In this paper we provide empirical evidence on the riskiness of the VaR. Given its performance during the recent financial crisis, we feel that there is more need of taking stock and communicating its potential shortcomings before adding another VaR approach to the literature.
Our goal in this paper is to study the robustness of the standard VaR with respect to different risk factors when applied to forecast losses in periods when it is needed most, such as the current financial crisis. By means of a meta-study approach, we show that the performance of VaR estimates differs across type of stocks subject to model choice, distributional assumptions or the choice of estimation window. Thus we show that popular VaR measures manage to accurately forecast the risk in calm periods or when applied to large-cap stocks, but exhibit a relatively low power when applied in crisis times or to stocks with lower capitalization. Higher parametric approaches (e.g. GARCH) which account for past shocks (e.g. market crash from 1987) within extreme value distributional settings lead to accurate loss forecasts ac-cording to Basel II rules during the recent financial crisis.
In order to improve the performance of standard risk measures, we propose a data-driven methodology of accurately estimating VaR based on the principle of forecast combination. The optimal loadings of VaR measures are driven by the maximiza-tion of condimaximiza-tional coverage rates (CCOM) or by the minimizamaximiza-tion of the distance between the population quantiles and the VaR’s combinations (CQOM). Using an empirical example, we show that optimal combinations of VaR forecasts radically improve the performance of stand-alone estimates during the recent crisis. All com-binations exhibit very good “out-of-sample” performance by generating independent exceedances within the limits imposed by Basel II rules.
Future research should aim at assessing and developing robust risk measures in real time settings, which are essential in the field of risk management, where investors face the continuous challenge of taking spontaneous decisions. The new risk measures should be able to instantaneously incorporate all relevant new information related to the underlying asset, market conditions, or other economical and financial variables, which affect the market price risk, such as market liquidity shortages.
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Appendix A.4: Tables
Table A.4.1: Backtest results for the calm period: nonparametric methods. Percentage rate of violations for VaR at p = 1% for the period from January 1st, 2007 to July 17th, 2007 (total of 143 days). ∗∗ refers to p-values of conditional coverage test smaller than 0.05,∗ to p-values between and 0.05 and 0.10 and no mark refers to p-values larger than 0.10. Bold type entries are in the “red zone”, italic type entries are in the “yellow zone” and no typeface entries are in the “green zone”.
Table A.4.2: Backtest results for the crisis period: nonparametric methods. Percentage rate of violations for VaR at p = 1% for the period from July 18th, 2007 to July 2nd, 2009 (total of 510 days). ∗∗ refers to p-values of conditional coverage test smaller than 0.05,∗ to p-values between and 0.05 and 0.10 and no mark refers to p-values larger than 0.10. Bold type entries are in the “red zone”, italic type entries are in the “yellow zone” and no typeface entries are in the “green zone”.
of violations for VaR at p = 1% for the period from September 1st, 2008 to July 2nd, 2009 (total of 129 days). ∗∗ refers to p-values of conditional coverage test smaller than 0.05,∗ to p-values between and 0.05 and 0.10 and no mark refers to p-values larger than 0.10. Bold type entries are in the “red zone”, italic type entries are in the “yellow zone” and no typeface entries are in the “green zone”.
Stock Number of HS Start HS FHS
Type observations date
small
1000 4.14** 1987 19.35** 1.84
750 6.91** 1996 16.59** 1.84
500 8.29** 2001 11.98** 2.76*
250 8.75** 2005 9.21** 1.38
middle
1000 2.76** 1987 13.36** 1.84
750 5.53** 1996 12.44** 2.30
500 7.37** 2001 10.13** 2.30
250 8.29** 2005 8.29** 1.84
large
1000 4.14** 1987 15.23** 1.84
750 5.53** 1996 14.28** 1.38
500 7.83** 2001 12.44** 1.38
250 8.75** 2005 9.67** 0.92
Appendix B.4: Figures
Figure B.4.1: Estimated d.f. of t-distribution. Left panel: small cap index, middle: middle cap index, right: large cap index; dotted lines correspond to d.f estimated on samples starting in 1987, dotted and dashed lines correspond to d.f estimated on samples starting in 1996, solid lines correspond to d.f estimated on samples starting in 2001 and dashed lines correspond to d.f estimated on samples starting in 2005.
Figure B.4.2: Estimated λ∗τ,0 through the crisis, CQOM, Part A: ARMA-GARCH. Left panels: small cap index, middle panels: middle cap index, right panels: large cap index; first row: sampling starting in 1987, second row: sampling starting in 1996, third row: sampling starting in 2001 and fourth row: sampling starting in 2005.
Figure B.4.3: Estimated λ∗τ,1 through the crisis, CQOM, Part A: ARMA-GARCH. Left panels: small cap index, middle panels: middle cap index, right panels: large cap index; first row: sampling starting in 1987, second row: sampling starting in 1996, third row: sampling starting in 2001 and fourth row: sampling starting in 2005.
Figure B.4.4: Estimated λ∗τ,0 through the crisis, CCOM, Part A: ARMA-GARCH. Left panels: small cap index, middle panels: middle cap index, right panels: large cap index; first row: sampling starting in 1987, second row: sampling starting in 1996, third row: sampling starting in 2001 and fourth row: sampling starting in 2005.
Figure B.4.5: Estimated λ∗τ,1 through the crisis, CCOM, Part A: ARMA-GARCH. Left panels: small cap index, middle panels: middle cap index, right panels: large cap index; first row: sampling starting in 1987, second row: sampling starting in 1996, third row: sampling starting in 2001 and fourth row: sampling starting in 2005.
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Ich versichere hiermit, dass ich die vorliegende Arbeit mit dem Thema
Four Essays on Measuring Financial Risks
ohne unzul¨assige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hil-fsmittel angefertigt habe. Die aus anderen Quellen direkt oder indirekt ¨ubernomme-nen Daten und Konzepte sind unter Angabe der Quelle gekennzeichnet. Weitere Personen, insbesondere Promotionsberater, waren an der inhaltlich materiellen Er-stellung dieser Arbeit nicht beteiligt.7 Die Arbeit wurde bisher weder im In- noch im Ausland in gleicher oder ¨ahnlicher Form einer anderen Pr¨ufungsbeh¨orde vorgelegt.
Konstanz, den 15. Februar 2010
(Roxana-Mihaela Chiriac)
Ich versichere hiermit, dass ich Kapitel 1 der vorliegenden Arbeit ohne Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe.
Kapitel 2 and 3 entstammen einer gemeinsamen Arbeit mit Herrn Valeri Voev (Uni-versit¨at Aarhus, CREATES). Meine individuelle Leistung bei der Erstellung dieser Arbeiten ist 50%.
Kapitel 4 entstammt einer gemeinsamen Arbeit mit Herrn Prof. Dr. Winfried Pohlmeier (Universit¨at Konstanz). Meine individuelle Leistung bei der Erstellung dieser Arbeit ist 75%.