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 ${\displaystyle D:{\mathcal {L}}^{2}\to [0,+\infty ]}$, where ${\displaystyle {\mathcal {L}}^{2}}$ is the L2 space of random portfolio returns, is a deviation risk measure if

1. Shift-invariant: ${\displaystyle D(X+r)=D(X)}$ for any $r\in {\mathbb {R}}$
2. Normalization: ${\displaystyle D(0)=0}$
3. Positively homogeneous: ${\displaystyle D(\lambda X)=\lambda D(X)}$ for any $X\in {\mathcal {L}}^{2}$ and $\lambda >0$
4. Sublinearity: ${\displaystyle D(X+Y)\leq D(X)+D(Y)}$ for any ${\displaystyle X,Y\in {\mathcal {L}}^{2}}$
5. Positivity: ${\displaystyle D(X)>0}$ for all nonconstant X, and ${\displaystyle D(X)=0}$ 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 $X\in {\mathcal {L}}^{2}$

• ${\displaystyle D(X)=R(X-\mathbb {E} [X])}$
• ${\displaystyle R(X)=D(X)-\mathbb {E} [X]}$.

R is expectation bounded if ${\displaystyle R(X)>\mathbb {E} [-X]}$ for any nonconstant X and ${\displaystyle R(X)=\mathbb {E} [-X]}$ for any constant X.

If ${\displaystyle D(X)<\mathbb {E} [X]-\operatorname {ess\inf } X}$ for every X (where ${\displaystyle \operatorname {ess\inf } }$ is the essential infimum), then there is a relationship between D and a coherent risk measure.^[1]

Examples

The standard deviation is clearly a deviation risk measure.

See also

• Unitized risk

References

1. Rockafellar, Tyrrell; Uryasev, Stanislav; Zabarankin, Michael (2002). "Deviation Measures in Risk Analysis and Optimization". Retrieved January 15, 2012.
2. Cheng, Siwei; Liu, Yanhui; Wang, Shouyang (2004). "Progress in Risk Measurement". Advanced Modelling and Optimization. 6 (1).