Doctoral School of Finance and Banking Academy of Economic Studies Bucharest

1. 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

2. Corina Lăpuşneanu - Introduction • Introduction 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. Basic Parameters of a VaR Model Advantages of VAR Limitations of VaR

3. Corina Lăpuşneanu - Introduction Basic Parameters of a VaR Model • 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.

4. Corina Lăpuşneanu - Introduction Advantages of VAR • 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.

5. Corina Lăpuşneanu - Introduction Limitations of VaR: • 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.

6. Corina Lăpuşneanu - Data and simulation methodology Data and simulation methodology • 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

7. Corina Lăpuşneanu - Data and simulation methodology Testing the normality assumption • Table 1.

8. Corina Lăpuşneanu - Data and simulation methodology Graph 1a: QQ-plots for exchange rates returns for USD

9. Corina Lăpuşneanu - Data and simulation methodology Graph 1b: QQ-plots for exchange rates returns for DEM

10. Corina Lăpuşneanu - Data and simulation methodology Homoskedasticity assumption Graph 2b: USD/ROL returns

11. Corina Lăpuşneanu - Data and simulation methodology Graph 2b: DEM/ROL returns

12. Corina Lăpuşneanu - Data and simulation methodology • Stationarity assumption: Table 2a

13. Corina Lăpuşneanu - Data and simulation methodology Table 2b

14. Corina Lăpuşneanu - Data and simulation methodology • Serial independence assumption: Graph 3a. Autocorrelation coefficients for returns (lags 1 to 36)

15. Corina Lăpuşneanu - Data and simulation methodology Graph 3b. Autocorrelation coefficients for squared returns (lags 1 to 36)

16. 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

17. 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

18. 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, • rt is the return of an asset on day t, • is the mean return of that asset.

19. Corina Lăpuşneanu - Value at Risk models and estimation results Graph 4. VaR estimation using Equally Weighted Moving Average

20. Corina Lăpuşneanu - Value at Risk models and estimation results Exponentially Weighted Moving Average Approach The parameter  is referred as “decay factor”.

21. Corina Lăpuşneanu - Value at Risk models and estimation results Graph 5. VaR estimation using Exponentially Weighted Moving Average

22. Corina Lăpuşneanu - Value at Risk models and estimation results Historical Simulation Graph 6. VaR estimation using Historical Simulation

23. 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:

24. 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

25. Corina Lăpuşneanu - Value at Risk models and estimation results • 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:

26. Corina Lăpuşneanu - Value at Risk models and estimation results Table 3.1. Estimation results with GARCH(1,1)

27. Corina Lăpuşneanu - Value at Risk models and estimation results Table 3.2. Estimation results with GARCH(1,1)

28. Corina Lăpuşneanu - Value at Risk models and estimation results Graph 7. VaR estimation results with GARCH(1,1)

29. Corina Lăpuşneanu - Value at Risk models and estimation results Table 4. Estimation results with GARCHFIT

30. Corina Lăpuşneanu - Value at Risk models and estimation results Graph 8. . VaR estimation results with GARCHFIT

31. 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

32. Corina Lăpuşneanu - Value at Risk models and estimation results The time-varying covariance matrix (Vt) is approximated by: • where • is the matrix of normalised factor weights • is the diagonal matrix of variances of principal components • The diagonal matrix Dt of variances of principal components is estimated using a GARCH model.

33. Corina Lăpuşneanu - Value at Risk models and estimation results Graph 9. VaR estimation results with Orthogonal GARCH

34. Corina Lăpuşneanu - Value at Risk models and estimation results Kernel Estimation • Estimating the pdf of portfolio returns • - Gaussian • - Epanechnikov , pentru • - Biweight , pentru • where

35. Corina Lăpuşneanu - Value at Risk models and estimation results Estimating the distribution of percentile or order statistic

36. Corina Lăpuşneanu - Value at Risk models and estimation results Graph 10a. VaR estimation results with Gaussian kernel

37. Corina Lăpuşneanu - Value at Risk models and estimation results Graph 10b. VaR estimation results with Epanechnikov kernel

38. Corina Lăpuşneanu - Value at Risk models and estimation results Graph 10c. VaR estimation results with biweight kernel

39. 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:

40. Corina Lăpuşneanu - Value at Risk models and estimation results • 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 .

41. Corina Lăpuşneanu - Value at Risk models and estimation results Graph 11. VaR estimation results using Monte Carlo Simulation

42. Corina Lăpuşneanu - Value at Risk models and estimation results Extreme value method Generalized Pareto Distribution:  = “shape parameter” or “tail index”  = “scaling parameter”

43. Corina Lăpuşneanu - Value at Risk models and estimation results Tail estimator:

44. Corina Lăpuşneanu - Value at Risk models and estimation results Graph 12. VaR estimation results using Extreme Value Method

45. Corina Lăpuşneanu - Estimation performance analysis Estimation performance analysis Measures of Relative Size and Variability Measures of Accuracy • Efficiency measures

46. Corina Lăpuşneanu - Estimation performance analysis Measures of Relative Size and Variability Mean Relative Bias Root Mean Squared Relative Bias Variability

47. Corina Lăpuşneanu - Estimation performance analysis Mean Relative Bias

48. Corina Lăpuşneanu - Estimation performance analysis Graph 13. Mean Relative Bias

49. Corina Lăpuşneanu - Estimation performance analysis Root Mean Squared Relative Bias The variability of a VaR estimate is computed as follows:

50. Corina Lăpuşneanu - Estimation performance analysis Graph 14. Root Mean Squared Relative Bias