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.{{cite journal|title=Extreme spectral risk measures: An application to futures clearinghouse margin requirements|first1=John|last1=Cotter|first2=Kevin|last2=Dowd|journal=Journal of Banking & Finance|volume=30|issue=12|date=December 2006|pages=3469–3485|doi=10.1016/j.jbankfin.2006.01.008|arxiv=1103.5653}}

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.{{cite journal|title=Spectral Risk Measures: Properties and Limitations|first1=Kevin|last1=Dowd|first2=John|last2=Cotter|first3=Ghulam|last3=Sorwar|year=2008|journal=CRIS Discussion Paper Series|issue=2|url=http://www.nottingham.ac.uk/business/cris/papers/2008-2.pdf|accessdate=October 13, 2011}}

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

Nonnegativity: $\phi_s\geq0$ for all $s=1, \dots, S$ ,
Normalization: $\sum_{s=1}^S\phi_s=1$ ,
Monotonicity : $\phi_s$ is non-increasing, that is $\phi_{s_1}\geq\phi_{s_2}$ if ${s_1}<{s_2}$ and ${s_1}, {s_2}\in\{1,\dots,S\}$ .{{Citation

| last=Acerbi

| first=Carlo

| author-link=

| publication-date=2002

| year=2002

| title=Spectral measures of risk: A coherent representation of subjective risk aversion

| periodical=Journal of Banking and Finance

| location=

| place=

| publisher=Elsevier

| volume=26

| issue=7

| pages=1505–1518

| url=

| doi=10.1016/S0378-4266(02)00281-9

| oclc=

| citeseerx=10.1.1.458.6645

}}

Properties

Spectral risk measures are also coherent. Every spectral risk measure $\rho: \mathcal{L} \to \mathbb{R}$ satisfies:

Positive Homogeneity: for every portfolio X and positive value $\lambda > 0$ , $\rho(\lambda X) = \lambda \rho(X)$ ;
Translation-Invariance: for every portfolio X and $\alpha \in \mathbb{R}$ , $\rho(X + a) = \rho(X) - a$ ;
Monotonicity: for all portfolios X and Y such that $X \geq Y$ , $\rho(X) \leq \rho(Y)$ ;
Sub-additivity: for all portfolios X and Y, $\rho(X+Y) \leq \rho(X) + \rho(Y)$ ;
Law-Invariance: for all portfolios X and Y with cumulative distribution functions $F_X$ and $F_Y$ respectively, if $F_X = F_Y$ then $\rho(X) = \rho(Y)$ ;
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 $\omega_1,\omega_2 \in \Omega: \; (X(\omega_2) - X(\omega_1))(Y(\omega_2) - Y(\omega_1)) \geq 0$ .{{cite journal|title=Spectral risk measures and portfolio selection|year=2007|first1=Alexandre|last1=Adam|first2=Mohamed|last2=Houkari|first3=Jean-Paul|last3=Laurent|url=http://laurent.jeanpaul.free.fr/Spectral_risk_measures_and_portfolio_selection.pdf|accessdate=October 11, 2011}}

In some texts{{which|date=October 2016}} 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

Category:Financial risk modeling

spectral risk measure

Definition

Properties

Examples

See also

References