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In a dynamic framework, we identify a new concept associated with the risk ofassessing the financial exposure by a measure that is not adequate to theactual time horizon of the position. This will be called horizon risk. Weclarify that dynamic risk measures are subject to horizon risk, so we proposeto use the fully-dynamic version. To quantify horizon risk, we introduceh-longevity as an indicator. We investigate these notions together with otherproperties of risk measures as normalization, restriction property, anddifferent formulations of time-consistency. We also consider these concepts forfully-dynamic risk measures generated by backward stochastic differentialequations (BSDEs), backward stochastic Volterra integral equations (BSVIEs),and families of these. Within this study, we provide new results for BSVIEssuch as a converse comparison theorem and the dual representation of theassociated risk measures.

Fully-dynamic risk measures: horizon risk, time-consistency, and relations with BSDEs and BSVIEs