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

Elisa Mastrogiacomo () and Emanuela Rosazza Gianin ()
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Elisa Mastrogiacomo: University of Insubria
Emanuela Rosazza Gianin: University of Milano-Bicocca

Decisions in Economics and Finance, 2019, vol. 42, issue 1, No 15, 287-317

Abstract: 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.

Keywords: Dynamic risk measures; Time-consistency; Quasi-convex risk measures; Cash-subadditive risk measures; Cocycle property; m-stability; 60G44; 91B30; 91G30; 46A20 (search for similar items in EconPapers)
JEL-codes: G11 G13 G22 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10203-019-00233-2

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