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.

Mathematical definition

A function , where is the L2 space of random variables (random portfolio returns), is a deviation risk measure if

  1. Shift-invariant: for any
  2. Normalization:
  3. Positively homogeneous: for any and
  4. Sublinearity: for any
  5. Positivity: for all nonconstant X, and for any constant X.[1][2]

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.

See also

References

  1. Rockafellar, Tyrrell; Uryasev, Stanislav; Zabarankin, Michael (2002). "Deviation Measures in Risk Analysis and Optimization". SSRN 365640. Cite journal requires |journal= (help)
  2. Cheng, Siwei; Liu, Yanhui; Wang, Shouyang (2004). "Progress in Risk Measurement". Advanced Modelling and Optimization. 6 (1).
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