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.
(denoting the portfolio payoff).
Then a spectral risk measure
is non-negative, non-increasing, right-continuous, integrable function defined on
is the cumulative distribution function for X.
equiprobable outcomes with the corresponding payoffs given by the order statistics
is a spectral measure of risk if
satisfies the conditions Spectral risk measures are also coherent.
Every spectral risk measure
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