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 $X$ (denoting the portfolio payoff). Then a spectral risk measure ${\displaystyle M_{\phi }:{\mathcal {L}}\to \mathbb {R} }$ where $\phi$ is non-negative, non-increasing, right-continuous, integrable function defined on $[0,1]$ 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 $F_{X}$ is the cumulative distribution function for X.^[2][3]

If there are $S$ 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 : $\phi_s$ 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 $\lambda >0$, ${\displaystyle \rho (\lambda X)=\lambda \rho (X)}$;
2. Translation-Invariance: for every portfolio X and $\alpha \in \mathbb {R}$, ${\displaystyle \rho (X+a)=\rho (X)-a}$;
3. Monotonicity: for all portfolios X and Y such that $X \geq Y$, ${\displaystyle \rho (X)\leq \rho (Y)}$;
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 $F_{X}$ and $F_Y$ 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 the input X is interpreted as losses rather than payoff of a portfolio. In this case the translation-invariance property would be given by ${\displaystyle \rho (X+a)=\rho (X)+a}$ 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

1. Cotter, John; Dowd, Kevin (December 2006). "Extreme spectral risk measures: An application to futures clearinghouse margin requirements". Journal of Banking & Finance. 30 (12): 3469–3485. doi:10.1016/j.jbankfin.2006.01.008.
2. Adam, Alexandre; Houkari, Mohamed; Laurent, Jean-Paul (2007). "Spectral risk measures and portfolio selection" (pdf). Retrieved October 11, 2011.
3. Dowd, Kevin; Cotter, John; Sorwar, Ghulam (2008). "Spectral Risk Measures: Properties and Limitations" (pdf). CRIS Discussion Paper Series (2). Retrieved October 13, 2011.
4. Acerbi, Carlo (2002), "Spectral measures of risk: A coherent representation of subjective risk aversion", Journal of Banking and Finance, Elsevier, 26 (7), pp. 1505–1518, doi:10.1016/S0378-4266(02)00281-9