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 D: \mathcal^2 \to ,+\infty/math>, where $\mathcal^2$ is the L2 space of random variables (random portfolio returns), is a deviation risk measure if
# Shift-invariant: $D(X + r) = D(X)$ for any $r \in \mathbb$
# Normalization: $D(0) = 0$
# Positively homogeneous: $D(\lambda X) = \lambda D(X)$ for any $X \in \mathcal^2$ and $\lambda > 0$
# Sublinearity: $D(X + Y) \leq D(X) + D(Y)$ for any $X, Y \in \mathcal^2$
# Positivity: $D(X) > 0$ for all nonconstant ''X'', and $D(X) = 0$ for any constant ''X''.

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 $X \in \mathcal^2$
* D(X) = R(X - \mathbb
* R(X) = D(X) - \mathbb /math>.
''R'' is expectation bounded if R(X) > \mathbb X/math> for any nonconstant ''X'' and R(X) = \mathbb X/math> for any constant ''X''.

If D(X) < \mathbb - \operatorname X for every ''X'' (where $\operatorname$ 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:
*Standard deviation $\sigma(X)=\sqrt$;
*Average absolute deviation $MAD(X)=E(, X-EX, )$;
*Lower and upper semideviations $\sigma_-(X)=\sqrt$ and $\sigma_+(X)=\sqrt$, where -:=\max\ and +:=\max\;
*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 $_\alpha^\Delta(X)\equiv ES_\alpha (X-EX)$, where $ES_\alpha(X)$ is Expected shortfall.

See also

* Unitized risk

References

{{Reflist

Financial risk modeling