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Risk Measures: Rationality and Diversification


  • Simone Cerreia-Vioglio
  • Fabio Maccheroni
  • Massimo Marinacci
  • Luigi Montrucchio


When there is uncertainty about interest rates (typically due to either illiquidity or defaultability of zero coupon bonds) the cash- additivity assumption on risk measures becomes problematic. When this assumption is weakened, to cash-subadditivity for example, the equivalence between convexity and the diversi cation principle no longer holds. In fact, this principle only implies (and it is implied by) quasiconvexity. For this reason, in this paper quasiconvex risk measures are studied. We provide a dual characterization of quasiconvex cash-subadditive risk measures and we establish necessary and sufficient conditions for their law invariance. As a byproduct, we obtain an alternative characterization of the actuarial mean value premium principle.

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  • Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci & Luigi Montrucchio, 2008. "Risk Measures: Rationality and Diversification," Carlo Alberto Notebooks 100, Collegio Carlo Alberto.
  • Handle: RePEc:cca:wpaper:100

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    References listed on IDEAS

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    Cited by:

    1. Francesca Biagini & Jean-Pierre Fouque & Marco Frittelli & Thilo Meyer-Brandis, 2015. "A Unified Approach to Systemic Risk Measures via Acceptance Sets," Papers 1503.06354,, revised Apr 2015.
    2. Simone Cerreia‐Vioglio & David Dillenberger & Pietro Ortoleva, 2015. "Cautious Expected Utility and the Certainty Effect," Econometrica, Econometric Society, vol. 83, pages 693-728, March.
    3. Davide La Torre & Marco Maggis, 2012. "A Goal Programming Model with Satisfaction Function for Risk Management and Optimal Portfolio Diversification," Papers 1201.1783,, revised Sep 2012.
    4. Walter Farkas & Pablo Koch-Medina & Cosimo Munari, 2014. "Beyond cash-additive risk measures: when changing the numéraire fails," Finance and Stochastics, Springer, vol. 18(1), pages 145-173, January.
    5. Marco Frittelli & Marco Maggis, 2010. "Dual Representation of Quasiconvex Conditional Maps," Papers 1001.3644,, revised Jan 2010.
    6. Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci & Luigi Montrucchio, 2011. "Complete Monotone Quasiconcave Duality," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 321-339, May.
    7. Enrico G. De Giorgi & Ola Mahmoud, 2016. "Diversification preferences in the theory of choice," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 39(2), pages 143-174, November.
    8. Cerreia-Vioglio, Simone & Maccheroni, Fabio & Marinacci, Massimo & Montrucchio, Luigi, 2012. "Probabilistic sophistication, second order stochastic dominance and uncertainty aversion," Journal of Mathematical Economics, Elsevier, vol. 48(5), pages 271-283.
    9. Dmitry B. Rokhlin, 2011. "On the game interpretation of a shadow price process in utility maximization problems under transaction costs," Papers 1112.2406,, revised Dec 2011.
    10. Giammarino, Flavia & Barrieu, Pauline, 2013. "Indifference pricing with uncertainty averse preferences," Journal of Mathematical Economics, Elsevier, vol. 49(1), pages 22-27.
    11. Samuel Drapeau & Michael Kupper & Antonis Papapantoleon, 2012. "A Fourier Approach to the Computation of CV@R and Optimized Certainty Equivalents," Papers 1212.6732,, revised Dec 2013.
    12. Enrico G. De Giorgi & David B. Brown & Melvyn Sim, 2010. "Dual representation of choice and aspirational preferences," University of St. Gallen Department of Economics working paper series 2010 2010-07, Department of Economics, University of St. Gallen.
    13. Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci & Aldo Rustichini, 2012. "Niveloids and Their Extensions:Risk Measures on Small Domains," Working Papers 458, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    14. Marco Frittelli & Marco Maggis & Ilaria Peri, 2012. "Risk Measures on $\mathcal{P}(\mathbb{R})$ and Value At Risk with Probability/Loss function," Papers 1201.2257,, revised Sep 2012.
    15. Dmitry Rokhlin, 2013. "On the game interpretation of a shadow price process in utility maximization problems under transaction costs," Finance and Stochastics, Springer, vol. 17(4), pages 819-838, October.
    16. Marco Frittelli & Ilaria Peri, 2012. "From Risk Measures to Research Measures," Papers 1205.1012,
    17. Marco Frittelli & Marco Maggis, 2012. "Complete duality for quasiconvex dynamic risk measures on modules of the $L^{p}$-type," Papers 1201.1788,, revised Sep 2012.
    18. Jaume Belles-Sampera & Montserrat Guillén & Miguel Santolino, 2013. "“Beyond Value-at-Risk: GlueVaR Distortion Risk Measures”," IREA Working Papers 201302, University of Barcelona, Research Institute of Applied Economics, revised Feb 2013.

    More about this item


    Risk Measures; Diversification; Cash-subadditivity; Quasiconvexity; Law-invariance; Mean Value Premium Principle;

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

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