Finance: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 [math]\displaystyle{ X }[/math] (denoting the portfolio payoff). Then a spectral risk measure [math]\displaystyle{ M_{\phi}: \mathcal{L} \to \mathbb{R} }[/math] where [math]\displaystyle{ \phi }[/math] is non-negative, non-increasing, right-continuous, integrable function defined on [math]\displaystyle{ [0,1] }[/math] such that [math]\displaystyle{ \int_0^1 \phi(p)dp = 1 }[/math] is defined by

[math]\displaystyle{ M_{\phi}(X) = -\int_0^1 \phi(p) F_X^{-1}(p) dp }[/math]

where [math]\displaystyle{ F_X }[/math] is the cumulative distribution function for X.

If there are [math]\displaystyle{ S }[/math] equiprobable outcomes with the corresponding payoffs given by the order statistics [math]\displaystyle{ X_{1:S}, ... X_{S:S} }[/math]. Let [math]\displaystyle{ \phi\in\mathbb{R}^S }[/math]. The measure [math]\displaystyle{ M_{\phi}:\mathbb{R}^S\rightarrow \mathbb{R} }[/math] defined by [math]\displaystyle{ M_{\phi}(X)=-\delta\sum_{s=1}^S\phi_sX_{s:S} }[/math] is a spectral measure of risk if [math]\displaystyle{ \phi\in\mathbb{R}^S }[/math] satisfies the conditions

1. Nonnegativity: [math]\displaystyle{ \phi_s\geq0 }[/math] for all [math]\displaystyle{ s=1, \dots, S }[/math],
2. Normalization: [math]\displaystyle{ \sum_{s=1}^S\phi_s=1 }[/math],
3. Monotonicity : [math]\displaystyle{ \phi_s }[/math] is non-increasing, that is [math]\displaystyle{ \phi_{s_1}\geq\phi_{s_2} }[/math] if [math]\displaystyle{ {s_1}\lt {s_2} }[/math] and [math]\displaystyle{ {s_1}, {s_2}\in\{1,\dots,S\} }[/math].

Properties

Spectral risk measures are also coherent. Every spectral risk measure [math]\displaystyle{ \rho: \mathcal{L} \to \mathbb{R} }[/math] satisfies:

1. Positive Homogeneity: for every portfolio X and positive value [math]\displaystyle{ \lambda \gt 0 }[/math], [math]\displaystyle{ \rho(\lambda X) = \lambda \rho(X) }[/math];
2. Translation-Invariance: for every portfolio X and [math]\displaystyle{ \alpha \in \mathbb{R} }[/math], [math]\displaystyle{ \rho(X + a) = \rho(X) - a }[/math];
3. Monotonicity: for all portfolios X and Y such that [math]\displaystyle{ X \geq Y }[/math], [math]\displaystyle{ \rho(X) \leq \rho(Y) }[/math];
4. Sub-additivity: for all portfolios X and Y, [math]\displaystyle{ \rho(X+Y) \leq \rho(X) + \rho(Y) }[/math];
5. Law-Invariance: for all portfolios X and Y with cumulative distribution functions [math]\displaystyle{ F_X }[/math] and [math]\displaystyle{ F_Y }[/math] respectively, if [math]\displaystyle{ F_X = F_Y }[/math] then [math]\displaystyle{ \rho(X) = \rho(Y) }[/math];
6. Comonotonic Additivity: for every comonotonic random variables X and Y, [math]\displaystyle{ \rho(X+Y) = \rho(X) + \rho(Y) }[/math]. Note that X and Y are comonotonic if for every [math]\displaystyle{ \omega_1,\omega_2 \in \Omega: \; (X(\omega_2) - X(\omega_1))(Y(\omega_2) - Y(\omega_1)) \geq 0 }[/math].

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 [math]\displaystyle{ \rho(X+a) = \rho(X) + a }[/math], and the monotonicity property by [math]\displaystyle{ X \geq Y \implies \rho(X) \geq \rho(Y) }[/math] 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