Spectral risk measure

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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 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

  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\}.[4]

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.[2]

Examples

See also

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. 
  2. ↑ 2.0 2.1 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): 1505–1518, doi:10.1016/S0378-4266(02)00281-9 


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