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 variables (random portfolio returns), is a deviation risk measure if
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
The most well-known examples of risk deviation measures are:[1]
- Standard deviation ;
- Average absolute deviation ;
- Lower and upper semideviations and , where and ;
- Range-based deviations, for example, and ;
- Conditional value-at-risk (CVaR) deviation, defined for any by , where is Expected shortfall.