Spectral risk measure

A Spectral risk measure is a risk measure given as a weighted average of outcomes (which are standardly assumed to be equiprobable) where bad outcomes are included with larger weights.

Definition

Consider a portfolio $X$. There are $S$ equiprobable outcomes with the corresponding payoffs given by the order statistics $X\_\{1:S\},\; ...\; X\_\{S:S\}$. Let $phiinmathbb\{R\}^S$. The measure$M\_\{phi\}:mathbb\{R\}^S\; ightarrow\; mathbb\{R\}$ defined by $M\_\{phi\}(X)=-deltasum\_\{s=1\}^Sphi\_sX\_\{s:S\}$ is a spectral measure of risk harv|Acerbi|2002 if $phiinmathbb\{R\}^S$ satisfies the conditions

# Nonnegativity: $phi\_sgeq0$ for all $s=1,\; dots,\; S$,
# Normalization: $sum\_\{s=1\}^Sphi\_s=1$,
# Monotonicity : $phi\_s$ is non-increasing, that is $phi\_\{s\_1\}geqphi\_\{s\_2\}$ if and $\{s\_1\},\; \{s\_2\}in\{1,dots,S\}$.

Properties

Spectral risk measures are also coherent.

Examples

The expected shortfall is a spectral measure of risk.

The expected value is -trivially- also a spectral measure of risk.

References

Citation
last=Acerbi
first=Carlo
author-link=
publication-date=2002
date=
year=2002
title=Spectral measures of risk: A coherent representation of subjective risk aversion
periodical=Journal of Banking and Finance
publication-place=
place=
publisher=Elsevier
volume=26
issue=
pages=1505-1518
url=
doi=10.1016/S0378-4266(02)00281-9
oclc=