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. A function , where is the L2 space of random variables (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. 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. The most well-known examples of risk deviation measures are: 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.