Spectral risk measure explained

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

X (denoting the portfolio payoff). Then a spectral risk measure M _{\phi} : l{L} \to R where \phi is non-negative, non-increasing, right-continuous, integrable function defined on [ 0 , 1 ] such that \int ^{1} _{0} \phi(p)dp = 1 is defined byM _{\phi} (X) = - \int ^{1} _{0} \phi(p) F ^{-1} _{X} (p) dp where F _{X} is the cumulative distribution function for X.^[2]

If there are

S equiprobable outcomes with the corresponding payoffs given by the order statistics X _{1:S} , ... X _{S:S} . Let \phi\in R ^{S} . The measureM _{\phi} : R ^{S →} R defined by M _{\phi} (X) =- \delta\sum ^{S\phi} _{sX} _{s:S} is a spectral measure of risk if \phi\in R ^{S} satisfies the conditions

1. Nonnegativity:

\phi _{s\geq0} for all s =1 , ... , S ,

1. Normalization:

\sum ^{S\phi} _{s =1} ,

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,...,S\}} .

Properties

Spectral risk measures are also coherent. Every spectral risk measure

\rho: l{L} \to R satisfies:

1. Positive Homogeneity: for every portfolio X and positive value

λ > 0 , \rho( λ X) = λ \rho(X) ;

1. Translation-Invariance: for every portfolio X and

\alpha \in R , \rho(X + a) = \rho(X) - a ;

1. Monotonicity: for all portfolios X and Y such that

X \geq Y , \rho(X) \leq \rho(Y) ;

1. Sub-additivity: for all portfolios X and Y,

\rho(X + Y) \leq \rho(X) + \rho(Y) ;

1. 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) ;

1. 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 .^[3] 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 , and the monotonicity property by 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

Notes and References

1. Extreme spectral risk measures: An application to futures clearinghouse margin requirements. John. Cotter. Kevin. Dowd. Journal of Banking & Finance. 30. 12. December 2006. 3469–3485. 10.1016/j.jbankfin.2006.01.008. 1103.5653.
2. Spectral Risk Measures: Properties and Limitations. Kevin. Dowd. John. Cotter. Ghulam. Sorwar. 2008. CRIS Discussion Paper Series. 2. October 13, 2011.
3. Spectral risk measures and portfolio selection. 2007. Alexandre. Adam. Mohamed. Houkari. Jean-Paul. Laurent. October 11, 2011.