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 $$X \in \mathcal{L}^2$$ R is expectation bounded if $$R(X) > \mathbb{E}[-X]$$ for any nonconstant X and $$R(X) = \mathbb{E}[-X]$$ for any constant X.
 * $$D(X) = R(X - \mathbb{E}[X])$$
 * $$R(X) = D(X) - \mathbb{E}[X]$$.

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 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 $${\rm CVaR}_\alpha^\Delta(X)\equiv ES_\alpha (X-EX)$$, where $$ES_\alpha(X)$$ is Expected shortfall.