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 , where is the L2 space of random portfolio returns, 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.[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
- .
R is expectation bounded if for any nonconstant X and for any constant X.
If for every X (where is the essential infimum), then there is a relationship between D and a coherent risk measure.[1]
Examples
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