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Deviation risk measure

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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 D : L 2 [ 0 , + ] , where L 2 is the L2 space of random portfolio returns, is a deviation risk measure if

  1. Shift-invariant: D ( X + r ) = D ( X ) for any r R
  2. Normalization: D ( 0 ) = 0
  3. Positively homogeneous: D ( λ X ) = λ D ( X ) for any X L 2 and λ > 0
  4. Sublinearity: D ( X + Y ) D ( X ) + D ( Y ) for any X , Y L 2
  5. Positivity: D ( X ) > 0 for all nonconstant X, and D ( 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 X L 2

  • D ( X ) = R ( X E [ X ] )
  • R ( X ) = D ( X ) E [ X ] .
  • R is expectation bounded if R ( X ) > E [ X ] for any nonconstant X and R ( X ) = E [ X ] for any constant X.

    If D ( X ) < E [ X ] e s s inf X for every X (where e s s inf is the essential infimum), then there is a relationship between D and a coherent risk measure.

    Examples

    The standard deviation is clearly a deviation risk measure.

    References

    Deviation risk measure Wikipedia