Coherent risk measures [ADEH], introduced to study both market and nonmarket risks, have four characteristic properties that lead to the term “coherent” present in their name. Coherent risk measures regarded as functionals on the space $L^{\infty }(\Omega ,ℱ,\mathbb{P})$ have been extensively studied [De] with respect to these four properties. In this paper we introduce CRM functionals, defined as isotonic Banach functionals [Al], and use them to characterize coherent risk measures on the space $L^{\infty }(\Omega ,ℱ,\mathbb{P})$ as order opposites of CRM functionals....