Risk measure

Set-valued

In a situation with ${\displaystyle {\mathbb{R}}^d}$-valued portfolios such that risk can be measured in ${\displaystyle m\le d}$ of the assets, then a set of portfolios is the proper way to depict risk. Set-valued risk measures are useful for markets with transaction costs.

Mathematically

A set-valued risk measure is a function ${\displaystyle R:L_d^p\to {\mathbb{F}}_M}$, 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:

Normalized

${\displaystyle K_M\subseteq R(0)\text{ and }R(0)\cap -\mathrm{int}{K}_M=\mathrm{\varnothing }}$

Translative in M

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

Monotone

${\displaystyle \mathrm{\forall }{X}_2-X_1\in L_d^p(K)\Rightarrow R(X_2)\supseteq R(X_1)}$

Examples

• Value at risk
• Expected shortfall
• Superposed risk measures
• Entropic value at risk
• Drawdown
• Tail conditional expectation
• Entropic risk measure
• Superhedging price
• Expectile

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, ${\displaystyle Var(X+a)=Var(X)\ne Var(X)-a}$ 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.

Relation to acceptance set

There is a one-to-one correspondence between an acceptance set and a corresponding risk measure. As defined below it can be shown that ${\displaystyle R_{A_R}(X)=R(X)}$ and ${\displaystyle A_{R_A}=A}$.

Risk measure to acceptance set

• If ${\displaystyle \rho }$ is a (scalar) risk measure then ${\displaystyle A_{\rho }=\{X\in L^p:\rho (X)\le 0\}}$ 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.

Relation with deviation risk measure

There is a one-to-one relationship between a deviation risk measure D and an expectation-bounded risk measure ${\displaystyle \rho }$ where for any ${\displaystyle X\in {\mathcal{L}}^2}$

• ${\displaystyle D(X)=\rho (X-\mathbb{E}[X])}$
• ${\displaystyle \rho (X)=D(X)-\mathbb{E}[X]}$.

${\displaystyle \rho }$ is called expectation bounded if it satisfies ${\displaystyle \rho (X)>\mathbb{E}[-X]}$ for any nonconstant X and ${\displaystyle \rho (X)=\mathbb{E}[-X]}$ for any constant X.