Introduction Data and simula- tion methodology VaR models and estimation results Estimation perfor- mance analysis Conclusions Appendix Doctoral School.

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Doctoral School of Finance and Banking Academy of Economic Studies Bucharest Testing and ComparingValue at Risk Models – an Approach to Measuring Foreign Exchange Exposure -dissertation paper- MSc student: Lapusneanu Corina Supervisor: Professor Moisa Altar Bucharest 2001

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix VaR is a method of assessing risk which measures the worst expected loss over a given time interval under normal market conditions at a given confidence level. Corina Lăpuşneanu - Introduction Basic Parameters of a VaR Model Advantages of VAR Limitations of VaRIntroduction

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix  For internal purposes the appropriate holding period corresponds to the optimal hedging or liquidation period.  These can be determined from traders knowledge or an economic model  The choice of significance level should reflect the manager’s degree of risk aversion. Corina Lăpuşneanu - Introduction Basic Parameters of a VaR Model

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Corina Lăpuşneanu - Introduction  VaR can be used to compare the market risks of all types of activities in the firm,  it provides a single measure that is easily understood by senior management,  it can be extended to other types of risk, notably credit risk and operational risk,  it takes into account the correlations and cross- hedging between various asset categories or risk factors. Advantages of VAR

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix  it only captures short-term risks in normal market circumstances,  VaR measures may be very imprecise, because they depend on many assumption about model parameters that may be very difficult to support,  it assumes that the portfolio is not managed over the holding period,  the almost all VaR estimates are based on historical data and to the extent that the past may not be a good predictor of the future, VaR measure may underpredict or overpredict risk. Corina Lăpuşneanu - Introduction Limitations of VaR:

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Statistical analysis of the financial series of exchange rates against ROL (first differences in logs): Testing the normality assumption Homoskedasticity assumption Stationarity assumption Serial independence assumption Corina Lăpuşneanu - Data and simulation methodology Data and simulation methodology

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Corina Lăpuşneanu - Data and simulation methodology Table 1. Testing the normality assumption

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Graph 1a: QQ-plots for exchange rates returns for USD Corina Lăpuşneanu - Data and simulation methodology

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Corina Lăpuşneanu - Data and simulation methodology Graph 1b: QQ-plots for exchange rates returns for DEM

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Corina Lăpuşneanu - Data and simulation methodology Graph 2b: USD/ROL returns Homoskedasticity assumption

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Corina Lăpuşneanu - Data and simulation methodology Graph 2b: DEM/ROL returns

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Stationarity assumption: Corina Lăpuşneanu - Data and simulation methodology Table 2a

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Corina Lăpuşneanu - Data and simulation methodology Table 2b

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Corina Lăpuşneanu - Data and simulation methodology Serial independence assumption: Graph 3a. Autocorrelation coefficients for returns (lags 1 to 36)

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Graph 3b. Autocorrelation coefficients for squared returns (lags 1 to 36) Corina Lăpuşneanu - Data and simulation methodology

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Value at Risk models and estimation results “Variance-covariance” approach Historical Simulation “GARCH models Kernel Estimation Structured Monte Carlo Extreme value method

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results “Variance-covariance” approach where Z(  ) is the 100  th percentile of the standard normal distribution Equally Weighted Moving Average Approach Exponentially Weighted Moving Average Approach

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Equally Weighted Moving Average Approach where represents the estimated standard deviation, represents the estimated covariance, T is the observation period, r t is the return of an asset on day t, is the mean return of that asset.

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Graph 4. VaR estimation using Equally Weighted Moving Average

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Exponentially Weighted Moving Average Approach The parameter is referred as “decay factor”.

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Graph 5. VaR estimation using Exponentially Weighted Moving Average

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Historical Simulation Graph 6. VaR estimation using Historical Simulation

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results GARCH models In the linear ARCH(q) model, the conditional variance is postulated to be a linear function of the past q squared innovations: GARCH(p,q) model:

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results GARCH (1,1) has the form: where the parameters , ,  are estimated using quasi maximum- likelihood methods

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix The constant correlation GARCH model estimates each diagonal element of the variance- covariance matrix using a univariate GARCH (1,1) and the risk factor correlation is time invariant: Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Table 3.1. Estimation results with GARCH(1,1) Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Table 3.2. Estimation results with GARCH(1,1)

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Graph 7. VaR estimation results with GARCH(1,1)

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Table 4. Estimation results with GARCHFIT

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Graph 8.. VaR estimation results with GARCHFIT

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Orthogonal GARCH X = data matrix X ’ X = correlation matrix W = matrix of eigenvectors of X ’ X The mth principal component of the system can be written: Principal component representation can be write: where

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results The time-varying covariance matrix (V t ) is approximated by: where is the matrix of normalised factor weights is the diagonal matrix of variances of principal components The diagonal matrix D t of variances of principal components is estimated using a GARCH model.

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Graph 9. VaR estimation results with Orthogonal GARCH

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Estimating the pdf of portfolio returns - Gaussian - Epanechnikov, pentru - Biweight, pentru where Kernel Estimation

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Estimating the distribution of percentile or order statistic Estimating the distribution of percentile or order statistic

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Graph 10a. VaR estimation results with Gaussian kernel

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Graph 10b. VaR estimation results with Epanechnikov kernel

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Graph 10c. VaR estimation results with biweight kernel

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Structured Monte Carlo If the variables are uncorrelated, the randomization can be performed independently for each variable:

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix But, generally, variables are correlated. To account or this correlation, we start with a set of independent variables , which are then transformed into the , using Cholesky decomposition. In a two- variable setting, we construct: where  is the correlation coefficient between the variables . Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Graph 11. VaR estimation results using Monte Carlo Simulation Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Generalized Pareto Distribution:  = “shape parameter” or “tail index”  = “scaling parameter” Extreme value method

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Tail estimator: Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Value at Risk models and estimation results Corina Lăpuşneanu - Value at Risk models and estimation results Graph 12. VaR estimation results using Extreme Value Method

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis Estimation performance analysis Measures of Relative Size and Variability Measures of Accuracy Efficiency measures

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Measures of Relative Size and Variability Mean Relative Bias Root Mean Squared Relative Bias Variability Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Mean Relative Bias Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis Graph 13. Mean Relative Bias

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis Root Mean Squared Relative Bias The variability of a VaR estimate is computed as follows:

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis Graph 14. Root Mean Squared Relative Bias

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis Variability Graph 15. Variability

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis Measures of Accuracy Binary Loss Function Quadratic Loss Function Multiple to Obtain Coverage Average Uncovered Losses to VaR Ratio Maximum Loss to VaR Ratio

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis The Binary Loss Function Graph 16. Binary Loss Function

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis Graph 17. Quadratic Loss Function Quadratic Loss Function

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis Multiple to Obtain Coverage  = the confidence level

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis Graph 18. Multiple to Obtain Coverage

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis Average Uncovered Losses to VaR Ratio M is the number of excesses

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Graph 19. Average Uncovered Loss to VaR Ratio Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis Maximum Loss to VaR Ratio Graph 20. Maximum Loss to VaR Ratio

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis Mean Relative Scaled Bias Correlation Efficiency measures

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis Mean Relative Scaled Bias Graph 21. Mean Related Scaled Bias

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Estimation performance analysis Corina Lăpuşneanu - Estimation performance analysis Correlation Graph 22. Correlation

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Conclusions Corina Lăpuşneanu - Conclusions Equally Weighted Moving Average is a conservative risk measure, which produce the second great average estimation of risk, with a medium variability, good accuracy and a medium efficiency. As the window length is increased (Appendix D), the conservatism and variability will increase. Exponentially Weighted Moving Average tends to produce estimates over all model average, a low variability, good accuracy and medium efficiency. This method is more efficient when calibrated on smaller data window lengths. Conclusions

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix GARCH models produce estimates over all model average, a medium variability. good accuracy and efficiency. Historical Simulation tends to produce estimates below all model average, a low variability, medium accuracy and efficiency. It is more efficient when calibrated on smaller data window. Conclusions Corina Lăpuşneanu - Conclusions

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Structured Monte Carlo Simulation: presents the highest level of conservatism, high variability, the least accurate estimates, and low efficiency. Kernel density estimation: produce estimates below all model average, a high variability, a good accuracy and efficiency except the Gaussian kernel that has a low accuracy and efficiency. Conclusions Corina Lăpuşneanu - Conclusions

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Conclusions Corina Lăpuşneanu - Conclusions Extreme Value: produce the least conservatives VaR estimates except 5% threshold with produce estimates over all model average, low variability, good accuracy except 15% threshold, the most efficient models after that was scaling with the multiple to obtain coverage. It’s preferred a model with a low threshold (5%).

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Appendix Corina Lăpuşneanu - AppendixAppendix

Introduction Data and simulation methodology VaR models and estimation results Estimation performance analysis Conclusions Appendix Appendix Corina Lăpuşneanu - Appendix

Introduction Data and simula- tion methodology VaR models and estimation results Estimation perfor- mance analysis Conclusions Appendix Doctoral School.

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Introduction Data and simula- tion methodology VaR models and estimation results Estimation perfor- mance analysis Conclusions Appendix Doctoral School.

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