Theory and Decision

, Volume 78, Issue 2, pp 289–304 | Cite as

General dual measures of riskiness

  • Klaas Schulze


Aumann and Serrano (J Political Econ 116(5):810–836, 2008) introduce the axiom of duality, which ensures that risk measures respect comparative risk aversion. This paper characterizes all dual risk measures by a simple equivalent condition. This equivalence provides a decomposition result and a construction method, which is used to analyze concrete dual measures. Moreover, this paper aims to extend this characterization to the most general setting. Compared with Aumann and Serrano (2008), it, therefore, relaxes the axiom of positive homogeneity, and allows for risk-neutral and risk-seeking agents, as well as for all integrable gambles.


Risk measures Duality Riskiness Expected utility  Decision-making under risk 



The author thanks the two anonymous referees, Damir Filipović, Eva Lüktebohmert-Holtz, Traian Pirvu, Frank Riedel, and numerous seminar participants for many helpful comments and discussions. The views expressed in this article are the author’s personal opinions and do not necessarily reflect the views of the Deutsche Bundesbank or its staff. This study is supported by the National Centre of Competence in Research “Financial Valuation and Risk Management” (NCCR FINRISK).


  1. Arrow, K. (1965). Aspects of the theory of risk-bearing. Helsinki: Yrjö Jahnssonin Säätiö.Google Scholar
  2. Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.CrossRefGoogle Scholar
  3. Aumann, R., & Serrano, R. (2008). An economic index of riskiness. Journal of Political Economy, 116(5), 810–836.CrossRefGoogle Scholar
  4. Bühlmann, H. (1970). Mathematical methods in risk theory. Berlin: Springer.Google Scholar
  5. Dhaene, J., Goovaerts, M., & Kaas, R. (2003). Economic capital allocation derived from risk measures. North American Actuarial Journal, 7(2), 44–59.CrossRefGoogle Scholar
  6. Diamond, P., & Stiglitz, J. (1974). Increasing risk and risk aversion. Journal of Economic Theory, 8, 337–360.CrossRefGoogle Scholar
  7. Filipović, D., & Svindland, G. (2012). The canonical model space for law-invariant convex risk measures is l1. Mathematical Finance, 22(3), 585–589.CrossRefGoogle Scholar
  8. Föllmer, H., & Schied, A. (2002). Convex measures of risk and trading constraints. Finance and Stochastics, 6(4), 429–447.CrossRefGoogle Scholar
  9. Foster, D., & Hart, S. (2009). An operational measure of riskiness. Journal of Political Economy, 117(5), 785–814.CrossRefGoogle Scholar
  10. Hart, S. (2011). Comparing risks by acceptance and rejection. Journal of Political Economy, 119(4), 617–638.CrossRefGoogle Scholar
  11. Homm, U., & Pigorsch, C. (2012a). An operational interpretation and existence of the Aumann–Serrano index of riskiness. Economics Letters, 114(3), 265–267.CrossRefGoogle Scholar
  12. Homm, U., & Pigorsch, C. (2012b). Beyond the Sharpe ratio: An application of the Aumann–Serrano index to performance measurement. Journal of Banking and Finance, 36(8), 2274–2284.CrossRefGoogle Scholar
  13. Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263–291.CrossRefGoogle Scholar
  14. Pratt, J. (1964). Risk aversion in the small and in the large. Econometrica, 32, 122–136.CrossRefGoogle Scholar
  15. Schnytzer, A., & Westreich, S. (2013). A global index of riskiness. Economics Letters, 118(3), 493–496.CrossRefGoogle Scholar
  16. Schulze, K. (2014). Existence and computation of the Aumann–Serrano index of riskiness and its extension. Journal of Mathematical Economics, 50(1), 219–224.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Deutsche BundesbankDepartment of Banking and Financial SupervisionFrankfurtGermany

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