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 X (denoting the portfolio payoff). Then a spectral risk measure where \phi is non-negative, non-increasing, right-continuous, integrable function defined on [0,1] such that is defined by where F_X is the cumulative distribution function for X. If there are S 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

Properties

Spectral risk measures are also coherent. Every spectral risk measure 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, and the monotonicity property by instead of the above.

Examples