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 $M_{\varphi }:ℒ\to ℝ$ where $\varphi$ is non-negative, non-increasing, right-continuous, integrable function defined on $[0,1]$ such that $\underset{0}{\overset{1}{\int }}\varphi (p)dp=1$ is defined by

$M_{\varphi }(X)=-\underset{0}{\overset{1}{\int }}\varphi (p)F_X^{-1}(p)dp$

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 $\varphi \in {ℝ}^S$. The measure $M_{\varphi }:{ℝ}^S\to ℝ$ defined by $M_{\varphi }(X)=-\delta \sum _{s=1}^S{\varphi }_s{X}_{s:S}$ is a spectral measure of risk if $\varphi \in {ℝ}^S$ satisfies the conditions

1. Nonnegativity: ${\varphi }_s\ge 0$ for all $s=1,\dots ,S$,
2. Normalization: $\sum _{s=1}^S{\varphi }_s=1$,
3. Monotonicity : ${\varphi }_s$ is non-increasing, that is ${\varphi }_{s_1}\ge {\varphi }_{s_2}$ 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 $\rho :ℒ\to ℝ$ 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 $\alpha \in ℝ$, $\rho (X+a)=\rho (X)-a$;
3. Monotonicity: for all portfolios X and Y such that $X\ge Y$, $\rho (X)\le \rho (Y)$;
4. Sub-additivity: for all portfolios X and Y, $\rho (X+Y)\le \rho (X)+\rho (Y)$;
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 ${\omega }_1,{\omega }_2\in \mathrm{\Omega }:(X({\omega }_2)-X({\omega }_1))(Y({\omega }_2)-Y({\omega }_1))\ge 0$.

In some textsTemplate: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 $\rho (X+a)=\rho (X)+a$, and the monotonicity property by $X\ge Y⟹\rho (X)\ge \rho (Y)$ 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