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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.
Mathematical definition
A function
, where
is the L2 space of random variables (random portfolio returns), is a deviation risk measure if
- Shift-invariant:
for any ![{\displaystyle r\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/79ada4fd070acfeeee62081f256be1c106346359)
- Normalization:
![{\displaystyle D(0)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d267140ceee12def9d079184d24eafe982eda7de)
- Positively homogeneous:
for any
and ![{\displaystyle \lambda >0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eea25afc0351140f919cf791c49c1964b8b081de)
- Sublinearity:
for any ![{\displaystyle X,Y\in {\mathcal {L}}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7895f3dddcae60ebee9346682ce6fa4440e2592)
- 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
![{\displaystyle D(X)=R(X-\mathbb {E} [X])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dca13fe97d3397ecd1204a749f14955602493248)
.
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
- ^ a b c Rockafellar, Tyrrell; Uryasev, Stanislav; Zabarankin, Michael (2002). "Deviation Measures in Risk Analysis and Optimization". SSRN 365640.
- ^ Cheng, Siwei; Liu, Yanhui; Wang, Shouyang (2004). "Progress in Risk Measurement". Advanced Modelling and Optimization. 6 (1).