Time-Consistency of Risk Measures: How Strong is Such a Property?

Abstract

Quite recently, a great interest has been devoted to time-consistency of risk measures in its different formulations (see Delbaen, Follmer and Penner, Bion-Nadal, Delbaen et al., Laeven and Stadje, 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 and Cerreia-Vioglio et al. 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

JEL Classification: G11, G13, G22