{{Short description|Risk measure derived by applying a distortion function to a loss distribution}} In financial mathematics and economics, 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 <math>\rho_g: L^p \to \mathbb{R}</math> associated with the distortion function <math>g: [0,1] \to [0,1]</math> is a ''distortion risk measure'' if for any random variable of gains <math>X \in L^p</math> (where <math>L^p</math> is the L<sup>p</sup> space) then : <math>\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</math> where <math>F_{-X}</math> is the cumulative distribution function for <math>-X</math> and <math>\tilde{g}</math> is the dual distortion function <math>\tilde{g}(u) = 1 - g(1-u)</math>.<ref name="PortfolioOpt"/>
If <math>X \leq 0</math> almost surely then <math>\rho_g</math> is given by the Choquet integral, i.e. <math>\rho_g(X) = -\int_0^{\infty} g(1 - F_{-X}(x)) dx.</math><ref name="PortfolioOpt">{{Cite book | last1 = Sereda | first1 = E. N. | last2 = Bronshtein | first2 = E. M. | last3 = Rachev | first3 = S. T. | last4 = Fabozzi | first4 = F. J. | last5 = Sun | first5 = W. | last6 = Stoyanov | first6 = S. V. | chapter = Distortion Risk Measures in Portfolio Optimization | doi = 10.1007/978-0-387-77439-8_25 | title = Handbook of Portfolio Construction | pages = 649 | year = 2010 | isbn = 978-0-387-77438-1 | citeseerx = 10.1.1.316.1053 }}</ref><ref name="Wirch">{{cite web|title=Distortion Risk Measures: Coherence and Stochastic Dominance|author=Julia L. Wirch|author2=Mary R. Hardy|url=http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf|access-date=March 10, 2012|archive-url=https://web.archive.org/web/20160705041252/http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf|archive-date=July 5, 2016|url-status=dead}}</ref> Equivalently, <math>\rho_g(X) = \mathbb{E}^{\mathbb{Q}}[-X]</math><ref name="Wirch"/> such that <math>\mathbb{Q}</math> is the monotone and normalized set function generated by <math>g</math>, i.e. for any <math>A \in \mathcal{F}</math> the sigma-algebra then <math>\mathbb{Q}(A) = g(\mathbb{P}(A))</math>.<ref name="PropertiesDRM">{{Cite journal | last1 = Balbás | first1 = A. | last2 = Garrido | first2 = J. | last3 = Mayoral | first3 = S. | doi = 10.1007/s11009-008-9089-z | title = Properties of Distortion Risk Measures | journal = Methodology and Computing in Applied Probability | volume = 11 | issue = 3 | pages = 385 | year = 2008 | hdl = 10016/14071 | s2cid = 53327887 | hdl-access = free }}</ref>
=== Properties === In addition to the properties of general risk measures, distortion risk measures also have: # ''Law invariant'': If the distribution of <math>X</math> and <math>Y</math> are the same then <math>\rho_g(X) = \rho_g(Y)</math>. # ''Monotone'' with respect to first order stochastic dominance. ## If <math>g</math> is a concave distortion function, then <math>\rho_g</math> is monotone with respect to second order stochastic dominance. # <math>g</math> is a concave distortion function if and only if <math>\rho_g</math> is a coherent risk measure.<ref name="PortfolioOpt"/><ref name="Wirch"/>
== Examples == * Value at risk is a distortion risk measure with associated distortion function <math>g(x) = \begin{cases}0 & \text{if }0 \leq x < 1-\alpha\\ 1 & \text{if }1-\alpha \leq x \leq 1\end{cases}.</math><ref name="Wirch"/><ref name="PropertiesDRM"/> * Conditional value at risk is a distortion risk measure with associated distortion function <math>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}.</math><ref name="Wirch"/><ref name="PropertiesDRM"/> * The negative expectation is a distortion risk measure with associated distortion function <math>g(x) = x</math>.<ref name="PortfolioOpt"/>
== See also == * Risk measure * Coherent risk measure * Deviation risk measure * Spectral risk measure
== References == {{reflist}} *{{cite journal|last=Wu|first=Xianyi|author2=Xian Zhou|title=A new characterization of distortion premiums via countable additivity for comonotonic risks|journal=Insurance: Mathematics and Economics|date=April 7, 2006|volume=38|issue=2|pages=324–334|doi=10.1016/j.insmatheco.2005.09.002}}
Category:Financial risk modeling