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?

  • $\begingroup$ I believe they are different, but I would like to see a good explanation. $\endgroup$
    – nbbo2
    Feb 2, 2016 at 17:51

2 Answers 2


Compare these two links:



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.

  • $\begingroup$ Do you have examples for each kind of risk measure? $\endgroup$
    – Richi Wa
    Feb 3, 2016 at 7:33
  • $\begingroup$ @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}.$ $\endgroup$
    – emcor
    Feb 3, 2016 at 8:27
  • $\begingroup$ Great example .. your answer could benefit from this example too. Thanks $\endgroup$
    – Richi Wa
    Feb 3, 2016 at 8:33
  • $\begingroup$ @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 $F_X$. $\endgroup$ Feb 3, 2016 at 11:17
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    $\begingroup$ And what is the precise meaning of the dual distortion function? What does dual mean and why does it have to be dual? $\endgroup$ Feb 3, 2016 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.


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