Spectral risk measure

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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]


Consider a portfolio (denoting the portfolio payoff). Then a spectral risk measure where is non-negative, non-increasing, right-continuous, integrable function defined on such that is defined by

where is the cumulative distribution function for X.[2][3]

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

  1. Nonnegativity: for all ,
  2. Normalization: ,
  3. Monotonicity : is non-increasing, that is if and .[4]


Spectral risk measures are also coherent. Every spectral risk measure satisfies:

  1. Positive Homogeneity: for every portfolio X and positive value , ;
  2. Translation-Invariance: for every portfolio X and , ;
  3. Monotonicity: for all portfolios X and Y such that , ;
  4. Sub-additivity: for all portfolios X and Y, ;
  5. Law-Invariance: for all portfolios X and Y with cumulative distribution functions and respectively, if then ;
  6. Comonotonic Additivity: for every comonotonic random variables X and Y, . Note that X and Y are comonotonic if for every .[2]

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.


See also[edit]


  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. ^ a b 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), pp. 1505–1518, doi:10.1016/S0378-4266(02)00281-9