1. Measuring market risk:
Academy of Economic Studies Bucharest Doctoral School of Finance and Banking
Measuring market risk:
a copula and extreme value approach
Supervisor
Professor Moisă Altăr
M. Sc. Student
Alexandru Stângă
July 2007

2. Estimation and results Conclusion
Contents
Goal
Literature review
Methodology
Estimation and results
Conclusion

3. Goal
Measuring the risk of a portfolio composed of 5 Romanian stocks traded on the Bucharest Stock Exchange
Modelling individual return series using GARCH methods and extreme value theory and the dependence structure using the notion of copula in order to simulate a portfolio returns distribution
Accurately capturing the data generating process for each return series in order to efficiently estimate VAR and ES values
Backtesting for precision of the risk measure selected

4. Literature review
Main sources:
McNeil, A.J. and R.Frey (2000) „Estimation of Tail-Related Risk Measures for Heteroscedastic Financial Time Series: an Extreme Value Approach”
Nyström, K. and J. Skoglund (2002a), „A Framework for Scenariobased Risk Management”

5. Methodology - GARCH conditional mean equation
conditional variance equation
Leverage coefficient introduced by Glosten, Jagannathan and Runkle (1993)

6. Methodology – Extreme Value Theory
Peak-over-threshold model
For a sample of observations, rt, t = 1, 2, . . , n with a distribution function F(x) = Pr{rt ≤ x} and a high-threshold u, the exceedances over this threshold occur when rt > u for any t = 1, 2, . . , n. An excess over u is defined by y = rt − u.
The theorem of Balkema and de Haan (1974) and Pickands (1975) shows that for sufficiently high threshold u, the distribution function of the excess may be approximated by the Generalized Pareto Distribution (GPD):
ξ - shape parameter; σ scale parameter; υ location parameter

7. Methodology – Copulas
A joint distribution can be decomposed into marginal distributions and a dependence structure represented by a copula function.
Multivariate Gaussian Copula:
where ΦR is the standard multivariate normal distribution with correlation matrix R; Φ-1(u) is the inverse of the normal cumulative distribution function
Multivariate Student’s t Copula
where TR,v denotes the standard multivariate Student’s t distribution with correlation matrix R and v degrees of freedom; tv-1(u) denotes the inverse of the Student’s t cumulative distribution function

8. Methodology – Measures of risk
Value-at-Risk
Measures the worst loss to be expected of a portfolio over a given time
horizon at a given confidence level
Advantages:
simple and intuitive method of evaluating risk
Disadvantages:
gives only an upper limit on the losses given a confidence level
tells nothing about the potential size of the loss if this upper limit is exceeded
not a coherent measure of risk (Artzner et al. 1997, 1998)
Expected Shortfall
Measures the average loss to be expected of a portfolio over a given time
horizon provided that VaR has been exceeded.

9. Estimation and results - Data
Five Romanian equities traded on the Bucharest Stock Exchange (symbols: SIF1, SIF2, SIF3, SIF4, SIF5)
Selection criteria
high market liquidity
long time series with few missing values
high volatility periods
Period: – ; 1564 observations
The price series are adjusted for corporate events

10. Estimation and results - Overview
GARCH coefficients estimation
Construction of semi-parametric distributions for the standardized residuals (zt)
Extreme value modelling of the tails (Generalized Pareto Distributions)
Kernel smoothing of the interior
Student’s t Copula calibration
Simulation of the conditional portfolio distribution
Value-at-risk and Expected Shortfall estimation
Value-at-Risk backtesting

11. Estimation and results - GARCH
Testing for the autocorrelation of returns and the presence of a volatility clustering effect
Sample autocorrelation function plot (returns and squared returns)

12. Estimation and results - GARCH
Testing for the autocorrelation of returns and the presence of
a volatility clustering effect
Ljung Box test for randomness (returns and squared returns)
Null Hypothesis: none of the autocorrelation coefficients up to lag 20 are different from zero
Ljung Box Test for returns, 20 lags, 5% significance level
Null Hypotesis: Data is random
Series H
pValue
Statistic
Critical
Value
SIF1 1
0,031399
33,284
31,4104
SIF2
0,019863
35,045
SIF3
0,19554
25,156
SIF4 0,
38,036
SIF5
0,031252
33,302
Ljung Box Test for squared returns, 20 lags, 5% significance level
Null Hypotesis: Data is random
Series H
pValue
Statistic
Critical Value
SIF1 1
300,04
31,4104
SIF2
248,48
SIF3
221,78
SIF4
245,23
SIF5
113,9

13. Estimation and results - GARCH
Initial model
 Initial
estimation C AR K
GARCH
ARCH
Leverage
DoF
SIF1
Value
0,001362
0,006147
4,60E-05
0,72498
0,30953
-0,077775
3,6836
Std Err
0,000465
0,026416
1,17E-05
0,033663
0,059267
0,060984
0,39126
T-Stat
2,9317
0,2327
3,9245
21,5365
5,2226
-1,2753
9,4147
SIF2
0,001992
0,040896
8,98E-05
0,68387
0,3013
-0,11796
4,0824
0,000538
0,026685
2,19E-05
0,043946
0,06246
0,0608
0,45051
3,704
1,5326
4,0994
15,5614
4,8239
-1,9402
9,0616
SIF3
0,001685
-0,02353
5,33E-05
0,73855
0,24825
-0,037494
3,6475
0,000474
0,026172
1,44E-05
0,03692
0,053417
0,055747
0,38714
3,5581
-0,8989
3,7037
20,0037
4,6474
-0,6726
9,4218
SIF4
0,001599
0,060422
6,83E-05
0,71545
0,2714
-0,080635
3,8946
0,000505
0,025919
1,71E-05
0,040135
0,057564
0,05978
0,41809
3,169
2,3312
3,9845
17,826
4,7147
-1,3489
9,3153
SIF5
0,001963
0,007492
9,23E-05
0,71299
0,26942
-0,12281
3,5881
0,00052
0,025493
2,37E-05
0,046853
0,063197
0,059662
0,34257
3,7788
0,2939
3,893
15,2174
4,2631
-2,0584
10,4741

14. Estimation and results - GARCH
Final
estimation C AR K
GARCH
ARCH
DoF
SIF1
Value
0,00124
N/A
4,59E-05
0,72085
0,27915
3,6882
Std Err
0,000461
1,17E-05
0,033774
0,048701
0,39026
T-Stat
2,6908
3,9195
21,3436
5,7319
9,4506
SIF2
0,001797
0,034957
8,69E-05
0,68862
0,24198
4,1105
0,000539
0,026676
2,13E-05
0,042992
0,043716
0,45594
3,3341
1,3104
4,0744
16,0174
5,5352
9,0155
SIF3
0,001599
5,40E-05
0,73498
0,23442
3,6559
0,000466
1,46E-05
0,037461
0,045878
0,38638
3,4353
3,7031
19,6198
5,1097
9,4617
SIF4
0,001464
0,058049
7,02E-05
0,70692
0,2431
3,8641
0,000501
0,026091
1,76E-05
0,040694
0,047402
0,41209
2,9238
2,2248
3,998
17,3718
5,1285
9,377
SIF5
0,001778
8,36E-05
0,72883
0,20305
3,5868
0,000516
2,18E-05
0,044148
0,043688
0,34209
3,4479
3,8425
16,509
4,6478
10,485

15. Ljung BoxTest for std residuals, 20 lags, 5% significance level
Estimation and results - GARCH
Testing for the autocorrelation of the standardized residuals
( ) and the presence of a volatility clustering effect
Ljung Box test for randomness - standardized residuals ( ) and squared standardized residuals ( )
Null Hypothesis: none of the autocorrelation coefficients up to lag 20 are different from zero
Ljung BoxTest for std residuals, 20 lags, 5% significance level
Null Hypotesis: Data is random
Series H
pValue
Statistic
Critical
Value
SIF1
0,10089
28,371
31,4104
SIF2 1
0,015045
36,082
SIF3
0,77333
15,054
SIF4
0,36896
21,487
SIF5
0,043424
31,988
Ljung BoxTest for squared std residuals, 20 lags, 5% significance level
Null Hypotesis: Data is random
Series H
pValue
Statistic
Critical Value
SIF1
0,99529
7,3687
31,4104
SIF2
0,84678
13,671
SIF3
0,8839
12,846
SIF4
0,95327
10,716
SIF5
0,99314
7,7947

16. Estimation and results – Extreme Value
Assumptions:
a skewed standardized residual distribution
an overestimation of the tail heaviness by the Student’s t distribution
 GPD estimation
Tail
Tail Shape
Std Error
T-Stat
Scale
SIF1
Lower
0,2022
0,088745
2,2785
0,52983
0,062743
8,4444
Upper
0,012265
0,094734
0,12947
0,68873
0,085424
8,0625
SIF2
0,21153
0,089722
2,3576
0,5177
0,061633
8,3997
-0,
-0,00097
0,67981
0,078891
8,6171
SIF3
0,13105
0,093611
1,4
0,52577
0,064529
8,1478
0,11445
0,079746
1,4352
0,65982
0,074316
8,8786
SIF4
0,30175
0,10018
3,012
0,44784
0,056405
7,9397
0,10545
0,096637
1,0912
0,59198
0,074122
7,9865
SIF5
0,44799
0,12018
3,7278
0,39466
0,055205
7,149
0,076981
0,088776
0,86714
0,61118
0,072967
8,3761

17. Estimation and results – Extreme Value
Peak-over-threshold method fits the tails better than the Student’s t distribution estimated by the GARCH model
Asymmetric standardized residual distribution with a heavier lower tail.

18. Estimation and results – Extreme Value
Construction of the semi-parametric distributions
Generalized Pareto fitted tails
Kernel Smoothed interior
Building of pseudo cumulative distribution functions (CDF) and inverse cumulative distribution functions (ICDF) for Monte Carlo simulation

19. Estimation and results – Copula
Calibrating the parameters of the Student’s t copula with canonical
maximum likelihood (CML) method.
CML method allows for an estimation of the copula parameters without an assumption about the marginal distributions
The standardized residuals X = (X1t,…, Xnt)t=1T are transformed into uniform variates using the marginal distribution functions (pseudo-CDF):
ut = (ut1,….., utn) = [F1(X1t),…., Fn(Xnt)].
The vector of copula parameters α are estimated via the following relation:

20. Estimation and results – Copula
DoF parameter estimated with the profile log likelihood method
Positive correlation of the series
Low degrees of freedom parameter implies a high tail dependence.
Correlation Matrix
SIF1
SIF2
SIF3
SIF4
SIF5 1
0.7118
0.6822
0.6673
0.6994
0.6615
0.6693
0.7701
0.6469
0.6408
0.6798
DoF
Std Error

21. Estimation and results – Simulation
Simulation of a conditional distribution for the portfolio with the semi-parametric
marginal distributions and the dependence structure given by the t-copula
3000 trials are generated from a multivariate Student’s t distribution with the same correlation matrix and degrees of freedom parameters as those estimated with the t-copula
transformation of each simulated series into the corresponding semi-parametrical distribution.
building the conditional distribution of the portfolio by reintroducing the volatility with the GARCH models

22. VaRα = qα Estimation and results – Risk measures
Value at Risk is estimated by taking the relevant quantile qα of
the conditional portfolio distribution*:
VaRα = qα
Expected Shortfall is estimated by using the following formula**:
 1-day horizon
VaR ES
90%
95%
99%
SIF1
-1.86%
-2.58%
-5.40%
-3.22%
-4.30%
-7.78%
SIF2
-2.02%
-2.86%
-5.68%
-3.55%
-4.73%
-8.58%
SIF3
-2.50%
-3.49%
-6.39%
-4.11%
-5.37%
-8.63%
SIF4
-2.40%
-3.09%
-6.08%
-3.80%
-4.92%
-8.33%
SIF5
-1.83%
-2.41%
-4.31%
-2.94%
-3.78%
-6.65%
Portfolio
-1.81%
-2.52%
-4.87%
-3.03%
-3.93%
-6.40%
* if the losses are marked with a positive sign and gains with a negative sign
** where n represents the number of trials

23. Estimation and results – Value-at-risk backtesting
1 day horizon
estimation for the last 500 days of the series
fixed data sample of 1000 observations
for each day all the parameters are re-estimated
Backtesting Results
VaR - 90%
VaR - 95%
VaR - 99%
Expected 50 25 5
SIF1 56 29 4
SIF2 55 32 2
SIF3 47 28 3
SIF4 46 6
SIF5 45 24
Portfolio 51

24. Estimation and results – Value-at-risk backtesting

25. Estimation and results – Value-at-risk backtesting

26. Conclusion
the GARCH models explain well the autocorrelation found in the return series and the volatility clustering effect
the distributions of the innovations are asymmetric with heavy lower tails and thin upper tails
the GPD describes the tails of the standardized residuals better than the Student’s t-distribution
the backtesting results for Value-at-Risk are not conclusive but give an indication of a possible underestimation of the risk at 95% confidence level
Further research:
estimation of risk for different risk factors
methodology improvement