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
[edit]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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle D(X + Y) \leq D(X) + D(Y)} for any
- Positivity: for all nonconstant X, and for any constant X.[1][2]
Relation to risk measure
[edit]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
[edit]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
[edit]References
[edit]- ^ a b c Rockafellar, Tyrrell; Uryasev, Stanislav; Zabarankin, Michael (2002). "Deviation Measures in Risk Analysis and Optimization". SSRN 365640.
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(help) - ^ Cheng, Siwei; Liu, Yanhui; Wang, Shouyang (2004). "Progress in Risk Measurement". Advanced Modelling and Optimization. 6 (1).