Skip to content
Quantitative Risk Management / Loss Distributions, VaR, and Coherent Risk Measures

Spectral and Distortion Risk Measures

Expected Shortfall averages the quantiles of the loss distribution in the worst (1−α)(1-\alpha) fraction of outcomes, using equal weight 1/(1−α)1/(1-\alpha) on each. But why equal weight? A risk-averse decision maker might prefer to weight the very worst outcomes (quantiles close to 1) more heavily than losses that are merely "in the tail." Spectral risk measures generalise ES by allowing an arbitrary non-decreasing weighting function ϕ\phi over quantiles.
This framework — introduced by Acerbi (2002) and Kusuoka (2001) — turns out to be exactly the class of law-invariant coherent risk measures that are also comonotone additive. The result is surprisingly clean: there is a one-to-one correspondence between (i) choices of risk-aversion spectrum ϕ\phi, (ii) distortion functions DD, and (iii) mixtures of ES over confidence levels. Every member of this class has a simple dual representation and inherits all four coherence axioms from ES.
The lesson closes with the expectile risk measure — an alternative to ES based on asymmetric least-squares, which has the statistical advantage of being elicitable (directly backtestable from point forecasts).
Learning objectives. By the end of this note you should be able to:
  • define a distortion risk measure via a convex distortion function DD and state three equivalent representations;
  • compute the spectrum ϕ(u)=D′(u)\phi(u) = D'(u) from a given distortion function;
  • show that ES is a distortion risk measure with distortion Dα(u)=(u−α)+/(1−α)D_\alpha(u) = (u-\alpha)^+/(1-\alpha);
  • state and interpret Proposition 8.18: every distortion risk measure is a weighted average of ES;
  • explain why the non-decreasing condition on ϕ\phi is the condition that makes a spectral measure coherent;
  • define the α\alpha-expectile, explain its connection to asymmetric least squares, and state its coherence for α≥0.5\alpha \ge 0.5.

In this lesson

    1. Distortion risk measures
    1. Spectral risk measures
    1. Three representations and their relationship
    1. The complete characterisation theorem
    1. Parametric families of distortion functions
    1. The expectile risk measure

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or browse the Quantitative Risk Management syllabus.