  In this paper, we study general monetary risk measures (without any convexity
or weak convexity). A monetary (respectively, positively homogeneous) risk
measure can be characterized as the lower envelope of a family of convex
(respectively, coherent) risk measures. The proof does not depend on but easily
leads to the classical representation theorems for convex and coherent risk
measures. When the law-invariance and the SSD (second-order stochastic
dominance)-consistency are involved, it is not the convexity (respectively,
coherence) but the comonotonic convexity (respectively, comonotonic coherence)
of risk measures that can be used for such kind of lower envelope
characterizations in a unified form. The representation of a law-invariant risk
measure in terms of VaR is provided.
