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

Properties

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.
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