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Value at risk estimation by quantile regression and kernel estimator

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Abstract

Risk management has attracted a great deal of attention, and Value at Risk (VaR) has emerged as a particularly popular and important measure for detecting the market risk of financial assets. The quantile regression method can generate VaR estimates without distributional assumptions; however, empirical evidence has shown the approach to be ineffective at evaluating the real level of downside risk in out-of-sample examination. This paper proposes a process in VaR estimation with methods of quantile regression and kernel estimator which applies the nonparametric technique with extreme quantile forecasts to realize a tail distribution and locate the VaR estimates. Empirical application of worldwide stock indices with 29 years of data is conducted and confirms the proposed approach outperforms others and provides highly reliable estimates.

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Notes

  1. In 1994, JP Morgan published its risk-measured program RiskMetrics which, for the first time, systematically developed detailed methodologies for VaR. In 1996, the Basel committee on banking supervision amended the Basel Capital Accord and obliged their member banks to reserve capital requirements calculated based on VaR.

  2. Baixauli and Alvarez (2006) showed that accurate VaR estimates can be produced with correct characterization of a left-tail distribution.

  3. For examples, see Gourieroux et al. (2000), Costello et al. (2008), Chen (2008), Chen and Liao (2009), and Lu et al. (2012).

  4. See Table 1 in page 372 of Engle and Manganelli (2004).

  5. See Table 5 in page 81 of Kuester et al. (2006).

  6. In empirical applications of this study, the process is applied rolling through the entire out-of-sample, so the tail distributions are generated for each day and are different from each other. The demonstration in the figure is done as an illustration from two selected dates: August 31, 2009 (the last sampling day) and September 1, 2007 (2 years prior to the last date).

  7. Both settings are empirically rational choices rather than selections according to statistical inferences. Thousandth quantile forecasts in empirical applications are considered very frequent and the threshold of 0.02 is reasonably extreme for determining the 1 % unconditional quantile.

  8. The two indices are extracted with different sample periods due to data availability and stability.

  9. Samples of emerging markets also end at August 31, 2009.

  10. Both VaR estimate series are transformed into percentage format to coincide with return series in same scale.

  11. The only exception is that the QR-symmetric model also has a favorable unconditional likelihood ratio test at a 10 % significance level for the FTSE100.

  12. With only one exception where both QR-asymmetric and KQ-asymmetric models have the same back-testing outcome of 38 for the Russian stock index with out-of-samples of 2,858.

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Acknowledgments

This research is supported in part by the National Science Council of Taiwan (NSC100-2632-H-155-001-MY2). The author thanks Wei-Hwa Peng and Li-Chiang Chen for their excellent research assistance.

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  1. College of Management, Yuan Ze University, 135 Yuan-Tung Road, Chung-Li, Taoyuan, 32003, Taiwan

    Alex YiHou Huang

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Correspondence to Alex YiHou Huang.

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Huang, A.Y. Value at risk estimation by quantile regression and kernel estimator. Rev Quant Finan Acc 41, 225–251 (2013). https://doi.org/10.1007/s11156-012-0308-x

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