Deviation risk measure

In financial mathematics, a deviation risk measure is a function to quantify financial risk in a different method than a general risk measure. Deviation risk measures generalize the concept of standard deviation.

Mathematical definition

A function, where is the L2 space of random variables, is a deviation risk measure if

Shift-invariant: for any
Normalization:
Positively homogeneous: for any and
Sublinearity: for any
Positivity: for all nonconstant X, and 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

R is expectation bounded if for any nonconstant X and for any constant X.
If for every X, 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 ;
Average absolute deviation ;
Lower and upper semi-deviations and, where and ;
Range-based deviations, for example, and ;
Conditional value-at-risk deviation, defined for any by, where is Expected shortfall.