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
  1. Shift-invariant: for any
  2. Normalization:
  3. Positively homogeneous: for any and
  4. Sublinearity: for any
  5. 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: