Shortfall-dependant Risk Measures and Previsions

Because of their simplicity, risk measures are often employed in financial risk evaluations and related decisions. In fact, the risk measure ρ(X) of a random variable X is a real number customarily determining the amount of money needed to face the potential losses X might cause. At a sort of second-order level, the adequacy of ρ(X) may be investigated considering the part of the losses it does not cover (its shortfall). This may suggest employing a further, more prudential risk measure, taking the shortfall of ρ(X) into account. In this paper a family of shortfalldependant risk measures is proposed, investigating its consistency properties and its utilization in insurance pricing. These results are obtained and subsequently extended within the framework of imprecise previsions, of which risk measures are an instance. This also leads us to investigate properties of a rather weak consistency notion for imprecise previsions, termed 1–convexity.