Risk measure

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 (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.

• 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.

• 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.