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.
Contents
Mathematical definition
A function
- Shift-invariant:
D ( X + r ) = D ( X ) for anyr ∈ R - Normalization:
D ( 0 ) = 0 - Positively homogeneous:
D ( λ X ) = λ D ( X ) for anyX ∈ L 2 λ > 0 - Sublinearity:
D ( X + Y ) ≤ D ( X ) + D ( Y ) for anyX , Y ∈ L 2 - Positivity:
D ( X ) > 0 for all nonconstant X, andD ( X ) = 0 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
If
Examples
The standard deviation is clearly a deviation risk measure.