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.^[1]

Definition

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

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

where ${\displaystyle F_{X}}$ is the cumulative distribution function for X.^[2][3]

If there are ${\displaystyle S}$ equiprobable outcomes with the corresponding payoffs given by the order statistics ${\displaystyle X_{1:S},...X_{S:S}}$. Let ${\displaystyle \phi \in \mathbb {R} ^{S}}$. The measure ${\displaystyle M_{\phi }:\mathbb {R} ^{S}\rightarrow \mathbb {R} }$ defined by ${\displaystyle M_{\phi }(X)=-\delta \sum _{s=1}^{S}\phi _{s}X_{s:S}}$ is a spectral measure of risk if ${\displaystyle \phi \in \mathbb {R} ^{S}}$ satisfies the conditions

1. Nonnegativity: ${\displaystyle \phi _{s}\geq 0}$ for all ${\displaystyle s=1,\dots ,S}$,
2. Normalization: ${\displaystyle \sum _{s=1}^{S}\phi _{s}=1}$,
3. Monotonicity : ${\displaystyle \phi _{s}}$ is non-increasing, that is ${\displaystyle \phi _{s_{1}}\geq \phi _{s_{2}}}$ if ${\displaystyle {s_{1}}<{s_{2}}}$ and ${\displaystyle {s_{1}},{s_{2}}\in \{1,\dots ,S\}}$.^[4]

Properties

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

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

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

References