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How accurate are modern Value-at-Risk estimators derived from extreme value theory?

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Abstract

In this study, we compare the out-of-sample forecasting performance of several modern Value-at-Risk (VaR) estimators derived from extreme value theory (EVT). Specifically, in a multi-asset study covering 30 years of stock, bond, commodity and currency market data, we analyse the accuracy of the classic generalised Pareto peak over threshold approach and three recently proposed methods based on the Box–Cox transformation, L-moment estimation and the Johnson system of distributions. We find that, in their unconditional form, some of the estimators may be acceptable under current regulatory assessment rules but none of them can continuously pass more advanced tests of forecasting accuracy. In their conditional forms, forecasting power is significantly increased and the Box–Cox method proves to be the most promising estimator. However, it is also important to stress that the traditional historical simulation approach, which is currently the most frequently used VaR estimator in commercial banks, can not only keep up with the EVT-based methods but occasionally even outperforms them (depending on the setting: unconditional versus conditional). Thus, recent claims to generally replace this simple method by theoretically more advanced EVT-based methods may be premature.

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Notes

  1. The VaR is not a coherent risk measure (see Artzner et al. 1999). It can lead to Pareto-inferior allocations if agents are risk averse. In addition, the VaR can fail to account appropriately for portfolio risk diversification (see Wong et al. 2012; Yamai and Yoshiba 2002, 2005).

  2. For another strand of the literature dealing with EVT in VaR copulas, see Hsu et al. (2012).

  3. For example, McNeil and Frey (2000) compare their peak over threshold estimator to historical simulation and Brooks et al. (2005) evaluate the performance of their simulation-based EVT approach relative to classic tail estimators (e.g., the Hill estimator).

  4. With an infinite right endpoint, we would allow the possibility of unreasonably large outcomes. Also note that the form of (7) requires to multiply the empirical long-position returns by \(-1\) in order to model the correct tail.

  5. Of course this is not mathematically complete because we do not exactly say what we mean by ‘a large class’. For this article, it is sufficient to know that the class contains all the common continuous distributions of statistics and actuarial science (normal, lognormal, \(\chi ^2\), Student-t, F, gamma, exponential, uniform, beta, etc.).

  6. Similar to the GEVD, the GPD nests other distributions. The special cases \(\xi = 0\) and \(\xi =1\) yield, respectively, the exponential distribution with mean \(\delta \) and the uniform distribution on \([0,\delta ]\). Classic Pareto distributions are obtained when \(\xi < 0\) (see Hosking and Wallis 1987).

  7. Alternative methods for threshold determination are described in El-Aroui and Diebold (2002) and Scarrott and MacDonald (2012).

  8. This is because \(\underset{{\lambda \rightarrow 0}}{\lim } \frac{x^\lambda -1}{\lambda } = \ln x\).

  9. The detailed results for the variants (ii) and (iii) are available upon request.

  10. Random GARCH returns have been produced by estimating the model, drawing randomly (with replacement) from the sample standardised residuals and then using the GARCH equations to construct a simulated return path.

  11. The reason why the GPD distribution is used for the tails rather than the empirical distribution throughout is that the number of observations in the tails may be insufficient to obtain accurate results without using an appropriate fitted distribution.

  12. In the case of the traditional approaches based on specific distributional assumptions, we use the quantile of the theoretical distribution (standard normal, Student-t) instead of (1) and (2).

  13. In our robustness checks in Sect. 5.2, we summarise the results for more general GARCH variants.

  14. This is because the approach suggested by Härdle and Tsybakov (1997) has several undesirable properties. For example, the procedure for estimating conditional variance suffers from significant bias and does not produce estimates that are constrained to be positive. Furthermore, it is sensitive to how well \(\mu (x)\) is estimated.

  15. In the simulation study, they use an interesting stochastic process with Hansen (1994) skewed-t errors, for which the true VaR can be directly calculated. Thus, in repeated sampling, they can answer the question of how close the estimates of different VaR methods are to the true VaR.

  16. See http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

  17. The codes of the series are S&PCOMP, GSCITOT, GOLDBLN, BMUS10Y and BOECGBP.

  18. The fact that heavy-tailed distributions may not possess low-order moments implies that usual significance tests for skewness and kurtosis are most likely unreliable and are not worth reporting (see Paolella 2001).

  19. Note that the statistics take high values partially because of the large sample size (see Bali 2007).

  20. We leave the performance of multi-step-ahead forecasts for future research because problems with the square-root-of-time scaling rule and related techniques must to be resolved first (see McNeil and Frey 2000).

  21. Brooks et al. (2005) use only one out-of-sample period of 250 days and calculate the percentage of days for which the VaRs were exceeded by actual trading losses. Bali (2007) defines a 10-year rolling sample (in one-year increments) to estimate parameters and sets a one-year holdout sample (subsequent to the estimation) to evaluate performance. Kuester et al. (2006) use our approach.

  22. Of course, in a regime of negative interest rates, this point of view can change because then reserves are subject to capital depreciation.

  23. Underpredictions have potentially serious solvency implications in the context of futures margin systems because margin setting is known to be sensitive to the occurrence of large price changes (see Brooks et al. 2005).

  24. This is consistent with the results of Kuester et al. (2006) for the NASDAQ Composite index.

  25. Berkowitz et al. (2011) summarises tests which focus on the duration between violations because, under the null that VaR forecasts are correctly specified, this duration should be completely unpredictable.

  26. In effect, the null hypothesis of the unconditional coverage test will be tested against the alternative of the independence test.

  27. Thus, at a 1% level, the tests require the critical values \(\chi ^2(4) = 13.28\) and \(\chi ^2(5) = 15.09\), respectively (see Domitrescu et al. 2012).

  28. The Basel three-zone framework deems a VaR model acceptable (green zone) if the number of violations of the 99%-VaR is below the 1%-binomial 95% quantile. A model is disputable (yellow zone) up to the 99.99% quantile and is deemed seriously flawed (red zone) whenever more violations occur (see Kuester et al. 2006; Campbell 2007). Thus, with the decision rule ‘reject the null hypothesis of a valid VaR model whenever the model scores red’, the procedure can be interpreted as a significance test, i.e., basically as a one-sided version of the unconditional coverage test (see Ziggel et al. 2014).

  29. Our results indicate that, at least for our dataset, the 5% quantile is still not large enough for the normal assumption to be adequate.

  30. A detailed investigation of this issue is bejond the scope of this article but might be an interesting topic for future research.

  31. Detailed results are available upon request.

  32. This approach is comparable to Bali et al. (2008) where the parameters of the skewed generalised t distribution (which nests the Hansen (1994) skewed-t distribution) are modelled to be time-varying.

  33. Thus, not only the means and the dispersion of extremes are time-varying but also the tail index which measures the fatness of the distribution (or the weight of the tails).

  34. Note that we have also experimented with alternative threshold quotas q. However, both increasing and decreasing our initial value of 10% negatively influences VaR estimator performance. This indicates that the 10% suggestion of McNeil and Frey (2000) is a quite good guideline for a variety of different time series.

  35. Kuester et al. (2006) also show that conditional historical simulation is quite robust to the choice of window length but also that there are some non-EVT parametric methods that outperform this method even as the sample size shrinks.

  36. We have also implemented the bootstrap-based procedures of Escanciano and Olmo (2010) which are designed to address the issue that classic coverage tests are affected by model misspecification in conditional VaR models.

  37. For an nice summary of the past 40 years in financial crises and a review of literature that attempts to identify, classify and explain such episodes, see Anderson (2013) and Claessens and Kose (2013), respectively.

  38. Reinhart and Rogoff (2008), Bartram and Bodnar (2009) and Bordo and Landon-Lane (2010) provide comparisons of the global financial crisis to other crises. Ben-David et al. (2012), Fratzscher (2012) and Flannery et al. (2013) discuss its consequences for hedge fund stock trading, international capital flows and bank opaqueness, respectively.

  39. Nevertheless, an application of the tests of Sect. 4 (and 5.2.1) confirms the results of our descriptive analysis.

  40. Tables 16 and 17 of the Appendix report the corresponding (absolute) averages of daily VaR estimates produced by our different methods.

  41. Nevertheless, ES has its own shortcomings. For example, it is not consistent with right tail risk as measured by the convex order of degree three (see Hürlimann 2004).

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Acknowledgements

We thank an anonymous reviewer for valuable comments and suggestions. Generous financial support was provided by the Deutsche Bundesbank (Hauptverwaltung in Sachsen und Thüringen).

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Authors and Affiliations

  1. Department of Finance, University of Leipzig, Grimmaische Str. 12, 04109, Leipzig, Germany

    Benjamin Mögel & Benjamin R. Auer

  2. Chair of Financial Services, University of Bremen, Hochschulring 4, 28359, Bremen, Germany

    Benjamin R. Auer

  3. Research Network Area Macro, Money and International Finance, CESifo Munich, Schackstr. 4, 80539, Munich, Germany

    Benjamin R. Auer

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  1. Benjamin Mögel
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Correspondence to Benjamin R. Auer.

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Appendix

Appendix

See Tables 16 and 17.

Table 16 Mean VaR estimates—Dotcom crash
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Table 17 Mean VaR estimates—Global financial crisis
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Mögel, B., Auer, B.R. How accurate are modern Value-at-Risk estimators derived from extreme value theory?. Rev Quant Finan Acc 50, 979–1030 (2018). https://doi.org/10.1007/s11156-017-0652-y

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