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 $D:{\mathcal {L}}^{2}\to [0,+\infty ]$, where ${\mathcal {L}}^{2}$ 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 $X\in {\mathcal {L}}^{2}$ and $\lambda >0$
4. Sublinearity: $D(X+Y)\leq D(X)+D(Y)$ for any $X,Y\in {\mathcal {L}}^{2}$
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 $X\in {\mathcal {L}}^{2}$

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

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 $D(X)<\mathbb {E} [X]-\operatorname {ess\inf } X$ for every X (where $\operatorname {ess\inf }$ 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 semi-deviations $\sigma _{-}(X)={\sqrt {{E[(X-EX)_{-}}^{2}]}}$ 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 ${\rm {CVaR}}_{\alpha }^{\Delta }(X)\equiv ES_{\alpha }(X-EX)$, where $ES_{\alpha }(X)$ is Expected shortfall.

See also

• Unitized risk