Distortion risk measure

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

: $\backslash rho\_g(X)\; =\; -\backslash int\_0^1\; F\_\{-X\}^\{-1\}(p)\; d\backslash tilde\{g\}(p)\; =\; \backslash int\_\{-\backslash infty\}^0\; \backslash tilde\{g\}(F\_\{-X\}(x))dx\; -\; \backslash int\_0^\{\backslash infty\}\; g(1\; -\; F\_\{-X\}(x))\; dx$

where $F\_\{-X\}$ is the cumulative distribution function for $-X$ and $\backslash tilde\{g\}$ is the dual distortion function $\backslash tilde\{g\}(u)\; =\; 1\; -\; g(1-u)$.

If $X\; \backslash leq\; 0$ almost surely then $\backslash rho\_g$ is given by the Choquet integral, i.e. $\backslash rho\_g(X)\; =\; -\backslash int\_0^\{\backslash infty\}\; g(1\; -\; F\_\{-X\}(x))\; dx.${{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 }}{{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}} Equivalently, $\backslash rho\_g(X)\; =\; \backslash mathbb\{E\}^\{\backslash mathbb\{Q\}\}[-X]$ such that $\backslash mathbb\{Q\}$ is the probability measure generated by $g$, i.e. for any $A\; \backslash in\; \backslash mathcal\{F\}$ the sigma-algebra then $\backslash mathbb\{Q\}(A)\; =\; g(\backslash mathbb\{P\}(A))$.{{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 }}

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 $\backslash rho\_g(X)\; =\; \backslash rho\_g(Y)$.
2. Monotone with respect to first order stochastic dominance.
3. If $g$ is a concave distortion function, then $\backslash rho\_g$ is monotone with respect to second order stochastic dominance.
4. $g$ is a concave distortion function if and only if $\backslash rho\_g$ is a coherent risk measure.

• Value at risk is a distortion risk measure with associated distortion function
• Conditional value at risk is a distortion risk measure with associated distortion function
• The negative expectation is a distortion risk measure with associated distortion function $g(x)\; =\; x$.

• Risk measure
• Coherent risk measure
• Deviation risk measure
• Spectral risk measure

{{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