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 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(λX) = λD(X) for any $X \in \mathcal{L}^2$ and λ > 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.^[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. ^ ^{a b} Rockafellar, Tyrrell; Uryasev, Stanislav; Zabarankin, Michael (2002) (pdf). Deviation Measures in Risk Analysis and Optimization. http://www.ise.ufl.edu/uryasev/Deviation_measures_wp.pdf. Retrieved October 13, 2011. 
2. ^ Cheng, Siwei; Liu, Yanhui; Wang, Shouyang (2004). "Progress in Risk Measurement". Advanced Modelling and Optimization 6 (1).