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.
