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

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

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

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”

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

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

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

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.

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: 01.2001 – 06.2007; 1564 observations The price series are adjusted for corporate events

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

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)

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 0,0087669 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

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

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

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

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 -0,0000811 0.084005 -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

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 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:

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 5.425141 0.328318

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

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

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

Estimation and results – Value-at-risk backtesting

Estimation and results – Value-at-risk backtesting

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