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[edit]

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]


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]


See also[edit]


  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. ^ a b c d e 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.