Deviation risk measure

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) 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.

References

References

1. (2002). "Deviation Measures in Risk Analysis and Optimization".
2. (2004). "Progress in Risk Measurement". Advanced Modelling and Optimization.