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 $M_: \mathcal \to \mathbb$ where $\phi$ is non-negative, non-increasing, right-continuous, integrable function defined on ,1/math> such that $\int_0^1 \phi(p)dp = 1$ is defined by
:$M_(X) = -\int_0^1 \phi(p) F_X^(p) dp$
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 $X_, ... X_$. Let $\phi\in\mathbb^S$. The measure
$M_:\mathbb^S\rightarrow \mathbb$ defined by $M_(X)=-\delta\sum_^S\phi_sX_$ is a spectral measure of risk if $\phi\in\mathbb^S$ satisfies the conditions

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

Properties

Spectral risk measures are also coherent. Every spectral risk measure $\rho: \mathcal \to \mathbb$ 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 $\alpha \in \mathbb$, $\rho(X + a) = \rho(X) - a$;
# 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 $\omega_1,\omega_2 \in \Omega: \; (X(\omega_2) - X(\omega_1))(Y(\omega_2) - Y(\omega_1)) \geq 0$.
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

References

{{Reflist

Financial risk modeling