Distortion risk measure

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

ρ g ( X ) = − ∫ 0 1 F − X − 1 ( p ) d g ~ ( p ) = ∫ − ∞ 0 g ~ ( F − X ( x ) ) d x − ∫ 0 ∞ g ( 1 − F − X ( x ) ) d x {\displaystyle \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} {\displaystyle \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 {\displaystyle F_{-X}} {\displaystyle F_{-X}} is the cumulative distribution function for − X {\displaystyle -X} {\displaystyle -X} and g ~ {\displaystyle {\tilde {g}}} {\displaystyle {\tilde {g}}} is the dual distortion function g ~ ( u ) = 1 − g ( 1 − u ) {\displaystyle {\tilde {g}}(u)=1-g(1-u)} {\displaystyle {\tilde {g}}(u)=1-g(1-u)}.[1]

If X ≤ 0 {\displaystyle X\leq 0} {\displaystyle X\leq 0} almost surely then ρ g {\displaystyle \rho _{g}} {\displaystyle \rho _{g}} is given by the Choquet integral, i.e. ρ g ( X ) = − ∫ 0 ∞ g ( 1 − F − X ( x ) ) d x . {\displaystyle \rho _{g}(X)=-\int _{0}^{\infty }g(1-F_{-X}(x))dx.} {\displaystyle \rho _{g}(X)=-\int _{0}^{\infty }g(1-F_{-X}(x))dx.}[1][2] Equivalently, ρ g ( X ) = E Q [ − X ] {\displaystyle \rho _{g}(X)=\mathbb {E} ^{\mathbb {Q} }[-X]} {\displaystyle \rho _{g}(X)=\mathbb {E} ^{\mathbb {Q} }[-X]}[2] such that Q {\displaystyle \mathbb {Q} } {\displaystyle \mathbb {Q} } is the monotone and normalized set function generated by g {\displaystyle g} {\displaystyle g}, i.e. for any A ∈ F {\displaystyle A\in {\mathcal {F}}} {\displaystyle A\in {\mathcal {F}}} the sigma-algebra then Q ( A ) = g ( P ( A ) ) {\displaystyle \mathbb {Q} (A)=g(\mathbb {P} (A))} {\displaystyle \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 {\displaystyle X} {\displaystyle X} and Y {\displaystyle Y} {\displaystyle Y} are the same then ρ g ( X ) = ρ g ( Y ) {\displaystyle \rho _{g}(X)=\rho _{g}(Y)} {\displaystyle \rho _{g}(X)=\rho _{g}(Y)}.
  2. Monotone with respect to first order stochastic dominance.
    1. If g {\displaystyle g} {\displaystyle g} is a concave distortion function, then ρ g {\displaystyle \rho _{g}} {\displaystyle \rho _{g}} is monotone with respect to second order stochastic dominance.
  3. g {\displaystyle g} {\displaystyle g} is a concave distortion function if and only if ρ g {\displaystyle \rho _{g}} {\displaystyle \rho _{g}} is a coherent risk measure.[1][2]

Examples

  • Value at risk is a distortion risk measure with associated distortion function g ( x ) = { 0 if  0 ≤ x < 1 − α 1 if  1 − α ≤ x ≤ 1 . {\displaystyle g(x)={\begin{cases}0&{\text{if }}0\leq x<1-\alpha \\1&{\text{if }}1-\alpha \leq x\leq 1\end{cases}}.} {\displaystyle 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 ) = { x 1 − α if  0 ≤ x < 1 − α 1 if  1 − α ≤ x ≤ 1 . {\displaystyle 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}}.} {\displaystyle 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 {\displaystyle g(x)=x} {\displaystyle g(x)=x}.[1]

See also

References

  1. 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). Archived from the original (PDF) on July 5, 2016. Retrieved March 10, 2012.
  3. 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. hdl:10016/14071. S2CID 53327887.
  • Wu, Xianyi; Xian Zhou (April 7, 2006). "A new characterization of distortion premiums via countable additivity for comonotonic risks". Insurance: Mathematics and Economics. 38 (2): 324–334. doi:10.1016/j.insmatheco.2005.09.002.