## From isotonic Banach functionals to coherent risk measures

### Volume 28 / 2001

#### Abstract

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 $L^{\infty }({\mit \Omega } , {\cal F},{\Bbb P})$ 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 $L^{\infty }({\mit \Omega } ,{\cal F},{\Bbb P})$ as order opposites of CRM functionals. The characterization involves only three axioms and leaves room for a larger class of functionals that can be related to a larger class of possible risks. We show that every CRM functional, when restricted to constant functions, is represented by a convex real function on ${\Bbb R}$ which is linear for nonnegative and nonpositive arguments separately. Next, we show that those CRM functionals which are extensions of the map $ {\Bbb R}\ni t\mapsto \beta t\in {\Bbb R}$, with $\beta >0$, are represented as maxima over a set of positive linear extensions.