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 formula_1 (denoting the portfolio payoff). Then a spectral risk measure formula_2 where formula_3 is non-negative, non-increasing, right-continuous, integrable function defined on formula_4 such that formula_5 is defined by where formula_7 is the cumulative distribution function for "X". If there are formula_8 equiprobable outcomes with the corresponding payoffs given by the order statistics formula_9. Let formula_10. The measure formula_11 defined by formula_12 is a spectral measure of risk if formula_10 satisfies the conditions Properties. Spectral risk measures are also coherent. Every spectral risk measure formula_21 satisfies: 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 formula_35, and the monotonicity property by formula_36 instead of the above.