Spectral risk measure

From Wikipedia, the free encyclopedia

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.

1 Definition
2 Properties
3 Examples
4 References

[edit] 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 $\phi\in\mathbb{R}^S$ . The measure $M_{\phi}:\mathbb{R}^S\rightarrow \mathbb{R}$ defined by $M_{\phi}(X)=-\delta\sum_{s=1}^S\phi_sX_{s:S}$ is a spectral measure of risk (Acerbi 2002) if $\phi\in\mathbb{R}^S$ satisfies the conditions

Nonnegativity: $\phi_s\geq0$ for all $s=1, \dots, S$ ,
Normalization: $\sum_{s=1}^S\phi_s=1$ ,
Monotonicity : $φ s$ is non-increasing, that is $\phi_{s_1}\geq\phi_{s_2}$ if $s 1 < s 2$