Spectral risk measure

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.

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_sX_{s:S} is a spectral measure of risk if \phi\in\mathbb{R}^S satisfies the conditions

1. Nonnegativity: \phi_s\geq0 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} and {s_1}, {s_2}\in{1,\dots,S}.{{Citation | author-link= | publication-date=2002

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. 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.

References

References

1. (December 2006). "Extreme spectral risk measures: An application to futures clearinghouse margin requirements". Journal of Banking & Finance.
2. (2008). "Spectral Risk Measures: Properties and Limitations". CRIS Discussion Paper Series.
3. (2007). "Spectral risk measures and portfolio selection".