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 $M_{\phi }:{\mathcal {L}}\to \mathbb {R}$ 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.^[2]^[3]

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

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

Properties

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

Positive Homogeneity: for every portfolio X and positive value $\lambda >0$ , $\rho (\lambda X)=\lambda \rho (X)$ ;
Translation-Invariance: for every portfolio X and $\alpha \in \mathbb {R}$ , $\rho (X+a)=\rho (X)-a$ ;
Monotonicity: for all portfolios X and Y such that $X \geq Y$ , $\rho (X)\leq \rho (Y)$ ;
Sub-additivity: for all portfolios X and Y, $\rho (X+Y)\leq \rho (X)+\rho (Y)$ ;
Law-Invariance: for all portfolios X and Y with cumulative distribution functions $F_{X}$ and $F_Y$ respectively, if $F_{X}=F_{Y}$ then $\rho (X)=\rho (Y)$ ;
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 $\omega _{1},\omega _{2}\in \Omega :\;(X(\omega _{2})-X(\omega _{1}))(Y(\omega _{2})-Y(\omega _{1}))\geq 0$ .^[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 $\rho (X+a)=\rho (X)+a$ instead of the above.

Examples

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

References

↑ 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.
1 2 Adam, Alexandre; Houkari, Mohamed; Laurent, Jean-Paul (2007). "Spectral risk measures and portfolio selection" (pdf). Retrieved October 11, 2011.
↑ Dowd, Kevin; Cotter, John; Sorwar, Ghulam (2008). "Spectral Risk Measures: Properties and Limitations" (pdf). CRIS Discussion Paper Series (2). Retrieved October 13, 2011.
↑ 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

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Spectral risk measure

Definition

Properties

Examples

See also

References