Mathematically

A set-valued risk measure is a function ${\displaystyle R:L_{d}^{p}\rightarrow \mathbb {F} _{M}}$, where ${\displaystyle L_{d}^{p}}$ is a ${\displaystyle d}$-dimensional Lp space, ${\displaystyle \mathbb {F} _{M}=\{D\subseteq M:D=cl(D+K_{M})\}}$, and ${\displaystyle K_{M}=K\cap M}$ where ${\displaystyle K}$ is a constant solvency cone and ${\displaystyle M}$ is the set of portfolios of the ${\displaystyle m}$ reference assets. ${\displaystyle R}$ must have the following properties:^[3]

Normalized

${\displaystyle K_{M}\subseteq R(0)\;\mathrm {and} \;R(0)\cap -\mathrm {int} K_{M}=\emptyset }$

Translative in M

${\displaystyle \forall X\in L_{d}^{p},\forall u\in M:R(X+u1)=R(X)-u}$

Monotone

${\displaystyle \forall X_{2}-X_{1}\in L_{d}^{p}(K)\Rightarrow R(X_{2})\supseteq R(X_{1})}$

Variance

Variance (or standard deviation) is not a risk measure in the above sense. This can be seen since it has neither the translation property nor monotonicity. That is, ${\displaystyle Var(X+a)=Var(X)\neq Var(X)-a}$ for all ${\displaystyle a\in \mathbb {R} }$, and a simple counterexample for monotonicity can be found. The standard deviation is a deviation risk measure. To avoid any confusion, note that deviation risk measures, such as variance and standard deviation are sometimes called risk measures in different fields.

Risk measure to acceptance set

• If ${\displaystyle \rho }$ is a (scalar) risk measure then ${\displaystyle A_{\rho }=\{X\in L^{p}:\rho (X)\leq 0\}}$ is an acceptance set.
• If ${\displaystyle R}$ is a set-valued risk measure then ${\displaystyle A_{R}=\{X\in L_{d}^{p}:0\in R(X)\}}$ is an acceptance set.

Acceptance set to risk measure

• If ${\displaystyle A}$ is an acceptance set (in 1-d) then ${\displaystyle \rho _{A}(X)=\inf\{u\in \mathbb {R} :X+u1\in A\}}$ defines a (scalar) risk measure.
• If ${\displaystyle A}$ is an acceptance set then ${\displaystyle R_{A}(X)=\{u\in M:X+u1\in A\}}$ is a set-valued risk measure.