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 $X$ 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 $M_{\phi}(X)=-\delta\sum_{s=1}^S\phi_sX_{s:S}$ 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 ${s_1}, {s_2}\in\{1,\dots,S\}$.^[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

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

See also

• Distortion risk measure

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. doi:10.1016/j.jbankfin.2006.01.008.
2. 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), pp. 1505–1518, doi:10.1016/S0378-4266(02)00281-9