Spectral risk measure

A Spectral risk measure is a risk measure given as a weighted average of outcomes where bad outcomes are, typically, included with larger weights. A spectral risk measure is a function of portfolio returns and outputs the amount of the numeraire (typically a currency) to be kept in reserve. A spectral risk measure is always a coherent risk measure, but the converse does not always hold. An advantage of spectral measures is the way in which they can be related to risk aversion, and particularly to a utility function, through the weights given to the possible portfolio returns.

Definition

Consider a portfolio $X$ (denoting the portfolio payoff). Then a spectral risk measure $M_{\phi }:{\mathcal {L}}\to \mathbb {R}$ where $\phi$ is non-negative, non-increasing, right-continuous, integrable function defined on $[0,1]$ such that $\int _{0}^{1}\phi (p)dp=1$ is defined by

$M_{\phi }(X)=-\int _{0}^{1}\phi (p)F_{X}^{-1}(p)dp$

where $F_{X}$ is the cumulative distribution function for X.

If 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 $M_{\phi }(X)=-\delta \sum _{s=1}^{S}\phi _{s}X_{s:S}$ is a spectral measure of risk if $\phi \in \mathbb {R} ^{S}$ satisfies the conditions

1. Nonnegativity: $\phi _{s}\geq 0$ for all $s=1,\dots ,S$,
2. Normalization: $\sum _{s=1}^{S}\phi _{s}=1$,
3. Monotonicity : $\phi _{s}$ is non-increasing, that is $\phi _{s_{1}}\geq \phi _{s_{2}}$ if ${s_{1}}<{s_{2}}$ and ${s_{1}},{s_{2}}\in \{1,\dots ,S\}$.

Properties

Spectral risk measures are also coherent. Every spectral risk measure $\rho :{\mathcal {L}}\to \mathbb {R}$ satisfies:

1. Positive Homogeneity: for every portfolio X and positive value $\lambda >0$, $\rho (\lambda X)=\lambda \rho (X)$;
2. Translation-Invariance: for every portfolio X and $\alpha \in \mathbb {R}$, $\rho (X+a)=\rho (X)-a$;
3. Monotonicity: for all portfolios X and Y such that $X\geq Y$, $\rho (X)\leq \rho (Y)$;
4. Sub-additivity: for all portfolios X and Y, $\rho (X+Y)\leq \rho (X)+\rho (Y)$;
5. Law-Invariance: for all portfolios X and Y with cumulative distribution functions $F_{X}$ and $F_{Y}$ respectively, if $F_{X}=F_{Y}$ then $\rho (X)=\rho (Y)$;
6. Comonotonic Additivity: for every comonotonic random variables X and Y, $\rho (X+Y)=\rho (X)+\rho (Y)$. Note that X and Y are comonotonic if for every $\omega _{1},\omega _{2}\in \Omega :\;(X(\omega _{2})-X(\omega _{1}))(Y(\omega _{2})-Y(\omega _{1}))\geq 0$.

In some texts the input X is interpreted as losses rather than payoff of a portfolio. In this case, the translation-invariance property would be given by $\rho (X+a)=\rho (X)+a$, and the monotonicity property by $X\geq Y\implies \rho (X)\geq \rho (Y)$ instead of the above.

Examples

• The expected shortfall is a spectral measure of risk.
• The expected value is trivially a spectral measure of risk.

See also

• Distortion risk measure