We consider financial positions belonging to the Banach lattice of bounded
measurable functions on a given measurable space. We discuss risk measures
generated by general acceptance sets allowing for capital injections to be
invested in a pre-specified eligible asset with an everywhere positive payoff.
Risk measures play a key role when defining required capital for a financial
institution. We address the three critical questions: when is required capital
a well-defined number for any financial position? When is required capital a
continuous function of the financial position? Can the eligible asset be chosen
in such a way that for every financial position the corresponding required
capital is lower than if any other asset had been chosen? In contrast to most
of the literature our discussion is not limited to convex or coherent
acceptance sets and allows for eligible assets that are not necessarily bounded
away from zero. This generality uncovers some unexpected phenomena and opens up
the field for applications to acceptance sets based both on Value-at-Risk and
on Tail Value-at-Risk.