Deviation risk measure

A function ${\displaystyle D:{\mathcal {L}}^{2}\to [0,+\infty ]}$, where ${\displaystyle {\mathcal {L}}^{2}}$ is the L2 space of random variables (random portfolio returns), is a deviation risk measure if

1. Shift-invariant: ${\displaystyle D(X+r)=D(X)}$ for any ${\displaystyle r\in \mathbb {R} }$
2. Normalization: ${\displaystyle D(0)=0}$
3. Positively homogeneous: ${\displaystyle D(\lambda X)=\lambda D(X)}$ for any ${\displaystyle X\in {\mathcal {L}}^{2}}$ and ${\displaystyle \lambda >0}$
4. Sublinearity: ${\displaystyle D(X+Y)\leq D(X)+D(Y)}$ for any ${\displaystyle X,Y\in {\mathcal {L}}^{2}}$
5. Positivity: ${\displaystyle D(X)>0}$ for all nonconstant X, and ${\displaystyle D(X)=0}$ for any constant X.[1][2]

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

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

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

If ${\displaystyle D(X)<\mathbb {E} [X]-\operatorname {ess\inf } X}$ for every X (where ${\displaystyle \operatorname {ess\inf } }$ is the essential infimum), then there is a relationship between D and a coherent risk measure.[1]

The most well-known examples of risk deviation measures are:[1]

• Standard deviation ${\displaystyle \sigma (X)={\sqrt {E[(X-EX)^{2}]}}}$;
• Average absolute deviation ${\displaystyle MAD(X)=E(|X-EX|)}$;
• Lower and upper semideviations ${\displaystyle \sigma _{-}(X)={\sqrt {{E[(X-EX)_{-}}^{2}]}}}$ and ${\displaystyle \sigma _{+}(X)={\sqrt {{E[(X-EX)_{+}}^{2}]}}}$, where ${\displaystyle [X]_{-}:=\max\{0,-X\}}$ and ${\displaystyle [X]_{+}:=\max\{0,X\}}$;
• Range-based deviations, for example, ${\displaystyle D(X)=EX-\inf X}$ and ${\displaystyle D(X)=\sup X-\inf X}$;
• Conditional value-at-risk (CVaR) deviation, defined for any ${\displaystyle \alpha \in (0,1)}$ by ${\displaystyle {\rm {CVaR}}_{\alpha }^{\Delta }(X)\equiv ES_{\alpha }(X-EX)}$, where ${\displaystyle ES_{\alpha }(X)}$ is Expected shortfall.

• Unitized risk