  In the literature of risk measures, cash subadditivity was proposed to
replace cash additivity, motivated by the presence of stochastic or ambiguous
interest rates and defaultable contingent claims. Cash subadditivity has been
traditionally studied together with quasi-convexity, in a way similar to cash
additivity with convexity. In this paper, we study cash-subadditive risk
measures without quasi-convexity. One of our major results is that a general
cash-subadditive risk measure can be represented as the lower envelope of a
family of quasi-convex and cash-subadditive risk measures. Representation
results of cash-subadditive risk measures with some additional properties are
also examined. The notion of quasi-star-shapedness, which is a natural analogue
of star-shapedness, is introduced and we obtain a corresponding representation
result.
