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.

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 $${s_1}<{s_2}$$ and $${s_1}, {s_2}\in\{1,\dots,S\}$$.

Properties
Spectral risk measures are also coherent. Every spectral risk measure $$\rho: \mathcal{L} \to \mathbb{R}$$ satisfies: 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.
 * 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 $$\omega_1,\omega_2 \in \Omega: \; (X(\omega_2) - X(\omega_1))(Y(\omega_2) - Y(\omega_1)) \geq 0$$.

Examples

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