Doctoral School of Finance and Banking

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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
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Corina Lapusneanu - 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
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Corina Lapusneanu - 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 managers degree of risk aversion.

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Corina Lapusneanu - 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.

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Corina Lapusneanu - 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.

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Corina Lapusneanu - 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
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Corina Lapusneanu - Data and simulation
methodology
Testing the normality assumption
Table 1.
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Corina Lapusneanu - Data and simulation
methodology
Graph 1a QQ-plots for exchange rates returns for
USD
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Corina Lapusneanu - Data and simulation
methodology
Graph 1b QQ-plots for exchange rates returns for
DEM
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Corina Lapusneanu - Data and simulation
methodology
Homoskedasticity assumption
Graph 2b USD/ROL returns
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Corina Lapusneanu - Data and simulation
methodology
Graph 2b DEM/ROL returns
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Corina Lapusneanu - Data and simulation
methodology
Stationarity assumption
Table 2a
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Corina Lapusneanu - Data and simulation
methodology
Table 2b
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Corina Lapusneanu - Data and simulation
methodology
Serial independence assumption
Graph 3a. Autocorrelation coefficients for
returns (lags 1 to 36)
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Corina Lapusneanu - Data and simulation
methodology
Graph 3b. Autocorrelation coefficients for
squared returns (lags 1 to 36)
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Corina Lapusneanu - 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
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Corina Lapusneanu - 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
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Corina Lapusneanu - 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.
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Corina Lapusneanu - Value at Risk models and
estimation results
Graph 4. VaR estimation using Equally Weighted
Moving Average
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Corina Lapusneanu - Value at Risk models and
estimation results
Exponentially Weighted Moving Average Approach
The parameter ? is referred as decay factor.
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Corina Lapusneanu - Value at Risk models and
estimation results
Graph 5. VaR estimation using Exponentially
Weighted Moving Average
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Corina Lapusneanu - Value at Risk models and
estimation results
Historical Simulation
Graph 6. VaR estimation using Historical
Simulation
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Corina Lapusneanu - 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
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Corina Lapusneanu - Value at Risk models and
estimation results
GARCH (1,1) has the form
where the parameters ?, ?, ? are estimated using
quasi maximum- likelihood methods
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Corina Lapusneanu - 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
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Corina Lapusneanu - Value at Risk models and
estimation results
Table 3.1. Estimation results with GARCH(1,1)
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Corina Lapusneanu - Value at Risk models and
estimation results
Table 3.2. Estimation results with GARCH(1,1)
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Corina Lapusneanu - Value at Risk models and
estimation results
Graph 7. VaR estimation results with GARCH(1,1)
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Corina Lapusneanu - Value at Risk models and
estimation results
Table 4. Estimation results with GARCHFIT
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Corina Lapusneanu - Value at Risk models and
estimation results
Graph 8. . VaR estimation results with GARCHFIT
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Corina Lapusneanu - Value at Risk models and
estimation results
Orthogonal GARCH
X data matrix XX correlation matrix W
matrix of eigenvectors of XX The mth
principal component of the system can be written
Principal component representation can be write
where
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Corina Lapusneanu - 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.
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Corina Lapusneanu - Value at Risk models and
estimation results
Graph 9. VaR estimation results with Orthogonal
GARCH
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Corina Lapusneanu - Value at Risk models and
estimation results
Kernel Estimation
Estimating the pdf of portfolio returns -
Gaussian -
Epanechnikov , pentru
- Biweight ,
pentru where
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Corina Lapusneanu - Value at Risk models and
estimation results
Estimating the distribution of
percentile or order statistic
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Corina Lapusneanu - Value at Risk models and
estimation results
Graph 10a. VaR estimation results with Gaussian
kernel
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Corina Lapusneanu - Value at Risk models and
estimation results
Graph 10b. VaR estimation results with
Epanechnikov kernel
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Corina Lapusneanu - Value at Risk models and
estimation results
Graph 10c. VaR estimation results with biweight
kernel
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Corina Lapusneanu - Value at Risk models and
estimation results
Structured Monte Carlo
If the variables are uncorrelated,
the randomization can be performed independently
for each variable
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Corina Lapusneanu - 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 ?.
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Corina Lapusneanu - Value at Risk models and
estimation results
Graph 11. VaR estimation results using Monte
Carlo Simulation
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Corina Lapusneanu - Value at Risk models and
estimation results
Extreme value method
Generalized Pareto Distribution
? shape parameter or tail index ?
scaling parameter
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Corina Lapusneanu - Value at Risk models and
estimation results
Tail estimator
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Corina Lapusneanu - Value at Risk models and
estimation results
Graph 12. VaR estimation results using Extreme
Value Method
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Corina Lapusneanu - Estimation performance
analysis
Estimation performance analysis
Measures of Relative Size and Variability
Measures of Accuracy
Efficiency measures
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Corina Lapusneanu - Estimation performance
analysis
Measures of Relative Size and Variability
Mean Relative Bias
Root Mean Squared Relative Bias
Variability
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Corina Lapusneanu - Estimation performance
analysis
Mean Relative Bias
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Corina Lapusneanu - Estimation performance
analysis
Graph 13. Mean Relative Bias
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Corina Lapusneanu - Estimation performance
analysis
Root Mean Squared Relative Bias
The variability of a VaR estimate is
computed as follows
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Corina Lapusneanu - Estimation performance
analysis
Graph 14. Root Mean Squared Relative Bias
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Corina Lapusneanu - Estimation performance
analysis
Variability
Graph 15. Variability
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Corina Lapusneanu - 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
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Corina Lapusneanu - Estimation performance
analysis
The Binary Loss Function
Graph 16. Binary Loss Function
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Corina Lapusneanu - Estimation performance
analysis
Quadratic Loss Function
Graph 17. Quadratic Loss Function
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Corina Lapusneanu - Estimation performance
analysis
Multiple to Obtain Coverage
? the confidence level
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Corina Lapusneanu - Estimation performance
analysis
Graph 18. Multiple to Obtain Coverage
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Corina Lapusneanu - Estimation performance
analysis
Average Uncovered Losses to VaR Ratio
M is the number of excesses
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Corina Lapusneanu - Estimation performance
analysis
Graph 19. Average Uncovered Loss to VaR Ratio
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Corina Lapusneanu - Estimation performance
analysis
Maximum Loss to VaR Ratio
Graph 20. Maximum Loss to VaR Ratio
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Corina Lapusneanu - Estimation performance
analysis
Efficiency measures
Mean Relative Scaled Bias
Correlation
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Corina Lapusneanu - Estimation performance
analysis
Mean Relative Scaled Bias
Graph 21. Mean Related Scaled Bias
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Corina Lapusneanu - Estimation performance
analysis
Correlation
Graph 22. Correlation
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Corina Lapusneanu - Conclusions
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.
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Corina Lapusneanu - Conclusions
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.
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Corina Lapusneanu - Conclusions
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.
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Corina Lapusneanu - 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.
Its preferred a model with a low threshold (5).
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Corina Lapusneanu - Appendix
Appendix
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Corina Lapusneanu - Appendix
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Corina Lapusneanu - Appendix
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Corina Lapusneanu - Appendix
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Corina Lapusneanu - Appendix
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Corina Lapusneanu - Appendix
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Corina Lapusneanu - Appendix
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Corina Lapusneanu - Appendix
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Corina Lapusneanu - Appendix
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Corina Lapusneanu - Appendix