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 (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. 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 . Properties Spectral risk measures are also coherent. Every spectral risk measure satisfies: Positive Homogeneity: for every portfolio X and positive value