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.
If there are equiprobable outcomes with the corresponding payoffs given by the order statistics . Let . The measure defined by is a spectral measure of risk if satisfies the conditions
- Nonnegativity: for all ,
- Normalization: ,
- Monotonicity : is non-increasing, that is if and .
Spectral risk measures are also coherent. Every spectral risk measure satisfies:
- Positive Homogeneity: for every portfolio X and positive value , ;
- Translation-Invariance: for every portfolio X and , ;
- Monotonicity: for all portfolios X and Y such that , ;
- Sub-additivity: for all portfolios X and Y, ;
- Law-Invariance: for all portfolios X and Y with cumulative distribution functions and respectively, if then ;
- Comonotonic Additivity: for every comonotonic random variables X and Y, . Note that X and Y are comonotonic if for every .
In some texts[which?] the input X is interpreted as losses rather than payoff of a portfolio. In this case the translation-invariance property would be given by instead of the above.
- The expected shortfall is a spectral measure of risk.
- The expected value is trivially a spectral measure of risk.
- 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.
- Adam, Alexandre; Houkari, Mohamed; Laurent, Jean-Paul (2007). "Spectral risk measures and portfolio selection" (pdf). Retrieved October 11, 2011.
- Dowd, Kevin; Cotter, John; Sorwar, Ghulam (2008). "Spectral Risk Measures: Properties and Limitations" (pdf). CRIS Discussion Paper Series (2). Retrieved October 13, 2011.
- 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