Finance:Deviation risk measure

In financial mathematics, a deviation risk measure is a function to quantify financial risk (and not necessarily downside risk) in a different method than a general risk measure. Deviation risk measures generalize the concept of standard deviation.

Mathematical definition

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

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

Relation to risk measure

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

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

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

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

Examples

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

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

See also

• Unitized risk