How Spectral and Distortion Risk Measures Are Related

Summary

The note compares distortion risk measures with spectral risk measures through the functions used to weight outcomes across a distribution. It states that the two frameworks differ in the assumptions imposed on their distortion or weighting functions. Distortion measures are described as law-invariant and monotone, while spectral measures add conditions that make them coherent, including sub-additivity, positive homogeneity, and translation invariance.

A second answer qualifies the distinction: certain coherent distortion measures are related to spectral measures, so familiar examples can be effectively equivalent within that class. The relationship does not extend to non-coherent distortion measures. The discussion is conceptual and points to formal definitions and a proof, but does not work through a specific risk measure or provide derivations. The precise function-space conditions in the source are incomplete, so the note should be read as a high-level distinction rather than a complete mathematical characterization.

Key ideas

• Both measure families use functions to transform or weight distribution outcomes.
• Distortion measures are described as law-invariant and monotone, but may lack coherence.
• Spectral measures impose conditions that yield coherence, including sub-additivity.
• Certain coherent distortion measures correspond to spectral measures.
• The stated relationship does not cover non-coherent distortion measures.

Tags

# Spectral and distortion risk measures


# 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?

## Answer by emcor (score 2, accepted)

https://quant.stackexchange.com/a/23068

Compare these two links:

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







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.

## Answer by T-at-R (score 1)

https://quant.stackexchange.com/a/31103

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.