Before revision:  We characterize the class of dynamically consistent measures that provide a tight approximation for a given inconsistent measure , and discuss how the approximation factors can be computed. For the case where the consistent measures are given by Average Value-at-Risk, we derive a polynomial-time algorithm for approximating an arbitrary inconsistent distortion measure.
After revision:  We summarize the relative strengths of the two methods, characterize several necessary and sufficient conditions under which one of the measurements always dominates the other, and introduce a metric to quantify how close the two risk measures are. Using this notion, we address the question of how tightly a given coherent measure can be approximated by lower or upper-bounding compositional measures. We exhibit an interesting asymmetry between the two cases: the tightest possible upper-bound can be exactly characterized, and corresponds to a popular construction in the literature, while the tightest-possible lower bound is not readily available. We show that testing domination and computing the approximation factors is generally NP-hard, even when the risk measures in question are comonotonic and law-invariant. However, we characterize conditions and discuss several examples where polynomial-time algorithm for approximating an arbitrary inconsistent distortion measure.