Advertisement

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

  • Benjamin Mögel
  • Benjamin R. Auer
Original Research
  • 276 Downloads

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.

Keywords

Value-at-Risk Extreme value theory Historical simulation Backtest Financial crisis 

JEL Classification

G10 G11 G17 

Notes

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).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Human and Animals rights

Our research does not involve human participants and/or animals.

Informed consent

Informed consent is not relevant in the context of our study.

References

  1. Abad P, Benito S, López C (2014) A comprehensive review of value at risk methodologies. Span Rev Financ Econ 12(1):15–32CrossRefGoogle Scholar
  2. Alexander C (2008) Market risk analysis. Vol. IV—value-at-risk models. Wiley, ChichesterGoogle Scholar
  3. Anderson S (2013) A history of the past 40 years in financial crises. Int Financ Rev 2000:48–52Google Scholar
  4. Angelidis T, Benos A, Degiannakis S (2004) The use of GARCH models in VaR estimation. Stat Methodol 1(1–2):105–128CrossRefGoogle Scholar
  5. Artzner P, Delbaen F, Eber J, Heath D (1999) Coherent measures of risk. Math Financ 9(3):203–228CrossRefGoogle Scholar
  6. Auer B (2015) Does the choice of performance measure influence the evaluation of commodity investments? Int Rev Financ Anal 38:142–150CrossRefGoogle Scholar
  7. Auer B, Schuhmacher F (2015) Liquid betting against beta in Dow Jones Industrial Average stocks. Financ Anal J 71(6):30–43CrossRefGoogle Scholar
  8. Baker GA (1934) Transformation of non-normal frequency distributions into normal distributions. Ann Math Stat 5(2):113–123CrossRefGoogle Scholar
  9. Bali T (2003) The generalized extreme value distribution. Econ Lett 79(3):423–427CrossRefGoogle Scholar
  10. Bali T (2007) A generalized extreme value approach to financial risk measurement. J Money Credit Bank 39(7):1613–1649CrossRefGoogle Scholar
  11. Bali T, Neftci S (2003) Disturbing extremal behavior of spot rate dynamics. J Empir Financ 10(4):455–477CrossRefGoogle Scholar
  12. Bali T, Theodossiou P (2007) A conditional-SGT-VaR-approach with alternative GARCH models. Ann Oper Res 151(1):241–267CrossRefGoogle Scholar
  13. Bali T, Weinbaum D (2007) A conditional extreme value volatility estimator based on high frequency data. J Econ Dyn Control 31(2):361–397CrossRefGoogle Scholar
  14. Bali T, Mo H, Tang Y (2008) The role of autoregressive conditional skewness and kurtosis in the estimation of conditional VaR. J Bank Financ 32(2):269–282CrossRefGoogle Scholar
  15. Balkema A, de Haan L (1974) Residual life time at great age. Ann Probab 2(5):792–804CrossRefGoogle Scholar
  16. Barone-Adesi G, Giannopoulos K (2001) Non-parametric VaR techniques: myths and realities. Econ Notes 30(2):167–181CrossRefGoogle Scholar
  17. Bartram S, Bodnar G (2009) No place to hide: the global crisis in equity markets in 2008/2009. J Int Money Financ 28(8):1246–1292CrossRefGoogle Scholar
  18. Basel Committee on Banking Supervision (1995) An internal model-based approach to market risk capital requirements. www.bis.org
  19. Basel Committee on Banking Supervision (1996a) Overview of the amendment to the capital accord to incorporate market risks. www.bis.org
  20. Basel Committee on Banking Supervision (1996b) Supervisory framework for the use of “backtesting” in conjunction with the internal models approach to market risk capital requirements. www.bis.org
  21. Basel Committee on Banking Supervision (2009) Revisions to the Basel II market risk framework. www.bis.org
  22. Basel Committee on Banking Supervision (2011a) Basel III: a global regulatory framework for more resilient banks and banking systems. www.bis.org
  23. Basel Committee on Banking Supervision (2011b) Messages from the academic literature on risk measurement for the trading book. www.bis.org
  24. Baur D, Lucey B (2010) Is gold a hedge or a safe haven? An analysis of stocks, bonds and gold. Financ Rev 45(2):217–229CrossRefGoogle Scholar
  25. Baur D, McDermott T (2010) Is gold a safe haven? International evidence. J Bank Financ 34(8):1886–1898CrossRefGoogle Scholar
  26. Ben-David I, Franzoni F, Moussawi R (2012) Hedge fund stock trading in the financial crisis of 2007–2009. Rev Financ Stud 25(1):1–54CrossRefGoogle Scholar
  27. Berkowitz J (2001) Testing density forecasts with applications to risk management. J Bus Econ Stat 19(4):465–474CrossRefGoogle Scholar
  28. Berkowitz J, O’Brien J (2002) How accurate are value-at-risk models at commercial banks? J Financ 57(3):1093–1111CrossRefGoogle Scholar
  29. Berkowitz J, Christoffersen P, Pelletier D (2011) Evaluating value-at-risk models with desk-level data. Manag Sci 57(12):2213–2227CrossRefGoogle Scholar
  30. Bianchi R, Drew M, Fan J (2015) Combining momentum with reversal in commodity futures. J Bank Financ 59:423–444CrossRefGoogle Scholar
  31. Bollerslev T (1986) Generalized autoregressive conditional heteroskedasticity. J Econom 31(3):307–327CrossRefGoogle Scholar
  32. Bollerslev T, Chou R, Kroner K (1992) ARCH-modeling in finance: a review of the theory and empirical evidence. J Econom 52(1–2):5–59CrossRefGoogle Scholar
  33. Bollerslev T, Engle R, Nelson D (1994) ARCH models. In: Engle R, McFadden D (eds) Handbook of econometrics, vol 4. Elsevier, Amsterdam, pp 2959–3038Google Scholar
  34. Bordo M, Landon-Lane J (2010) The global financial crisis of 2007-08: Is it unprecedented? NBER Working Paper No. 16589Google Scholar
  35. Boulter T, Wongchan V (2013) Thai hedging practices post-Asian financial crisis. Rev Pac Basin Financ Mark Polic 16(1):1350003:1–1350003:25Google Scholar
  36. Bowman K, Shenton L (2004) Johnson’s system of distributions. In: Encyclopedia of statictical sciences. Wiley, New YorkGoogle Scholar
  37. Box G, Cox D (1964) An analysis of transformations. J Roy Stat Soc B 26(2):211–252Google Scholar
  38. Brooks C, Clare AD, Dalle Molle JW, Persand G (2005) A comparison of extreme value theory approaches for determining value at risk. J Empir Financ 12(2):339–352CrossRefGoogle Scholar
  39. Campbell S (2007) A review of backtesting and backtesting procedures. J Risk 9(2):1–17CrossRefGoogle Scholar
  40. Campbell J, Grossman S, Wang J (1993) Trading volume and serial correlation in stock returns. Q J Econ 108(4):905–939CrossRefGoogle Scholar
  41. Campbell J, Lo A, MacKinlay A (1997) The econometrics of financial markets. Princeton University Press, PrincetonGoogle Scholar
  42. Campbell R, Huisman R, Koedijk K (2001) Optimal portfolio selection in a value-at-risk framework. J Bank Financ 25(9):1789–1804CrossRefGoogle Scholar
  43. Candelon B, Colletaz G, Hurlin C, Tokpavi S (2011) Backtesting value-at-risk: a GMM duration-based test. J Financ Econom 9(2):314–343CrossRefGoogle Scholar
  44. Carroll R, Härdle W, Mammen E (2002) Estimation in an additive model when the components are linked parametrically. Econom Theory 18(4):886–912CrossRefGoogle Scholar
  45. Castillo E (1988) Extreme value theory in enginering. Academic Press, New YorkGoogle Scholar
  46. Chen J (2014) Measuring market risk under the basel accords: VaR, stressed VaR, and expected shortfall. IEB Int J Financ 8:184–201Google Scholar
  47. Chen Y, Lu J (2012) Value at risk estimation. In: Duan J, Härdle W, Gentle J (eds) Handbook of computational finance. Springer, Berlin, pp 307–334CrossRefGoogle Scholar
  48. Cheng W, Hung J-C (2011) Skewness and leptokurtosis in GARCH-type VaR estimation of petroleum and metal asset returns. J Empir Financ 18(1):160–173CrossRefGoogle Scholar
  49. Christoffersen P (1998) Evaluating interval forecasts. Int Econ Rev 39(4):841–862CrossRefGoogle Scholar
  50. Christoffersen P (2003) Elements of financial risk management. Academic Press, New YorkGoogle Scholar
  51. Ciner C, Gurdgiev V, Lucey B (2013) Hedges and safe havens: an examination of stocks, bonds, gold, oil and exchange rates. Int Rev Financ Anal 29:202–211CrossRefGoogle Scholar
  52. Claessens S, Kose M (2013) Financial crises: explanations, types, and implications. IMF Working Paper No. 13/28Google Scholar
  53. Cont R (2001) Empirical properties of asset returns: stylized facts and statistical issues. Quant Financ 1(2):223–236CrossRefGoogle Scholar
  54. Cox D, Hinkley D (1974) Theoretical statistics. Chapman and Hall, LondonCrossRefGoogle Scholar
  55. Danielsson J, de Vries C (2000) Value-at-risk and extreme returns. Ann Econ Stat 60:239–270Google Scholar
  56. Danielsson J, Morimoto Y (2000) Forecasting extreme financial risk: a critical analysis of practical methods for the Japanese market. Monet Econ Stud 18(2):25–48Google Scholar
  57. De Haan L, Resnick S (1980) A simple asymptotic estimate for the index of a stable distribution. J Roy Stat Soc 42(1):83–87Google Scholar
  58. Diebold J, Guégan D (1993) Tail behaviour of the stationary density of general non-linear autoregressive processes of order 1. J Appl Probab 30(2):315–329CrossRefGoogle Scholar
  59. Diebold FX, Schuermann T, Stroughair J (2000) Pitfalls and opportunities in the use of extreme value theory in risk management. J Risk Financ 1(2):30–35CrossRefGoogle Scholar
  60. Domitrescu E, Hurlin C, Pham V (2012) Backtesting value-at-risk: from dynamic quantile to dynamic binary tests. Finance 33(1):79–122Google Scholar
  61. Dwyer G (September 2009) Stock prices in the financial crisis. Notes from the Vault, Federal Reserve Bank of AtlantaGoogle Scholar
  62. Efron B (1982) The jackknife, the bootstrap, and other resampling plans. Society for Industrial and Applied Mathematics, PhiladelphiaGoogle Scholar
  63. El-Aroui M, Diebold J (2002) On the use of the peaks over thresholds method for estimating out-of-sample quantiles. Comput Stat Data Anal 39(4):453–475CrossRefGoogle Scholar
  64. Eling M, Schuhmacher F (2007) Does the choice of performance measure influence the evaluation of hedge funds? J Bank Financ 31(9):2632–2647CrossRefGoogle Scholar
  65. Embrechts P, Klppelberg C, Mikosch T (1997) Modelling extremal events: for insurance and finance. Springer, BerlinCrossRefGoogle Scholar
  66. Embrechts P, Resnick S, Samorodnitsky G (1999) Extreme value theory as a risk management tool. N Am Actuar J 3(2):30–41CrossRefGoogle Scholar
  67. Engle R, Manganelli S (2004) CAViaR: conditional autoregressive value at risk by regression quantiles. J Bus Econ Stat 22(4):367–381CrossRefGoogle Scholar
  68. Engle R, Patton A (2001) What good is a volatility model? Quant Financ 1(2):237–245CrossRefGoogle Scholar
  69. Escanciano J, Olmo J (2010) Backtesting parametric value-at-risk with estimation risk. J Bus Econ Stat 28(1):36–51CrossRefGoogle Scholar
  70. Fan J (1992) Design-adaptive nonparametric regression. J Am Stat Assoc 87:998–1004CrossRefGoogle Scholar
  71. Fan J, Yao Q (1998) Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85(3):645–660CrossRefGoogle Scholar
  72. Ferreira A, De Haan L (2015) On the block maxima method in extreme value theory: PWM estimators. Ann Stat 43(1):276–298CrossRefGoogle Scholar
  73. Fisher RA, Tippett LHC (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Math Proc Camb Philos Soc 24(2):180–190CrossRefGoogle Scholar
  74. Flannery M, Kwan S, Nimalendran M (2013) The 2007–2009 financial crisis and bank opaqueness. J Financ Intermed 22(1):55–84CrossRefGoogle Scholar
  75. Florence G, Ramachandran K (2011) Estimation of parameters of Johnson’s system of distributions. J Mod Appl Stat Methods 10(2):494–504CrossRefGoogle Scholar
  76. Fratzscher M (2012) Capital flows, push versus pull factors and the global financial crisis. J Int Econ 88(2):341–356CrossRefGoogle Scholar
  77. Gaglianone W, Lima L, Linton O, Smith D (2011) Evaluating value-at-risk models via quantile regressions. J Bus Econ Stat 29(1):150–160CrossRefGoogle Scholar
  78. Gençay R, Selçuk F (2004) Extreme value theory and value-at-risk: relative performance in emerging markets. Int J Forecast 20(2):287–303CrossRefGoogle Scholar
  79. Gilli M, Këllezi E (2006) An application of extreme value theory for measuring financial risk. Comput Econ 27(2):207–228CrossRefGoogle Scholar
  80. Gnedenko B (1943) Sur La Distribution Limite Du Terme D’une Série Aléatoire. Ann Math 4(3):423–453CrossRefGoogle Scholar
  81. Gumbel E (1958) Statistics of extremes. Columbia University Press, New YorkGoogle Scholar
  82. Hafner C (1998) Nonlinear time series analysis with applications to foreign exchange rate volatility. Physica, HeidelbergCrossRefGoogle Scholar
  83. Hansen B (1994) Autoregressive conditional density estimation. Int Econ Rev 35(3):705–730CrossRefGoogle Scholar
  84. Härdle W, Tsybakov A (1997) Local polynomial estimators of the volatility function in nonparametric autoregression. J Econom 81(1):223–242CrossRefGoogle Scholar
  85. Hill B (1975) A simple general approach to inference about the tail of a distribution. Ann Stat 3(5):1163–1174CrossRefGoogle Scholar
  86. Ho L, Burridge P, Cadle J, Theobald M (2000) Value-at-risk: applying the extreme value approach to Asian markets in the recent financial turmoil. Pac Basin Financ J 8(2):249–275CrossRefGoogle Scholar
  87. Hoel P (1954) A test for Markoff chains. Biometrika 41(3/4):430–433CrossRefGoogle Scholar
  88. Hosking J (1989) Some theoretical results concerning L-moments. IBM Research Division, Technical ReportGoogle Scholar
  89. Hosking J (1990) L-moments: analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc B 52(1):105–124Google Scholar
  90. Hosking J, Wallis J (1987) Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29(3):339–349CrossRefGoogle Scholar
  91. Hosking J, Wallis J, Wood E (1985) Estimation of the generalized extreme value distribution by the method of probability weighted moments. Technometrics 27(3):251–261CrossRefGoogle Scholar
  92. Hosking J, Wallis J (1997) Regional frequency analysis: an approach based on L-moments. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  93. Hsieh D (1993) Implications of nonlinear dynamics for financial risk management. J Financ Quant Anal 28(1):41–64CrossRefGoogle Scholar
  94. Hsu C, Huang C, Chiou W (2012) Effectiveness of copula-extreme value theory in estimating value-at-risk: empirical evidence from Asian emerging markets. Rev Quant Financ Acc 39(4):447–468CrossRefGoogle Scholar
  95. Huang Y, Lin B (2004) Value-at-risk analysis for Taiwan stock index futures: fat tails and conditional asymmetries in return innovations. Rev Quant Financ Account 22(2):79–95CrossRefGoogle Scholar
  96. Huisman R, Koedijk K, Kool C, Palm F (2001) Tail index estimates in small samples. J Bus Econ Stat 19(2):208–216CrossRefGoogle Scholar
  97. Hürlimann W (2004) Distortion risk measures and economic capital. N Am Actuar J 8(1):86–96CrossRefGoogle Scholar
  98. Hwang S, Vallis Pereira P (2006) Small sample properties of GARCH estimates and persistence. Eur J Financ 12(6–7):473–494CrossRefGoogle Scholar
  99. Jenkinson A (1955) The frequency distribution of the annual maximum (minimum) values of meteorological elements. Q J R Meteorol Soc 81(348):158–171CrossRefGoogle Scholar
  100. Johnson N (1949) Systems of frequency curves generated by methods of translation. Biometrika 36(1/2):149–176CrossRefGoogle Scholar
  101. Jorion P (2007) Value at risk: the new benchmark for managing financial risk, 3rd edn. Mc-Graw-Hill, New YorkGoogle Scholar
  102. Jou Y, Wang C, Chiu W (2013) Is the realized volatility good for option pricing during the recent financial crisis? Rev Quant Financ Acc 40(1):171–188CrossRefGoogle Scholar
  103. Kinateder H (2016) Basel II versus III—a comparative assessment of minimum capital requirements for internal model approaches. J Risk 18(3):25–45CrossRefGoogle Scholar
  104. Kuester K, Mittnik S, Paolella M (2006) Value-at-risk prediction: a comparison of alternative strategies. J Financ Econom 4(1):53–89CrossRefGoogle Scholar
  105. Kupiec P (1995) Techniques for verifying the accuracy of risk measurement models. J Deriv 3(2):73–84CrossRefGoogle Scholar
  106. Landwehr J, Matalas N, Wallis J (1979) Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles. Water Resour Res 15(5):1055–1064CrossRefGoogle Scholar
  107. Lange K, Little R, Taylor J (1989) Robust statistical modeling using the t distribution. J Am Stat Assoc 84(408):881–896Google Scholar
  108. Leadbetter M, Lindgren G, Rootzén H (1983) Extremes and related properties of random sequences and processes. Springer, New YorkCrossRefGoogle Scholar
  109. Lee C, Su J (2012) Alternative statistical distributions for estimating value-at-risk: theory and evidence. Rev Quant Financ Account 39(3):309–331CrossRefGoogle Scholar
  110. Linsmeier T, Pearson N (2000) Value at risk. Financ Anal J 56(2):47–67CrossRefGoogle Scholar
  111. Linton O, Nielsen J (1995) A Kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82(1):93–100CrossRefGoogle Scholar
  112. Ljung G, Box G (1978) On a measure of lack of fit in time series models. Biometrika 65(2):297–303CrossRefGoogle Scholar
  113. Longin F (2000) From value at risk to stress testing: the extreme value approach. J Bank Financ 24(7):1097–1130CrossRefGoogle Scholar
  114. Lopez J (1997) Regulatory evaluation of value-at-risk models. Federal Reserve Bank of New York, Staff Report No. 33Google Scholar
  115. Lopez J (1999) Methods for evaluating value-at-risk estimates. Fed Reserve Bank San Franc Econ Rev 2:3–17Google Scholar
  116. Marimoutou V, Raggad B, Trabelsi A (2009) Extreme value theory and value at risk: application to oil market. Energy Econ 31(4):519–530CrossRefGoogle Scholar
  117. Martins-Filho C, Yao F (2006) Estimation of value-at-risk and expected shortfall based on nonlinear models of return dynamics and extreme value theory. Stud Nonlinear Dyn Econom 10(2):1–41 Article 4Google Scholar
  118. McNeil A (1999) Extreme value theory for risk managers. In: Internal modeling and CAD II. Risk Books, London, pp 93–113Google Scholar
  119. McNeil A, Frey R (2000) Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. J Empir Financ 7(3–4):271–300CrossRefGoogle Scholar
  120. Nadarajah S, Zhang B, Chan S (2014) Estimation methods for expected shortfall. Quant Financ 14(2):271–291CrossRefGoogle Scholar
  121. Neftci S (2000) Value at risk calculations, extreme events, and tail estimation. J Deriv 7(3):23–37CrossRefGoogle Scholar
  122. Ofek E, Richardson M (2003) DotCom mania: the rise and fall of internet stock prices. J Financ 58(3):1113–1137CrossRefGoogle Scholar
  123. Oja H (1981) On location, scale, skewness and kurtosis of univariate distributions. Scand J Stat 8(3):154–168Google Scholar
  124. Paolella M (2001) Testing the stable Paretian assumption. Math Comput Modell 34:1095–1112CrossRefGoogle Scholar
  125. Patton A (2004) On the out-of-sample importance of skewness and asymmetric dependence for asset allocation. J Financ Econom 2(1):130–168CrossRefGoogle Scholar
  126. Pérignon C, Smith D (2010) The level and quality of value-at-risk disclosure by commercial banks. J Bank Financ 34(9–11):362–377CrossRefGoogle Scholar
  127. Pflug G, Römisch W (2007) Modeling, measuring and managing risk. World Scientific, SingaporeCrossRefGoogle Scholar
  128. Pickands J (1975) Statistical inference using extreme order statistics. Ann Stat 3(1):119–131CrossRefGoogle Scholar
  129. Pritsker M (2006) The hidden dangers of historical simulation. J Bank Financ 30(2):561–582CrossRefGoogle Scholar
  130. Reinhart C, Rogoff K (2008) Is the 2007 US-sub-prime financial crisis so different? An international comparison. Am Econ Rev 98(2):339–344CrossRefGoogle Scholar
  131. Resnick S (1987) Extreme values, regular variation, and point processes. Springer, New YorkCrossRefGoogle Scholar
  132. Rocco M (2014) Extreme value theory in finance: a survey. J Econ Surv 28(1):82–108CrossRefGoogle Scholar
  133. Ruppert S, Wand M (1995) An effective bandwidth selection for local least squares regression. J Am Stat Assoc 90(432):1257–1270CrossRefGoogle Scholar
  134. Scarrott C, MacDonald A (2012) A review of extreme value threshold estimation and uncertainty quantification. REVSTAT Stat J 10(1):33–60Google Scholar
  135. Silvennoinen A, Teräsvirta T (2009) Multivariate GARCH models. In: Anderson T, Davis R, Kreiß J, Mikosch T (eds) Handbook of financial time series. Springer, Berlin, pp 201–229CrossRefGoogle Scholar
  136. Smith R (1984) Threshold methods for sample extremes. In: de Oliveira J (ed) Statistical extremes and applications. Springer, Dordrecht, pp 621–638CrossRefGoogle Scholar
  137. Smith R (1987) Estimating tails of probability distributions. Ann Stat 15(3):1174–1207CrossRefGoogle Scholar
  138. Tauchen G (2001) Notes on financial econometrics. J Econom 100(1):57–64CrossRefGoogle Scholar
  139. Taylor N (2014) The rise and fall of technical trading rule success. J Bank Financ 40:286–302CrossRefGoogle Scholar
  140. Tsay R (2005) Analysis of financial time series, 2nd edn. Wiley, HobokenCrossRefGoogle Scholar
  141. Vogel R, Fennessey N (1993) L moment diagrams should replace product moment diagrams. Water Resour Res 29(6):1745–1752CrossRefGoogle Scholar
  142. von Mises R (1954) La Distribution De La Plus Grande de n Valeurs. In: AMS Selected Papers. Vol. 2. American Statistical Society, Providence, pp 271–294Google Scholar
  143. Wang Q (1990) Estimation of the GEV distribution from censored samples by method of partial probability weighted moments. J Hydrol 120(1–4):103–114CrossRefGoogle Scholar
  144. Wong W (2008) Backtesting trading risk of commercial banks using expected shortfall. J Bank Financ 32(7):1404–1415CrossRefGoogle Scholar
  145. Wong K, Fan G, Zeng Y (2012) Capturing tail risks beyond VaR. Rev Pac Basin Financ Mark Polic 15(3):1250015:1–1250015:25Google Scholar
  146. Yamai Y, Yoshiba T (2002) Comparative analyses of expected shortfall and value-at-risk: their estimation error, decomposition, and optimization. Monet Econ Stud 20(1):87–122Google Scholar
  147. Yamai Y, Yoshiba T (2005) Value-at-risk versus expected shortfall: a practical perspective. J Bank Financ 29(4):997–1015CrossRefGoogle Scholar
  148. Ziggel D, Berens T, Weiß G, Wied D (2014) A new set of improved value-at-risk backtests. J Bank Financ 48:29–41CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of FinanceUniversity of LeipzigLeipzigGermany
  2. 2.Chair of Financial ServicesUniversity of BremenBremenGermany
  3. 3.Research Network Area Macro, Money and International FinanceCESifo MunichMunichGermany

Personalised recommendations