1932
0
  1. Home
  2. A-Z Publications
  3. Annual Review of Financial Economics
  4. Volume 7, 2015
  5. Article

Annual Review of Financial Economics

Volume 7, 2015

The Axiomatic Approach to Risk Measures for Capital Determination

Abstract

The quantification of downside risk in terms of capital requirements is a key issue for both regulators and the financial industry. This review presents the axiomatic approach, which is based on monetary risk measures. These provide a unifying mathematical framework for the determination of capital requirements, for economic indices of riskiness, and for the analysis of preferences in the face of risk and Knightian uncertainty. In the special case of distribution-based risk measures, we review recent advances in characterizing their statistical properties such as elicitability and robustness.

Keyword(s): capital requirements, elicitability, indices of riskiness, monetary risk measures, robustness, variational preferences
Loading

Article metrics loading...

/content/journals/10.1146/annurev-financial-111914-042031
2015-12-07
2025-04-23
Loading full text...

Full text loading...

/deliver/fulltext/financial/7/1/annurev-financial-111914-042031.html?itemId=/content/journals/10.1146/annurev-financial-111914-042031&mimeType=html&fmt=ahah

Literature Cited

  1. Acerbi C, Szekely B. 2014. Back-testing expected shortfall. Risk Nov 17 [Google Scholar]
  2. Acerbi C, Tasche D. 2002. On the coherence of expected shortfall. J. Bank. Finance 26:1487–503 [Google Scholar]
  3. Anscombe F, Aumann R. 1963. A definition of subjective probability. Ann. Math. Stat. 34:199–205 [Google Scholar]
  4. Artzner P, Delbaen F, Eber JM, Heath D. 1999. Coherent measures of risk. Math. Finance 9:203–28 [Google Scholar]
  5. Artzner P, Delbaen F, Koch-Medina P. 2009. Risk measures and efficient use of capital. ASTIN Bull. 39:101–16 [Google Scholar]
  6. Aumann RJ, Serrano R. 2008. An economic index of riskiness. J. Polit. Econ. 116:810–36 [Google Scholar]
  7. BCBS (Basel Comm. Bank. Superv.) 2009. Revisions to the Basel II Market Risk Framework. Basel, Switz: Bank Int. Settl http://www.bis.org/publ/bcbs158.pdf [Google Scholar]
  8. Bellini F, Bignozzi V. 2015. Elicitable risk measures. Quant. Finance 15:725–33 [Google Scholar]
  9. Bellini F, Klar B, Müller A, Rosazza Gianin E. 2014. Generalized quantiles as risk measures. Insur. Math. Econ. 54:41–48 [Google Scholar]
  10. Ben-Tal A, Teboulle M. 1987. Penalty functions and duality in stochastic programming via ϕ-divergence functionals. Math. Oper. Res. 12:224–40 [Google Scholar]
  11. Ben-Tal A, Teboulle M. 2007. An old-new concept of convex risk measures: the optimized certainty equivalent. Math. Finance 17:449–76 [Google Scholar]
  12. Biagini S, Frittelli M. 2009. On the extension of the Namioka–Klee theorem and on the Fatou property for risk measures. Optimality and Risk—Modern Trends in Mathematical Finance F Delbaen, M Rásonyi, C Stricker 1–28 Berlin: Springer [Google Scholar]
  13. Biagini F, Fouque J-P, Frittelli M, Meyer-Brandis T. 2015. A unified approach to systemic risk measures via acceptance sets. arXiv:1503.06354
  14. Brown D, De Giorgi E, Sim M. 2012. Aspirational preferences and their representation by risk measures. Manag. Sci. 58:2095–113 [Google Scholar]
  15. Cerreia-Vioglio S, Maccheroni F, Marinacci M, Montrucchio L. 2011. Risk measures: rationality and diversification. Math. Finance 21:743–74 [Google Scholar]
  16. Cheridito P, Li T. 2009. Risk measures on Orlicz hearts. Math. Finance 19:189–214 [Google Scholar]
  17. Cherny A, Madan D. 2009. New measures for performance evaluation. Rev. Financ. Stud. 22:2571–606 [Google Scholar]
  18. Cont R, Deguest R, He XD. 2013. Loss-based risk measures. Stat. Risk Model. 30:133–67 [Google Scholar]
  19. Cont R, Deguest R, Scandolo G. 2010. Robustness and sensitivity analysis of risk measurement procedures. Quant. Finance 10:593–606 [Google Scholar]
  20. Dana R. 2005. A representation result for concave Schur concave functions. Math. Finance 15:613–34 [Google Scholar]
  21. Davis M. 2013. Consistency of risk measure estimates. arXiv:1410.4382
  22. de Finetti B. 1931. Sul concetto di media. G. Ist. Ital. Attuari 2:369–96 [Google Scholar]
  23. Delbaen F. 2007. Coherent Risk Measures Pisa, Italy: Ed. Sc. Norm. Super. [Google Scholar]
  24. Delbaen F. 2002. Coherent risk measures on general probability spaces. Advances in Finance and Stochastics K Sandmann, PJ Schönbucher 1–37 Berlin: Springer [Google Scholar]
  25. Delbaen F, Bellini F, Bignozzi V, Ziegel JF. 2014. Risk measures with the CxLS property. arXiv:1411.0426
  26. Dhaene J, Kukush A, Pupashenko M. 2006. On the characterization of premium principle with respect to pointwise comonotonicity. Theory Stoch. Process. 12:26–42 [Google Scholar]
  27. Dhaene J, Vanduffel S, Goovaerts MJ, Kaas R, Tang Q, Vyncke D. 2006. Risk measures and comonotonicity: a review. Stoch. Models 22:573–606 [Google Scholar]
  28. Drapeau S. 2010. Risk preferences and their robust representation PhD Thesis, Humboldt-Univ. Berlin, Berlin. http://edoc.hu-berlin.de/dissertationen/drapeau-samuel-2010-04-30/PDF/drapeau.pdf [Google Scholar]
  29. Drapeau S, Kupper M. 2013. Risk preferences and their robust representation. Math. Oper. Res. 38:28–62 [Google Scholar]
  30. Drapeau S, Kupper M, Papapantoleon A. 2014. A Fourier approach to the computation of cv@r and optimized certainty equivalents. J. Risk 16:3–29 [Google Scholar]
  31. Drapeau S, Kupper M, Reda R. 2011. A note on robust representations of law-invariant quasi-convex functions. Adv. Math. Econ. 15:27–39 [Google Scholar]
  32. Dunkel J, Weber S. 2010. Stochastic root finding and efficient estimation of convex risk measures. Oper. Res. 58:1505–21 [Google Scholar]
  33. Embrechts P, Hofert M. 2014. Statistics and quantitative risk management for banking and insurance. Annu. Rev. Stat. Appl. 1:493–514 [Google Scholar]
  34. Embrechts P, McNeil A, Straumann D. 2002. Correlation and dependence in risk management: properties and pitfalls. Risk Management: Value at Risk and Beyond M Dempster 176–223 Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  35. Emmer S, Kratz M, Tasche D. 2013. What is the best risk measure in practice? A comparison of standard measures Work. Pap. 1322, ESSEC Bus. Sch., Paris [Google Scholar]
  36. Feinstein Z, Rudloff B, Weber S. 2015. Measures of systemic risk. arXiv:1502.07961
  37. Filipović D, Svindland G. 2012. The canonical model space for law-invariant convex risk measures is L1. Math. Finance 22:585–89 [Google Scholar]
  38. Filipović D, Vogelpoth N. 2008. A note on the Swiss solvency test risk measure. Insur. Math. Econ. 42:897–902 [Google Scholar]
  39. Fissler T, Ziegel JF. 2015. Higher order elicitability and Osband's principle. arXiv:1503.08123
  40. Fissler T, Ziegel JF, Gneiting T. 2015. Expected Shortfall is jointly elicitable with Value at Risk—implications for backtesting. arXiv:1507.00244
  41. Föllmer H, Klüppelberg C. 2014. Spatial risk measures: local specification and boundary risk. Stochastic Analysis and Applications 2014 D Crisan, B Hambly, T Zariphopoulou 307–326 Springer Proc. Math. Stat. 100 New York: Springer [Google Scholar]
  42. Föllmer H, Knispel T. 2013. Convex risk measures: basic facts, law-invariance and beyond, asymptotics for large portfolios. Handbook of the Fundamentals of Financial Decision Making LC MacLean, WT Ziemba 507–54 [Google Scholar]
  43. Föllmer H, Schied A. 2002. Convex measures of risk and trading constraints. Finance Stoch. 6:429–48 [Google Scholar]
  44. Föllmer H, Schied A. 2011. Stochastic Finance: An Introduction in Discrete Time. Berlin: de Gruyter, 3rd ed.. [Google Scholar]
  45. Föllmer H, Schied A, Weber S. 2009. Robust preferences and robust portfolio choice. Handbook of Numerical Analysis, Mathematical Modeling and Numerical Methods in Finance A Bensoussan, Q. Zhang 29–88 Amsterdam: North-Holland [Google Scholar]
  46. Foster D, Hart S. 2009. An operational measure of riskiness. J. Polit. Econ. 117:758–814 [Google Scholar]
  47. Frittelli M, Rosazza Gianin E. 2002. Putting order in risk measures. J. Bank. Finance 26:1473–86 [Google Scholar]
  48. Frittelli M, Rosazza Gianin E. 2005. Law invariant convex risk measures. Adv. Math. Econ. 7:33–46 [Google Scholar]
  49. FSA (Financ. Serv. Auth.) 2009. The Turner Review: A Regulatory Response to the Global Banking Crisis. London: Financ. Serv. Auth http://www.fsa.gov.uk/pubs/other/turner_review.pdf [Google Scholar]
  50. Gerber H, Goovaerts M. 1981. On the representation of additive principles of premium calculation. Scand. Actuar. J. 4:221–27 [Google Scholar]
  51. Gilboa I, Schmeidler D. 1989. Maxmin expected utility with nonunique prior. J. Math. Econ. 18:141–53 [Google Scholar]
  52. Gneiting T. 2011. Making and evaluating point forecasts. J. Am. Stat. Assoc. 106:746–62 [Google Scholar]
  53. Goovaerts M, Kaas R, Laeven R, Tang Q. 2004. A comonotonic image of independence for additive risk measures. Insur. Math. Econ. 35:581–94 [Google Scholar]
  54. Goovaerts M, Laeven R. 2008. Actuarial risk measures for financial derivative pricing. Insur. Math. Econ. 42:540–47 [Google Scholar]
  55. Hamm AM, Salfeld T, Weber S. 2013. Stochastic root finding for optimized certainty equivalents Proc. 2013 Winter Simul. Conf, pp. 922–32. http://informs-sim.org/wsc13papers/includes/files/080.pdf [Google Scholar]
  56. Hampel FR. 1971. A general qualitative definition of robustness. Ann. Math. Stat. 42:1887–96 [Google Scholar]
  57. Hart S. 2011. Comparing risks by acceptance and rejection. J. Polit. Econ. 119:617–38 [Google Scholar]
  58. Hattendorff K. 1868. Über die Berechnung der Reserven und des Risico bei der Lebensversicherung. Masius' Rundsch. Versicher. 18:169–83 [Google Scholar]
  59. Heath D. 2000. Back to the future Plenary Lect., World Congr. Bachelier Finance Soc., 1st, June 28–July 1, Paris [Google Scholar]
  60. Huber PJ. 1964. Robust estimation of a location parameter. Ann. Math. Stat. 35:73–101 [Google Scholar]
  61. Huber PJ. 1981. Robust Statistics Wiley Ser. Probab. Math. Stat. New York: Wiley [Google Scholar]
  62. Jarrow RA. 2002. Put option premiums and coherent risk measures. Math. Finance 12:135–42 [Google Scholar]
  63. Jouini E, Schachermayer W, Touzi N. 2006. Law invariant risk measures have the Fatou property. Adv. Math. Econ. 9:49–71 [Google Scholar]
  64. Kaina M, Rüschendorf L. 2009. On convex risk measures on Lp-spaces. Math. Methods Oper. Res. 69:475–95 [Google Scholar]
  65. Koenker R. 2005. Quantile Regression. Econometric Society Monographs. New York: Cambridge Univ. Press [Google Scholar]
  66. Kou S, Peng X. 2014. On the measurement of economic tail risk Work. Pap., Dep. Math., Hong Kong Univ. [Google Scholar]
  67. Kou S, Peng X, Heyde CC. 2013. External risk measures and Basel accords. Math. Oper. Res. 38:393–417 [Google Scholar]
  68. Krätschmer V, Schied A, Zähle H. 2014. Comparative and qualitative robustness for law-invariant risk measures. Finance Stoch. 18:271–95 [Google Scholar]
  69. Kreps D. 1988. Notes on the Theory of Choice Boulder, CO: Westview Press [Google Scholar]
  70. Kunze M. 2003. Verteilungsinvariante konvexe Risikomaße Diploma Thesis, Humboldt-Univ. Berlin, Berlin [Google Scholar]
  71. Kusuoka S. 2001. On law invariant coherent risk measures. Adv. Math. Econ. 3:83–95 [Google Scholar]
  72. Maccheroni F, Marinacci M, Rustichini A. 2006. Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74:1447–98 [Google Scholar]
  73. Madan D, Cherny A. 2010. Markets as a counterparty: an introduction to conic finance. Int. J. Theor. Appl. Finance 13:1149–77 [Google Scholar]
  74. Newey WK, Powell JL. 1987. Asymmetric least squares estimation and testing. Econometrica 55:819–47 [Google Scholar]
  75. Osband KH. 1985. Providing incentives for better cost forecasting PhD Thesis, Univ. Calif., Berkeley [Google Scholar]
  76. Savage L. 1954. The Foundations of Statistics New York: Wiley [Google Scholar]
  77. Scandolo G. 2003. Risk measures in a dynamic setting PhD Thesis, University of Milan, Milan, Italy [Google Scholar]
  78. Schmeidler D. 1986. Integral representation without additivity. Proc. Am. Math. Soc. 97:255–61 [Google Scholar]
  79. Song Y, Yan JA. 2009. Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders. Insur. Math. Econ. 45:459–65 [Google Scholar]
  80. Uryasev S, Rockafellar RT. 2001. Conditional value-at-risk: optimization approach. Stochastic Optimization: Algorithms and Applications S. Uryasev, P. Pardalos 411–35 Dordrecht, Neth: Kluwer [Google Scholar]
  81. von Neumann J, Morgenstern O. 1944. Theory of Games and Economic Behavior Princeton, NJ: Princeton Univ. Press [Google Scholar]
  82. Weber S. 2006. Distribution-invariant risk measures, information, and dynamic consistency. Math. Finance 16:419–42 [Google Scholar]
  83. Weber S, Anderson W, Hamm AM, Knispel T, Liese M, Salfeld T. 2013. Liquidity-adjusted risk measures. Math. Financ. Econ. 7:69–91 [Google Scholar]
  84. Yaari M. 1987. The dual theory of choice under risk. Econometrica 55:95–115 [Google Scholar]
  85. Ziegel JF. 2014. Coherence and elicitability. Math. Finance [Google Scholar]
/content/journals/10.1146/annurev-financial-111914-042031
Loading
  • Article Type: Review Article

Most Cited Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error