Infinancial mathematics, adeviation risk measure is a function to quantifyfinancial risk (and not necessarilydownside risk) in a different method than a generalrisk measure. Deviation risk measures generalize the concept ofstandard deviation.

A function${\displaystyle D:{\mathcal{L}}^2\to [0,+\mathrm{\infty }]}$, where${\displaystyle {\mathcal{L}}^2}$ is theL2 space ofrandom variables (randomportfolio 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:${\displaystyle D(X+Y)\le D(X)+D(Y)}$ for any${\displaystyle X,Y\in {\mathcal{L}}^2}$
5. Positivity:${\displaystyle D(X)>0}$ for all nonconstantX, and${\displaystyle D(X)=0}$ for any constantX.^[1][2]

There is aone-to-one relationship between a deviation risk measureD and an expectation-boundedrisk measureR 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 nonconstantX and${\displaystyle R(X)=\mathbb{E}[-X]}$ for any constantX.

If for everyX (where${\displaystyle \mathrm{e}\mathrm{s}\mathrm{s}inf}$ is theessential infimum), then there is a relationship betweenD and acoherent 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 semi-deviations${\displaystyle {\sigma }_{-}(X)=\sqrt{{E[(X-EX{)}_{-}}^2]}}$ 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,${\displaystyle D(X)=EX-infX}$ and${\displaystyle D(X)=supX-infX}$;
• Conditional value-at-risk (CVaR) deviation, defined for any${\displaystyle \alpha \in (0,1)}$ by${\displaystyle {\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }^{\mathrm{\Delta }}(X)\equiv E{S}_{\alpha }(X-EX)}$, where${\displaystyle E{S}_{\alpha }(X)}$ isExpected shortfall.

• Unitized risk – Relative measure of dispersion expressed as the ratio of standard deviation to the meanPages displaying short descriptions of redirect targets

• Financial risk modeling

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