Spectral and distortion risk measures

Is there any difference between the spectral and distortion risk measure? Or is it just a different name for the same kind of risk measure?

• I believe they are different, but I would like to see a good explanation. Feb 2 '16 at 17:51

https://en.wikipedia.org/wiki/Distortion_risk_measure

https://en.wikipedia.org/wiki/Spectral_risk_measure

Then these risk measures are only different by their assumptions on the distortion function:

$$\tilde{g}$$ is the dual distortion function $$\tilde{g}(u) = 1 - g(1-u)$$ with $$g: [0,1] \to [0,1]$$.

$$\phi$$ is non-negative, non-increasing, right-continuous, integrable function defined on $$[0,1]$$ such that $$\int_0^1 \phi(p)dp = 1$$ and $$\phi\in\mathbb{R}^S$$ satisfies the conditions

• Nonnegativity: $$\phi_s\geq$$ for all $$s=1, \dots, S$$,
• Normalization: $$\sum_{s=1}^S\phi_s=1$$,
• Monotonicity : $$\phi_s$$ is non-increasing, that is $$\phi_{s_1}\geq\phi_{s_2}$$ if $${s_1}<{s_2}$$ and $${s_1}, {s_2}\in\{1,\dots,S\}$$.

Then distortion risk measures are Law-Invariant and Monotone, but not coherent.

Spectral risk measures are fully coherent (Positive Homogeneity, Translation-Invariance, Monotonicity, Sub-additivity, Law-Invariance).

I think that would be the main difference.

• Do you have examples for each kind of risk measure?
– Ric
Feb 3 '16 at 7:33
• @Richard VaR is Distortion risk measure (non-coherent) with $g(x) = \begin{cases}0 & \text{if }0 \leq x < 1-\alpha\\ 1 & \text{if }1-\alpha \leq x \leq 1\end{cases}.$, CVaR is spectral risk measure (coherent) and coherent risk measure with $g(x) = \begin{cases}\frac{x}{1-\alpha} & \text{if }0 \leq x < 1-\alpha\\ 1 & \text{if }1-\alpha \leq x \leq 1\end{cases}.$ Feb 3 '16 at 8:27
• Great example .. your answer could benefit from this example too. Thanks
– Ric
Feb 3 '16 at 8:33
• @emcor, by CVaR you mean the Expected Shortfall? Because $CVaR_\alpha = -\mathbb{E}[X|X \leq F_X^{-1}(\alpha) ]$ is not coherent in general and it coincides with the Expected Shortfall $ES_\alpha = - \frac{1}{\alpha} \int_0^\alpha F_X^{-1}(p) dp$ only under certain conditions such as the continuity of $F_X$. Feb 3 '16 at 11:17
• And what is the precise meaning of the dual distortion function? What does dual mean and why does it have to be dual? Feb 3 '16 at 11:21

There is actually a proof by Gzyl and Mayoral that relates certain distortion risk measures, namely the coherent ones to spectral risk measures. See Spectral and Distortion Risk for details.

So yes for a large class of well known risk measures they are essentially the same. As pointed out this relationship does not hold for non-coherent risk measures.