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Time-consistency of risk measures: how strong is such a property?

Author

Listed:
  • Elisa Mastrogiacomo

    (University of Insubria)

  • Emanuela Rosazza Gianin

    (University of Milano-Bicocca)

Abstract

Quite recently, a great interest has been devoted to time-consistency of risk measures in its different formulations (see Delbaen in Memoriam Paul-André Meyer, Lecture notes in mathematics, vol 1874, pp 215–258, 2006; Föllmer and Penner in Stat Decis 14(1):1–15, 2006; Bion-Nadal in Stoch Process Appl 119:633–654, 2009; Delbaen et al. in Finance Stoch 14(3):449–472, 2010; Laeven and Stadje in Math Oper Res 39:1109–1141, 2014, among many others). However, almost all the papers address to coherent or convex risk measures satisfying cash-additivity. In the present work, we study time-consistency for more general dynamic risk measures where either only cash-invariance or both cash-invariance and convexity are dropped. This analysis is motivated by the recent papers of El Karoui and Ravanelli (Math Finance 19:561–590, 2009) and Cerreia-Vioglio et al. (Math Finance 21(4):743–774, 2011) who discussed and weakened the axioms above by introducing cash-subadditivity and quasi-convexity. In particular, we investigate and discuss whether the notion of time-consistency is too restrictive, when considered in the general framework of quasi-convex and cash-subadditive risk measures. Finally, we provide some conditions guaranteeing time-consistency in this more general framework.

Suggested Citation

  • Elisa Mastrogiacomo & Emanuela Rosazza Gianin, 2019. "Time-consistency of risk measures: how strong is such a property?," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(1), pages 287-317, June.
    • Handle: RePEc:spr:decfin:v:42:y:2019:i:1:d:10.1007_s10203-019-00233-2
    DOI: 10.1007/s10203-019-00233-2
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    References listed on IDEAS

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    More about this item

    Keywords

    Dynamic risk measures; Time-consistency; Quasi-convex risk measures; Cash-subadditive risk measures; Cocycle property; m-stability;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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