Measuring market risk:

a copula and extreme value approach

Supervisor

Professor Moisă Altăr

Academy of Economic Studies BucharestDoctoral School of Finance and Banking

M. Sc. Student

Alexandru Stângă

July 2007

• Goal

• Literature review

• Methodology

• Estimation and results

• Conclusion

Contents

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

Goal

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”

Literature review

• conditional mean equation

• conditional variance equation

• Leverage coefficient introduced by Glosten, Jagannathan and Runkle

(1993)

t

n

jjtj

m

iitit ycy

11

q

jjtj

p

iitit AGk

1

2

1

22

q

jjtjtj

q

jjtj

p

iitit SgnLAGk

1

2

1

2

1

22

0ε0

0ε1Sgn

jt

jt

jt

Methodology - GARCH

nceunit varia and zeromean with ddistributey identicall andt independen isz

z ε

t

tt t

Peak-over-threshold modelFor 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

Methodology – Extreme Value Theory

0ξif,e1

0ξif,σ

υyξ11

G

σ

υy

ξ

1

υσ,ξ,

• 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

Methodology – Copulas

nRn uuuRuuuC 12

11

12,1 ,.....,,;,....,

nRn utututTRuuuC 12

11

1.2,1 ,....,,,;,....,

Value-at-RiskMeasures the worst loss to be expected of a portfolio over a given timehorizon 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 ShortfallMeasures the average loss to be expected of a portfolio over a given timehorizon provided that VaR has been exceeded.

Methodology – Measures of risk

)](|[)( VaRrrEES

• 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: 01.2001 – 06.2007; 1564 observations

• The price series are adjusted for corporate events

Estimation and results - Data

• 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

Estimation and results - Overview

• Testing for the autocorrelation of returns and the presence of a volatility clustering effect– Sample autocorrelation function plot (returns and squared returns)

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

Estimation and results - GARCH

Ljung Box Test for returns, 20 lags, 5% significance level

Null Hypotesis: Data is random

Series H pValue StatisticCritical Value

SIF1 1 0,031399 33,284 31,4104

SIF2 1 0,019863 35,045 31,4104

SIF3 0 0,19554 25,156 31,4104

SIF4 1 0,0087669 38,036 31,4104

SIF5 1 0,031252 33,302 31,4104

Ljung Box Test for squared returns, 20 lags, 5% significance level

Null Hypotesis: Data is random

Series H pValue StatisticCritical Value

SIF1 1 0 300,04 31,4104

SIF2 1 0 248,48 31,4104

SIF3 1 0 221,78 31,4104

SIF4 1 0 245,23 31,4104

SIF5 1 0 113,9 31,4104

Initial model

Estimation and results - GARCH

211

21

21

2 ttttt LSgnAGk

ttt rcr 1

 Initialestimation 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

Value 0,001992 0,040896 8,98E-05 0,68387 0,3013 -0,11796 4,0824

Std Err 0,000538 0,026685 2,19E-05 0,043946 0,06246 0,0608 0,45051

T-Stat 3,704 1,5326 4,0994 15,5614 4,8239 -1,9402 9,0616

SIF3

Value 0,001685 -0,02353 5,33E-05 0,73855 0,24825 -0,037494 3,6475

Std Err 0,000474 0,026172 1,44E-05 0,03692 0,053417 0,055747 0,38714

T-Stat 3,5581 -0,8989 3,7037 20,0037 4,6474 -0,6726 9,4218

SIF4

Value 0,001599 0,060422 6,83E-05 0,71545 0,2714 -0,080635 3,8946

Std Err 0,000505 0,025919 1,71E-05 0,040135 0,057564 0,05978 0,41809

T-Stat 3,169 2,3312 3,9845 17,826 4,7147 -1,3489 9,3153

SIF5

Value 0,001963 0,007492 9,23E-05 0,71299 0,26942 -0,12281 3,5881

Std Err 0,00052 0,025493 2,37E-05 0,046853 0,063197 0,059662 0,34257

T-Stat 3,7788 0,2939 3,893 15,2174 4,2631 -2,0584 10,4741

ddistribute- tbe toassumedz

z ε

t

tt t

Estimation and results - GARCH

Finalestimation 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 N/A 1,17E-05 0,033774 0,048701 0,39026

T-Stat 2,6908 N/A 3,9195 21,3436 5,7319 9,4506

SIF2

Value 0,001797 0,034957 8,69E-05 0,68862 0,24198 4,1105

Std Err 0,000539 0,026676 2,13E-05 0,042992 0,043716 0,45594

T-Stat 3,3341 1,3104 4,0744 16,0174 5,5352 9,0155

SIF3

Value 0,001599 N/A 5,40E-05 0,73498 0,23442 3,6559

Std Err 0,000466 N/A 1,46E-05 0,037461 0,045878 0,38638

T-Stat 3,4353 N/A 3,7031 19,6198 5,1097 9,4617

SIF4

Value 0,001464 0,058049 7,02E-05 0,70692 0,2431 3,8641

Std Err 0,000501 0,026091 1,76E-05 0,040694 0,047402 0,41209

T-Stat 2,9238 2,2248 3,998 17,3718 5,1285 9,377

SIF5

Value 0,001778 N/A 8,36E-05 0,72883 0,20305 3,5868

Std Err 0,000516 N/A 2,18E-05 0,044148 0,043688 0,34209

T-Stat 3,4479 N/A 3,8425 16,509 4,6478 10,485

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

Estimation and results - GARCH

tz

2tz

tz

Ljung BoxTest for std residuals, 20 lags, 5% significance level

Null Hypotesis: Data is random

Series H pValue StatisticCritical Value

SIF1 0 0,10089 28,371 31,4104

SIF2 1 0,015045 36,082 31,4104

SIF3 0 0,77333 15,054 31,4104

SIF4 0 0,36896 21,487 31,4104

SIF5 1 0,043424 31,988 31,4104

Ljung BoxTest for squared std residuals, 20 lags, 5% significance level

Null Hypotesis: Data is random

Series H pValue StatisticCritical Value

SIF1 0 0,99529 7,3687 31,4104

SIF2 0 0,84678 13,671 31,4104

SIF3 0 0,8839 12,846 31,4104

SIF4 0 0,95327 10,716 31,4104

SIF5 0 0,99314 7,7947 31,4104

Assumptions:- a skewed standardized residual distribution- an overestimation of the tail heaviness by the Student’s t distribution

Estimation and results – Extreme Value

 GPD estimation Tail Tail Shape Std Error T-Stat Scale Std Error T-Stat

SIF1Lower 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

SIF2Lower 0,21153 0,089722 2,3576 0,5177 0,061633 8,3997

Upper -0,0000811 0.084005 -0,00097 0,67981 0,078891 8,6171

SIF3Lower 0,13105 0,093611 1,4 0,52577 0,064529 8,1478

Upper 0,11445 0,079746 1,4352 0,65982 0,074316 8,8786

SIF4Lower 0,30175 0,10018 3,012 0,44784 0,056405 7,9397

Upper 0,10545 0,096637 1,0912 0,59198 0,074122 7,9865

SIF5Lower 0,44799 0,12018 3,7278 0,39466 0,055205 7,149

Upper 0,076981 0,088776 0,86714 0,61118 0,072967 8,3761

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.

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

Estimation and results – Copula

Calibrating the parameters of the Student’s t copula with canonicalmaximum 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,….., ut

n) = [F1(X1t),…., Fn(Xnt)].

– The vector of copula parameters α are estimated via the following relation:

T

t

c1

tn

t1CML );u ..,,u(lnmaxarg

• DoF parameter estimated with the profile log likelihood method

• Positive correlation of the series

• Low degrees of freedom parameter implies a high tail dependence.

Estimation and results – Copula

Correlation Matrix

  SIF1 SIF2 SIF3 SIF4 SIF5

SIF1 1 0.7118 0.6822 0.6673 0.6994

SIF2 0.7118 1 0.6615 0.6693 0.7701

SIF3 0.6822 0.6615 1 0.6469 0.6408

SIF4 0.6673 0.6693 0.6469 1 0.6798

SIF5 0.6994 0.7701 0.6408 0.6798 1

DoF Std Error

5.425141 0.328318

Simulation of a conditional distribution for the portfolio with the semi-parametricmarginal 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

Estimation and results – Simulation

Value at Risk is estimated by taking the relevant quantile qα ofthe conditional portfolio distribution*:

VaRα = qα

Expected Shortfall is estimated by using the following formula**:

Estimation and results – Risk measures

])[/()()|(][

)( nanXVaRXXEESn

nii

* if the losses are marked with a positive sign and gains with a negative sign

** where n represents the number of trials

 1-day horizon VaR ES

  90% 95% 99% 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%

Estimation and results – Value-at-risk backtesting

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 32 6

SIF5 45 24 4

Portfolio 51 32 6

Estimation and results – Value-at-risk backtesting

Estimation and results – Value-at-risk backtesting

• 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

Conclusion