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

1. Shift-invariant: [math]\displaystyle{ D(X + r) = D(X) }[/math] for any [math]\displaystyle{ r \in \mathbb{R} }[/math]
2. Normalization: [math]\displaystyle{ D(0) = 0 }[/math]
3. Positively homogeneous: [math]\displaystyle{ D(\lambda X) = \lambda D(X) }[/math] for any [math]\displaystyle{ X \in \mathcal{L}^2 }[/math] and [math]\displaystyle{ \lambda \gt 0 }[/math]
4. Sublinearity: [math]\displaystyle{ D(X + Y) \leq D(X) + D(Y) }[/math] for any [math]\displaystyle{ X, Y \in \mathcal{L}^2 }[/math]
5. Positivity: [math]\displaystyle{ D(X) \gt 0 }[/math] for all nonconstant X, and [math]\displaystyle{ D(X) = 0 }[/math] 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 [math]\displaystyle{ X \in \mathcal{L}^2 }[/math]

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

R is expectation bounded if [math]\displaystyle{ R(X) \gt \mathbb{E}[-X] }[/math] for any nonconstant X and [math]\displaystyle{ R(X) = \mathbb{E}[-X] }[/math] for any constant X.

If [math]\displaystyle{ D(X) \lt \mathbb{E}[X] - \operatorname{ess\inf} X }[/math] for every X (where [math]\displaystyle{ \operatorname{ess\inf} }[/math] 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 [math]\displaystyle{ \sigma(X)=\sqrt{E[(X-EX)^2]} }[/math];
• Average absolute deviation [math]\displaystyle{ MAD(X)=E(|X-EX|) }[/math];
• Lower and upper semideviations [math]\displaystyle{ \sigma_-(X)=\sqrt{{E[(X-EX)_-}^2]} }[/math] and [math]\displaystyle{ \sigma_+(X)=\sqrt{{E[(X-EX)_+}^2]} }[/math], where [math]\displaystyle{ [X]_-:=\max\{0,-X\} }[/math] and [math]\displaystyle{ [X]_+:=\max\{0,X\} }[/math];
• Range-based deviations, for example, [math]\displaystyle{ D(X)=EX-\inf X }[/math] and [math]\displaystyle{ D(X)=\sup X-\inf X }[/math];
• Conditional value-at-risk (CVaR) deviation, defined for any [math]\displaystyle{ \alpha\in(0,1) }[/math] by [math]\displaystyle{ {\rm CVaR}_\alpha^\Delta(X)\equiv ES_\alpha (X-EX) }[/math], where [math]\displaystyle{ ES_\alpha(X) }[/math] is Expected shortfall.

See also

• Unitized risk