Optimizing static risk-averse objectives in Markov decision processes is challenging because they do not readily admit dynamic programming decompositions. Prior work has proposed to use a dynamic decomposition of risk measures that help to formulate dynamic programs on an augmented state space. This paper shows that several existing decompositions are inherently inexact, contradicting several claims in the literature. In particular, we give examples that show that popular decompositions for CVaR and EVaR risk measures are strict overestimates of the true risk values. However, an exact decomposition is possible for VaR, and we give a simple proof that illustrates the fundamental difference between VaR and CVaR dynamic programming properties.

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