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Abstract

This article reviews methods from extreme value analysis with applications to risk assessment in finance. It covers three main methodological paradigms: the classical framework for independent and identically distributed data with application to risk estimation for market and operational loss data, the multivariate framework for cross-sectional dependent data with application to systemic risk, and the methods for stationary serially dependent data applied to dynamic risk management. The article is addressed to statisticians with interest and possibly experience in financial risk management who are not familiar with extreme value analysis.

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2021-03-07
2024-06-15
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Literature Cited

  1. Acharya VV, Pedersen LH, Philippon T, Richardson M 2017. Measuring systemic risk. Rev. Financ. Stud. 30:2–47
    [Google Scholar]
  2. Adrian T, Brunnermeier MK. 2016. CoVaR. Am. Econ. Rev. 106:71705–41
    [Google Scholar]
  3. Bali TG. 2003. An extreme value approach to estimating volatility and value at risk. J. Bus. 76:83–108
    [Google Scholar]
  4. book 2013. Fundamental review of the trading book: a revised marked risk framework Consult. Doc., Bank Int. Settl Basel, Switz: http://www.bis.org/publ/bcbs265.pdf
    [Google Scholar]
  5. book 2014. Fundamental review of the trading book: outstanding issues Consult. Doc., Bank Int. Settl Basel, Switz: http://www.bis.org/bcbs/publ/d305.pdf
    [Google Scholar]
  6. Barbe P, Fougères A-L, Genest C 2006. On the tail behavior of sums of dependent risks. ASTIN Bull 36:361–73
    [Google Scholar]
  7. book 2006. International convergence of capital measurement and capital standards: a revised framework Tech. Rep., Bank Int. Settl Basel, Switz:.
    [Google Scholar]
  8. Basrak B, Davis RA, Mikosch TV 2002. A characterization of multivariate regular variation. Ann. Appl. Probab. 12:3908–20
    [Google Scholar]
  9. Beirlant J, Goegebeur Y, Segers J, Teugels JL 2006. Statistics of Extremes: Theory and Applications New York: Wiley
    [Google Scholar]
  10. Benoit S, Colliard JE, Hurlin C, Pérignon C 2017. Where the risks lie: a survey on systemic risk. Rev. Financ. 21:109–52
    [Google Scholar]
  11. Berghaus B, Bücher A. 2018. Weak convergence of a pseudo maximum likelihood estimator for the extremal index. Ann. Stat. 46:52307–35
    [Google Scholar]
  12. Brechmann EC, Czado C, Paterlini S 2014. Flexible dependence modeling of operational risk losses and its impact on total capital requirements. J. Bank. Financ. 40:271–85
    [Google Scholar]
  13. Brownlees CT, Engle RF. 2017. SRISK: a conditional capital shortfall measure of systemic risk. Rev. Financ. Stud. 30:48–79
    [Google Scholar]
  14. Cai J-J, Einmahl JHJ, de Haan L 2011. Estimation of extreme risk regions under multivariate regular variation. Ann. Stat. 39:31803–26
    [Google Scholar]
  15. Cai J-J, Einmahl JHJ, de Haan L, Zhou C 2015. Estimation of the marginal expected shortfall: the mean when a related variable is extreme. J. R. Stat. Soc. Ser. B 77:417–42
    [Google Scholar]
  16. Cai J-J, Musta E. 2020. Estimation of the marginal expected shortfall under asymptotic independence. Scand. J. Stat. 47:56–83
    [Google Scholar]
  17. Chan KF, Gray P. 2006. Using extreme value theory to measure value-at-risk for daily electricity spot prices. Int. J. Forecast. 22:283–300
    [Google Scholar]
  18. Chavez-Demoulin V, Embrechts P. 2010. Revisiting the edge, ten years on. Commun. Stat. Theory Methods 39:1674–88
    [Google Scholar]
  19. Chavez-Demoulin V, Embrechts P, Hofert M 2016. An extreme value approach for modeling operational risk losses depending on covariates. J. Risk Insur. 83:735–76
    [Google Scholar]
  20. Chavez-Demoulin V, Guillou A. 2018. Extreme quantile estimation for β-mixing time series and applications. Insur. Math. Econ. 83:59–74
    [Google Scholar]
  21. Coles SG. 2001. An Introduction to Statistical Modeling of Extreme Values New York: Springer
    [Google Scholar]
  22. Cooley D, Thibaud E. 2019. Decompositions of dependence for high-dimensional extremes. Biometrika 106:3587–604
    [Google Scholar]
  23. Cruz MG, Peters GW, Shevchenko PV 2015. Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk New York: Wiley
    [Google Scholar]
  24. Cuberos A, Masiello E, Maume-Deschamps V 2015. High level quantile approximations of sums of risks. Depend. Model. 3:141–58
    [Google Scholar]
  25. Danielsson J, de Haan L, Peng L, de Vries CG 2001. Using a bootstrap method to choose the sample fraction in tail index estimation. J. Multivar. Anal. 76:226–48
    [Google Scholar]
  26. Danielsson J, De Vries CG 1997. Tail index and quantile estimation with very high frequency data. J. Empir. Financ. 4:241–57
    [Google Scholar]
  27. Das B, Fasen-Hartmann V. 2018. Risk contagion under regular variation and asymptotic tail independence. J. Multivar. Anal. 165:194–215
    [Google Scholar]
  28. Das B, Fasen-Hartmann V. 2019. Conditional excess risk measures and multivariate regular variation. Stat. Risk Model. 36:1–41–23
    [Google Scholar]
  29. Das B, Resnick SI. 2011. Conditioning on an extreme component: model consistency with regular variation on cones. Bernoulli 17:226–52
    [Google Scholar]
  30. Davison AC, Huser R. 2015. Statistics of extremes. Annu. Rev. Stat. Appl. 2:203–35
    [Google Scholar]
  31. de Haan L, Ferreira A 2006. Extreme Value Theory: An Introduction New York: Springer
    [Google Scholar]
  32. de Haan L, Mercadier C, Zhou C 2016. Adapting extreme value statistics to financial time series: dealing with bias and serial dependence. Financ. Stoch. 20:321–54
    [Google Scholar]
  33. de Haan L, Sinha AK 1999. Estimating the probability of a rare event. Ann. Stat. 27:2732–59
    [Google Scholar]
  34. de Haan L, Zhou C 2011. Extreme residual dependence for random vectors and processes. Adv. Appl. Probab. 43:1217–42
    [Google Scholar]
  35. de Haan L, Zhou C 2020. Trends in extreme value indices. J. Am. Stat. Assoc. https://doi.org/10.1080/01621459.2019.1705307
    [Crossref] [Google Scholar]
  36. De Jonghe O. 2010. Back to the basics in banking? A micro-analysis of banking system stability. J. Financ. Intermed. 19:3387–417
    [Google Scholar]
  37. Diebold FX, Schuermann T, Stroughair JD 2000. Pitfalls and opportunities in the use of extreme value theory in risk management. J. Risk Financ. 1:30–35
    [Google Scholar]
  38. Draisma G, Drees H, Ferreira A, de Haan L 2004. Bivariate tail estimation: dependence in asymptotic independence. Bernoulli 10:251–80
    [Google Scholar]
  39. Drees H. 2003. Extreme quantile estimation for dependent data with applications to finance. Bernoulli 9:617–57
    [Google Scholar]
  40. Drees H, Janssen A. 2017. Conditional extreme value models: fallacies and pitfalls. Extremes 20:777–805
    [Google Scholar]
  41. Dutta K, Perry J. 2006. A tale of tails: an empirical analysis of loss distribution models for estimating operational risk capital Work. Pap. 06–13 Fed. Reserve Bank Boston, MA:
    [Google Scholar]
  42. Einmahl JHJ, de Haan L, Zhou C 2016. Statistics of heteroscedastic extremes. J. R. Stat. Soc. Ser. B 78:31–51
    [Google Scholar]
  43. Einmahl JHJ, Krajina A, Segers J 2008. A method of moments estimator of tail dependence. Bernoulli 14:1003–26
    [Google Scholar]
  44. Einmahl JHJ, Krajina A, Segers J 2012. An M-estimator for tail dependence in arbitrary dimensions. Ann. Stat. 40:1764–93
    [Google Scholar]
  45. Einmahl JHJ, Yang F, Zhou C 2020. Testing the multivariate regular variation model. J. Bus. Econ. Stat. https://doi.org/10.1080/07350015.2020.1737533
    [Crossref] [Google Scholar]
  46. Embrechts P, Klüppelberg C, Mikosch T 1997. Modelling Extremal Events for Insurance and Finance New York: Springer
    [Google Scholar]
  47. Embrechts P, Lambrigger DD, Wüthrich MV 2009. Multivariate extremes and the aggregation of dependent risks: examples and counter-examples. Extremes 12:107–27
    [Google Scholar]
  48. Engelke S, Hitz AS. 2020. Graphical models for extremes. arXiv:1812.01734 [math.ST]
  49. Escobar-Bach M, Goegebeur Y, Guillou A 2018. Local robust estimation of the Pickands dependence function. Ann. Stat. 46:6A2806–43
    [Google Scholar]
  50. Feller W. 1971. An Introduction to Probability Theory and Its Applications, Vol. II New York: Wiley, 2nd ed..
    [Google Scholar]
  51. Ferro CAT, Segers J. 2003. Inference for clusters of extreme values. J. R. Stat. Soc. Ser. B 65:2545–56
    [Google Scholar]
  52. Gençay R, Selçuk F. 2004. Extreme value theory and value-at-risk: relative performance in emerging markets. Int. J. Forecast. 20:2287–303
    [Google Scholar]
  53. Girard S, Stupfler G. 2015. Extreme geometric quantiles in a multivariate regular variation framework. Extremes 18:4629–63
    [Google Scholar]
  54. Girardi G, Ergün AT. 2013. Systemic risk measurement: multivariate GARCH estimation of CoVaR. J. Bank. Financ. 37:3169–80
    [Google Scholar]
  55. Gissibl N, Klüppelberg C. 2018. Max-linear models on directed acyclic graphs. Bernoulli 24:4A2693–720
    [Google Scholar]
  56. Goldie CM. 1991. Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1:126–66
    [Google Scholar]
  57. Hadley D, Joe H, Nolde N 2019. On the selection of loss severity distributions to model operational risk. J. Oper. Risk 14:73–94
    [Google Scholar]
  58. Hartmann P, Straetmans S, de Vries CG 2004. Asset market linkages in crisis periods. Rev. Econ. Stat. 86:313–26
    [Google Scholar]
  59. Hartmann P, Straetmans S, de Vries CG 2005. Banking system stability: a cross-Atlantic perspective NBER Work. Pap11698
    [Google Scholar]
  60. Hashorva E. 2019. Approximation of some multivariate risk measures for Gaussian risk. J. Multivariate Anal. 169:330–40
    [Google Scholar]
  61. Heffernan J, Resnick SI. 2007. Limit laws for random vectors with an extreme component. Ann. Appl. Probab. 17:537–71
    [Google Scholar]
  62. Heffernan J, Tawn J. 2004. A conditional approach for multivariate extreme values. J. R. Stat. Soc. B 66:497–546
    [Google Scholar]
  63. Hilal S, Poon S-H, Tawn JA 2014. Portfolio risk assessment using multivariate extreme value methods. Extremes 17:531–56
    [Google Scholar]
  64. Hill B. 1975. A simple general approach to inference about the tail of a distribution. Ann. Stat. 3:1163–74
    [Google Scholar]
  65. Hua L, Joe H. 2014. Strength of tail dependence based on conditional tail expectation. J. Multivar. Anal. 123:143–59
    [Google Scholar]
  66. Huang X. 1992. Statistics of Bivariate Extreme Values Amsterdam: Thesis Publ.
    [Google Scholar]
  67. Hyung N, De Vries CG 2005. Portfolio diversification effects of downside risk. J. Financ. Econom. 3:1107–25
    [Google Scholar]
  68. Jansen DW, de Vries CG 1991. On the frequency of large stock returns: putting booms and busts into perspective. Rev. Econ. Stat. 73:118–24
    [Google Scholar]
  69. Joe H, Li H. 2011. Tail risk of multivariate regular variation. Methodol. Comput. Appl. Probab. 13:4671–93
    [Google Scholar]
  70. Kesten H. 1973. Random difference equations and renewal theory for products of random matrices. Acta Math 131:207–48
    [Google Scholar]
  71. Klüppelberg C, Kuhn G, Peng L 2007. Estimating the tail dependence function of an elliptical distribution. Bernoulli 13:229–51
    [Google Scholar]
  72. Konstantinides D. 2017. Risk Theory: A Heavy Tail Approach Singapore: World Sci.
    [Google Scholar]
  73. Krajina A. 2012. A method of moments estimator of tail dependence in meta-elliptical models. J. Stat. Plann. Inference 142:1811–23
    [Google Scholar]
  74. Kulik R, Tong Z. 2019. Estimation of the expected shortfall given an extreme component under conditional extreme value model. Extremes 22:29–70
    [Google Scholar]
  75. Laurini F, Tawn JA. 2008. Regular variation and extremal dependence of GARCH residuals with application to market risk measures. Econom. Rev. 28:146–69
    [Google Scholar]
  76. Leadbetter MR, Rootzen H. 1988. Extremal theory for stochastic processes. Ann. Probab. 16:2431–78
    [Google Scholar]
  77. Ledford A, Tawn JA. 1997. Modelling dependence within joint tail regions. J. R. Stat. Soc. B 59:475–99
    [Google Scholar]
  78. Longin FM. 2000. From value at risk to stress testing: the extreme value approach. J. Bank. Financ. 24:1097–130
    [Google Scholar]
  79. Mainik G, Embrechts P. 2013. Diversification in heavy-tailed portfolios: properties and pitfalls. Ann. Actuar. Sci. 7:126–45
    [Google Scholar]
  80. Mainik G, Rüschendorf L. 2010. On optimal portfolio diversification with respect to extreme risks. Financ. Stochast. 14:4593–623
    [Google Scholar]
  81. McNeil AJ, Frey R. 2000. Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. J. Empir. Financ. 7:271–300
    [Google Scholar]
  82. McNeil AJ, Frey R, Embrechts P 2015. Quantitative Risk Management: Concepts, Techniques, Tools Princeton, NJ: Princeton Univ. Press. , Rev. ed..
    [Google Scholar]
  83. Moscadelli M. 2004. The modelling of operational risk: experiences with the analysis of the data collected by the Basel committee Work. Pap. 517 Bank of Italy Rome:
    [Google Scholar]
  84. Nguyen T, Samorodnitsky G. 2013. Multivariate tail estimation with application to analysis of CoVaR. ASTIN Bull 43:245–70
    [Google Scholar]
  85. Nolde N, Zhang J. 2018. Conditional extremes in asymmetric financial markets. J. Bus. Econ. Stat. 38:201–13
    [Google Scholar]
  86. Oh DH, Patton AJ. 2018. Time-varying systemic risk: evidence from a dynamic copula model of CDS spreads. J. Bus. Econ. Stat. 36:181–95
    [Google Scholar]
  87. Poon S-H, Rockinger M, Tawn JA 2004. Extreme-value dependence in financial markets: diagnostics, models and financial implications. Rev. Financ. Stud. 17:581–610
    [Google Scholar]
  88. Resnick SI. 1987. Extreme Values, Regular Variation, and Point Processes New York: Springer
    [Google Scholar]
  89. Resnick SI. 2002. Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5:303–36
    [Google Scholar]
  90. Resnick SI. 2007. Heavy-Tail Phenomena: Probabilistic and Statistical Modeling New York: Springer
    [Google Scholar]
  91. Schmidt R, Stadtmüller U. 2006. Non-parametric estimation of tail dependence. Scand. J. Stat. 33:2307–35
    [Google Scholar]
  92. Segoviano M, Goodhart C. 2009. Banking stability measures Work. Pap. 09/04, IMF Washington, DC:
    [Google Scholar]
  93. Smith RL. 1987. Estimating tails of probability distributions. Ann. Stat. 15:1174–207
    [Google Scholar]
  94. Smith RL, Weissman I. 1994. Estimating the extremal index. J. R. Stat. Soc. Ser. B 56:3515–28
    [Google Scholar]
  95. Strasburg J. 2018. Barclays to pay $2 billion to resolve mortgage-securities claims. Wall Street Journal Mar. 29. https://www.wsj.com/articles/barclays-to-pay-2-billion-to-resolve-mortgage-securities-claims-1522331649
    [Google Scholar]
  96. journal 2016. The final bill. The Economist Aug. 11. https://www.economist.com/finance-and-economics/2016/08/11/the-final-bill
    [Google Scholar]
  97. van Oordt M, Zhou C 2019a. Estimating systematic risk under extremely adverse market conditions. J. Financ. Econom. 17:3432–61
    [Google Scholar]
  98. van Oordt M, Zhou C 2019b. Systemic risk and bank business models. J. Appl. Econom. 34:3365–84
    [Google Scholar]
  99. Wadsworth JL, Tawn JA, Davison AC, Elton DM 2017. Modelling across extremal dependence classes. J. R. Stat. Soc. B 79:149–75
    [Google Scholar]
  100. Weissman I. 1978. Estimation of parameters and large quantiles based on the k largest observations. J. Am. Stat. Assoc. 73:812–15
    [Google Scholar]
  101. Zhou C. 2010a. Are banks too big to fail? Measuring systemic importance of financial institutions. Int. J. Cent. Bank. 6:205–50
    [Google Scholar]
  102. Zhou C. 2010b. Dependence structure of risk factors and diversification effects. Insur. Math. Econ. 46:531–40
    [Google Scholar]
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