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

Introduction

Data and simula-tion methodology

VaR models andestimation results

Estimation perfor-mance 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 CorinaSupervisor: Professor Moisa Altar

Bucharest 2001

Introduction




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 VaR

IntroductionIntroduction

Introduction




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.


Basic Parameters of a VaR ModelBasic Parameters of a VaR Model

Introduction




Conclusions

Appendix


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




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.


Limitations of VaR:

Introduction




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




Conclusions

Appendix

Series: LUSD Series: LDEM

Mean 0.001490Median 0.001108Maximum 0.059451Minimum -0.022983Std. Dev. 0.004495Skewness 2.917048Kurtosis 44.70935Jarque-Bera 57571.57Probability 0.000000

Mean 0.001297Median 0.000937Maximum 0.068797Minimum -0.026033Std. Dev. 0.007949Skewness 0.973316Kurtosis 9.842127Jarque-Bera 1642.524Probability 0.000000


Table 1.

Testing the normality assumption

Introduction




Conclusions

AppendixGraph 1a: QQ-plots for exchange rates returns for USD


Introduction




Conclusions

Appendix


Graph 1b: QQ-plots for exchange rates returns for DEM

Introduction




Conclusions

Appendix


Graph 2b: USD/ROL returns

Homoskedasticity assumption

Introduction




Conclusions

Appendix


Graph 2b: DEM/ROL returns

Introduction




Conclusions

Appendix

Stationarity assumption:


ADF Test Statistic -13.08055 1% Critical Value* -3.4414 5% Critical Value -2.8657 10% Critical Value -2.5690

*MacKinnon critical values for rejection of hypothesis of a unit root.

Augmented Dickey-Fuller Test EquationDependent Variable: D(LDEM)Method: Least SquaresDate: 06/30/01 Time: 19:24Sample(adjusted): 1/13/1998 12/29/2000Included observations: 774 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob.

LDEM(-1) -1.015504 0.077635 -13.08055 0.0000D(LDEM(-1)) 0.123205 0.069236 1.779493 0.0756D(LDEM(-2)) 0.038332 0.059546 0.643745 0.5199D(LDEM(-3)) 0.022510 0.048136 0.467624 0.6402D(LDEM(-4)) 0.040782 0.035954 1.134269 0.2570

C 0.001286 0.000301 4.277140 0.0000

R-squared 0.455810 Mean dependent var -3.40E-06Adjusted R-squared 0.452267 S.D. dependent var 0.010660S.E. of regression 0.007889 Akaike info criterion -6.838858Sum squared resid 0.047803 Schwarz criterion -6.802799Log likelihood 2652.638 F-statistic 128.6543Durbin-Watson stat 1.994196 Prob(F-statistic) 0.000000

Table 2a

Introduction




Conclusions

Appendix


ADF Test Statistic -12.14042 1% Critical Value* -3.4414 5% Critical Value -2.8657 10% Critical Value -2.5690

*MacKinnon critical values for rejection of hypothesis of a unit root.

Augmented Dickey-Fuller Test EquationDependent Variable: D(LUSD)Method: Least SquaresDate: 06/30/01 Time: 19:29Sample(adjusted): 1/13/1998 12/29/2000Included observations: 774 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob.

LUSD(-1) -0.763705 0.062906 -12.14042 0.0000D(LUSD(-1)) 0.096612 0.057071 1.692850 0.0909D(LUSD(-2)) 0.027387 0.050519 0.542119 0.5879D(LUSD(-3)) 0.009963 0.043113 0.231089 0.8173D(LUSD(-4)) -0.011338 0.035883 -0.315968 0.7521

C 0.001109 0.000179 6.197424 0.0000

R-squared 0.349533 Mean dependent var -6.95E-06Adjusted R-squared 0.345299 S.D. dependent var 0.005261S.E. of regression 0.004256 Akaike info criterion -8.073030Sum squared resid 0.013914 Schwarz criterion -8.036972Log likelihood 3130.263 F-statistic 82.53817Durbin-Watson stat 1.998644 Prob(F-statistic) 0.000000

Table 2b

Introduction




Conclusions

Appendix


Serial independence assumption:

Graph 3a. Autocorrelation coefficients for returns

(lags 1 to 36)

Introduction




Conclusions

Appendix Graph 3b. Autocorrelation coefficients for squared returns (lags 1 to 36)


Introduction




Conclusions

Appendix

Corina Lăpuşneanu - Value at Risk models and estimation resultsValue 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




Conclusions

Appendix


““Variance-covariance” approachVariance-covariance” approach

where Z() is the 100th percentile of the standard normal distribution

ZVAR

Equally Weighted Moving Average Approach

Exponentially Weighted Moving Average Approach

Introduction




Conclusions

Appendix


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.

ij

r

; 1

2

T

rrT

tt

T

rrrrT

tjjtiit

ij

1

Introduction




Conclusions

Appendix


Graph 4. VaR estimation using Equally Weighted Moving Average

Introduction




Conclusions

Appendix


Exponentially Weighted Moving Average Approach

T

tt

t rr1

211

T

tjjtiit

tij rrrr

1

11

The parameter is referred as “decay factor”.

Introduction




Conclusions

Appendix


Graph 5. VaR estimation using Exponentially Weighted

Moving Average

Introduction




Conclusions

Appendix


Historical Simulation

Graph 6. VaR estimation using Historical Simulation

Introduction




Conclusions

Appendix


GARCH modelsGARCH models In the linear ARCH(q) model, the conditional variance is postulated to be a linear function of the past q squared innovations:

21

1

22

t

q

iitit L

GARCH(p,q) model:

21

21

1

2

1

22

tt

p

jjtj

q

iitit

LL

Introduction




Conclusions

Appendix


GARCH (1,1) has the form:

21

21

2 ttt

where the parameters , , are estimated using quasi maximum- likelihood methods

Introduction




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:

211,, itiitiiiiiitii

tjjtiiijtij ,,,


Introduction




Conclusions

Appendix

Dependent Variable: LDEMMethod: ML - ARCHDate: 07/01/01 Time: 19:32Sample(adjusted): 1/06/1998 12/29/2000Included observations: 779 after adjusting endpointsConvergence achieved after 8 iterationsBollerslev-Wooldrige robust standard errors & covariance

Coefficient Std. Error z-Statistic Prob.

C 0.000963 0.000281 3.429349 0.0006DUMMY 0.066992 0.000852 78.62253 0.0000

D199 0.001102 0.000834 1.321575 0.1863

Variance Equation

C 4.73E-06 4.06E-06 1.163966 0.2444ARCH(1) 0.032776 0.023325 1.405153 0.1600

GARCH(1) 0.882684 0.088609 9.961515 0.0000

R-squared 0.095860 Mean dependent var 0.001297Adjusted R-squared 0.090012 S.D. dependent var 0.007949S.E. of regression 0.007582 Akaike info criterion -6.933546Sum squared resid 0.044442 Schwarz criterion -6.897669Log likelihood 2706.616 F-statistic 16.39123Durbin-Watson stat 1.885839 Prob(F-statistic) 0.000000

Table 3.1. Estimation results with GARCH(1,1)


Introduction




Conclusions

Appendix


Dependent Variable: LUSDMethod: ML - ARCHDate: 07/01/01 Time: 19:41Sample(adjusted): 1/27/1998 12/29/2000Included observations: 764 after adjusting endpointsConvergence achieved after 14 iterationsBollerslev-Wooldrige robust standard errors & covariance


C 0.000445 8.22E-05 5.416127 0.0000LUSD(-1) 0.348223 0.038563 9.029914 0.0000

DALT 0.014344 0.002912 4.925951 0.0000LUSD(-5) 0.145513 0.039341 3.698805 0.0002LUSD(-10) 0.091662 0.039170 2.340098 0.0193LUSD(-15) 0.060467 0.031869 1.897349 0.0578

Variance Equation

C 3.19E-08 1.51E-08 2.109582 0.0349ARCH(1) 0.197947 0.053888 3.673306 0.0002

GARCH(1) 0.780169 0.054744 14.25124 0.0000D98 4.06E-06 3.20E-06 1.267954 0.2048D199 1.97E-06 1.28E-06 1.541100 0.1233

DUMMY 0.000662 0.000293 2.260207 0.0238


Table 3.2. Estimation results with GARCH(1,1)

Introduction




Conclusions

Appendix


Graph 7. VaR estimation results with GARCH(1,1)

Introduction




Conclusions

Appendix


Dependent Variable: LUSDMethod: ML - ARCHDate: 07/01/01 Time: 19:51Sample(adjusted): 1/27/1998 12/29/2000Included observations: 764 after adjusting endpointsConvergence achieved after 13 iterationsBollerslev-Wooldrige robust standard errors & covariance


C 0.000439 8.03E-05 5.467788 0.0000LUSD(-1) 0.340662 0.037487 9.087487 0.0000

DALT 0.012951 0.003576 3.621597 0.0003LUSD(-5) 0.141505 0.037448 3.778702 0.0002LUSD(-10) 0.086646 0.038931 2.225640 0.0260LUSD(-15) 0.070710 0.031520 2.243316 0.0249

Variance Equation

C 1.86E-08 1.20E-08 1.549059 0.1214ARCH(1) 0.253414 0.077004 3.290925 0.0010ARCH(2) -0.089373 0.076501 -1.168267 0.2427

GARCH(1) 0.822235 0.048018 17.12342 0.0000D98 3.07E-06 2.42E-06 1.269957 0.2041D199 1.52E-06 1.10E-06 1.373354 0.1696

DUMMY 0.000445 0.000304 1.464928 0.1429


Table 4. Estimation results with GARCHFIT

Introduction




Conclusions

Appendix


Graph 8. . VaR estimation results with GARCHFIT

Introduction




Conclusions

Appendix


Orthogonal GARCH

X = data matrixX’X = correlation matrixW = matrix of eigenvectors of X’X

The mth principal component of the system can be written:

WXWX '

kkmmmm xwxwxwp ........2211 Principal component representation can be write:

mimiii ppy *1

*1 .......

where iijij w *

Introduction




Conclusions

Appendix


The time-varying covariance matrix (Vt) is approximated by:

'AADV tt 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.

*ijA

mpVpVdiagD ,......1

Introduction




Conclusions

Appendix


Graph 9. VaR estimation results with Orthogonal GARCH

Introduction




Conclusions

Appendix


Estimating the pdf of portfolio returns

- Gaussian

- Epanechnikov , pentru - Biweight , pentru

where

n

i n

i

nn h

xxK

nhxf

1

1

2exp

2

1 2t

203.015.05 t

2201.025.03 t

52 t

5t

h

xxt i

Kernel Estimation

Introduction




Conclusions

Appendix


Estimating the distribution of percentile or Estimating the distribution of percentile or order statisticorder statistic

jnjj xFxFxf

jnj

nxg

1

!!1

! 1

Introduction




Conclusions

Appendix


Graph 10a. VaR estimation results with Gaussian kernel

Introduction




Conclusions

Appendix


Graph 10b. VaR estimation results with Epanechnikov kernel

Introduction




Conclusions

Appendix


Graph 10c. VaR estimation results with biweight kernel

Introduction




Conclusions

Appendix


Structured Monte Carlo

dzSdtSdS ttttt If the variables are uncorrelated, the randomization can be performed independently for each variable:

ttSS tjjjtjtj ,1,,

Introduction




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:

2

2/1212

11

1

where is the correlation coefficient between the variables .


Introduction




Conclusions

AppendixGraph 11. VaR estimation results using Monte Carlo Simulation


Introduction




Conclusions

Appendix


Generalized Pareto Distribution:

0 /exp1

0 /111

,

y

yyG

= “shape parameter” or “tail index” = “scaling parameter”

yGyFu ,

Extreme value method

Introduction




Conclusions

Appendix

Tail estimator:

uux

n

NxF u

for x ,ˆ

ˆ11ˆˆ/1

F(u)q where, 11ˆ

ˆVAR

ˆ^

qN

nu

u

q


Introduction




Conclusions

Appendix


Graph 12. VaR estimation results using Extreme Value Method

Introduction




Conclusions

Appendix

Corina Lăpuşneanu - Estimation performance analysisEstimation performance analysis

Estimation performance analysisEstimation performance analysis

Measures of Relative Size and Variability

Measures of Accuracy

Efficiency measures

Introduction




Conclusions

Appendix

Measures of Relative Size and Variability

Mean Relative Bias

Root Mean Squared Relative Bias

Variability


Introduction




Conclusions

Appendix

Mean Relative Bias

T

t t

titi

VAR

VARVAR

TMRB

1

1

N

iitt VAR

NVARwhere

1

1


Introduction




Conclusions

Appendix


Graph 13. Mean Relative Bias

Introduction




Conclusions

Appendix


Root Mean Squared Relative Bias

T

t t

titi

VAR

VARVAR

TRMSRB

1

21

2

1

1

T

t iti VART

The variability of a VaR estimate is computed as follows:

Introduction




Conclusions

Appendix


Graph 14. Root Mean Squared Relative Bias

Introduction




Conclusions

Appendix


Variability

Graph 15. Variability

Introduction




Conclusions

Appendix


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




Conclusions

Appendix


The Binary Loss Function

if 0

if 1

tt

ttt VARP

VARPL

Graph 16. Binary Loss Function

Introduction




Conclusions

Appendix


tt

ttttt

VARPif

VARPifVARPL

0

1 2

Graph 17. Quadratic Loss Function

Quadratic Loss Function

Introduction




Conclusions

Appendix


Multiple to Obtain Coverage

T

t itit

ititi VARXPif

VARXPifF

1 0

1

1 where TFi

= the confidence level

Introduction




Conclusions

Appendix


Graph 18. Multiple to Obtain Coverage

Introduction




Conclusions

Appendix


Average Uncovered Losses to VaR Ratio

M

mmi X

MAM

1

1

ittti

tm VARPif

VAR

PX

where

,

M is the number of excesses

Introduction




Conclusions

Appendix Graph 19. Average Uncovered Loss to VaR Ratio


Introduction




Conclusions

Appendix


Maximum Loss to VaR Ratio

Graph 20. Maximum Loss to VaR Ratio

Introduction




Conclusions

Appendix


Mean Relative Scaled Bias

Correlation

Efficiency measures

Introduction




Conclusions

Appendix


Mean Relative Scaled Bias

Graph 21. Mean Related Scaled Bias

Introduction




Conclusions

Appendix


Correlation

Graph 22. Correlation

Introduction




Conclusions

Appendix

Corina Lăpuşneanu - ConclusionsConclusions

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.

ConclusionsConclusions

Introduction




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.


Introduction




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.


Introduction




Conclusions

Appendix


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




Conclusions

Appendix

Corina Lăpuşneanu - AppendixAppendix

AppendixAppendix

MRB100 days 175 days 250 days

EqWMA 2.03E-01 2.61E-01 4.07E-01ExpWMA92 9.23E-02 2.61E-01 1.42E-01ExpWMA95 1.15E-01 1.57E-01 1.79E-01ExpWMA98 7.19E-02 1.85E-01 2.72E-01HIST -3.84E-02 -2.31E-02 -2.99E-03MC 2.12E-01 6.63E-02 4.88E-03KGAUSS -1.66E-01 -1.78E-01 -3.54E-01KEPAN -6.49E-02 -2.37E-01 -2.16E-01KBIW -9.54E-02 -2.02E-01 -1.67E-01EV15 -4.75E-01 -4.78E-01 -4.85E-01EV10 -2.27E-01 -3.02E-01 -2.81E-01EV5 4.22E-02 5.45E-02 7.37E-02GCH(1,1) 1.08E-01 1.45E-01 1.78E-01GCH(2,1) 1.15E-01 1.48E-01 1.78E-01OGCH 1.07E-01 1.43E-01 7.05E-02

Introduction




Conclusions

Appendix


RMSRB100 days 175 days 250 days

EqWMA 4.72E-01 4.91E-01 6.13E-01ExpWMA92 3.40E-01 4.91E-01 4.21E-01ExpWMA95 3.09E-01 3.89E-01 4.06E-01ExpWMA98 2.94E-01 3.32E-01 4.22E-01HIST 2.99E-01 2.69E-01 2.77E-01MC 8.48E-01 6.60E-01 5.64E-01KGAUSS 7.17E-01 7.10E-01 6.31E-01KEPAN 6.51E-01 5.97E-01 5.29E-01KBIW 7.41E-01 6.56E-01 6.04E-01EV15 5.10E-01 5.19E-01 5.24E-01EV10 3.36E-01 3.85E-01 3.66E-01EV5 3.11E-01 2.81E-01 2.94E-01GCH(1,1) 4.34E-01 4.09E-01 3.58E-01GCH(2,1) 4.43E-01 4.19E-01 3.51E-01OGCH 6.44E-01 6.68E-01 5.87E-01

Introduction




Conclusions

Appendix


Variability100 days 175 days 250 days


Introduction




Conclusions

Appendix


Binary Loss Function100 days 175 days 250 days


Introduction




Conclusions

Appendix


Quadratic Loss Function100 days 175 days 250 days


Introduction




Conclusions

Appendix


Multiple to Obtain Coverage100 days 175 days 250 days

EqWMA 9.42E-01 9.03E-01 8.07E-01ExpWMA92 9.40E-01 9.03E-01 9.98E-01ExpWMA95 9.19E-01 9.24E-01 9.71E-01ExpWMA98 1.00E+00 9.52E-01 9.58E-01HIST 1.03E+00 1.00E+00 1.11E+00MC 1.41E+00 1.59E+00 1.71E+00KGAUSS 2.14E+00 1.40E+00 1.42E+00KEPAN 1.02E+00 1.04E+00 8.88E-01KBIW 8.90E-01 8.51E-01 7.99E-01EV15 1.21E+00 1.20E+00 1.36E+00EV10 8.88E-01 1.00E+00 9.73E-01EV5 7.16E-01 7.16E-01 7.11E-01GCH(1,1) 1.01E+00 8.71E-01 8.51E-01GCH(2,1) 1.01E+00 8.86E-01 8.56E-01OGCH 1.03E+00 9.60E-01 9.91E-01

Introduction




Conclusions

Appendix


Average Uncovered Losses100 days 175 days 250 days

EqWMA 1.41E+00 1.50E+00 1.54E+00ExpWMA92 1.23E+00 1.21E+00 1.19E+00ExpWMA95 1.22E+00 1.25E+00 1.20E+00ExpWMA98 1.35E+00 1.30E+00 1.22E+00HIST 1.71E+00 1.90E+00 1.94E+00MC 2.23E+00 2.23E+00 2.26E+00KGAUSS 2.14E+00 2.00E+00 2.35E+00KEPAN 2.05E+00 2.42E+00 2.16E+00KBIW 2.02E+00 2.29E+00 2.42E+00EV15 1.78E+00 1.57E+00 1.63E+00EV10 1.57E+00 1.78E+00 1.47E+00EV5 1.32E+00 1.35E+00 1.84E+00GCH(1,1) 1.48E+00 1.60E+00 1.42E+00GCH(2,1) 1.48E+00 1.65E+00 1.41E+00OGCH 1.48E+00 1.59E+00 1.42E+00

Introduction




Conclusions

Appendix


Maximum Loss100 days 175 days 250 days

EqWMA 2.67E+00 2.86E+00 2.68E+00ExpWMA92 1.50E+00 1.58E+00 1.65E+00ExpWMA95 1.72E+00 1.60E+00 1.62E+00ExpWMA98 2.36E+00 2.10E+00 1.96E+00HIST 4.53E+00 5.32E+00 6.53E+00MC 9.10E+00 9.10E+00 9.10E+00KGAUSS 9.11E+00 9.47E+00 8.33E+00KEPAN 8.41E+00 9.86E+00 9.25E+00KBIW 8.35E+00 9.64E+00 8.49E+00EV15 5.07E+00 4.56E+00 5.53E+00EV10 3.98E+00 7.12E+00 3.81E+00EV5 2.08E+00 1.83E+00 2.90E+00GCH(1,1) 3.44E+00 3.25E+00 3.53E+00GCH(2,1) 3.44E+00 3.26E+00 3.49E+00OGCH 3.44E+00 3.25E+00 3.53E+00

Introduction




Conclusions

Appendix


MRSB100 days 175 days 250 days

EqWMA 8.13E-02 1.50E-01 1.42E-01ExpWMA92 -2.27E-02 1.50E-01 1.44E-01ExpWMA95 -2.33E-02 7.79E-02 1.50E-01ExpWMA98 2.26E-02 1.39E-01 2.24E-01HIST -5.97E-02 -1.52E-02 1.05E-01MC 5.89E-01 6.72E-01 7.00E-01KGAUSS 6.13E-01 1.35E-01 -8.51E-02KEPAN -1.33E-01 -2.15E-01 -3.04E-01KBIW -2.64E-01 -3.28E-01 -3.35E-01EV15 -3.98E-01 -3.71E-01 -2.97E-01EV10 -3.48E-01 -2.96E-01 -2.98E-01EV5 -2.90E-01 -2.39E-01 -2.33E-01GCH(1,1) 7.05E-02 5.96E-03 7.80E-03GCH(2,1) 7.07E-02 2.68E-02 1.24E-02OGCH 9.23E-02 1.09E-01 6.71E-02

Introduction




Conclusions

Appendix


Correlation100 days 175 days 250 days


Documents

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