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. 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 D : L 2 → [ 0 , + ∞ ] {displaystyle D:{mathcal {L}}^{2} o } , where L 2 {displaystyle {mathcal {L}}^{2}} is the L2 space of random variables (random portfolio returns), is a deviation risk measure if There is a one-to-one relationship between a deviation risk measure D and an expectation-bounded risk measure R where for any X ∈ L 2 {displaystyle Xin {mathcal {L}}^{2}} R is expectation bounded if R ( X ) > E [ − X ] {displaystyle R(X)>mathbb {E} } for any nonconstant X and R ( X ) = E [ − X ] {displaystyle R(X)=mathbb {E} } for any constant X. If D ( X ) < E [ X ] − e s s inf ⁡ X {displaystyle D(X)<mathbb {E} -operatorname {essinf } X} for every X (where e s s inf {displaystyle operatorname {essinf } } 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: