Extreme Risk and Value-at-Risk in the German Stock Market

Abstract

Extreme Value Theory methods are used to investigate the distribution of the extreme minima in the German stock market over the period 1973 to 2001. Innovative aspects of this paper include (i) a wide set of distributions considered, (ii) L-moment diagrams employed to identify the most appropriate distribution/s, (iii) ‘probability weighted moments’ used to estimate the parameters of these distribution/s and (iv) the Anderson–Darling goodness of fit test employed to test the adequacy of fit. The ‘generalized logistic’ distribution is found to provide adequate descriptions of the extreme minima of the German stock market over the period studied. VaR analysis results show that the EVT methods used in this study can be particularly useful for market risk measurement since they produce estimates that outperform those derived by traditional methods at high confidence levels.

Keywords:

• Extreme value theory
• value-at-risk
• L-moments
• probability weighted moments
• Anderson–Darling goodness of fit test
• generalized extreme value distribution
• generalized logistic distribution

Notes

¹ See Aparicio and Estrada (2001) for a comprehensive review of the literature regarding the empirical distributions of financial returns.

² For example, Beder (1995) applied eight commonly used variations of the HS and MCS methods to three hypothetical portfolios and she found that VaR estimates can vary significantly from one method to another; sometimes even by as much as 14 times. Hendricks (1996) evaluated the exponential weighted moving average (EWMA), VC and HS models using 1000 randomly selected foreign exchange portfolios. He found that none of the models used and their possible variations is consistently superior to the others and that the choice of confidence level can have a substantial effect on the VaR estimates. These findings are in line with those of Brooks and Persand (2000) and Marshall and Siegel (1997).

³ The RiskMetrics model is based on EWMA estimates of volatility.

⁴ An alternative way to analyse the behaviour of the extremes is known as the ‘peaks over threshold’ (POT) method, according to which, extremes are defined as excesses over a threshold (see Davison and Smith, 1990 for a detailed description of the POT). However, financial returns tend to cluster and this could lead to considerably serial dependence in the time series of the extremes. Additionally, there is an issue regarding the choice of the threshold. A low one will result in many central observations entering the sample while a high one will leave so few in the sample that could lead to inaccurate estimates. On the other hand, the definition of extremes adopted in this paper means that some of the minimum extremes will not be large negative returns or they might be even positive returns. Regrettably, there is a decision to be made and in order to avoid or reduce as much as possible the problem of serial dependence in the time series of extremes it was decided to collect the extremes as the minimum daily returns over non-overlapping time intervals of pre-specified length.

⁵ In addition, the series of the data will be divided into sub-periods and moving window techniques will be used to estimate VaR. It is believed that these approaches will help to mitigate the problem with non-iid data since it is likely to capture some of the non-stationary behaviour. Another alternatively would be to fit the tail of the conditional distribution of returns by using an autoregressive volatility model (e.g. GARCH), standardize the returns by the estimated conditional volatility and proceed in EVT analysis. This approach has received attention by McNeil and Frey (2000) and Byström (2004). However, additional parameters have to be estimated which make this approach subject to increased estimation standard error and model risk.

⁶ As noted by the editor and the referee the definition of the L-moment ratios is not scale free and that it would be possible to adopt a definition that makes the L-moment ratios scale free. However, in order to be in line with the existing literature it was decided to keep the definition of the L-moment ratios unchanged.

⁷ On such a diagram, a three-parameter distribution (e.g. the Weibull) is represented by a curve whereas a two-parameter distribution (e.g. the normal) is represented by a single point.

⁸ The calculation of conventional moments, like skewness and kurtosis, involves third and fourth powers. Thus, greater weight is given to outliers, which can lead to considerable bias and variance.

⁹ Hosking et al. (1985) showed that for the GEV distribution, parameters and quantiles made using the L-moments version of the PWM method are estimated with at least 70% efficiency. For example, when the shape parameter of the GEV is −0.2, the asymptotic bias of the 0.01 quantile estimated by the PWM and ML methods is found to be −0.2 and 1.6, respectively. They also demonstrated that for shape parameter values in the range −0.5 to 0.5 and samples of up to 100 observations, PWM estimates have lower root-mean-square error than estimates generated by the ML method. Similar results are reported in the literature for the GP distribution (Hosking and Wallis, 1987; Roótzen and Tajvidi, 1997).

¹⁰ For example if we test a VaR model at the 95% confidence level we expect that the actual returns will be larger than the VaR forecasts only 5% of the time (e.g. if we use 1000 past daily returns we expect to record 50 VaR violations).

¹¹ The corresponding Datastream code is TOTMKBD and the index is composed of 250 of the most heavily traded shares that aim to cover 70–80% of the total market capitalization. Prices take account of capital changes and in order to retain some familiarity this index is denoted as DAX-DS. The main advantage of using this index instead of the DAX is that it is available for a much longer period.

¹² For example the eight daily returns which occurred in the cluster and exceeded the μ−4σ threshold, occurred on 19/10/87 (−6.84%), 26/10/87 (−5.12%), 28/10/87 (−4.66%), 9/11/87 (−5.87%), 10/11/87 (−6.57%), 1/10/98 (−5.33%), 2/10/98 (−5.30%), 11/9/01 (−7.21%) and 20/9/01 (−5.54%).

¹³ This period contains the large negative daily returns of −6.84% on 19/10/87, −4.28% on 20/10/87, −5.12% on 26/10/87, −4.66% on 28/10/87 and −6.57% on 10/11/87.

¹⁴ This period contains the largest negative daily return of −12.14% on 16/10/89.

¹⁵ This period contains the large negative daily returns of −5.00% on 6/8/90, −3.08% on 17/8/90, −4.40% on 21/8/90 and −3.19% on 23/8/90.

¹⁶ This period contains the large negative daily return of −9.29% on 19/8/91.

¹⁷ These period contains the large negative daily returns of −3.97% on 23/10/97, −3.60% on 27/10/97, −7.21% on 28/10/97, −3.69% on 11/8/98, −5.24% on 21/8/98, −4.97% on 10/9/98, −3.30% on 17/9/98, −3.29% on 21/9/98, −5.33% on 1/10/98, −5.30% on 2/10/98 and −3.91% on 8/10/98.

¹⁸ This period contains the large negative daily returns of −7.21% on 11/9/01, −4.12% on 14/9/01, −3.04% on 19/9/01 and −5.54% on 20/9/01.

¹⁹ The signed and squared L-skewness, instead of the L-skewness, is used because it almost restores linearity in the L-moments ratios diagram and thus, provides a clearer view of the distribution location.

²⁰ In the interest of brevity these diagrams are not included in the paper; however, they are available from the authors upon request.

²¹ Similar indications for the behaviour of extremes in the German stock market were obtained by Lux (2001). This characteristic of the extremes has also been noted by McNeil and Frey (2000) and Pownall and Koedijk (1999).

²² The same VaR analysis was also carried out by using the series of monthly and semi-annual extremes. However, the series of weekly extremes found to give the best results and therefore, only these are reported. For the traditional VaR methods 250, 500, 1000 and 1500 past daily returns were used. However, to make the comparison fair, the best results from these methods were also used.

²³ There have been attempts to take into account the time varying distributional characteristics of the extremes by using autoregressive processes (McNeil and Frey, 2000; Pownall and Koedijk, 1999) or quantile regression techniques (Engle and Manganelli, 2004). However, these approaches introduce yet more parameters in to the modelling procedure and this is likely to result in larger estimation errors and possibly even more inaccurate VaR estimates.

²⁴ The first period was a volatile period for the German stock market because of the effects of the stock markets' crash in 1987, the turbulence due to the political uncertainty surrounding German unification, the Gulf crisis and the coup against the Soviet leader Michael Gorbachev. The daily standard deviation was 1.24%, skewness was −1.353 and kurtosis was 13.489. Normality was firmly rejected by the Shapiro–Wilk test. The second period was even more volatile with a daily standard deviation of 1.40%, a skewness of −0.444 and a kurtosis of 1.963. This period contains the turbulence during the Asian and Russian crises and the negative market sentiment after the events of 11 September 2001 that drove the stock markets downwards. In addition, these time periods contain 1248 and 1260 daily returns, respectively, sizes which can be considered adequate for statistical evaluation.

²⁵ At the 99.75% confidence level, the 10 unexpected VaR violations occurred in 28/1/87 (−5.06%), 19/10/87 (−6.84%), 26/10/87 (−5.12%), 28/10/87 (−4.66%), 9/11/87 (−5.86%), 10/11/87 (−6.57%), 16/10/89 (−12.14%), 6/8/90 (−5.00%), 21/8/90 (−4.40%) and 19/8/91 (−9.29%).

²⁶ For example, the 14 largest unexpected VaR violations occurred in 23/10/97 (−3.97%), 27/10/97 (−3.60%), 28/10/97 (−7.20%), 21/8/98 (−5.23%), 10/9/98 (−4.96%), 1/10/98 (−5.33), 2/10/98 (−5.30%), 13/1/99 (−4.52%), 31/1/00 (−3.64%), 14/3/01 (−3.53%), 22/3/01 (−4.32%), 11/9/01 (−7.21%), 14/9/01 (−4.12%) and 20/9/01 (−5.47%).