Deviation risk measure explained

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

D: l{L} ^{2} \to [ 0 , +inf ty] , where l{L} ^{2} is the L2 space of random variables (random portfolio returns), is a deviation risk measure if

1. Shift-invariant:

D(X + r) = D(X) for any r \in R

1. Normalization:

D( 0 ) = 0

1. Positively homogeneous:

D( λ X) = λ D(X) for any X \in l{L} ^{2} and λ > 0

1. Sublinearity:

D(X + Y) \leq D(X) + D(Y) for any X, Y \in l{L} ^{2}

1. Positivity:

D(X) > 0 for all nonconstant X, and D(X) = 0 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

X \in l{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] - \operatorname{ess inf } X for every X (where \operatorname{ess inf } is the essential infimum), then there is a relationship between D and a coherent risk measure.

Examples

The most well-known examples of risk deviation measures are:^[1]

\sigma(X) = \sqrt{E[(X - EX) ^{2]}} ;MAD(X) = E(|X - EX|) ;

• Lower and upper semideviations

\sigma _{-(X)=\sqrt{{E[(X-EX)} ^{2 ]}} ^{- }} and \sigma _{+(X)=\sqrt{{E[(X-EX)} ^{2 ]}} ^{+ }} , where [X] _{-:=max\{0,-X\}} and [X] _{+:=max\{0,X\}} ;

• Range-based deviations, for example,

D(X) = EX -inf X and D(X) = \sup X -inf X ;

• Conditional value-at-risk (CVaR) deviation, defined for any

\alpha\in( 0 , 1 ) by {\rm CVaR} ^{\Delta(X)\equiv} _{\alpha} ES _{\alpha} (X - EX) , where ES _{\alpha(X)} is Expected shortfall.

See also

• Unitized risk

Notes and References

1. Deviation Measures in Risk Analysis and Optimization. Tyrrell. Rockafellar. Stanislav. Uryasev. Michael. Zabarankin. 2002. 365640.
2. Progress in Risk Measurement. Siwei. Cheng. Yanhui. Liu. Shouyang. Wang. 2004. Advanced Modelling and Optimization. 6. 1.