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

See also

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. ↑ 2.0 2.1 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): 1505–1518, doi:10.1016/S0378-4266(02)00281-9