Coherent risk measure

In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent risk measure is a function ϱ {displaystyle varrho } that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance. In the fields of actuarial science and financial economics there are a number of ways that risk can be defined; to clarify the concept theoreticians have described a number of properties that a risk measure might or might not have. A coherent risk measure is a function ϱ {displaystyle varrho } that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance. Consider a random outcome X {displaystyle X} viewed as an element of a linear space L {displaystyle {mathcal {L}}} of measurable functions, defined on an appropriate probability space. A functional ϱ : L {displaystyle varrho :{mathcal {L}}} → R ∪ { + ∞ } {displaystyle mathbb {R} cup {+infty }} is said to be coherent risk measure for L {displaystyle {mathcal {L}}} if it satisfies the following properties: