Deviation risk measure

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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 portfolio returns, is a deviation risk measure if

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

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.^[1]

Examples

The standard deviation is clearly a deviation risk measure.

References

↑ 1.0 1.1 Rockafellar, Tyrrell; Uryasev, Stanislav; Zabarankin, Michael (2002). Deviation Measures in Risk Analysis and Optimization. Retrieved January 15, 2012.
↑ Cheng, Siwei; Liu, Yanhui; Wang, Shouyang (2004). "Progress in Risk Measurement". Advanced Modelling and Optimization 6 (1).

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