# Distortion risk measure

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In financial mathematics, a distortion risk measure is a type of risk measure which is related to the cumulative distribution function of the return of a financial portfolio.

## Mathematical definition

The function $\rho_g: L^p \to \mathbb{R}$ associated with the distortion function $g: [0,1] \to [0,1]$ is a distortion risk measure if for any random variable of gains $X \in L^p$ (where $L^p$ is the Lp space) then

$\rho_g(X) = -\int_0^1 F_{-X}^{-1}(p) d\tilde{g}(p) = \int_{-\infty}^0 \tilde{g}(F_{-X}(x))dx - \int_0^{\infty} g(1 - F_{-X}(x)) dx$

where $F_{-X}$ is the cumulative distribution function for $-X$ and $\tilde{g}$ is the dual distortion function $\tilde{g}(u) = 1 - g(1-u)$.[1]

If $X \leq 0$ almost surely then $\rho_g$ is given by the Choquet integral, i.e. $\rho_g(X) = -\int_0^{\infty} g(1 - F_{-X}(x)) dx.$[1][2] Equivalently, $\rho_g(X) = \mathbb{E}^{\mathbb{Q}}[-X]$[2] such that $\mathbb{Q}$ is the probability measure generated by $g$, i.e. for any $A \in \mathcal{F}$ the sigma-algebra then $\mathbb{Q}(A) = g(\mathbb{P}(A))$.[3]

### Properties

In addition to the properties of general risk measures, distortion risk measures also have:

1. Law invariant: If the distribution of $X$ and $Y$ are the same then $\rho_g(X) = \rho_g(Y)$.
2. Monotone with respect to first order stochastic dominance.
1. If $g$ is a concave distortion function, then $\rho_g$ is monotone with respect to second order stochastic dominance.
3. $g$ is a concave distortion function if and only if $\rho_g$ is a coherent risk measure.[1][2]

## Examples

• Value at risk is a distortion risk measure with associated distortion function $g(x) = \begin{cases}0 & \text{if }0 \leq x < 1-\alpha\\ 1 & \text{if }1-\alpha \leq x \leq 1\end{cases}.$[2][3]
• Conditional value at risk is a distortion risk measure with associated distortion function $g(x) = \begin{cases}\frac{x}{1-\alpha} & \text{if }0 \leq x < 1-\alpha\\ 1 & \text{if }1-\alpha \leq x \leq 1\end{cases}.$[2][3]
• The negative expectation is a distortion risk measure with associated distortion function $g(x) = x$.[1]

## References

1. ^ a b c d Sereda, E. N.; Bronshtein, E. M.; Rachev, S. T.; Fabozzi, F. J.; Sun, W.; Stoyanov, S. V. (2010). "Distortion Risk Measures in Portfolio Optimization". Handbook of Portfolio Construction. p. 649. doi:10.1007/978-0-387-77439-8_25. ISBN 978-0-387-77438-1.
2. Julia L. Wirch; Mary R. Hardy. "Distortion Risk Measures: Coherence and Stochastic Dominance" (pdf). Retrieved March 10, 2012.
3. ^ a b c Balbás, A.; Garrido, J.; Mayoral, S. (2008). "Properties of Distortion Risk Measures". Methodology and Computing in Applied Probability 11 (3): 385. doi:10.1007/s11009-008-9089-z.