Risk Measuring of Chosen Stocks of the Prague Stock Exchange

Risk Measuring of Chosen Stocks of the Prague Stock Exchange Ing. Mgr. Radim Gottwald, Department of Finance, Faculty of Business and Economics, Mendelu University in Brno, radim.gottwald@mendelu.cz Abstract The paper is focused on risk measuring of stocks. Author presents the Value at Risk model, which is often used for risk analyses mostly in the banking industry. The model is legally defined. Author assesses the application of the Value at Risk model to chosen stocks from the Prague Stock Exchange within the 2011 period. In detail, three sub-methods, namely the historical simulation method, variance and covariance method and Monte Carlo method are applied. Respective confidence interval, hold period, historical period and other parameters related to the sub-methods are gradually chosen. Then standard deviations, mean values, variance coefficients, covariance matrices, correlation matrices and other types of matrices are calculated. Author compares diversified and non-diversified Value at Risk, calculated by the sub-methods. Author comments conclusions resulting from certain differences of calculated values. Attention is also paid to the relative and marginal Value at Risk, back testing and stress testing, possibility of reducing the Value at Risk and other options how to use this model in the practice. Key Words Historical simulation method, variance covariance method, Monte Carlo method, risk measurement, Value at Risk model Introduction Every investment is unique for its particular risk level, which is a degree of uncertainty that the financial instrument chosen will not reach such profit levels as expected by the investor from it. The only way how to calculate the risk level for any investment in financial assets subject to an assessment is the adoption of the Value at Risk model. In the context of the implications of the economic downturn, strong emphasis has been laid on the adoption of high-quality risk management mechanisms in the banking sector over the past five years or so. The numerous directives and other recent documents covering the Value at Risk topic are evidence of how pressing this matter has become. As banking regulation and supervision are being made even stricter, this model is also enjoying growing popularity among bank analysis. The Value at Risk model, also called "internal model of Value at Risk", is generally applied to measurements and management of the overall market

risk in a portfolio. The Value at Risk model is currently widely used in the banking industry in connection with capital adequacy requirements placed on certain banks. This fact inspired author to write this paper. The Value at Risk model allows a more flexible form of banking supervision. Besides the Basel committee on banking supervision, also central banks and other financial institutions are dealing with this bank risk management model. Within EU legislation, this model is represented mostly in documents BASEL I of 1996, BASEL II of 2004, BASEL III of 2010, CAD II of 1998 and CNB Measure No. 3 of 1999 (capital adequacy measure). The documents BASEL I, BASEL II and BASEL III contain recommendations by the Basel committee on banking supervision regarding banking right and regulation. Together the standardised methods, the Value at Risk model is mentioned here within the first pillar as an instrument through which the banks can lay down their capital requirements on the market risk of the trade portfolio. The Value at Risk expresses the maximum potential loss calculated with a certain probability over the following chosen hold period, determined based on a chosen historical period which the financial entity can have in its portfolio in case of unfavourable changes on the market. When applying the Value at Risk model, the confidence interval, hold period and historical period are chosen as the first thing. The Value at Risk is actually a single-sided quantile (e.g. 95 %) from distributed portfolio profits and losses during a chosen hold period (e.g. one day), determined based on the chosen historical period (e.g. 3 years). It is the maximum potential, not maximum possible loss. The Value at Risk is therefore an estimation and the hold period expresses the time horizon in this estimation. Three sub-methods are applied within the Value at Risk model: the historical simulation method, variance and covariance method and Monte Carlo method. Different financial assets can be chosen as risk factors in this historical simulation method as part of empirical investigations. Maximum losses expected calculated by Value at Risk in moments of systematic crisis are analyzed by Fernandes, Lustosa and Paulo (2010). The Value at Risk is calculated daily, during the period from 1993 to 2010, using three methodologies: historical simulation, normal linear and Monte Carlo simulation. Non-parametric historical simulation approaches are used by Aussenegg and Miazhynskaia (2006) that focuse on uncertainty in Value at Risk estimates under parametric and non-parametric modelling. They evaluate a set of parametric and non-parametric Value at Risk models that quantify the uncertainty in Value at Risk estimates in form of a Value at Risk distribution. Sharma (2012) surveys 38 papers to understand the comparative performance of historical simulation and the performance

measures for Value at Risk Methods. The measures are divided into conditional coverage and unconditional coverage measures. Term structure model with macro Value at Risk in a stochastic volatility petting is proposed by Niu (2007). Risk premium of yields is directly driven by the time-varying variance covariance of the Value at Risk innovations. Steelyana (2011) compare historical simulation with variance covariance approach. Value at Risk summarizes the worst loss over a target horizon that will not be exceeded with a given level of confidence. Proposing Monte Carlo simulation, GARCH regression and wavelets decomposition algorithm, Giovanis (2011) examines four different approaches in trading rules for stock returns. Other approaches relate to the moving average oscillator, the moving average convergence divergence oscillator and the simple random walk autoregressive model. Using various tests, Colucci and Brandolini (2011) present results on both Value at Risk 5% and Value at Risk 1% on one day horizon for the two Value at Risk models for the following indices: Standard&Poors 500, MSCI United Kingdom, MSCI France, MSCI Canada, MSCI Emerging Markets, Topix, Dax, Italy Comit Globale, RJ/CRB. Capriotti (2007) describes least squares importance sampling strategy for Monte Carlo simulation security pricing. Using several numerical examples, he shows that this strategy provides efficiency gains comparable to the state of the art techniques, when the latter are known to perform well. Detemple, Garcia and Rindisbacher (2000) focuse on the performance of Monte Carlo simulations for optimal portfolios. This performance in the numerical implementation of portfolio rules derived on the basis of probabilistic arguments is improved. Monte Carlo simulation portfolio optimization for general investor risk-return objectives and arbitrary return distributions is realized by Shaw (2010). To solve portfolio investment problems, he uses Monte Carlo simulation sampling of random portfolios. He also presents Monte Carlo simulation approach for long-only and bounded short portfolios with optional robustness and a simplified approach to covariance matching. Tracking of the US interest rate by sequential Monte Carlo simulation is realized by Lombardi and Sgherri (2007). They attempt a real-time evaluation of the US monetary policy stance while ensuring consistency between the specification of price adjustments and the evolution of the economy under flexible prices. The model's likelihood function is evaluated using a sequential Monte Carlo algorithm. The objective of this paper is to assess the application of the Value at Risk model to real data. Specifically, the historical simulation method, variance and covariance method and Monte Carlo method shall be applied.

Materials and Methods Through an empirical analysis and by using descriptive analysis methods, theoretical procedures of three methods are applied to a particular market and resulting values are compared. These methods will enable a closer identification of a particular market. Based on the results, the maximum potential loss can be calculated with a certain probability over the following chosen hold period, determined based on a chosen historical period which the financial entity can have in its portfolio in case of unfavourable changes on the market. Characteristics shared by all the three methods are chosen as the first thing. The portfolio value structure and volume are determined. Chosen risk factors to which the methods are applied are stocks. A market where the methods are applied is chosen. It is the Prague Stock Exchange. Four stocks from different industrial sectors - signed as CETV, ČEZ, ERSTE BANK and VIG - are chosen. The historical period over which required data was collected is 2011 period and the Value at Risk is calculated to 30.12.2011. One day is chosen as the hold period. Input data includes a time line with a one-day periodicity. The confidence interval is 95 %. The Value at Risk is calculated with the probability on this level. The source of data is the company Patria Finance, a.s. The input data necessary for the implementation of the empirical analysis, obtained from Patria Online, a.s. (2013), are historical stock prices. They are the closing stock prices and are always quoted in CZK. Based on the stock prices as of 30.12.2011, the weights of the different stocks in the portfolio are calculated. Results Shared Input Data for Application of All Three Methods Tab. I shows the historical stock prices being in fact the absolute historical values of risk factors. I: Historical Stock Prices Date CETV ČEZ ERSTE BANK VIG 30.12.2011 127.95 786.00 347.00 799.90 29.12.2011 126.79 790.00 346.00 778.10 28.12.2011 128.00 780.00 347.00 798.80 05.01.2011 396.50 821.70 879.00 985.00 04.01.2011 399.00 820.10 902.00 988.00 03.01.2011 400.50 800.00 898.40 994.60

Tab. II presents indicators related to historical stock prices standard deviation, mean value and the variance coefficient. II: Indicators Related to Historical Stock Prices Indicator CETV ČEZ ERSTE BANK VIG Standard deviation 89.39 70.12 217.50 127.25 Mean value 279.92 820.09 689.01 867.69 Variance coefficient 0.32 0.09 0.32 0.15 Tab. III shows changes in the historical stock prices being in fact the relative historical values of risk factors. This determines the volatility of stocks in the portfolio. III: Changes in Historical Stock Prices Date CETV ČEZ ERSTE BANK VIG 30.12.2011 0.00 0.00 0.00 0.00 29.12.2011-0.01 0.01 0.00-0.03 28.12.2011 0.01-0.01 0.00 0.03 05.01.2011 0.00-0.02 0.00 0.00 04.01.2011 0.01 0.00 0.03 0.00 03.01.2011 0.00-0.03 0.00 0.01 Applying the Historical Simulation Method Tab. IV shows the converted stock prices being in fact the current values of risk factors. These values are calculated as the product of the stock prices as of 30.12.2011 and of changes to historical stock prices. IV: Converted Stock Prices Date CETV ČEZ ERSTE BANK VIG Total sum 30.12.2011 0.00 0.00 0.00 0.00 0.00 29.12.2011-1.17 3.98-1.00-22.41-20.60 28.12.2011 1.21-10.08 1.00 20.73 12.86 05.01.2011-0.13-13.68-0.39-3.25-1.,45 04.01.2011 0.80-1.53 8.85 2.43 10.55 03.01.2011 0.48-19.75-1.39 5.31-15.35

From the values in the last column in Tab. V the 5 % percentile (single-sided quantile) being -49.47 is calculated against chosen confidence interval 95 %. This is the diversified Value at Risk. Tab. V shows the 5 % percentile calculated extra for each stock from converted stock prices. V: 5 % Percentile Indicator CETV ČEZ ERSTE BANK VIG 5 % percentile -6.78-19.03-14.31-28.70 The total of -68.82 is calculated from the values in the table. This is the non-diversified Value at Risk. Tab. V shows clearly what are the shares of stocks in the non-diversified Value at Risk. Applying the Variance and Covariance Method Standard deviations are calculated from changes to historical stock prices. Normal distribution is chosen as the probability distribution. Then, the V matrix is calculated being the diagonal volatilities matrix calculated from the products of standard deviations and 1.64 values. The 1.64 value is one of normalised normal distribution for the confidence interval of 95 %. Then the VC matrix is calculated as the product of matrices V and C. Subsequently, the VCV matrix is calculated being the variations and covariations matrix. Then the VCVW is calculated. This is a column vector. The W matrix is a column vector the values of which are made of stock price as of 30.12.2011. This W weights vector contains cash flow values. Calculated is also the W T VCVW matrix. The W T matrix is a row vector the values of which are made of stock prices as of 30.12.2011. The W T VCVW matrix is made only of number 4927.67 being the square of the diversified Value at Risk. After extracting the root, the diversified Value at Risk, being -70.20 in this case, is calculated. The negative value is chosen by taking into account the fact that the Value at Risk expresses the loss, not the profit. Tab. VI shows the Value at Risk component and individual matrix. The component matrix is calculated as the product of matrices VCVW and W, subsequently divided by the diversified Value at Risk in the amount of -70.20. This is a column vector (the values are stated in the row).

VI: Component and Individual Matrix Stock CETV ČEZ ERSTE BANK VIG The first value 6.84 18.45 16.90 28.02 The second value 6.98 20.46 17.69 28.80 The total of 70.20 is calculated from the values in the first row in the table. The diversified Value at Risk is a value opposite to this figure, i.e. of the already calculated - 70.20. The table shows clearly what are the shares of stocks in the diversified Value at Risk. The individual Value at Risk matrix is calculated as the product of matrices W T and V. It is a row vector in the second row in the table. The total of 73.93 is calculated from the values in the second row in the table. The non-diversified Value at Risk is a value opposite to this figure, i.e. -73.93. The table shows clearly what are the shares of stocks in the non-diversified Value at Risk. Applying the Monte Carlo Method The covariance matrix is calculated from the changes to historical stock prices. Normal distribution is chosen as the probability distribution. Also the number of simulations and the form of value simulation are chosen. 2000 simulations are implemented by using the NtRand 3.2 software of Numerical Technologies. Tab. VII presents converted simulations of stock prices. These converted simulations are calculated as the products of stock prices as of 30.12.2011 and of simulations of stock prices. VII: Converted Simulations of Stock Prices Number of ERSTE CETV ČEZ simulation BANK VIG Total sum 1-2.27 4.49 1.23-8.75-5.30 2 3.28-4.58 1.05 9.75 9.51 3 4.70 9.73 15.29 18.16 47.88 1998 0.22-0.35 1.60-2.15-0.67 1999 3.95-4.72 0.80-22.34-22.30 2000-2.94 4.63 1.48 23.34 26.50 The 5 % percentile being -55.13 - is calculated from the values in the last column in Tab. VII. This is the diversified Value at Risk. Tab. VIII shows the 5 % percentile calculated extra for each stock from converted simulations of stock prices.

VIII: 5 % Percentil Indicator CETV ČEZ ERSTE BANK VIG 5 % percentile -6.57-20.33-16.40-28.48 The total of -71.77 is calculated from the values in the table. This is the non-diversified Value at Risk. The Tab. VIII shows clearly what are the stocks of stocks in the nondiversified Value at Risk. Discussion The Value at Risk calculated under the historical simulation method, variance and covariance method and Monte Carlo method can be compared. This comparison is presented in Tab. IX. IX: Comparison of Calculated Value at Risk Method Value at Risk Historical simulation method (nondiversified Value at Risk) -68.82 Historical simulation method (diversified -49.47 Value at Risk) Variance and covariance method (nondiversified Value at Risk) -73.93 Variance and covariance method (diversified -70.20 Value at Risk) Monte Carlo method (non-diversified Value -71.77 at Risk) Monte Carlo method (diversified Value at -55.13 Risk) The Value at Risk expresses in this case the maximum potential loss calculated with a probability of 95 % over the following day. The probability is determined based on the 2011 period which the financial entity can have in its portfolio during unfavourable changes on the market. When calculating the non-diversified Value at Risk, no correlation between stocks is considered. When calculating the diversified Value at Risk, however, correlation between stocks is considered. Any losses related to the non-diversified Value at Risk are always higher within each of the three methods than the loss related to the diversified Value at Risk. Values calculated based on the three methods can be also compared with the results which can be achieved with a different value of some of the characteristics. Further investigation in this area can therefore continue in several directions. The characteristics applied for these methods count the confidence interval, hold period, historical period and

portfolio structure and volume. It is therefore possible to use the Value at Risk calculated with the confidence interval not 95 % but 99 % for the comparison. Similarly, also the values of other characteristics applied can be changed. When making the decision which of the three methods is to be applied, a variety of factors must be taken into account. Each method is specific and rests upon different presumptions. The methods differ from each other in the way in which input data is processed particularly in whether the parameters of the probability distribution of risk factors of the correlation, volatility and other types need to be estimated beforehand, whether the method may be applied to non-linear relations between the portfolio value and the risk factor level, i.e. to instruments with non-linear value progress such as to non-linear option portfolios, whether the method entails demanding calculations, whether specialised software is necessary for the application of the method and whether random numbers shall be made use of. Besides the absolute Value at Risk presented above, also the relative Value at Risk and marginal Value at Risk are used in practice. The relative Value at Risk is a risk with a lower performance in respect of a specific standard such as the market index. The marginal Value at Risk measures the degree by which the relative or absolute Value at Risk grows when assets are added to or removed from a portfolio. The Value at Risk model also enables back testing: the results of the Value at Risk model are compared with real future results. In back testing, the anticipated losses of a portfolio are compared with real losses of the portfolio, so that it can be determined how accurate the model is i.e. to what extent the results calculated based on the Value at Risk model reflect reality. Also stress testing can be implemented. Here, the Value at Risk model is tested on a given portfolio for a specific stress scenario of the stock market, commodity prices, foreign exchange rates and interest rates. The Value at Risk can be reduced by changing the structure of the original portfolio. Based on the risk factor volatilities calculated, the risk factor with the lowest volatility value can be strengthened and the risk factor with the highest volatility weakened by changing the weights in the portfolio but by maintaining the same overall portfolio value. This model is adopted to analyse portfolios, find the optimum portfolio value, evaluate financial derivatives including options, evaluate investments and for securing techniques. The Value at Risk model can be applied by banks, insurance companies, pension funds, investment funds and non-financial institutions such as businesses. In the banking sector this model is adopted also for determining the capital adequacy of banks. As for banks, the implications of the Value at Risk on the capital adequacy can be tracked, but some banks

currently have significant trade portfolio and only some of them directly adopt the Value at Risk model. The contribution of this paper is firstly the introduction of new knowledge of market risk measurements. The Value at Risk model is relatively popular and the topic of this paper is certainly very up-to-date. To define the space which this paper can take in area of the systematic market risk measurements and management, this paper applies several methods simultaneously: the historical simulation method, variance and covariance method and Monte Carlo method. Similarly as with some empirical investigations quoted above, the risk factors chosen are stocks. Conclusion The Value at Risk model belongs to models adopted for market risk measurements and management and the author is dealing with this model. Following its theoretical definition, the model is applied to real data. Specifically, three sub-methods are applied to the stocks from the Prague Stock Exchange. The values calculated define the maximum potential losses during the day following 30.12.2011 and calculated probability of 95 % which the financial entity can have in its portfolio in case of unfavourable changes on the market. Historical stock prices from 2011 period were used. The Value at Risk results differ depending on the method applied and on whether it is a diversified or non-diversified Value at Risk. Potential investors can estimate the future market development in the short term by using past data. They can predict the losses of a stock investment which makes orientation on the market easier. The Value at Risk model is a frequent topic for a number of empirical investigations, which goes hand in hand with the wide application possibilities of this model not only in the banking industry. This paper has been created within the research project IGA 28/2013 of Mendel University in Brno Stock markets sensitivity to information in period after the financial crisis. References AUSSENEGG, W., MIAZHYNSKAIA, T., 2006: Uncertainty in Value-at-Risk Estimates under Parametric and Non-parametric Modeling. Social Science Research Network. [cit. 2013-10-01]. Cited from http://papers.ssrn.com/sol3/papers.cfm?abstract_id=673662

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