distribution (RNHD) model. Labuschagne et al. (2015) [8] concludethat the smoother skew obtained from the GARCH models are more reliable. Theessence of the argument is that the simulation obtained from the GARCH modelsuse four calibrated parameters, which is more reliable than the one parameter usedin the RNHD framework.To estimate VaR for Dutch bond portfolios, Vlaar (2000) [10] used historical sim-ulation, variance-covariance, and Monte Carlo Simulation methods. Vlaar (2000)[10] mentions that when implementing Monte Carlo simulation, it is necessary tomake certain assumptions regarding the distribution of the assets. Vlaar (2000) [10]assumed a zero mean return, and a GARCH process for the conditional variance.The most reliable VaR estimate was obtained using a combination of the variance-covariance method, and Monte Carlo simulation. The standard GARCH model as-Simulation Techniques and Value-At-Risk of the CARBS Indices 3sumes that positive and negative shocks lead to the same rise in volatility, which isunrealistic in certain cases. Therefore, asymmetric GARCH models will be includedin the numerical implementation of this paper.3 Methodology3.1 Monte Carlo VaR (standard normal random variates)If the returns are assumed to be normally distributed with mean µ and standarddeviation σ , Chan (2015) [4] argues that a typical model isRt = µ+σY, where Y ∼N (0,1). (1)If the above return dynamics are assumed, it is easy to derive thatVaRX = yXσ −µ,where yX is the X-quantile of the standard normal distribution. In terms of imple-mentation, the following algorithm is a slightly modified version of the algorithm inChan (2015) [4] in order to include the back-testing procedure.• Step 1: Generate N independent standard normal random variables, i.e.Yj ∼N (0,1)i.i.d. for j = 1, ...,N.• Step 2: Compute R j = µ+σYj.• Step 3: Rank R j in ascending order to obtain R∗j .• Step 4: VaR is set equal to the value of −R∗k where k is equal to (X×N)• Step 5: Create the back-testing vector, which takes a value of one if the actualloss (based on historical data) is greater than VaR, and zero otherwise.As part of the empirical analysis, the above algorithm will be implemented usingthe historical mean and standard deviation of each returns series, with the number ofsimulated variables (N) set equal to 50000. Furthermore, the same algorithm will beimplemented using the out of sample forecast of the conditional standard deviationof the returns using three GARCH family models. The different univariate GARCHmodels will be compared to the use of the historical parameters on the basis of aback-testing procedure.4 Coenraad CA Labuschagne, Niel Oberholzer, and Pierre J Venter3.2 Monte Carlo VaR with GARCH modelsFinancial returns series are usually not normally distributed, there are signs of lep-tokurtosis. Furthermore, financial returns series exhibit signs of volatility clustering.Therefore, assuming that return dynamics are normally distributed can be consid-ered unreasonable.According to Carmona (2014) [3], fitted GARCH models provide an efficientway to generate Monte Carlo samples which can be used as possible scenarios ofa trading day. Alexander (2008) [1] argues that the main distinguishing feature ofthe use of GARCH models in Monte Carlo simulation is that it is possible to simu-late returns with volatility clustering. The procedure for simulating from a GARCHmodel is explained by the algorithm below, which is illustrated by Alexander (2008)[1].• Step 1: Fix an initial value for the conditional variance (σ21 )• Step 2: Generate Yt ∼N (0,1).• Step 3: Compute εˆt = σtYt .• Step 4: Find σt+1 from σt and εˆt using the optimal parameters of the GARCHmodel.• Step 5: Return to step 2, t = t +1.After simulating a return series that exhibits volatility clustering, steps (3) to (5) ofMonte Carlo VaR algorithm are performed in order to compute VaR.This procedure is tested numerically. For each return series, 100 sample pathsare generated using the three GARCH family models, each sample path is of length1000, 99% VaR is then computed for each sample path. The average VaR is con-sidered for each respective model. Finally, the reliability of the estimated VaR isthen tested using a back-testing procedure. The back-testing procedure is completedusing 1000 of the most recent data points.4 Empirical resultsIn what follows, details of the Monte Carlo VaR and the VaR computed usingGARCH simulation are provided. This section is divided into two subsections. Foreach simulation technique, an example is shown the the simulated return distributionand compared to the normal density.Simulation Techniques and Value-At-Risk of the CARBS Indices 54.1 Monte Carlo VaRFigure 1 below illustrates the histograms of the simulated returns using standardnormal random variables. When the simulated return distribution is compared to theFig. 1: Histograms of simulated returns6 Coenraad CA Labuschagne, Niel Oberholzer, and Pierre J Venternormal density, it is evident that the simulated returns seem normally distributed.Furthermore, when the figure above is compared to figure a financial returns distri-bution, it is clear that the simulated returns do not resemble the same distribution.Most importantly, the simulated returns do not exhibit signs of leptokurtosis, andtherefore contradicts one of the stylised facts of financial times series.VaR was computed using the simulated returns. The mean in all cases is assumedto be equal to the mean of the historical returns. Four different assumptions weretested regarding the variance, namely: the historical variance of returns, and the outof sample (90 day) forecast of the conditional variance using the GARCH familymodels. The reliability of the computed VaR was tested using a back-testing proce-dure, the results are reported in the table below. The lowest percentage of exceptionsis shown in bold.Table 1: Monte Carlo VaR: percentage of exceptionsCanada Australia Russia Brazil South Africa GMVPHistorical 1.10% 1.60% 1.30% 1.60% 2.10% 1.70%GARCH 0.70% 0.30% 1.10% 1.60% 1.30% 0.70%GJR-GARCH 1.00% 0.40% 1.30% 1.60% 1.60% 1.40%EGARCH 1.00% 1.40% 1.30% 1.50% 1.90% 1.70%It is evident that the GARCH model forecast produces the VaR with the lowestnumber of exceptions. When the returns are simulated using the volatility predictedby the GARCH(1,1) model, VaR is overestimated for Canada, Australia, and theGMVP, VaR is underestimated for the other variables. When the historical varianceis used when computing Monte Carlo VaR, risk is underestimated for every variableincluded in this study. This clearly shows the shortcoming of assuming a mesokurticdistribution, when financial returns are slightly more peaked at the mean and havefat tails.The graphical representation of the back-testing procedure shows the exceptionsin red, the figure includes a graphical back-test of the assumption that produces thelowest percentage of exceptions only. The Y-axis denotes the log returns, and theX-axis denotes time. It is evident that the VaR estimated for Russia is the highest,and the VaR estimated for the GMVP is the lowest.4.2 Monte Carlo VaR with GARCH modelsWhen returns are assumed to follow a GARCH process, it is possible to simulatereturn series that exhibit signs of volatility clustering. As shown previously, theEGARCH(1,1) model is the best fit for the the return series in this study, thereforesample paths of simulated EGARCH(1,1) return series are plotted below.Simulation Techniques and Value-At-Risk of the CARBS Indices 7Fig. 2: Back-testing: Monte Carlo VaR(a) Canada: GARCH(1,1) (b) Australia: GARCH(1,1)(c) Russia: GARCH(1,1) (d) Brazil: EGARCH(1,1)(e) South Africa: GARCH(1,1) (f) GMVP: GARCH(1,1)The sample paths above show signs of volatility clustering, this is consistentwith the stylised facts of financial time series. Moreover, there also seems to sig-nificant asymmetry when the simulated returns are considered. The leverage effectwas found to be significant for all the variables included. To give an indication ofthe distribution, histograms of the simulated sample paths are plotted below.The histograms exhibit evidence of leptokurtosis and therefore making the as-sumption of a simulated GARCH process more realistic. Because the distributionsabove have fat tails, it reduces the chance of underestimating VaR. The results of8 Coenraad CA Labuschagne, Niel Oberholzer, and Pierre J VenterFig. 3: EGARCH simulated returns: sample pathsthe back-testing procedure of the Monte Carlo VaR using GARCH models are illus-trated below.Table 2: GARCH Monte Carlo VaR: percentage of exceptionsCanada Australia Russia Brazil South Africa GMVPGARCH 0.90% 1.40% 0.70% 1.40% 0.90% 1.50%GJR-GARCH 1.20% 1.60% 0.90% 1.20% 1.30% 1.70%EGARCH 0.70% 1.20% 0.70% 1.00% 0.90% 1.30%It is evident that the simulated EGARCH(1,1) process results in the lowest per-centage of exceptions for all the return series considered. The simulated GARCH(1,1)process leads to the same percentage of exceptions when compared to the EGARCHprocess for Russia and South Africa. Of all the assumptions and methods imple-Simulation Techniques and Value-At-Risk of the CARBS Indices 9Fig. 4: Histograms of EGARCH simulated returnsmented in this study, the simulated EGARCH process is the only assumption thatreduces the percentage of exceptions of VaR estimated for Brazil to 1%, all the othermethods underestimate VaR for Brazil.The graphical back-test of the EGARCH Monte Carlo VaR shows that this modelperforms fairly well when estimating VaR and is reliable when it comes to riskmeasurement.10 Coenraad CA Labuschagne, Niel Oberholzer, and Pierre J VenterFig. 5: Back-testing: GARCH Monte Carlo VaR(a) Canada: EGARCH(1,1) (b) Australia: EGARCH(1,1)(c) Russia: EGARCH(1,1) (d) Brazil: EGARCH(1,1)(e) South Africa: EGARCH(1,1) (f) GMVP: EGARCH(1,1)5 ConclusionMonte Carlo simulation can be defined as the practice of simulating possible scenar-ios by making use of computer generated random numbers (Danielsson (2011) [5]).It is evident that the use of Monte Carlo methods in finance have been invaluable tothe pricing of securities and scenario analysis when it comes to risk measurement.In this study, two algorithms were implemented to estimate VaR. The MonteCarlo VaR Algorithm makes use ofN (0,1) random variables to generate normallydistributed returns, the mean is set equal to the mean of the historical returns, andSimulation Techniques and Value-At-Risk of the CARBS Indices 114 assumptions are considered for the variance. More specifically, the variance is setequal to the historical variance of the returns, and the 3 forecasts of conditional vari-ance obtained from the GARCH models. The GARCH simulation algorithm makesuse of the fitted univariate GARCH models to simulate log returns with volatilityclustering.When the histograms obtained by implementing the Monte Carlo VaR algorithmare compared to the histograms of the historical returns, it is evident that normallydistributed returns is an unrealistic assumption. The sample paths of the GARCHsimulated log returns seem more realistic when compared to the actual historicallog returns.When the empirical results are considered, the percentage of exceptions producedusing simulation techniques is less than the percentage of exceptions using a rollingforecast. When implementing the GARCH simulation algorithm, assuming that thevariance is equal to the conditional variance predicted using a GARCH(1,1) modelleads to the lowest percentage of exceptions for 5 out of 6 variables. Finally, whenthe GARCH processes are used to simulate returns, the EGARCH model is the mostreliable for the CARBS indices and the GMVP.References1. Alexander, C., (2008) Market risk analysis: Practical financial econometrics (Vol. II). JohnWiley & Sons.2. Boyle, P., Broadie, M. and Glasserman, P., (1997) Monte Carlo methods for security pricing.Journal of economic dynamics and control, 21(8), pp.1267-1321.3. Carmona, R., (2014) Statistical analysis of financial data in R (Vol. 2). New York: Springer.4. Chan, N.H. and Wong, H.Y., (2015) Simulation techniques in financial risk management. JohnWiley & Sons.5. Danelsson, J., (2011) Financial risk forecasting: the theory and practice of forecasting marketrisk with implementation in R and Matlab. John Wiley & Sons.6. Ghalanos, A., (2014) rugarch: Univariate GARCH models, R package version 1.3-3.7. Embrechts, P., Frey, R. and McNeil, A., (2005) Quantitative risk management. PrincetonSeries in Finance, Princeton, 10.8. Labuschagne, C.C., Venter, P.J. and von Boetticher, S.T., (2015) A comparison of Risk Neu-tral Historic Distribution-, E-GARCH-and GJR-GARCH model generated volatility skews forBRICS Securities Exchange indexes. Procedia Economics and Finance, 24, pp.344-352.9. Oberholzer, N. and Venter, P.J., (2015) Simulating South African equity Index returns usingGARCH models.10. Vlaar, P.J., (2000) Value at risk models for Dutch bond portfolios. Journal of banking &finance, 24(7), pp.1131-1154.

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